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1 |
+
1
|
2 |
+
Model-Driven Deep Learning for Non-Coherent
|
3 |
+
Massive Machine-Type Communications
|
4 |
+
Zhe Ma, Wen Wu, Senior Member, IEEE, Feifei Gao, Fellow, IEEE, and Xuemin
|
5 |
+
(Sherman) Shen, Fellow, IEEE
|
6 |
+
Abstract
|
7 |
+
In this paper, we investigate the joint device activity and data detection in massive machine-type
|
8 |
+
communications (mMTC) with a one-phase non-coherent scheme, where data bits are embedded in
|
9 |
+
the pilot sequences and the base station simultaneously detects active devices and their embedded data
|
10 |
+
bits without explicit channel estimation. Due to the correlated sparsity pattern introduced by the non-
|
11 |
+
coherent transmission scheme, the traditional approximate message passing (AMP) algorithm cannot
|
12 |
+
achieve satisfactory performance. Therefore, we propose a deep learning (DL) modified AMP network
|
13 |
+
(DL-mAMPnet) that enhances the detection performance by effectively exploiting the pilot activity
|
14 |
+
correlation. The DL-mAMPnet is constructed by unfolding the AMP algorithm into a feedforward neural
|
15 |
+
network, which combines the principled mathematical model of the AMP algorithm with the powerful
|
16 |
+
learning capability, thereby benefiting from the advantages of both techniques. Trainable parameters
|
17 |
+
are introduced in the DL-mAMPnet to approximate the correlated sparsity pattern and the large-scale
|
18 |
+
fading coefficient. Moreover, a refinement module is designed to further advance the performance by
|
19 |
+
utilizing the spatial feature caused by the correlated sparsity pattern. Simulation results demonstrate that
|
20 |
+
the proposed DL-mAMPnet can significantly outperform traditional algorithms in terms of the symbol
|
21 |
+
error rate performance.
|
22 |
+
Index Terms
|
23 |
+
Massive machine-type communication (mMTC), non-coherent transmission, grant-free random ac-
|
24 |
+
cess, deep learning, model-driven.
|
25 |
+
Z. Ma and F. Gao are with the Institute for Artificial Intelligence Tsinghua University, State Key Lab of Intelligent Technologies
|
26 |
+
and Systems, Beijing National Research Center for Information Science and Technology, Department of Automation, Tsinghua
|
27 |
+
University, Beijing 100084, China (e-mail: maz16@mails.tsinghua.edu.cn; feifeigao@ieee.org).
|
28 |
+
W. Wu is with the Frontier Research Center, Peng Cheng Laboratory, Shenzhen, Guangdong 518055, China (email:
|
29 |
+
wuw02@pcl.ac.cn).
|
30 |
+
X. Shen is with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON N2L 3G1,
|
31 |
+
Canada (e-mail: sshen@uwaterloo.ca).
|
32 |
+
arXiv:2301.00516v1 [cs.IT] 2 Jan 2023
|
33 |
+
|
34 |
+
2
|
35 |
+
I. INTRODUCTION
|
36 |
+
To embrace the forthcoming era of Internet of Things (IoT), the 3rd Generation Partnership
|
37 |
+
Project (3GPP) has specified massive machine-type communications (mMTC) as one of the
|
38 |
+
three main service classes for fifth-generation (5G) network and beyond [1]. In a typical mMTC
|
39 |
+
scenario, a massive number of IoT devices are required to establish uplink-dominated commu-
|
40 |
+
nication with a single base station (BS) [2]. The uplink transmission is usually sporadic and
|
41 |
+
has a short packet size, so only a small and random subset of devices are active for a short
|
42 |
+
while [3]- [4]. As a result, conventional grant-based random access protocols are inappropriate
|
43 |
+
for the mMTC scenarios. To better support mMTC services, one potential solution is to develop
|
44 |
+
novel multiple-access schemes that can accomplish user activity and data detection in a timely
|
45 |
+
and accurate manner.
|
46 |
+
Grant-free (GF) random access is a promising solution for mMTC and IoT, as it eliminates the
|
47 |
+
signaling overhead required for the coordination between the BS and massive devices [5]. In the
|
48 |
+
GF-random access, the user activity and data detection are usually conducted through a two-phase
|
49 |
+
coherent scheme. Specifically, each activated device directly transmits a unique pilot sequence
|
50 |
+
followed by data packets without a prior scheduling assignment. After receiving the superimposed
|
51 |
+
signal from these devices, the BS first detects the active devices and estimates the channel, based
|
52 |
+
on which the corresponding transmitted data bits are then decoded. However, due to the massive
|
53 |
+
number of devices, it is impossible to assign orthogonal pilot sequences to each device, which
|
54 |
+
inevitably leads to collisions among devices and results in performance degradation [6]. Thanks
|
55 |
+
to the sporadic mMTC traffic pattern, the device activity detection and channel estimation can
|
56 |
+
be formulated as a compressed sensing (CS) problem [7]. Consequently, various CS techniques
|
57 |
+
have been considered for device detection in mMTC, and they have been shown to outperform
|
58 |
+
traditional methods by mitigating pilot contamination [8]- [10]. Nevertheless, the two-phase
|
59 |
+
coherent scheme incurs non-negligible overhead for channel training. Thus it may not be suitable
|
60 |
+
for mMTC where devices usually transmit small packets intermittently, prompting researchers
|
61 |
+
to consider the non-coherent schemes [11]- [16].
|
62 |
+
Several existing works have attempted to investigate the one-phase non-coherent scheme [13]-
|
63 |
+
[16]. In contrast to the coherent scheme, explicit channel estimation is not required in the
|
64 |
+
non-coherent scheme. The intuition behind the one-phase non-coherent scheme is to allocate
|
65 |
+
multiple distinct pilot sequences to each device. When transmitting, each device selects only
|
66 |
+
|
67 |
+
3
|
68 |
+
one pilot sequence based on its data, and the BS detects the user activity and data jointly by
|
69 |
+
determining which pilot sequence is received. The paper [13] proposes a novel method for
|
70 |
+
embedding 1 bit in pilot sequences, which outperforms the two-phase coherent scheme. The
|
71 |
+
work [14] considers the case when multiple bits are embedded and conducts joint user activity
|
72 |
+
and data detection using the approximate message passing (AMP) algorithm. In [15], a modified-
|
73 |
+
AMP algorithm is proposed, where the soft-thresholding function is utilized to decide on one
|
74 |
+
of the possible pilot sequences while suppressing the other ones. In [16], a covariance-based
|
75 |
+
detection scheme is developed to acquire the indices of the transmitted pilot sequences. However,
|
76 |
+
all the aforementioned works assume that the activity of each pilot sequence is independently
|
77 |
+
and identically distributed. Although the i.d.d. assumption produces an analytically tractable
|
78 |
+
solution, it neglects the correlation among the pilot sequence activity in each user and thus may
|
79 |
+
not be optimal. In this work, we investigate the possibility of applying the deep learning method
|
80 |
+
to explore the correlation structure of the sparsity pattern and improve the joint user activity and
|
81 |
+
data detection.
|
82 |
+
Thanks to the strong capability of solving intricate and intractable problems, machine learning
|
83 |
+
has become a favorable research topic for future wireless communications [17]- [24]. In particu-
|
84 |
+
lar, as a major branch in machine learning, deep learning has been extensively investigated for sig-
|
85 |
+
nal detection [20], channel estimation [21], and constellation design [22] to improve performance
|
86 |
+
while reducing computational complexity. Among vast techniques that employ deep learning in
|
87 |
+
wireless communication, the “deep unfolding” method that unfolds iterative algorithms into deep
|
88 |
+
neural networks (DNN) is especially attractive [23]. By incorporating communication expert
|
89 |
+
knowledge into DNN, “deep unfolding” inherits the mathematical models of classic algorithms
|
90 |
+
and enables the interpretation of network topology design [24]. Meanwhile, by exploiting the
|
91 |
+
powerful learning capability of DL, “deep unfolding” compensates the imperfections resulting
|
92 |
+
from the inaccuracy of the model and predetermined parameters.
|
93 |
+
Motivated by existing works, we propose a model-driven DL algorithm, namely DL-modified
|
94 |
+
AMP network (DL-mAMPnet), for the joint device activity and data detection in mMTC with
|
95 |
+
single-phase non-coherent scheme. DL-mAMPnet is constructed by unfolding the AMP algorithm
|
96 |
+
while adding trainable parameters and a refinement module to explore the correlated sparsity
|
97 |
+
pattern of the pilot sequence activity. Simulation results validate the superior symbol error rate
|
98 |
+
(SER) performance of the proposed DL-mAMPnet. The main contributions can be summarized
|
99 |
+
as follows.
|
100 |
+
|
101 |
+
4
|
102 |
+
• We formulate the joint device activity and data detection in mMTC with single-phase non-
|
103 |
+
coherent scheme as a hierarchical CS problem with two-level sparsity, where the device
|
104 |
+
activity sparsity and transmitted pilot sequence sparsity are modeled as the system-level
|
105 |
+
sparsity and the device-level sparsity, respectively.
|
106 |
+
• We propose an AMP-based algorithm to solve the formulated CS problem. On this basis,
|
107 |
+
we discuss the limitations of the AMP-based algorithm, which serves as the underlying
|
108 |
+
motivation for designing the DL-based algorithm.
|
109 |
+
• We propose a DL-based algorithm, termed DL-mAMPnet, to conduct the device activity
|
110 |
+
and data detection jointly. DL-mAMPnet is composed of multiple AMP layers and one
|
111 |
+
refinement module. The AMP layers are obtained by unfolding the AMP algorithm into
|
112 |
+
a feedforward DNN, where trainable parameters are introduced to compensate for the
|
113 |
+
inaccurate i.d.d model of the traditional AMP algorithm. The refinement module exploits
|
114 |
+
the unique spatial feature of the two-level sparsity structure to refine the output of the AMP
|
115 |
+
layers.
|
116 |
+
The remainder of the paper is organized as follows. In Section II, we present the system model
|
117 |
+
and briefly introduce the non-coherent scheme. In Section III, we formulate a hierarchical CS
|
118 |
+
problem with two-level sparsity and correspondingly derive an AMP-based algorithm. In Section
|
119 |
+
IV, we elaborate the structure of the proposed DL-mAMPnet. In Section V, we present the
|
120 |
+
parameter initialization and training method of the proposed DL-mAMPnet. Simulation results
|
121 |
+
are presented in Section VI, and conclusions are made in Section VII.
|
122 |
+
Notations: We use normal lower-case, bold lower-case, and bold upper-case letters to denote
|
123 |
+
scalars, vectors, and matrices, respectively. For matrix X, XT denotes its transpose, XH denotes
|
124 |
+
its Hermitian transpose, |X| denotes its determinant, and ||X||F denotes its Frobenius norm. For
|
125 |
+
vector x, ||x||p denotes its lp-norm. E{·} denotes the expectation operation. RM×N and CM×N
|
126 |
+
denote the M × N dimensional real space and complex space, respectively. CN(µ, Σ) denotes
|
127 |
+
the multivariate complex Gaussian distribution with mean µ and covariance Σ.
|
128 |
+
II. SYSTEM MODEL
|
129 |
+
A. Uplink Massive Access Scenario in mMTC Systems
|
130 |
+
We consider a typical uplink massive access scenario in mMTC systems, where a set of
|
131 |
+
randomly distributed single-antenna devices, denoted by N = {1, · · · , N}, communicate with
|
132 |
+
|
133 |
+
5
|
134 |
+
a BS equipped with M antennas. The uplink channel from device n to the BS is denoted by
|
135 |
+
hn ∈ CM×1 and modeled as
|
136 |
+
hn =
|
137 |
+
�
|
138 |
+
βngn, ∀n ∈ N,
|
139 |
+
(1)
|
140 |
+
where βn is the large-scale fading component and gn denotes the small-scale fading component.
|
141 |
+
We assume gn is distributed as CN(0, IM), and accordingly we have hn ∼ CN(0, βnIM).
|
142 |
+
This paper adopts a block-fading channel model, where hn remains unchanged within channel
|
143 |
+
coherence time but is independent from block to block.
|
144 |
+
Due to the sporadic activity pattern of mMTC, only a small fraction of devices are active in
|
145 |
+
each block. We assume that the devices are synchronized, and each device independently decides
|
146 |
+
whether to access the channel with probability ϵ in each block. Consequently, the device activity
|
147 |
+
indicator for device n ∈ N is defined as
|
148 |
+
αn =
|
149 |
+
�
|
150 |
+
�
|
151 |
+
�
|
152 |
+
�
|
153 |
+
�
|
154 |
+
1,
|
155 |
+
if device n is active,
|
156 |
+
0,
|
157 |
+
otherwise,
|
158 |
+
(2)
|
159 |
+
where Pr(αn = 1) = ϵ and Pr(αn = 0) = 1 − ϵ. We further define the set of active devices
|
160 |
+
within a block as
|
161 |
+
K = {n ∈ N : αn = 1},
|
162 |
+
(3)
|
163 |
+
and the number of active devices is K = |K|. The received signal y ∈ CM×1 at the BS is given
|
164 |
+
by
|
165 |
+
y =
|
166 |
+
�
|
167 |
+
n∈N
|
168 |
+
αnhnxn + n =
|
169 |
+
�
|
170 |
+
k∈K
|
171 |
+
hkxk + n,
|
172 |
+
(4)
|
173 |
+
where xn ∈ C is the transmitted signal of device n, and n ∈ CM×1 is the additive white Gaussian
|
174 |
+
noise (AWGN) distributed as CN(0, σ2IM).
|
175 |
+
B. One-Phase Non-Coherent Scheme
|
176 |
+
To successfully transmit the messages of the active devices, two schemes have been proposed
|
177 |
+
in the literature, namely the two-phase coherent scheme and the one-phase non-coherent scheme.
|
178 |
+
The two-phase coherent scheme divides each coherence block into two contiguous phases. In
|
179 |
+
the first phase, the active devices send their pilot sequences to the BS synchronously, and the
|
180 |
+
BS jointly detects the device activity, i.e., αn, as well as their corresponding channels, i.e.,
|
181 |
+
hn, ∀n ∈ K. In the second phase, the active devices send their messages to the BS using the
|
182 |
+
|
183 |
+
6
|
184 |
+
remaining coherence block, and the BS decodes these messages based on the knowledge of
|
185 |
+
device activity and channels obtained in the first phase.
|
186 |
+
Unlike the two-phase coherent scheme, the one-phase non-coherent scheme considered in this
|
187 |
+
paper can jointly detect the active devices and the corresponding messages without explicit chan-
|
188 |
+
nel estimation. Specifically, in the non-coherent scheme, the transmitted messages are embedded
|
189 |
+
in the index of the transmitted pilot sequence of each active device. To this end, each device
|
190 |
+
maintains a unique set of pre-assigned Q = 2J pilot sequences. When a device is active, it sends
|
191 |
+
a J-bit message by transmitting one sequence from the set. By detecting which sequences are
|
192 |
+
received, the BS acquires both the identity of the active devices as well as the J-bit message
|
193 |
+
from each of the active devices. We define the pilot sequences allocated for device n as:
|
194 |
+
Sn = {s1
|
195 |
+
n, s2
|
196 |
+
n, · · · , sQ
|
197 |
+
n },
|
198 |
+
(5)
|
199 |
+
where sq
|
200 |
+
n = [sq
|
201 |
+
n1, sq
|
202 |
+
n2, · · · , sq
|
203 |
+
nL]T ∈ CL×1, 1 ≤ q ≤ Q, and L is the sequence length. Note that
|
204 |
+
the total number of pilot sequences is usually much larger than the length of pilot sequence (or
|
205 |
+
the length of a coherence block), i.e., NQ ≫ L. As such, it is impossible to assign mutually
|
206 |
+
orthogonal sequences to all devices. Following the pioneering work [25], we adopt the random
|
207 |
+
Gaussian sequences in this paper. Specifically, each entry of the pilot sequences is generated from
|
208 |
+
i.i.d complex Gaussian distribution with zero mean and variance 1/L, i.e., sq
|
209 |
+
nl ∼ CN(0, 1/L),
|
210 |
+
so that each pilot sequence has a unit norm, i.e., ||sq
|
211 |
+
n||2 = 1, ∀n ∈ N and q = 1, · · · , Q.
|
212 |
+
For transmission, each active device selects exactly only one sequence from Sn based on its
|
213 |
+
message. Then, the composite received signal Y ∈ CL×M of the non-coherent scheme can be
|
214 |
+
expressed as
|
215 |
+
Y =
|
216 |
+
N
|
217 |
+
�
|
218 |
+
n=1
|
219 |
+
Q
|
220 |
+
�
|
221 |
+
q=1
|
222 |
+
αq
|
223 |
+
nsq
|
224 |
+
nhT
|
225 |
+
n + N =
|
226 |
+
N
|
227 |
+
�
|
228 |
+
n=1
|
229 |
+
SnXn + N,
|
230 |
+
(6)
|
231 |
+
where Xn = [α1
|
232 |
+
nhn, α2
|
233 |
+
nhn, · · · , αQ
|
234 |
+
n hn]T ∈ CQ×M and αq
|
235 |
+
n ∈ {0, 1} indicates whether or not
|
236 |
+
sequence q of device n is transmitted, with a slight abuse of notation. Recall that each device
|
237 |
+
is active with probability ϵ, we have
|
238 |
+
Q
|
239 |
+
�
|
240 |
+
q=1
|
241 |
+
αq
|
242 |
+
n =
|
243 |
+
�
|
244 |
+
�
|
245 |
+
�
|
246 |
+
�
|
247 |
+
�
|
248 |
+
1,
|
249 |
+
with probability ϵ;
|
250 |
+
0,
|
251 |
+
with probability 1 − ϵ.
|
252 |
+
(7)
|
253 |
+
By further concatenating all sequences of N devices as S = [S1, S2, · · · , SN] ∈ CL×NQ, the
|
254 |
+
received signal in (6) can be simplified as
|
255 |
+
Y = SX + N,
|
256 |
+
(8)
|
257 |
+
|
258 |
+
7
|
259 |
+
where X = [XT
|
260 |
+
1 , XT
|
261 |
+
2 , · · · , XT
|
262 |
+
N]T ∈ CNQ×M. The pictorial form of (8) is sketched in Fig. 1,
|
263 |
+
which intuitively shows that X has a hierarchical sparse structure. The hierarchical sparse
|
264 |
+
structure comprises two levels of sparsity, including the system-level sparsity and the device-
|
265 |
+
level sparsity. The system-level sparsity means that most rows in X are zero, which is due to
|
266 |
+
the sporadic traffic pattern. The device-level sparsity enforces that there is at most one non-zero
|
267 |
+
row exists in Xn, ∀n, because each active device only transmits one pilot sequence from its pilot
|
268 |
+
set.
|
269 |
+
=
|
270 |
+
+
|
271 |
+
Device-level
|
272 |
+
Sparsity
|
273 |
+
System-level
|
274 |
+
Sparsity
|
275 |
+
At most one non-zero
|
276 |
+
row exists in each
|
277 |
+
Most rows in are zero
|
278 |
+
Fig. 1. Pictorial form of the signal model.
|
279 |
+
III. PROBLEM FORMULATION AND AMP-BASED JOINT DETECTION ALGORITHM
|
280 |
+
A. Problem Formulation
|
281 |
+
Our goal is to detect the binary variable αq
|
282 |
+
n that indicates both the activity of device n and its
|
283 |
+
transmitted message, which can be achieved by recovering X from the received signal Y . Once
|
284 |
+
X is recovered, αq
|
285 |
+
n can be determined by the rows of X. Due to the hierarchical sparse structure
|
286 |
+
of X, such problem is a classic CS problem with known measurement matrix S. Therefore, we
|
287 |
+
can formulate the problem as follows:
|
288 |
+
P1 : min
|
289 |
+
X
|
290 |
+
||Y − SX||2
|
291 |
+
F
|
292 |
+
(9)
|
293 |
+
s.t.
|
294 |
+
N
|
295 |
+
�
|
296 |
+
n=1
|
297 |
+
Q
|
298 |
+
�
|
299 |
+
q=1
|
300 |
+
I(Xnq,:) ≤ K,
|
301 |
+
(10)
|
302 |
+
Q
|
303 |
+
�
|
304 |
+
q=1
|
305 |
+
I(Xnq,:) ≤ 1, ∀n,
|
306 |
+
(11)
|
307 |
+
|
308 |
+
8
|
309 |
+
where Xnq,: is the qth row of Xn and I(·) is the indicator function defined as
|
310 |
+
I(x) =
|
311 |
+
�
|
312 |
+
�
|
313 |
+
�
|
314 |
+
�
|
315 |
+
�
|
316 |
+
1,
|
317 |
+
if x has non-zero elements;
|
318 |
+
0,
|
319 |
+
otherwise.
|
320 |
+
(12)
|
321 |
+
The constraint (10) comes from the system-level sparsity and the constraint (11) ensures the
|
322 |
+
device-level sparsity. However, it is challenging to solve P1 directly due to the non-smooth
|
323 |
+
constraints. Hence, we relax (11) into a l2,1-norm regularized least-square problem by replacing
|
324 |
+
the indicator function with l2 norm as [26]
|
325 |
+
min
|
326 |
+
X
|
327 |
+
1
|
328 |
+
2||Y − SX||2
|
329 |
+
F + λ
|
330 |
+
N
|
331 |
+
�
|
332 |
+
n=1
|
333 |
+
Q
|
334 |
+
�
|
335 |
+
q=1
|
336 |
+
||Xnq,:||2,
|
337 |
+
(13)
|
338 |
+
where λ is the tunable parameter that balances the the sparsity of the solution and the mean
|
339 |
+
square error (MSE) ||Y − SX||2
|
340 |
+
F. Although conventional CS algorithms such as orthogonal
|
341 |
+
matching pursuit (OMP) and sparse Bayesian learning (SBL) can be directly used to solve (13),
|
342 |
+
they suffer high computational complexity due to the matrix inverse operation, especially in
|
343 |
+
mMTC system with massive devices. In view of this, this paper utilizes the computationally
|
344 |
+
efficient AMP algorithm as the main technique [27].
|
345 |
+
B. Review of the AMP Algorithm
|
346 |
+
AMP refers to a class of efficient algorithms for statistical estimation in high-dimensional
|
347 |
+
problems such as linear regression and low-rank matrix estimation. The goal of the AMP
|
348 |
+
algorithm is to obtain an estimate of X with the minimum MSE based on Y . Starting with
|
349 |
+
X0 = 0 and R0 = Y , the AMP algorithm can be described as follows:
|
350 |
+
Xt+1,n = ηt,n(SH
|
351 |
+
n Rt + Xt,n), ∀n,
|
352 |
+
(14)
|
353 |
+
Rt+1 = Y − SXt+1 + btRt,
|
354 |
+
(15)
|
355 |
+
where t = 0, 1, · · · is the index of the iteration, ηt,n(·) is the shrinkage function for device n
|
356 |
+
that shrinks some items of its input to zero, and Rt is the corresponding residual. The residual
|
357 |
+
in (15) is updated with the “Onsager correction” term btRt, which substantially improves the
|
358 |
+
performance of the AMP algorithm [28]. Note that ηt,n(·) is assumed to be Lipschitz-continuous
|
359 |
+
and bt can be written as
|
360 |
+
bt = 1
|
361 |
+
L
|
362 |
+
N
|
363 |
+
�
|
364 |
+
n=1
|
365 |
+
η
|
366 |
+
′
|
367 |
+
t,n(SH
|
368 |
+
n Rt + Xt,n),
|
369 |
+
(16)
|
370 |
+
|
371 |
+
9
|
372 |
+
where η
|
373 |
+
′
|
374 |
+
t,n(·) is the first-order derivative of ηt,n(·). In addition to improving the performance, the
|
375 |
+
Onsager correction also enables the AMP algorithm to be analyzed by a set of state evolution
|
376 |
+
equations in the asymptotic regime [29]. The asymptotic regime is when L, N → ∞, while their
|
377 |
+
ratio converges to a positive constant, i.e., N/L → ρ where ρ ∈ (0, ∞), and while keeping the
|
378 |
+
data length J fixed. To facilitate the theoretical analysis, this paper considers a certain asymptotic
|
379 |
+
regime where N → ∞, and the empirical distribution of the large-scale fading components βn’s
|
380 |
+
converges to a fixed distribution pβ.
|
381 |
+
Define β ∼ pβ and Xβ ∈ CQ×M as a random matrix distributed as (1 − ϵ
|
382 |
+
Q) �Q
|
383 |
+
i=1 δxβ,i +
|
384 |
+
ϵ
|
385 |
+
Q
|
386 |
+
�Q
|
387 |
+
i=1 Phβ
|
388 |
+
�
|
389 |
+
j̸=i δxβ,j, where δxβ,i is the Dirac delta at zero corresponding to the element xβ,i
|
390 |
+
and Phβ denotes the distribution hβ ∼ CN(0, βIM). The state evolution equations can be written
|
391 |
+
as the following recursions for t ≥ 0 [29]
|
392 |
+
Σ0 = σ2IM + ρEβ{XH
|
393 |
+
β Xβ},
|
394 |
+
(17)
|
395 |
+
Σt+1 = σ2IM + ρEβ{(ηt(Xβ + V Σ
|
396 |
+
1
|
397 |
+
2
|
398 |
+
t ) − Xβ)H(ηt(Xβ + V Σ
|
399 |
+
1
|
400 |
+
2
|
401 |
+
t ) − Xβ)},
|
402 |
+
(18)
|
403 |
+
where V ∈ CQ×M is a random matrix independent with Xβ, of which the rows are i.i.d. and
|
404 |
+
each follows the distribution CN(0, IM). It can be observed from (14) and (18) that applying
|
405 |
+
ηt,n(·) to SH
|
406 |
+
n Rt + Xt,n is statistically equivalent to applying ηt,n(·) to Xt,n + V Σ
|
407 |
+
1
|
408 |
+
2
|
409 |
+
t . Therefore,
|
410 |
+
the input to the shrinkage function ηt,n(·) can be modeled as an AWGN-corrupted signal, i.e.,
|
411 |
+
Zt,n = Xt,n + SH
|
412 |
+
n Rt = Xt,n + V Σ
|
413 |
+
1
|
414 |
+
2
|
415 |
+
t ,
|
416 |
+
(19)
|
417 |
+
In this case, the update given by (14) is statistically equivalent to a denosing problem, and
|
418 |
+
thus ηt(·) can also be called “denoiser”. Hereafter, we use “shrinkage function” and “denoiser”
|
419 |
+
interchangeably for convenience.
|
420 |
+
C. AMP-Based Joint Device Activity and Date Detection Algorithm
|
421 |
+
The core idea behind the joint detection algorithm is to first estimate X from Y , based on
|
422 |
+
which αq
|
423 |
+
n is determined according to the norm of each rows in X. To this end, we first derive the
|
424 |
+
denoiser ηt,n(·) under the MMSE-optimal criterion. After that, we observe that ηt,n(·) exhibits
|
425 |
+
an asymptotic property, which motivates us to design a threshold-based strategy to extract αq
|
426 |
+
n
|
427 |
+
from X.
|
428 |
+
|
429 |
+
10
|
430 |
+
1) Derivation of ηt,n(·): For notational simplicity, we omit the iteration index t in the fol-
|
431 |
+
lowing. According to (19), the likelihood of Zn given Xn takes the form of
|
432 |
+
PZn|Xn =
|
433 |
+
Q
|
434 |
+
�
|
435 |
+
q=1
|
436 |
+
exp(−(zq
|
437 |
+
n − xq
|
438 |
+
n)HΣ−1(zq
|
439 |
+
n − xq
|
440 |
+
n))
|
441 |
+
πM|Σ|
|
442 |
+
.
|
443 |
+
(20)
|
444 |
+
Accordingly, the MMSE-optimal denoiser is given by the conditional expectation E{Xn|Zn}
|
445 |
+
and can be expressed as
|
446 |
+
ηn(Zn) = E{Xn|Zn} = [φ1
|
447 |
+
nΩnz1
|
448 |
+
n, · · · , φQ
|
449 |
+
n ΩnzQ
|
450 |
+
n ],
|
451 |
+
(21)
|
452 |
+
where
|
453 |
+
Ωn = βn(βnIM + Σ)−1,
|
454 |
+
(22)
|
455 |
+
φq
|
456 |
+
n =
|
457 |
+
1
|
458 |
+
1 + Q−ϵ
|
459 |
+
ϵ
|
460 |
+
exp(M(ψn − πq
|
461 |
+
n)),
|
462 |
+
(23)
|
463 |
+
ψn = log(|IM + βnΣ−1|)
|
464 |
+
M
|
465 |
+
,
|
466 |
+
(24)
|
467 |
+
and
|
468 |
+
πq
|
469 |
+
n = zq
|
470 |
+
n
|
471 |
+
H(Σ−1 − (Σ + βnIM)−1)zq
|
472 |
+
n
|
473 |
+
M
|
474 |
+
.
|
475 |
+
(25)
|
476 |
+
Proof: Please refer to Appendix A.
|
477 |
+
It is important to realize that the MMSE-optimal denoiser ηn(·) is rather complicated as it
|
478 |
+
involves the computation of the state evolution matrix Σ, where the matrix multiplication and
|
479 |
+
expectation are needed. Hence, we simplify ηn(·) by using the following theorem.
|
480 |
+
Theorem 1: Considering the asymptotic regime where both the number of devices N and
|
481 |
+
the length of the pilot sequences L go to the infinity with their ratio converging to some fixed
|
482 |
+
positive values, i.e., N/L → ρ where ρ ∈ (0, ∞), the state evolution matrix Σt always remains
|
483 |
+
as a diagonal matrix with identical diagonal entries after each iteration, i.e.,
|
484 |
+
Σt = τ 2
|
485 |
+
t IM, ∀t ≥ 0.
|
486 |
+
(26)
|
487 |
+
Correspondingly, the signal model given in (19) reduces to
|
488 |
+
Zt,n = Xt,n + SH
|
489 |
+
n Rt = Xt,n + τtV ,
|
490 |
+
(27)
|
491 |
+
and the MMSE-optimal dnoiser given in (21)-(25) is simplified as
|
492 |
+
ηn(Zn) = E{Xn|Zn} = [φ1
|
493 |
+
nωnz1
|
494 |
+
n, · · · , φQ
|
495 |
+
n ωnzQ
|
496 |
+
n ],
|
497 |
+
(28)
|
498 |
+
|
499 |
+
11
|
500 |
+
where
|
501 |
+
ωn =
|
502 |
+
βn
|
503 |
+
βn + τ 2,
|
504 |
+
(29)
|
505 |
+
φq
|
506 |
+
n =
|
507 |
+
1
|
508 |
+
1 + Q−ϵ
|
509 |
+
ϵ
|
510 |
+
exp(M(ψn − πq
|
511 |
+
n)),
|
512 |
+
(30)
|
513 |
+
ψn = log(1 + βn
|
514 |
+
τ 2 ),
|
515 |
+
(31)
|
516 |
+
and
|
517 |
+
πq
|
518 |
+
n =
|
519 |
+
βnzq
|
520 |
+
n
|
521 |
+
Hzq
|
522 |
+
n
|
523 |
+
τ 2(βn + τ 2)M .
|
524 |
+
(32)
|
525 |
+
Finally, τ 2
|
526 |
+
t can be obtained using the following recursions for t ≥ 0:
|
527 |
+
τ 2
|
528 |
+
0 = σ2 + ρϵEβ{β},
|
529 |
+
(33)
|
530 |
+
τ 2
|
531 |
+
t+1 = σ2 + ρ
|
532 |
+
Q
|
533 |
+
�
|
534 |
+
q=1
|
535 |
+
Eβ{ φq
|
536 |
+
ββτ 2
|
537 |
+
t
|
538 |
+
β + τ 2
|
539 |
+
t
|
540 |
+
} + ρ
|
541 |
+
Q
|
542 |
+
�
|
543 |
+
q=1
|
544 |
+
Eβ{φq
|
545 |
+
β(1 − φq
|
546 |
+
β) β2zq
|
547 |
+
n
|
548 |
+
Hzq
|
549 |
+
n
|
550 |
+
(β + τ 2
|
551 |
+
t )2M }.
|
552 |
+
(34)
|
553 |
+
We omit the detailed proof here for brevity. Interested readers can refer to theorem 1 in [8],
|
554 |
+
where a similar derivation is provided. It should be mentioned that the proposed Theorem 1 in
|
555 |
+
this paper is essentially a generalization of Theorem 1 in [8]. When each device is assigned with
|
556 |
+
only one pilot sequence, i.e., Q = 1, the proposed Theorem 1 reduces to Theorem 1 in [8].
|
557 |
+
2) Threshold-Based Strategy: It can be seen from (28)-(30) that for large M, we have φq
|
558 |
+
n → 1
|
559 |
+
if πq
|
560 |
+
n > ψn and φq
|
561 |
+
n → 0 if πq
|
562 |
+
n < ψn. The asymptotic behavior of φq
|
563 |
+
n indicates that it is reasonable
|
564 |
+
to adopt a threshold-based strategy for solution refinement. Meanwhile, considering the device
|
565 |
+
sparsity in (7), an element selection operation is necessitated to enforce all the elements except
|
566 |
+
the one with the largest magnitude in each Xn to be zeros. Consequently, the proposed threshold-
|
567 |
+
based strategy should be able to perform the following two operations.
|
568 |
+
Element Selection Operation: To surely guarantee the sparsity constraint in (11), we choose
|
569 |
+
the largest row in each Xn = [x1
|
570 |
+
n, x2
|
571 |
+
n, · · · , xQ
|
572 |
+
n ] and define the index of the largest element as
|
573 |
+
i∗
|
574 |
+
n = arg max
|
575 |
+
i
|
576 |
+
xi
|
577 |
+
n
|
578 |
+
Hxi
|
579 |
+
n, ∀n ∈ N.
|
580 |
+
(35)
|
581 |
+
Threshold-based Decisive Operation: After obtaining i∗
|
582 |
+
n, the binary variable vector αn =
|
583 |
+
{α1
|
584 |
+
n, · · · , αQ
|
585 |
+
n } can be given as
|
586 |
+
αn =
|
587 |
+
�
|
588 |
+
�
|
589 |
+
�
|
590 |
+
ei∗n,
|
591 |
+
if κi∗
|
592 |
+
n
|
593 |
+
n > 0;
|
594 |
+
0,
|
595 |
+
otherwise,
|
596 |
+
(36)
|
597 |
+
|
598 |
+
12
|
599 |
+
where ei∗n is a one-hot vector of length Q with only the i∗
|
600 |
+
nth element equal 1 and the others
|
601 |
+
equal 0, and the corresponding threshold is computed using (31) and (32) as
|
602 |
+
κi∗
|
603 |
+
n
|
604 |
+
n =
|
605 |
+
zi∗
|
606 |
+
n
|
607 |
+
n
|
608 |
+
Hzi∗
|
609 |
+
n
|
610 |
+
n βn
|
611 |
+
τ 2
|
612 |
+
t (βn + τ 2
|
613 |
+
t )M − log
|
614 |
+
�
|
615 |
+
1 + βn
|
616 |
+
τ 2
|
617 |
+
t
|
618 |
+
�
|
619 |
+
.
|
620 |
+
(37)
|
621 |
+
3) Limitation: Although the traditional AMP-based algorithm can successfully recover aq
|
622 |
+
n
|
623 |
+
from Y , it has some inherent limitations: (i) The traditional AMP algorithm implicitly assumes
|
624 |
+
Xn has a prior distribution with i.i.d. entries, which neglects the dependencies among the rows
|
625 |
+
of Xn imposed by the device-level sparsity; (ii) The calculation of the denoiser ηt,n(·) and the
|
626 |
+
threshold κn requires the exact value of βn, which is costly to obtain in a large-scale mMTC
|
627 |
+
system with massive devices.
|
628 |
+
AMP
|
629 |
+
layer 1
|
630 |
+
AMP
|
631 |
+
layer 2
|
632 |
+
AMP
|
633 |
+
layer t
|
634 |
+
AMP
|
635 |
+
layer T
|
636 |
+
. . .
|
637 |
+
. . .
|
638 |
+
. . .
|
639 |
+
. . .
|
640 |
+
. . .
|
641 |
+
. . .
|
642 |
+
AMP Layers
|
643 |
+
Refinement Module
|
644 |
+
…
|
645 |
+
…
|
646 |
+
Fig. 2. Network architecture of the proposed DL-mAMPnet.
|
647 |
+
IV. DEEP LEARNING MODIFIED AMP NETWORK
|
648 |
+
To address the aforementioned limitations, we propose a deep learning modified AMP network
|
649 |
+
(DL-mAMPnet). The DL-mAMPnet is constructed by unfolding the AMP algorithm into a
|
650 |
+
feedforward DNN, which inherits the mathematical model and structure of the AMP algorithm,
|
651 |
+
thereby avoiding the requirements for accurate modeling. On this basis, we introduce a few
|
652 |
+
trainable parameters into the DL-mAMPnet to learn the active probability and the large-scale
|
653 |
+
fading. By making the active probability trainable, we compensate for the inaccuracy caused
|
654 |
+
by the i.i.d. assumption in the traditional AMP algorithm. By making the large-scale fading
|
655 |
+
coefficient trainable, we bypass the statistical measurements for the large-scale fadings of mas-
|
656 |
+
sive devices. According to the threshold-based strategy in Section III-C, we further design a
|
657 |
+
refinement module to guarantee the device-level sparsity and obtain the desired aq
|
658 |
+
n.
|
659 |
+
|
660 |
+
13
|
661 |
+
As depicted in Fig. 2, the proposed DL-mAMPnet consists of T uniform AMP layers and
|
662 |
+
one refinement module. For the sake of clarity, each part of the DL-mAMPnet is elaborated
|
663 |
+
respectively in the following subsection.
|
664 |
+
A. Input and Output
|
665 |
+
To facilitate the learning process of DL-mAMPnet, the complex matrices need to be converted
|
666 |
+
into the real domain and then vectorized. To do this, we first express (8) as
|
667 |
+
�
|
668 |
+
� ℜ(Y )
|
669 |
+
ℑ(Y )
|
670 |
+
�
|
671 |
+
� =
|
672 |
+
�
|
673 |
+
� ℜ(S)
|
674 |
+
−ℑ(S)
|
675 |
+
ℑ(S)
|
676 |
+
ℜ(S)
|
677 |
+
�
|
678 |
+
�
|
679 |
+
�
|
680 |
+
� ℜ(X)
|
681 |
+
ℑ(X)
|
682 |
+
�
|
683 |
+
� +
|
684 |
+
�
|
685 |
+
� ℜ(N)
|
686 |
+
ℑ(N)
|
687 |
+
�
|
688 |
+
� ,
|
689 |
+
(38)
|
690 |
+
where ℜ(·) and ℑ(·) denote the real and imaginary parts, respectively. The real and imaginary
|
691 |
+
parts are then concatenated together and vectorized as
|
692 |
+
˜Y = vec([ℜ(Y )T, ℑ(Y )T]T) ∈ R2LM×1,
|
693 |
+
(39)
|
694 |
+
˜S =
|
695 |
+
�
|
696 |
+
[ℜ(S), −ℑ(S)]T, [ℑ(S), ℜ(S)]T�T ⊗ IM ∈ R2LM×2NQM,
|
697 |
+
(40)
|
698 |
+
˜
|
699 |
+
X = vec([ℜ(X)T, ℑ(X)T]T) ∈ R2NQM×1,
|
700 |
+
(41)
|
701 |
+
˜
|
702 |
+
N = vec([ℜ(N)T, ℑ(N)T]T) ∈ R2LM×1,
|
703 |
+
(42)
|
704 |
+
where vec(·) is the vectorize operation that flattens a matrix into a vector in the order of columns,
|
705 |
+
and ⊗ is the Kronecker product operator. Consequently, (8) can be rewritten as
|
706 |
+
˜Y = ˜S ˜
|
707 |
+
X + ˜
|
708 |
+
N.
|
709 |
+
(43)
|
710 |
+
According to the recursive formula in (14)-(15), the input to the DL-mAMPnet is chosen to
|
711 |
+
be the the received signal, the estimated signal, and the residual, which are initialized as ˜
|
712 |
+
X0 = 0
|
713 |
+
and ˜R0 = ˜Y . Meanwhile, unlike the existing AMP-inspired network that uses ˜
|
714 |
+
X [30], we adopt
|
715 |
+
α = [α1
|
716 |
+
1, · · · , αQ
|
717 |
+
1 , α1
|
718 |
+
2, · · · , αQ
|
719 |
+
N]T ∈ {0, 1}NQ×1 as the output of DL-mAMPnet, such that αq
|
720 |
+
n can
|
721 |
+
be directly obtained once DL-mAMPnet is well-trained.
|
722 |
+
|
723 |
+
14
|
724 |
+
×
|
725 |
+
×
|
726 |
+
-
|
727 |
+
×
|
728 |
+
×
|
729 |
+
Fig. 3. Detailed structure of the tth AMP layer.
|
730 |
+
B. AMP Layer
|
731 |
+
Since each layer has the same structure, we focus on the tth AMP layer of the DL-mAMPnet,
|
732 |
+
of which the detailed structure is illustrated in Fig. 3. Define the input as ˜
|
733 |
+
Xt−1, ˜Rt−1 and the
|
734 |
+
output as ˜
|
735 |
+
Xt, ˜Rt, the tth AMP layer proceeds as follows
|
736 |
+
˜
|
737 |
+
Xt = ηt( ˜
|
738 |
+
Xt−1 + Bt ˜Rt−1; Θt),
|
739 |
+
(44)
|
740 |
+
˜Rt = ˜Y − At ˜
|
741 |
+
Xt +
|
742 |
+
˜Rt−1
|
743 |
+
LM
|
744 |
+
2NQM
|
745 |
+
�
|
746 |
+
j=1
|
747 |
+
[ηt( ˜
|
748 |
+
Xt−1 + Bt ˜Rt−1; Θt)]
|
749 |
+
′
|
750 |
+
j,
|
751 |
+
(45)
|
752 |
+
where At and Bt are trainable matrices that acts as the matched filter and Θt = {θt,1, θt,2} is
|
753 |
+
the trainable parameter set of ηt(·).
|
754 |
+
It should be mentioned that the denoiser in (28)-(32) cannot be applied in the AMP layer, as
|
755 |
+
the complex-to-real transformation and vectorization in (39)-(43) have changed the dimension
|
756 |
+
and distribution of the corresponding matrices. Following the same derivation in Appendix A
|
757 |
+
but considering ˜
|
758 |
+
X as a real-valued Bernoulli Gaussian variable and changing the dimension,
|
759 |
+
ηt(·) in (28) can be expressed as
|
760 |
+
[ηt( ˜Z)]j =
|
761 |
+
β ˜Zj
|
762 |
+
(β + τ 2
|
763 |
+
t )
|
764 |
+
�
|
765 |
+
1 + Q−ϵ
|
766 |
+
ϵ
|
767 |
+
exp(log(1 + β
|
768 |
+
τ 2
|
769 |
+
t )1/2 −
|
770 |
+
˜
|
771 |
+
Z2
|
772 |
+
j β
|
773 |
+
2(β+τ 2
|
774 |
+
t )τ 2
|
775 |
+
t )
|
776 |
+
�,
|
777 |
+
=
|
778 |
+
˜Zj
|
779 |
+
(1 + τ 2
|
780 |
+
t
|
781 |
+
β )
|
782 |
+
�
|
783 |
+
1 +
|
784 |
+
�
|
785 |
+
1 + β
|
786 |
+
τ 2
|
787 |
+
t exp(log( Q−ϵ
|
788 |
+
ϵ ) −
|
789 |
+
˜
|
790 |
+
Z2
|
791 |
+
j
|
792 |
+
2(τ 2
|
793 |
+
t +τ 4
|
794 |
+
t /β))
|
795 |
+
�,
|
796 |
+
(46)
|
797 |
+
where ˜Zj is the jth element of ˜Z.
|
798 |
+
As discussed in Section II-D, ηt(·) exploits an i.i.d. assumption that fails to effectively explore
|
799 |
+
the correlated sparsity pattern. To tackle this issue, we replace log(Q−ϵ
|
800 |
+
ϵ ) with a trainable parameter
|
801 |
+
θt,1 = [θt,1,1, · · · , θt,1,2NQM]T ∈ R2NQM×1, such that the correlation among entries of ˜
|
802 |
+
X can be
|
803 |
+
|
804 |
+
15
|
805 |
+
learned and approximated. Meanwhile, to circumvent the need for the prior information of the
|
806 |
+
large-scale fading, we introduce a trainable parameter θt,2 = [θt,2,1, · · · , θt,2,2NQM]T ∈ R2NQM×1
|
807 |
+
and substitute it for β in (46). The trainable ηt(·) can then be defined as
|
808 |
+
[ηt( ˜Z)]j =
|
809 |
+
˜Zj
|
810 |
+
(1 +
|
811 |
+
τ 2
|
812 |
+
t
|
813 |
+
θt,2,j )
|
814 |
+
�
|
815 |
+
1 +
|
816 |
+
�
|
817 |
+
1 + θt,2,j
|
818 |
+
τ 2
|
819 |
+
t
|
820 |
+
exp(θt,1,j −
|
821 |
+
˜
|
822 |
+
Z2
|
823 |
+
j
|
824 |
+
2(τ 2
|
825 |
+
t +τ 4
|
826 |
+
t /θt,2,j))
|
827 |
+
�.
|
828 |
+
(47)
|
829 |
+
The derivative of ηt(·) is thus be given by
|
830 |
+
[ηt( ˜Z)]
|
831 |
+
′
|
832 |
+
j = [ηt( ˜Z)]j
|
833 |
+
∂ ˜Zj
|
834 |
+
=
|
835 |
+
1 +
|
836 |
+
�
|
837 |
+
1 + θt,2,j
|
838 |
+
τ 2
|
839 |
+
t
|
840 |
+
exp(θt,1,j −
|
841 |
+
˜
|
842 |
+
Z2
|
843 |
+
j
|
844 |
+
2(τ 2
|
845 |
+
t +τ 4
|
846 |
+
t /θt,2,j))(1 +
|
847 |
+
˜
|
848 |
+
Z2
|
849 |
+
j
|
850 |
+
(τ 2
|
851 |
+
t +τ 4
|
852 |
+
t /θt,2,j))
|
853 |
+
(1 +
|
854 |
+
τ 2
|
855 |
+
t
|
856 |
+
θt,2,j )
|
857 |
+
�
|
858 |
+
1 +
|
859 |
+
�
|
860 |
+
1 + θt,2,j
|
861 |
+
τ 2
|
862 |
+
t
|
863 |
+
exp(θt,1,j −
|
864 |
+
˜
|
865 |
+
Z2
|
866 |
+
j
|
867 |
+
2(τ 2
|
868 |
+
t +τ 4
|
869 |
+
t /θt,2,j))
|
870 |
+
�2 .
|
871 |
+
(48)
|
872 |
+
Note that to evade the computation of the expectation involved in τ 2, this paper adopts an
|
873 |
+
empirical result where τ 2 is estimated by the standard deviation of the corrupted noise in ˜Z,
|
874 |
+
i.e., τ 2
|
875 |
+
t = || ˜Rt||2/
|
876 |
+
√
|
877 |
+
2LM [30].
|
878 |
+
Remark 1: It is worth noting that the denoiser derived in (28) operates in a section-wise
|
879 |
+
manner, i.e., acts on Q rows of each Xn, while the ηt(·) in the AMP layer operates row-by-row
|
880 |
+
on X. Although the section-wise manner may exploit the correlations better than the row-wise
|
881 |
+
manner, it is quite challenging to be implemented in DNNs. This is because to realize such
|
882 |
+
section-wise manner, we have to either construct N sublayers or impose N iterations in each
|
883 |
+
AMP layer. The former will heavily expand the network size and trainable parameters, reducing
|
884 |
+
the scalability and stunting the training process of the DL-mAMPnet. The latter will greatly
|
885 |
+
increase the computational complexity of the DL-mAMPnet and negate the “deep unfolding”
|
886 |
+
advantage. It should also be noted that although the AMP layer can explore the correlated
|
887 |
+
sparsity pattern with the help of trainable parameters, the device-level sparsity constraint in (7)
|
888 |
+
is not surely guaranteed. Motivated by this consideration, we propose a felicitous method in
|
889 |
+
the refinement module that utilizes the Maxpool-MaxUnpool operation to ensure device-level
|
890 |
+
sparsity, as detailed in the subsection below.
|
891 |
+
C. Refinement Module
|
892 |
+
The refinement module should be capable of ensuring the device-level sparsity while extracting
|
893 |
+
aq
|
894 |
+
n from
|
895 |
+
˜
|
896 |
+
XT without explicit channel state information (CSI). To fulfil these functionalities,
|
897 |
+
two components are integrated in the refinement module, namely the soft-thresholding denois-
|
898 |
+
ing component and the hard-thresholding decision component. The soft-thresholding denoising
|
899 |
+
component is intended to further denoise ˜
|
900 |
+
XT by exploiting the hierarchical sparse structure. The
|
901 |
+
|
902 |
+
16
|
903 |
+
×
|
904 |
+
-
|
905 |
+
-
|
906 |
+
2NQ× M
|
907 |
+
2NQM × 1
|
908 |
+
2NQ × 1
|
909 |
+
2NQ × 1
|
910 |
+
2NQ × 1
|
911 |
+
2NQ × 1
|
912 |
+
2NQ × 1
|
913 |
+
2NQ × 1
|
914 |
+
2NQ × 1
|
915 |
+
2N × 1
|
916 |
+
2NQ × 1
|
917 |
+
2NQM × 1
|
918 |
+
1 × 1
|
919 |
+
(2NQM+1) × 1
|
920 |
+
2NQ × 1
|
921 |
+
NQ × 1
|
922 |
+
Reshape
|
923 |
+
Absolute
|
924 |
+
Conv
|
925 |
+
FC+ReLU
|
926 |
+
FC+ReLU
|
927 |
+
FC+ReLU
|
928 |
+
Sigmoid
|
929 |
+
Average
|
930 |
+
ReLU
|
931 |
+
MaxPool
|
932 |
+
Concatenate
|
933 |
+
MaxUnpool
|
934 |
+
FC+
|
935 |
+
Soft-thresholding Denosing
|
936 |
+
Hard-thresholding Decision
|
937 |
+
Fig. 4. Detailed architecture of the proposed refinement module.
|
938 |
+
hard-thresholding decision component is aimed at implementing the threshold-based strategy in
|
939 |
+
(35)-(37). The detailed structure of the refinement module is presented in Fig. 4 and elaborated
|
940 |
+
as follows.
|
941 |
+
Soft-Thresholding Denoising: As shown in Fig. 1, the two-level sparsity exhibits a unique
|
942 |
+
spatial structure that has not been utilized in the AMP layers. Here, the soft-thresholding
|
943 |
+
denoising aims to distill
|
944 |
+
˜
|
945 |
+
XT using such spatial feature, enhancing useful information while
|
946 |
+
removing noise information. To do this, we first de-vectorize ˜
|
947 |
+
XT and take the absolute value as
|
948 |
+
X = |Vec−1( ˜
|
949 |
+
XT)| = [|ℜ(X)T|, |ℑ(X)T|]T ∈ R+2NQ×M.
|
950 |
+
(49)
|
951 |
+
Then, a convolutional layer with 1 × M kernel size is applied to X to combine the information
|
952 |
+
from all M antennas and extract a coarse estimation of aq
|
953 |
+
n. This arrangement is motivated by the
|
954 |
+
fact that all M elements in each row of X share the same aq
|
955 |
+
n, as observed from (6) and Fig. 1.
|
956 |
+
The coarse estimation can be expressed as fθc(X), where fθc(·) is the function expression of the
|
957 |
+
convolutional layer with parameter θc. After that, an average pooling with 1 × M kernel size is
|
958 |
+
applied to X to get a 1-D average vector over M antennas. The 1-D vector ι =
|
959 |
+
1
|
960 |
+
M
|
961 |
+
�M
|
962 |
+
m=1 X:,m
|
963 |
+
is forwarded into a two-layer fully-connected (FC) network to obtain a scaling parameter, such
|
964 |
+
that the inner features of the average value among the 2NQ rows of X can be learned. The
|
965 |
+
scaling parameter is then scaled to the range of (0, 1) using a sigmoid function, which can be
|
966 |
+
written as follows
|
967 |
+
ϑ =
|
968 |
+
1
|
969 |
+
1 + e
|
970 |
+
−fθF C1 (ι),
|
971 |
+
(50)
|
972 |
+
|
973 |
+
17
|
974 |
+
where ϑ is the scaling vector and fθF C1(·) is the function expression of the two-layer FC network
|
975 |
+
with parameter θFC1. Next, ϑ is multiplied by ι to get the threshold as
|
976 |
+
κST = ϑ ⊙ ι,
|
977 |
+
(51)
|
978 |
+
where ⊙ is the Hadamard product operator. This operation is inspired by the fact that the
|
979 |
+
threshold for soft thresholding must be positive and not too large [31]. If the threshold is larger
|
980 |
+
than the largest value of fθc(X), then the output of soft thresholding will all be zeros, and thus
|
981 |
+
the useful information will be removed. Finally, the obtained threshold κST is subtracted by
|
982 |
+
fθc(X) and fed into a ReLU activation function as
|
983 |
+
o = max(0, fθc(X) − κST),
|
984 |
+
(52)
|
985 |
+
where o denotes the output of the soft-thresholding denoising component. We can observe from
|
986 |
+
(52) that by keeping κST in a reasonable range, the useful information can be preserved while
|
987 |
+
the noise information is eliminated. It is worth noting that, rather than being manually set
|
988 |
+
by experts, such a threshold can be learned automatically in the proposed soft-thresholding
|
989 |
+
denoising component, removing the need for the expertise of signal processing and the statistical
|
990 |
+
characteristic of X.
|
991 |
+
3
|
992 |
+
7
|
993 |
+
1
|
994 |
+
5
|
995 |
+
9
|
996 |
+
8
|
997 |
+
5
|
998 |
+
9
|
999 |
+
8
|
1000 |
+
0
|
1001 |
+
0
|
1002 |
+
0
|
1003 |
+
5
|
1004 |
+
9
|
1005 |
+
8
|
1006 |
+
Pooling Indices
|
1007 |
+
MaxPool
|
1008 |
+
MaxUnpool
|
1009 |
+
Filter
|
1010 |
+
Fig. 5. Illustration of the MaxPool-MaxUnpool process.
|
1011 |
+
Hard-thresholding Decision: It is challenging to directly implement the threshold-based
|
1012 |
+
strategy in DNNs, as (35) is non-differentiable and will stunt the backpropagation process.
|
1013 |
+
To tackle this issue, the hard-thresholding decision component elegantly uses the Maxpool
|
1014 |
+
and MaxUnpool procedures to ensure the device-level sparsity. Maxpool is a down-sampling
|
1015 |
+
technique that uses a max filter to non-overlapping subregions of the initial input [32]. For each
|
1016 |
+
region represented by the filter, we will take the max of that region and create a new output
|
1017 |
+
|
1018 |
+
18
|
1019 |
+
matrix where each element is the max of a region in the original input. Maxunpool, in contrast,
|
1020 |
+
expands the output of the maxpool operation to its original size by upsampling and padding
|
1021 |
+
with zeros. Except for the maximum position, all the rest elements in the unpooled matrix are
|
1022 |
+
supplemented with 0.
|
1023 |
+
For an intuitive explanation, we illustrate the process of Maxpool and MaxUnpool in Fig. 5.
|
1024 |
+
It can be observed from Fig. 5 that in each filter, except for the largest value that remains
|
1025 |
+
unchanged, all the rest elements become 0. Such manipulation perfectly executes the element
|
1026 |
+
selection operation in (18). By setting the filter size as Q × 1, we enforce that at most one
|
1027 |
+
non-zero row exists in the Q rows of Xn, and therefore the device-level sparsity constraint in
|
1028 |
+
(11) can be guaranteed. It should also be mentioned that the pooling procedure is only a module
|
1029 |
+
that alters the dimension size during the deep learning process, which has no parameters and
|
1030 |
+
thus has no impact on network training.
|
1031 |
+
After guaranteeing the device-level sparsity, the onus shifts to performing the threshold-based
|
1032 |
+
decisive operation in (36), i.e., determining the binary sequence α by comparing the threshold
|
1033 |
+
κi∗
|
1034 |
+
n
|
1035 |
+
n with the matrix obtained from the maxpool-maxunpool procedure Mp(Mup(o)). However,
|
1036 |
+
some issues exist when determining α. The first issue is that the threshold in (37) may not be
|
1037 |
+
precise sufficiently because it is derived under an mismatched i.i.d. assumption. To tackle this
|
1038 |
+
issue, we look afresh at (37) and find that the threshold is a function of β and τ. Since β has
|
1039 |
+
been represented by θ2 in (47), we concatenate θT,2 and τ 2
|
1040 |
+
T outputted from the last AMP layer
|
1041 |
+
and feed it into an FC layer with ReLU activation function to learn the accurate threshold, which
|
1042 |
+
is denoted by
|
1043 |
+
κHT = max(0, fθF C2(θT,2, τ 2
|
1044 |
+
T)),
|
1045 |
+
(53)
|
1046 |
+
where fθF C2 is the function expression of the FC network with parameter θFC2.
|
1047 |
+
Then, the learned threshold κHT is subtracted by Mp(Mup(o)) and forwarded into an FC
|
1048 |
+
layer with parameter θFC3 to fulfil the threshold-based decisive operation. The FC layer here
|
1049 |
+
has two functionalities: compressing the dimension from 2NQ × 1 to NQ × 1 and converting
|
1050 |
+
the κHT-Mp(Mup(o)) difference into a binary sequence. Mathematically, the optimal function
|
1051 |
+
for threshold-based binary decision is the signum function denoted as
|
1052 |
+
sng(x) =
|
1053 |
+
�
|
1054 |
+
�
|
1055 |
+
�
|
1056 |
+
1,
|
1057 |
+
x > 0;
|
1058 |
+
0,
|
1059 |
+
x ≤ 0.
|
1060 |
+
(54)
|
1061 |
+
|
1062 |
+
19
|
1063 |
+
However, since sng(x) is non-differentiable, it cannot be used in DNN, necessitating the devel-
|
1064 |
+
opment of a substitute function.
|
1065 |
+
-10
|
1066 |
+
-8
|
1067 |
+
-6
|
1068 |
+
-4
|
1069 |
+
-2
|
1070 |
+
0
|
1071 |
+
2
|
1072 |
+
4
|
1073 |
+
6
|
1074 |
+
8
|
1075 |
+
10
|
1076 |
+
Input
|
1077 |
+
0
|
1078 |
+
0.1
|
1079 |
+
0.2
|
1080 |
+
0.3
|
1081 |
+
0.4
|
1082 |
+
0.5
|
1083 |
+
0.6
|
1084 |
+
0.7
|
1085 |
+
0.8
|
1086 |
+
0.9
|
1087 |
+
1
|
1088 |
+
Output
|
1089 |
+
Region with positive output
|
1090 |
+
and negative input
|
1091 |
+
Sgn (Optimal)
|
1092 |
+
Sigmoid
|
1093 |
+
Proposed (m=1)
|
1094 |
+
Proposed (m=5)
|
1095 |
+
Proposed (m=10)
|
1096 |
+
Fig. 6. The curves of the optimal signum, sigmoid, and hard-thresholding decision functions.
|
1097 |
+
When it comes to DL-based binary decisions, the sigmoid function is a popular choice and
|
1098 |
+
has been widely used in the literature [33], as it can map the input to the interval within [0, 1].
|
1099 |
+
The sigmoid function, nevertheless, is still inapplicable to the hard-thresholding decision module.
|
1100 |
+
The reasons are as follows: (i) The sigmoid function returns a continuous value between 0 and
|
1101 |
+
1, implying that a threshold is further required to distinguish the outputted value as 0 or 1.
|
1102 |
+
However, it is usually non-trivial to design an appropriate threshold; (ii) According to (36), the
|
1103 |
+
output of the threshold-based decision should be strictly 0 with negative input. However, as
|
1104 |
+
shown in Fig. 6, there is a region where the output is still positive with negative input in the
|
1105 |
+
sigmoid function, which may introduce additional errors. To solve the above issues, we devise a
|
1106 |
+
novel hard-thresholding decision function, whose core idea is to cascade the ReLU function with
|
1107 |
+
tahn function and introduce a multiplier ϱ to approximate the cascaded function as a signum
|
1108 |
+
function. The proposed hard-thresholding decision function is given by
|
1109 |
+
fϱ(x) = max(0, eϱx − e−ϱx
|
1110 |
+
eϱx + e−ϱx).
|
1111 |
+
(55)
|
1112 |
+
By cascading the ReLU function with tahn function, we not only ensure that the output of
|
1113 |
+
the threshold-based decision is strictly 0 with negative input, but also guarantee the output
|
1114 |
+
with positive input approximates to 1 with the increment of ϱ. The optimal signum, sigmoid,
|
1115 |
+
and hard-thresholding decision functions are plotted in Fig. 6. The figure shows that with the
|
1116 |
+
increase of ϱ, fϱ(·) gradually approximates to sng(x), validating the rationality of the proposed
|
1117 |
+
hard-thresholding decision function.
|
1118 |
+
|
1119 |
+
20
|
1120 |
+
Remark 2: Although we restrict the application of the hard-thresholding decision component
|
1121 |
+
to the non-coherent transmission in mMTC, the proposed component can be used in any other
|
1122 |
+
scenarios where the signal has a special sparsity structure, such as the spatial modulation system.
|
1123 |
+
Meanwhile, the devised hard-thresholding decision function can also be used in any bit-level
|
1124 |
+
detector. That is, the hard-thresholding decision component is a plug-and-play module with a
|
1125 |
+
wide range of applications.
|
1126 |
+
V. THE IMPLEMENTATION OF DL-MAMPNET
|
1127 |
+
A. Parameter Initialization
|
1128 |
+
In deep learning, parameter initialization plays a critical role in speeding up convergence and
|
1129 |
+
achieving lower error rates. Choosing proper initialization values is especially important for the
|
1130 |
+
proposed DL-mAMPnet, as the DL-mAMPnet is built on the AMP algorithm and thus should
|
1131 |
+
preserve some essential features to ensure performance and interpretability. There are mainly
|
1132 |
+
three items needed to be considered for parametrization: the trainable matrices At and Bt, the
|
1133 |
+
denoiser parameter set Θt, and the refinement module parameters θRM = {θFC1, θFC2, θFC3, θC}.
|
1134 |
+
1) Initializing At and Bt: It can be observed from (44)-(45) that the DL-mAMPnet imple-
|
1135 |
+
ments a generalization of the AMP algorithm in (14)-(15), wherein the matched filters (S, SH
|
1136 |
+
n )
|
1137 |
+
manifest as (At, Bt) at iteration t. However, such generalization does not enforce Bt = AH
|
1138 |
+
t and
|
1139 |
+
thus may not preserve the independent-Gaussian nature of the denoiser input (19). According to
|
1140 |
+
the analysis in [30], the desired nature maintains when At = υtS with υt > 0. Therefore, At is
|
1141 |
+
parameterized as υtS and (44)-(45) can be rewritten as
|
1142 |
+
˜
|
1143 |
+
Xt = υtηt( ˜
|
1144 |
+
Xt−1 + Bt ˜Rt−1; Θt),
|
1145 |
+
(56)
|
1146 |
+
˜Rt = ˜Y − S ˜
|
1147 |
+
Xt + υt ˜Rt−1
|
1148 |
+
LM
|
1149 |
+
2NQM
|
1150 |
+
�
|
1151 |
+
j=1
|
1152 |
+
[ηt( ˜
|
1153 |
+
Xt−1 + Bt ˜Rt−1; Θt)]
|
1154 |
+
′
|
1155 |
+
j,
|
1156 |
+
(57)
|
1157 |
+
the derivation of which can be found in [30] and is omitted here for brevity. In this paper, we
|
1158 |
+
initialize Bt = ˜ST and υt = 1, since such initialization can greatly expedite the convergence of
|
1159 |
+
the training process [30].
|
1160 |
+
2) Initializing Θt: For θ1, we initialize each element as log(Q−ϵ
|
1161 |
+
ϵ ), i.e., initialize that each
|
1162 |
+
pilot sequence has the same active probability. This is because we have no prior information
|
1163 |
+
about the device activity and the transmitted pilot sequence index. By adopting such a uniform
|
1164 |
+
initialization, the initial θ1 will have the minimum Euclidean distance from the actual value.
|
1165 |
+
|
1166 |
+
21
|
1167 |
+
For example, consider a device with a 2-bit message and active indicator {1, 0, 0, 0}. If we start
|
1168 |
+
with a mismatched one-hot vector, then the Euclidean distance will be
|
1169 |
+
√
|
1170 |
+
2. If we initialize αn
|
1171 |
+
as {1
|
1172 |
+
4, 1
|
1173 |
+
4, 1
|
1174 |
+
4, 1
|
1175 |
+
4}, then the Euclidean distance will be
|
1176 |
+
�
|
1177 |
+
3
|
1178 |
+
4. Therefore, the uniform initialization can
|
1179 |
+
accelerate the convergence as a shorter Euclidean distance may lead to faster convergence.
|
1180 |
+
The initial value of θ2 can be computed from the received signal strength. Recall that each
|
1181 |
+
pilot sequence has a unit norm and hn ∼ CN(0, βnIM), each element of the initial θ2 is roughly
|
1182 |
+
given by || ˜Y ||2
|
1183 |
+
2/
|
1184 |
+
√
|
1185 |
+
2K.
|
1186 |
+
3) Initializing θRM: For all parameters in the refinement module, we adopt the He initial-
|
1187 |
+
ization [34] as it has been mathematically proved to be the best weight initialization strategy for
|
1188 |
+
the ReLU activation function [35].
|
1189 |
+
B. Parameter Training
|
1190 |
+
1) Training Algorithm: Aside from the network structure and parameter initialization, the
|
1191 |
+
training algorithm also determines the performance of the DL-mAMPnet. The standard training
|
1192 |
+
strategy is the end-to-end training where all the parameters are optimized simultaneously by
|
1193 |
+
following the back-propagation rule. However, the end-to-end training is not appropriate for the
|
1194 |
+
DL-mAMPnet due to the following reasons: (i) The AMP algorithm aims to provide an estimate
|
1195 |
+
ˆ
|
1196 |
+
X(Y ) based on Y that minimizes the MSE EXY || ˆ
|
1197 |
+
X(Y )−X||2
|
1198 |
+
2. If the DL-mAMPnet is trained
|
1199 |
+
to learn the direct mapping from Y to α, the MSE optimality of the AMP layers may not be
|
1200 |
+
achieved; (ii) Even if the AMP layers and the refinement module are trained separately, the AMP
|
1201 |
+
layers can still easily converge to a bad local optimal solution due to overfitting [36].
|
1202 |
+
For these reasons, we propose a layer-wise training strategy, the idea behind which is to
|
1203 |
+
decouple the training of each layer. The details are given in Algorithm 1. There are totally
|
1204 |
+
T + 2 phases in the layer-wise training. In the first phase, we train the learnable parameters of
|
1205 |
+
the first AMP layer. Then in the t phase, we train the first t AMP layers with the parameters
|
1206 |
+
of the first t − 1 AMP layers fixed as the parameters learned by the first t − 1 phases. In the
|
1207 |
+
T + 1 phase, we train the whole network with only the parameters of the refinement module
|
1208 |
+
is learnable, while the parameters of the AMP layers are fixed as the parameters learned by
|
1209 |
+
the first T phases. Finally, in the last phase, all the parameters are initialized as the parameters
|
1210 |
+
learned during the first T + 1 phases and then trained jointly.
|
1211 |
+
|
1212 |
+
22
|
1213 |
+
Algorithm 1 Parameter training of the DL-mAMPnet via layer-wise training strategy
|
1214 |
+
Input: Training dataset DAMP, DRM;
|
1215 |
+
Output: Trained parameter {υt, Bt, Θt}T
|
1216 |
+
t=1 and θRM;
|
1217 |
+
Initialize parameters according to Section IV-B;
|
1218 |
+
for t = 1 to T do
|
1219 |
+
Learn {υt, Bt, Θt}t with fixed {υt, Bt, Θt}t−1
|
1220 |
+
t=1 based on the loss function (58);
|
1221 |
+
end for
|
1222 |
+
Learn θRM with fixed {υt, Bt, Θt}T
|
1223 |
+
t=1 based on the loss function (59);
|
1224 |
+
Re-learn {υt, Bt, Θt}T
|
1225 |
+
t=1 and θRM based on the loss function (59);
|
1226 |
+
return {υt, Bt, Θt}T
|
1227 |
+
t=1 and θRM.
|
1228 |
+
The training dataset DAMP for the first T phases comprises 100, 000 pairs of ˜
|
1229 |
+
X and ˜Y , and
|
1230 |
+
the corresponding loss function is the MSE loss
|
1231 |
+
Lt( ˜Y ) = || ˜
|
1232 |
+
Xt( ˜Y ) − ˜
|
1233 |
+
X||2
|
1234 |
+
2, t = [1, · · · , T].
|
1235 |
+
(58)
|
1236 |
+
The training dataset DRM for the last 2 phases has 100, 000 pairs of α and ˜Y , and the loss
|
1237 |
+
function is the binary cross entropy loss
|
1238 |
+
Lt( ˜Y ) =
|
1239 |
+
1
|
1240 |
+
NQ
|
1241 |
+
NQ
|
1242 |
+
�
|
1243 |
+
i=1
|
1244 |
+
�
|
1245 |
+
α( ˜Y )i log αi + (1 − α( ˜Y )i) log(1 − αi)
|
1246 |
+
�
|
1247 |
+
, t = [T + 1, T + 2].
|
1248 |
+
(59)
|
1249 |
+
The DL-mAMPnet is trained epoch by epoch with the training dataset using the Adam optimizer,
|
1250 |
+
while within an epoch, the whole training dataset is shuffled and split into batches with the size
|
1251 |
+
of 500.1
|
1252 |
+
2) Training Dataset: The training dataset is synthetically generated as follows: (i) Generating
|
1253 |
+
αn: K active devices are randomly selected among N devices. Then, each active device is
|
1254 |
+
randomly assigned with a Q-dimensional one-hot vector, and each inactive device is assigned
|
1255 |
+
with a Q-dimensional zero vector; (ii) Generating Xn: The uplink channel of device n, i.e., hn, is
|
1256 |
+
first generated according to (1). Then Xn is obtained by multiplying hn and αn; (iii) Generating
|
1257 |
+
Y : The pilot sequence Sn is generated by sampling from complex Gaussian distribution with
|
1258 |
+
zero mean and variance. Given Xn and Sn, Y can be directly obtained according to (6).
|
1259 |
+
1It should be mentioned that the number of epochs and the learning rate are different for each phase, which are empirically
|
1260 |
+
determined in Section V.
|
1261 |
+
|
1262 |
+
23
|
1263 |
+
VI. SIMULATION RESULTS
|
1264 |
+
In this section, extensive simulations are provided to verify the effectiveness of the proposed
|
1265 |
+
algorithm. The setup is as follows unless otherwise stated. We consider a mMTC system with
|
1266 |
+
N = 100 devices for illustration purpose, although the proposed algorithm can be used for
|
1267 |
+
a much larger-scale system. Each device accesses the BS independently with probability ϵ =
|
1268 |
+
0.1 at each coherence block. The large-scale fading coefficient for device n is βn = 128.1 −
|
1269 |
+
36.7 log10(dn) in dB, where dn is the distance between device n and the BS that follows a uniform
|
1270 |
+
distribution within [0.05, 1] km. The small-scale fading coefficient for each device follows the
|
1271 |
+
i.i.d. multivariate complex Gaussian distribution with zero mean and unit variance. The power
|
1272 |
+
spectral density of the AWGN at the BS is assumed to be −169 dBm/Hz [8] and the bandwidth
|
1273 |
+
of the wireless channel is 1 MHz.
|
1274 |
+
The number of AMP layers in the DL-mAMPnet is set to be T = 4. The training epochs and
|
1275 |
+
learning rate for each training phase are set to be {2, 000, 1, 500, 1, 000, 1, 000, 1, 500, 5, 000} and
|
1276 |
+
{2 × 10−5, 2 × 10−5, 2 × 10−5, 2 × 10−5, 1 × 10−5, 1 × 10−5}.2 We train the DL-mAMPnet with
|
1277 |
+
80, 000 training samples and test with 20, 000 data samples, which are randomly drawn from
|
1278 |
+
DAMP for the first 4 phases and DRM for the last 2 phases. The DL-mAMPnet is trained and
|
1279 |
+
tested by on an x86 PC with one Nvidia GeForce GTX 1080 Ti graphics card, and Pytorch 1.1.0
|
1280 |
+
is employed as the backend. The traditional AMP-based algorithm with TAMP = 50 iterations
|
1281 |
+
and the covariance-based method with TCov = 50 iterations [16] are employed as the benchmark
|
1282 |
+
and evaluated on the same dataset. In addition, the SER is adopted as the performance metric:
|
1283 |
+
SER = 1
|
1284 |
+
N
|
1285 |
+
�N
|
1286 |
+
n=1 I( ˆαn ̸= αn), where ˆαn and αn denote the estimated pilot sequence activity for
|
1287 |
+
device n and its ground truth, respectively.
|
1288 |
+
A. Performance of the DL-mAMPnet
|
1289 |
+
Fig. 7 depicts the SER versus L with different values of M. It is observed that both the SER
|
1290 |
+
of the DL-mAMPnet and AMP-based algorithm decrease as L and M increase. Although the
|
1291 |
+
SER of the covariance-based algorithm is lowest when L is small, it becomes saturated when L
|
1292 |
+
exceeds some point, e.g., L = 40 when M = 16. This is mainly due to the suboptimality of the
|
1293 |
+
2All the parameters are empirically determined using the general workflow, where the training starts with relatively small
|
1294 |
+
values and increases the values until the learning performance cannot be further improved.
|
1295 |
+
|
1296 |
+
24
|
1297 |
+
10
|
1298 |
+
20
|
1299 |
+
30
|
1300 |
+
40
|
1301 |
+
50
|
1302 |
+
60
|
1303 |
+
70
|
1304 |
+
80
|
1305 |
+
90
|
1306 |
+
100
|
1307 |
+
Pilot Sequence Length: L
|
1308 |
+
10-4
|
1309 |
+
10-3
|
1310 |
+
10-2
|
1311 |
+
10-1
|
1312 |
+
100
|
1313 |
+
SER
|
1314 |
+
AMP, M=8
|
1315 |
+
AMP, M=16
|
1316 |
+
AMP, M=32
|
1317 |
+
Covariance, M=8
|
1318 |
+
Covariance, M=16
|
1319 |
+
Covariance, M=32
|
1320 |
+
DL-mAMPnet, M=8
|
1321 |
+
DL-mAMPnet, M=16
|
1322 |
+
DL-mAMPnet, M=32
|
1323 |
+
Fig. 7. SER performance versus the pilot sequence length L for J = 1 bit.
|
1324 |
+
0
|
1325 |
+
5
|
1326 |
+
10
|
1327 |
+
15
|
1328 |
+
20
|
1329 |
+
25
|
1330 |
+
30
|
1331 |
+
35
|
1332 |
+
40
|
1333 |
+
Number of Receiving Antennas: M
|
1334 |
+
10-4
|
1335 |
+
10-3
|
1336 |
+
10-2
|
1337 |
+
10-1
|
1338 |
+
100
|
1339 |
+
SER
|
1340 |
+
AMP, L=50
|
1341 |
+
AMP, L=60
|
1342 |
+
AMP, L=70
|
1343 |
+
Covariance, L=50
|
1344 |
+
Covariance, L=60
|
1345 |
+
Covariance, L=70
|
1346 |
+
DL-mAMPnet,L=50
|
1347 |
+
DL-mAMPnet,L=60
|
1348 |
+
DL-mAMPnet,L=70
|
1349 |
+
Fig. 8. SER performance versus the number of receiving antennas M for J = 1 bit.
|
1350 |
+
fixed threshold.3 Meanwhile, the proposed DL-mAMPnet notably outperforms the AMP-based
|
1351 |
+
algorithm by a large margin. For example, the proposed DL-mAMPnet achieves more than 10
|
1352 |
+
pilot length gain over the AMP-based algorithm when L is larger than 70, which indicates that
|
1353 |
+
the proposed DL-mAMPnet can reduce the required pilot sequence length, lowering the difficulty
|
1354 |
+
of pilot design and adapting to fast-changing channels. Moreover, although for any M, the SERs
|
1355 |
+
of both the DL-mAMPnet and AMP-based algorithm decrease over L, the reduction is faster
|
1356 |
+
3As observed from (37), the threshold is variable and related to system parameters such as signal power and receiving antenna
|
1357 |
+
numbers, whereas the covariance-based algorithm adopts a fixed threshold. Since there is no concrete method to design such a
|
1358 |
+
fixed threshold, we empirically set the threshold of the covariance-based algorithm to be βn/2 in this paper.
|
1359 |
+
|
1360 |
+
25
|
1361 |
+
10
|
1362 |
+
20
|
1363 |
+
30
|
1364 |
+
40
|
1365 |
+
50
|
1366 |
+
60
|
1367 |
+
70
|
1368 |
+
80
|
1369 |
+
90
|
1370 |
+
100
|
1371 |
+
Pilot Sequence Length: L
|
1372 |
+
10-3
|
1373 |
+
10-2
|
1374 |
+
10-1
|
1375 |
+
100
|
1376 |
+
SER
|
1377 |
+
AMP, M=16,J=1bit
|
1378 |
+
AMP, M=16,J=2bits
|
1379 |
+
Covariance, M=16,J=1bits
|
1380 |
+
Covariance, M=16,J=2bits
|
1381 |
+
DL-mAMPnet, M=16,J=1bit
|
1382 |
+
DL-mAMPnet,M=16,J=2bits
|
1383 |
+
Fig. 9. SER performance versus the pilot sequence length L with different lengths of transmitted messages J.
|
1384 |
+
when M is 32 as compared to that when M is 8, which shows that increasing the number of
|
1385 |
+
receiving antennas can further reduce the required pilot sequence length.
|
1386 |
+
Fig. 8 shows the SER versus M for various values of L. We observe that for the DL-mAMPnet
|
1387 |
+
and AMP-based algorithm, the SER drops effectively as M increases, whereas for the covariance-
|
1388 |
+
based algorithm, there are error floors in the SER. Moreover, the DL-mAMPnet needs fewer
|
1389 |
+
receiving antennas to achieve the same performance as the AMP-based algorithm, implying
|
1390 |
+
that the proposed DL-mAMPnet can reduce demand for receiving antennas, resulting in lower
|
1391 |
+
deployment cost and energy consumption.
|
1392 |
+
Fig. 9 plots the SER versus L, with 2 different lengths of transmitted messages, i.e., J = 1 bit
|
1393 |
+
and J = 2 bits. The number of receiving antennas is M = 16. It can be seen that the SERs of all
|
1394 |
+
three algorithms increase as the length of transmitted messages increases, which implies that the
|
1395 |
+
performance of both algorithms deteriorates when more messages are transmitted. An important
|
1396 |
+
point is that as the message length increases, the performance gap between the proposed DL-
|
1397 |
+
mAMPnet and the other two algorithms increases, indicating the potential of the DL-mAMPnet
|
1398 |
+
to handle long packet size.
|
1399 |
+
B. Visualization of the DL-mAMPnet
|
1400 |
+
To offer more insights of the proposed DL-mAMPnet, we present the visualization of the
|
1401 |
+
outputs of each component of a well-trained DL-mAMPnet. For clarity, we only present the
|
1402 |
+
case where N = 10 devices transmit 1-bit message with ϵ = 0.1 active probability, L = 10 pilot
|
1403 |
+
|
1404 |
+
26
|
1405 |
+
0123
|
1406 |
+
0
|
1407 |
+
4
|
1408 |
+
8
|
1409 |
+
12
|
1410 |
+
16
|
1411 |
+
20
|
1412 |
+
24
|
1413 |
+
28
|
1414 |
+
32
|
1415 |
+
36
|
1416 |
+
(a)
|
1417 |
+
0123
|
1418 |
+
0
|
1419 |
+
4
|
1420 |
+
8
|
1421 |
+
12
|
1422 |
+
16
|
1423 |
+
20
|
1424 |
+
24
|
1425 |
+
28
|
1426 |
+
32
|
1427 |
+
36
|
1428 |
+
(b)
|
1429 |
+
0
|
1430 |
+
0
|
1431 |
+
4
|
1432 |
+
8
|
1433 |
+
12
|
1434 |
+
16
|
1435 |
+
20
|
1436 |
+
24
|
1437 |
+
28
|
1438 |
+
32
|
1439 |
+
36
|
1440 |
+
(c)
|
1441 |
+
0
|
1442 |
+
0
|
1443 |
+
4
|
1444 |
+
8
|
1445 |
+
12
|
1446 |
+
16
|
1447 |
+
(d)
|
1448 |
+
0
|
1449 |
+
0
|
1450 |
+
4
|
1451 |
+
8
|
1452 |
+
12
|
1453 |
+
16
|
1454 |
+
(e)
|
1455 |
+
Fig. 10.
|
1456 |
+
A visualization of a well-trained DL-mAMPnet. (a) ˜
|
1457 |
+
XT , the output of AMP layers; (b) The ground truth ˜
|
1458 |
+
X; (c)
|
1459 |
+
Mp(Mup(o)), the output of the maxpool- maxunpool procedure; (d) ˆα, the output of the refinement module; (e) The ground
|
1460 |
+
truth α.
|
1461 |
+
sequence length, and M = 2 receiving antennas. For visualization, we transform the outputs of
|
1462 |
+
each component to the reverse grayscale images. Specifically, the elements of each output matrix
|
1463 |
+
are normalized to an interval within [0, 1], where 0 and 1 are represented by white color and
|
1464 |
+
black color, respectively. It should be mentioned that we take the absolute value of ˜
|
1465 |
+
X and ˜
|
1466 |
+
XT
|
1467 |
+
to show the signal strength difference more intuitively.
|
1468 |
+
The output of the AMP layers and its ground truth are shown in Fig. 10(a) and Fig. 10(b),
|
1469 |
+
respectively. It can be seen that the non-zero rows of ˜
|
1470 |
+
X are correctly recovered, paving the way
|
1471 |
+
for the subsequent refinement progress. Then, the output of the maxpool-maxunpool procedure
|
1472 |
+
is visualized in Fig. 10(c), where the largest of the two adjacent rows is retained and the other
|
1473 |
+
becomes 0, demonstrating the validity of the maxpool-maxunpool procedure in ensuring the
|
1474 |
+
device-level sparsity. Fig. 10(d) and Fig. 10(e) are the visualizations of ˆα and α, where we
|
1475 |
+
find that the pilot sequence activity is perfectly estimated by the well-trained DL-mAMPnet.
|
1476 |
+
Moreover, it is observed from Fig. 10(c) and Fig. 10(e) that the pilot sequence activity is correctly
|
1477 |
+
reserved in Fig. 10(c) (the 1st, 4th, 21st, and 24th rows), which indicates the effectiveness of
|
1478 |
+
the proposed soft-thresholding denoising component.
|
1479 |
+
C. Computational Complexity Analysis
|
1480 |
+
Finally, we analyze the computational complexities of the traditional AMP-based algorithm
|
1481 |
+
and DL-mAMPnet.
|
1482 |
+
|
1483 |
+
27
|
1484 |
+
For the traditional AMP-based algorithm, the computational complexity mainly comes from
|
1485 |
+
the matrix multiplication in (14)-(15) [8]. Since SH
|
1486 |
+
n
|
1487 |
+
∈ CQ×L, Rt ∈ CL×M, S ∈ CL×NQ,
|
1488 |
+
and Xt+1 ∈ CNQ×M, the computational complexity for N devices and TAMP iterations is
|
1489 |
+
O(4TAMP(NQLM + NQLM)) = O(8TAMPNQLM), where the proportional constant “4”
|
1490 |
+
appears because a complex multiplication requires 4 real multiplications, the former “NQLM”
|
1491 |
+
comes from the multiplication between SH
|
1492 |
+
n and Rt for N devices and the latter “NQLM”
|
1493 |
+
comes from the multiplication between S and Xt+1. After the iterative process, the AMP-based
|
1494 |
+
algorithm requires the element selection operation (i.e., (35)) whose computational complexity
|
1495 |
+
is O(4NQM), and the threshold calculation (i.e., (37)) whose computational complexity is
|
1496 |
+
O(4NQM). Taking all the operations into account, the computational complexity of the AMP-
|
1497 |
+
based algorithm is given by O(8TAMPNQLM).
|
1498 |
+
For the proposed DL-mAMPnet, we focus on the computational complexity of online imple-
|
1499 |
+
mentation. The computational complexity of the AMP layers comes from the matrix multipli-
|
1500 |
+
cation Bt ˜Rt−1 and At ˜
|
1501 |
+
Xt, which is O(8TDLNQLM 2) with TDL denoting the number of AMP
|
1502 |
+
layers. For the refinement module, the computational complexity is mainly resulted from the FC
|
1503 |
+
and convolutional layers. For a FC layer with Nl−1 input and N1 output, its computational
|
1504 |
+
complexity is given by O(Nl−1N1). For a convolutional layer with a H × W input and a
|
1505 |
+
Hf × Wf filter, its computational complexity can be expressed as O(HWHfWf). Therefore,
|
1506 |
+
the total computational complexity of the refinement module is O(4N 2Q2M). Consequently, the
|
1507 |
+
computational complexity of DL-mAMPnet is O(8TDLNQLM 2 + 4N 2Q2M).
|
1508 |
+
From the above discussions, it seems that the proposed DL-mAMPnet can achieve better
|
1509 |
+
performance at the expense of a higher computational complexity compared to the AMP-based
|
1510 |
+
algorithm. However, as observed in Fig. 7-Fig. 9, the DL-mAMPnet with TDL = 4 AMP layers
|
1511 |
+
outperforms the AMP-based algorithm with TAMP = 50 iterations, indicating that the proposed
|
1512 |
+
DL-mAMPnet may need less computational complexity to achieve the same SER performance
|
1513 |
+
with the AMP-based algorithm.
|
1514 |
+
VII. CONCLUSION
|
1515 |
+
This paper has proposed a novel DL-based algorithm, termed DL-mAMPnet, for the joint
|
1516 |
+
device activity and data detection in mMTC with a single-phase non-coherent scheme. Trainable
|
1517 |
+
parameters have been added in the DL-mAMPnet to compensate for the inaccuracy caused by
|
1518 |
+
the i.i.d. assumption in the traditional AMP algorithm. A refinement module has been further
|
1519 |
+
|
1520 |
+
28
|
1521 |
+
designed to enhance the SER performance and guarantee the device-level sparsity by exploiting
|
1522 |
+
the correlated sparsity pattern. The proposed algorithm can be applied to scenarios where massive
|
1523 |
+
users intermittently transmit small packets, e.g., smart home and industrial control. For the future
|
1524 |
+
work, we will investigate the pilot sequence design scheme to maintain orthogonality and mitigate
|
1525 |
+
the inter-device interference.
|
1526 |
+
APPENDIX A
|
1527 |
+
DERIVATION OF MMSE DENOISER (21)
|
1528 |
+
To enable the derivation of the conditional probability PXn|Zn, we assume xq
|
1529 |
+
n is independent
|
1530 |
+
with each other, and thus we have
|
1531 |
+
Pxq
|
1532 |
+
n =
|
1533 |
+
�
|
1534 |
+
1 − ϵ
|
1535 |
+
Q
|
1536 |
+
�
|
1537 |
+
δ + ϵ
|
1538 |
+
Q
|
1539 |
+
exp(−xq
|
1540 |
+
n
|
1541 |
+
H(βnIM)−1xq
|
1542 |
+
n)
|
1543 |
+
πM|βnIM|
|
1544 |
+
.
|
1545 |
+
(60)
|
1546 |
+
According to (20), the likelihood of observing zq
|
1547 |
+
n given xq
|
1548 |
+
n is
|
1549 |
+
Pzq
|
1550 |
+
n|xq
|
1551 |
+
n = exp(−(zq
|
1552 |
+
n − xq
|
1553 |
+
n)HΣ−1(zq
|
1554 |
+
n − xq
|
1555 |
+
n))
|
1556 |
+
πM|Σ|
|
1557 |
+
.
|
1558 |
+
(61)
|
1559 |
+
Denoting k as the proportional constant, Pxq
|
1560 |
+
n|zq
|
1561 |
+
n can be computed using the Bayes’ formula
|
1562 |
+
as follows
|
1563 |
+
Pxq
|
1564 |
+
n|zq
|
1565 |
+
n = kPzq
|
1566 |
+
n|xq
|
1567 |
+
nPxq
|
1568 |
+
n
|
1569 |
+
= k
|
1570 |
+
�
|
1571 |
+
(1 − ϵ
|
1572 |
+
Q)δ + ϵ
|
1573 |
+
Q
|
1574 |
+
exp(−xq
|
1575 |
+
n
|
1576 |
+
H(βnIM)−1xq
|
1577 |
+
n)
|
1578 |
+
πM|βnIM|
|
1579 |
+
� �exp(−(zq
|
1580 |
+
n − xq
|
1581 |
+
n)HΣ−1(zq
|
1582 |
+
n − xq
|
1583 |
+
n))
|
1584 |
+
πM|Σ|
|
1585 |
+
�
|
1586 |
+
= k
|
1587 |
+
�
|
1588 |
+
(1 − ϵ
|
1589 |
+
Q)exp(−zq
|
1590 |
+
n
|
1591 |
+
HΣ−1zq
|
1592 |
+
n)
|
1593 |
+
πM|Σ|
|
1594 |
+
δ + ϵ
|
1595 |
+
Q
|
1596 |
+
exp(−xq
|
1597 |
+
n
|
1598 |
+
H(βnIM)−1xq
|
1599 |
+
n − (zq
|
1600 |
+
n − xq
|
1601 |
+
n)HΣ−1(zq
|
1602 |
+
n − xq
|
1603 |
+
n))
|
1604 |
+
π2M|βnIM||Σ|
|
1605 |
+
�
|
1606 |
+
.
|
1607 |
+
(62)
|
1608 |
+
Note that
|
1609 |
+
xq
|
1610 |
+
n
|
1611 |
+
H(βnIM)−1xq
|
1612 |
+
n + (zq
|
1613 |
+
n − xq
|
1614 |
+
n)HΣ−1(zq
|
1615 |
+
n − xq
|
1616 |
+
n) = (xq
|
1617 |
+
n − ζ)HΞ−1(xq
|
1618 |
+
n − ζ) + zq
|
1619 |
+
n
|
1620 |
+
H∆−1zq
|
1621 |
+
n,
|
1622 |
+
where Ξ = ( 1
|
1623 |
+
βnIM + Σ−1), ζ = ΞΣ−1zq
|
1624 |
+
n, and ∆ = βnIM + Σ, (62) can be rewritten as
|
1625 |
+
Pxq
|
1626 |
+
n|zq
|
1627 |
+
n
|
1628 |
+
= k
|
1629 |
+
�
|
1630 |
+
(1 − ϵ
|
1631 |
+
Q)exp(−zq
|
1632 |
+
n
|
1633 |
+
HΣ−1zq
|
1634 |
+
n)
|
1635 |
+
πM|Σ|
|
1636 |
+
δ + ϵ
|
1637 |
+
Q
|
1638 |
+
exp
|
1639 |
+
�
|
1640 |
+
−(xq
|
1641 |
+
n − ζ)HΞ−1(xq
|
1642 |
+
n − ζ) − zq
|
1643 |
+
n
|
1644 |
+
H∆−1zq
|
1645 |
+
n
|
1646 |
+
�
|
1647 |
+
π2M|βnIM||Σ|
|
1648 |
+
�
|
1649 |
+
.
|
1650 |
+
(63)
|
1651 |
+
|
1652 |
+
29
|
1653 |
+
Since
|
1654 |
+
�
|
1655 |
+
Pxq
|
1656 |
+
n|zq
|
1657 |
+
n dxq
|
1658 |
+
n = 1, k can be obtained by integrating (63) out. Accordingly, we have
|
1659 |
+
k =
|
1660 |
+
�
|
1661 |
+
(1 − ϵ
|
1662 |
+
Q)exp(−zq
|
1663 |
+
n
|
1664 |
+
HΣ−1zq
|
1665 |
+
n)
|
1666 |
+
πM|Σ|
|
1667 |
+
+ ϵ
|
1668 |
+
Q
|
1669 |
+
exp
|
1670 |
+
�
|
1671 |
+
−zq
|
1672 |
+
n
|
1673 |
+
H∆−1zq
|
1674 |
+
n
|
1675 |
+
�
|
1676 |
+
|Ξ|
|
1677 |
+
πM|βnIM||Σ|
|
1678 |
+
�−1
|
1679 |
+
(a)
|
1680 |
+
=
|
1681 |
+
�
|
1682 |
+
(1 − ϵ
|
1683 |
+
Q)exp(−zq
|
1684 |
+
n
|
1685 |
+
HΣ−1zq
|
1686 |
+
n)
|
1687 |
+
πM|Σ|
|
1688 |
+
+ ϵ
|
1689 |
+
Q
|
1690 |
+
exp
|
1691 |
+
�
|
1692 |
+
−zq
|
1693 |
+
n
|
1694 |
+
H∆−1zq
|
1695 |
+
n
|
1696 |
+
�
|
1697 |
+
πM|∆|
|
1698 |
+
�−1
|
1699 |
+
,
|
1700 |
+
(64)
|
1701 |
+
where (a) holds because | 1
|
1702 |
+
βnIM + Σ−1| = |βnIM||Σ|/|βnIM + Σ|.
|
1703 |
+
Substituting (64) into (62), Pxq
|
1704 |
+
n|zq
|
1705 |
+
n can be determined as
|
1706 |
+
Pxq
|
1707 |
+
n|zq
|
1708 |
+
n = e−(xq
|
1709 |
+
n−ζ)HΞ−1(xq
|
1710 |
+
n−ζ)ϵ|Σ| + (Q − ϵ)e−zq
|
1711 |
+
n
|
1712 |
+
H(Σ−1−∆−1)zq
|
1713 |
+
nπM|Ξ||∆|δ
|
1714 |
+
ϵπM|Ξ||Σ| + (Q − ϵ)e−zq
|
1715 |
+
n
|
1716 |
+
H(Σ−1−∆−1)zq
|
1717 |
+
nπM|Ξ||∆|
|
1718 |
+
.
|
1719 |
+
(65)
|
1720 |
+
Hence, the conditional expectation E{xq
|
1721 |
+
n|zq
|
1722 |
+
n} is given by
|
1723 |
+
E{xq
|
1724 |
+
n|zq
|
1725 |
+
n} =
|
1726 |
+
�
|
1727 |
+
xq
|
1728 |
+
nPxq
|
1729 |
+
n|zq
|
1730 |
+
n dxq
|
1731 |
+
n =
|
1732 |
+
ζϵ|Σ|
|
1733 |
+
ϵ|Σ| + (Q − ϵ)e−zq
|
1734 |
+
n
|
1735 |
+
H(Σ−1−∆−1)zq
|
1736 |
+
n|∆|
|
1737 |
+
=
|
1738 |
+
βn(βnIM + Σ)−1zq
|
1739 |
+
n
|
1740 |
+
1 + Q−ϵ
|
1741 |
+
ϵ |IM + βnΣ−1|e−zq
|
1742 |
+
n
|
1743 |
+
H(Σ−1−(Σ+βnIM)−1)zq
|
1744 |
+
n .
|
1745 |
+
(66)
|
1746 |
+
The MMSE-optimal denoiser in (21)-(25) can be straightforwardly obtained from (66) through
|
1747 |
+
simple mathematical transformation.
|
1748 |
+
REFERENCES
|
1749 |
+
[1] N. H. Mahmood et al., “White paper on critical and massive machine type communication towards 6G,” arXiv preprint,
|
1750 |
+
arXiv:2004.14146, 2020.
|
1751 |
+
[2] X. Chen, D. W. K. Ng, W. Yu, E. G. Larsson, N. Al-Dhahir, and R. Schober, “Massive access for 5G and beyond,” IEEE
|
1752 |
+
J. Sel. Areas Commun., vol. 39, no. 3, pp. 615–637, Mar. 2021.
|
1753 |
+
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|
1754 |
+
massive connectivity: A future paradigm for random access protocols in the Internet of things,” IEEE Signal Process.
|
1755 |
+
Mag., vol. 35, no. 5, pp. 88–99, Sep. 2018.
|
1756 |
+
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|
1757 |
+
machine-type communications in 5G: Physical and MAC-layer solutions,” IEEE Commun. Mag., vol. 54, no. 9, pp. 59–65,
|
1758 |
+
Sep. 2016.
|
1759 |
+
[5] M. B. Shahab, R. Abbas, M. Shirvanimoghaddam, and S. J. Johnson, “Grant-free non-orthogonal multiple access for IoT:
|
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+
A survey,” IEEE Commun. Surveys Tuts., vol. 22, no. 3, pp. 1805–1838, May 2020.
|
1761 |
+
[6] E. Bjornson, E. de Carvalho, J. H. Sorensen, E. G. Larsson, and P. Popovski, “A random access protocol for pilot allocation
|
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|
1 |
+
Draft version January 3, 2023
|
2 |
+
Typeset using LATEX preprint style in AASTeX631
|
3 |
+
Disk Wind Feedback from High-mass Protostars. II. The Evolutionary Sequence
|
4 |
+
Jan E. Staff,1 Kei E. I. Tanaka,2 Jon P. Ramsey,3 Yichen Zhang,3 and Jonathan C. Tan4
|
5 |
+
1Department of Space, Earth & Environment, Chalmers University of Technology, Gothenburg, Sweden
|
6 |
+
and
|
7 |
+
College of Science and Math, University of the Virgin Islands, St Thomas, 00802, United States Virgin Islands
|
8 |
+
2Center for Astrophysics and Space Astronomy, Department of Astrophysical and Planetary Sciences, University of Colorado Boulder,
|
9 |
+
Boulder, CO 80309, USA
|
10 |
+
and
|
11 |
+
ALMA Project, National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan
|
12 |
+
3Department of Astronomy, University of Virginia, Charlottesville, VA 22904, USA
|
13 |
+
4Department of Space, Earth & Environment, Chalmers University of Technology, Gothenburg, Sweden
|
14 |
+
and
|
15 |
+
Department of Astronomy, University of Virginia, Charlottesville, VA 22904, USA
|
16 |
+
(Dated:)
|
17 |
+
ABSTRACT
|
18 |
+
Star formation is ubiquitously associated with the ejection of accretion-powered outflows that carve
|
19 |
+
bipolar cavities through the infalling envelope. This feedback is expected to be important for regulating
|
20 |
+
the efficiency of star formation from a natal pre-stellar core. These low-extinction outflow cavities
|
21 |
+
greatly affect the appearance of a protostar by allowing the escape of shorter wavelength photons.
|
22 |
+
Doppler-shifted CO line emission from outflows is also often the most prominent manifestation of
|
23 |
+
deeply embedded early-stage star formation. Here, we present 3D magneto-hydrodynamic simulations
|
24 |
+
of a disk wind outflow from a protostar forming from an initially 60 M⊙ core embedded in a high
|
25 |
+
pressure environment typical of massive star-forming regions. We simulate the growth of the protostar
|
26 |
+
from m∗ = 1 M⊙ to 26 M⊙ over a period of ∼100,000 years. The outflow quickly excavates a cavity
|
27 |
+
with half opening angle of ∼ 10◦ through the core. This angle remains relatively constant until the star
|
28 |
+
reaches 4 M⊙. It then grows steadily in time, reaching a value of ∼ 50◦ by the end of the simulation.
|
29 |
+
We estimate a lower limit to the star formation efficiency (SFE) of 0.43. However, accounting for
|
30 |
+
continued accretion from a massive disk and residual infall envelope, we estimate that the final SFE
|
31 |
+
may be as high as ∼ 0.7. We examine observable properties of the outflow, especially the evolution of
|
32 |
+
the cavity opening angle, total mass and momentum flux, and velocity distributions of the outflowing
|
33 |
+
gas, and compare with the massive protostars G35.20-0.74N and G339.88-1.26 observed by ALMA,
|
34 |
+
yielding constraints on their intrinsic properties.
|
35 |
+
1. INTRODUCTION
|
36 |
+
Low-mass stars and their associated accretion disks
|
37 |
+
form from gravitationally bound cores (Shu et al. 1987)
|
38 |
+
and are frequently associated with the launching of bipo-
|
39 |
+
lar jets and outflows (for reviews, see, e.g., Frank et al.
|
40 |
+
2014; Bally 2016). The magnetocentrifugal mechanism
|
41 |
+
(Blandford & Payne 1982; Pudritz & Norman 1983;
|
42 |
+
Konigl & Pudritz 2000) is widely thought to be respon-
|
43 |
+
Corresponding author: Jan E. Staff
|
44 |
+
jestaff.astro@gmail.com
|
45 |
+
sible for launching, accelerating and collimating these
|
46 |
+
protostellar outflows. In this scenario, the combination
|
47 |
+
of large-scale magnetic fields with gravity and rotation
|
48 |
+
results in the ejection, acceleration and then collima-
|
49 |
+
tion of gas originating from the surface of the accretion
|
50 |
+
disk.
|
51 |
+
A number of numerical simulation studies have
|
52 |
+
been performed to investigate this process across a va-
|
53 |
+
riety of different conditions and assumptions (e.g., Shi-
|
54 |
+
bata & Uchida 1985; Uchida & Shibata 1985; Ouyed
|
55 |
+
et al. 2003, 1997; Ouyed & Pudritz 1997; Romanova
|
56 |
+
et al. 1997; Krasnopolsky et al. 1999; Ramsey & Clarke
|
57 |
+
2011; Staff et al. 2010, 2015, 2019; Anderson et al. 2006;
|
58 |
+
Zanni et al. 2007; Te¸sileanu et al. 2012; Sheikhnezami
|
59 |
+
arXiv:2301.00749v1 [astro-ph.GA] 2 Jan 2023
|
60 |
+
|
61 |
+
2
|
62 |
+
et al. 2012; Stute et al. 2014; Stepanovs & Fendt 2014,
|
63 |
+
2016; Ramsey & Clarke 2019; Gressel et al. 2020; Mat-
|
64 |
+
tia & Fendt 2020a,b).
|
65 |
+
However, alternative theoreti-
|
66 |
+
cal scenarios have also been proposed as being relevant
|
67 |
+
for outflow launching, including: the X-wind model in-
|
68 |
+
volving the interaction of the protostellar magnetic field
|
69 |
+
with the inner disk (e.g., Lovelace et al. 1991; Shu et al.
|
70 |
+
2000); stellar wind driven outflows (e.g., Matt & Pudritz
|
71 |
+
2005); and magnetic pressure driven outflows (Lynden-
|
72 |
+
Bell 1996).
|
73 |
+
Observationally, support for the disk wind model in
|
74 |
+
low- and intermediate-mass systems has been provided
|
75 |
+
by high angular resolution observations of a handful of
|
76 |
+
systems, e.g., TMC1A (Bjerkeli et al. 2016), HH212 (Lee
|
77 |
+
et al. 2017), DG Tau B (de Valon et al. 2020), and IRAS
|
78 |
+
21078+5211 (Moscadelli et al. 2022). In each case, the
|
79 |
+
launching of the outflow can be traced to the accretion
|
80 |
+
disk, demonstrating a launching radius that extends out
|
81 |
+
to scales of up to ∼ 20 au from the central protostar.
|
82 |
+
The formation of high-mass stars is more difficult to
|
83 |
+
characterize observationally as there are fewer sources,
|
84 |
+
they are farther away and they are more obscured by
|
85 |
+
surrounding gas and dust. Nevertheless, massive star
|
86 |
+
formation is also typically observed to be associated with
|
87 |
+
the launching of bipolar jets and outflows (see, e.g., Arce
|
88 |
+
et al. 2007; Tan et al. 2014; Beltr´an & de Wit 2016;
|
89 |
+
Hirota et al. 2017).
|
90 |
+
For example, the central source
|
91 |
+
powering HH 80 and HH 81 (IRAS 18162-2048) (Marti
|
92 |
+
et al. 1993), and G339.88-1.26 (Zhang et al. 2019) are
|
93 |
+
both associated with highly collimated outflows.
|
94 |
+
An-
|
95 |
+
other massive protostar, G35.20-0.74N, has also been
|
96 |
+
found to launch a highly collimated jet (e.g., Fedriani
|
97 |
+
et al. 2019). Indeed, Caratti o Garatti et al. (2015) found
|
98 |
+
that outflows from a number of intermediate and high-
|
99 |
+
mass protostars appear as scaled-up versions of those
|
100 |
+
from low-mass protostars, while Sandell et al. (2020)
|
101 |
+
also found this to be the case for the outflow from the
|
102 |
+
massive protostar NGC 7538 IRS1. Wider angle molec-
|
103 |
+
ular outflows have also been observed from massive pro-
|
104 |
+
tostars (e.g. Beuther et al. 2002; Wu et al. 2004; Zhang
|
105 |
+
et al. 2013, 2014a; Maud et al. 2015). In general, the
|
106 |
+
trend is that higher luminosity, i.e., more massive, pro-
|
107 |
+
tostars tend to have more powerful and more massive
|
108 |
+
outflows with wider opening angles than their low-mass
|
109 |
+
counterparts.
|
110 |
+
McKee & Tan (2002) suggested that a combination
|
111 |
+
of turbulence and magnetic pressure provides most of
|
112 |
+
the support in a massive pre-stellar core against gravity.
|
113 |
+
In this “Turbulent Core Accretion” (TCA) model, high-
|
114 |
+
mass star formation is a scaled-up version of low-mass
|
115 |
+
star formation, with accretion rates expected to be ∼
|
116 |
+
10−4 to ∼ 10−3 M⊙ yr−1, compared to ∼ 10−6 to ∼
|
117 |
+
10−5 M⊙ yr−1 in lower-mass cores. If that is the case,
|
118 |
+
then outflows from forming massive stars can therefore
|
119 |
+
also be a scaled-up version of the outflows from lower-
|
120 |
+
mass forming stars.
|
121 |
+
Other formation scenarios for high-mass stars have
|
122 |
+
also been proposed. Bonnell et al. (1998) suggested that
|
123 |
+
high-mass stars form by the collision of multiple smaller
|
124 |
+
objects that formed close together.
|
125 |
+
Another possibil-
|
126 |
+
ity suggested by Bonnell et al. (2001) is that massive
|
127 |
+
stars form together in the central region of dense proto-
|
128 |
+
clusters, where most of the mass is accreted from a glob-
|
129 |
+
ally collapsing clump (see Tan et al. 2014 for a review
|
130 |
+
of these scenarios). This could lead high-mass stars to
|
131 |
+
accrete from smaller disks that change orientation over
|
132 |
+
time, leading to outflows that also keep changing direc-
|
133 |
+
tions (Goddi et al. 2020).
|
134 |
+
In contrast to outflows from low-mass protostars, it
|
135 |
+
is still debated whether or not strong magnetization is
|
136 |
+
required to drive an outflow from high-mass protostars.
|
137 |
+
For example, Machida & Hosokawa (2020) found that,
|
138 |
+
in their simulations, the outflow launching failed or was
|
139 |
+
much delayed unless the initial cloud was strongly mag-
|
140 |
+
netized. In contrast, Beuther et al. (2020), based on ob-
|
141 |
+
servations, argued for a weak magnetization in the case
|
142 |
+
of G327.3, despite it also having an outflow. The direc-
|
143 |
+
tion of the magnetic field in the core is also debated; in
|
144 |
+
some cases, it has been found to be parallel to the out-
|
145 |
+
flow and perpendicular to the disk (Carrasco-Gonz´alez
|
146 |
+
et al. 2010; Sanna et al. 2015), while other studies have
|
147 |
+
found that the outflow axis is randomly oriented with
|
148 |
+
respect to the core-field (Zhang et al. 2014a). From an
|
149 |
+
analysis of about 200 outflows, Xu et al.
|
150 |
+
(2022) find
|
151 |
+
evidence for preferential alignment of outflow directions
|
152 |
+
with large-scale B−fields, but with significant scatter
|
153 |
+
for any given outflow to B−field to orientation.
|
154 |
+
Staff et al. (2019) (hereafter Paper I) presented 3D
|
155 |
+
magneto-hydrodynamic (MHD) simulations of disk wind
|
156 |
+
outflows from a 60 M⊙ core, but with the protostellar
|
157 |
+
mass, accretion rate and mass outflow rate held at fixed
|
158 |
+
values representing various stages of the protostellar evo-
|
159 |
+
lution. The method was to run each simulation for a
|
160 |
+
roughly a local accretion time to infer the properties of
|
161 |
+
the outflow cavity - envelope system. However, because
|
162 |
+
of this approximation this method involved significant
|
163 |
+
uncertainties.
|
164 |
+
In this paper, i.e., Paper II, we present similar MHD
|
165 |
+
simulations as Paper I, but now following the protostel-
|
166 |
+
lar evolutionary sequence consistently, i.e., as its mass
|
167 |
+
grows from m∗ = 1 M⊙ to more than 24 M⊙. As in
|
168 |
+
Paper I, we assume that a star is growing from a 60 M⊙
|
169 |
+
core embedded in a clump with mass surface density of
|
170 |
+
Σcl = 1 g cm−2 within the framework of the Turbulent
|
171 |
+
|
172 |
+
3
|
173 |
+
Core Accretion model of McKee & Tan (2002, 2003). A
|
174 |
+
disk-wind (launched from the accretion disk) is injected
|
175 |
+
into the simulation box, where some envelope material
|
176 |
+
becomes entrained by the outflow. We simulate the out-
|
177 |
+
flow as it propagates through the envelope to investigate
|
178 |
+
the interaction between the wind and the envelope ma-
|
179 |
+
terial, and to investigate how much envelope material
|
180 |
+
is pushed away, providing us with an estimate of the
|
181 |
+
star formation efficiency. We also compare our simula-
|
182 |
+
tion results with observations of outflows from massive
|
183 |
+
protostars.
|
184 |
+
In §2 we describe our numerical methods. We present
|
185 |
+
our results in §3 and discuss their implications in §4.
|
186 |
+
Finally, we summarize our findings in §5.
|
187 |
+
2. METHODS
|
188 |
+
The goal of this work is to simulate a magnetically-
|
189 |
+
powered outflow from a massive, growing protostar. Us-
|
190 |
+
ing the ZEUS-MP code (Norman 2000), we conduct a
|
191 |
+
3D, ideal MHD simulation of an outflow from a massive
|
192 |
+
protostar in the framework of the turbulent core accre-
|
193 |
+
tion model (McKee & Tan 2003; Zhang et al. 2014b;
|
194 |
+
Zhang & Tan 2018).
|
195 |
+
As in Paper I, we consider an
|
196 |
+
initial core of mass of 60 M⊙. However, instead of sim-
|
197 |
+
ulating a sequence of separate models for different fixed
|
198 |
+
values of protostellar mass, m∗, here we follow the evo-
|
199 |
+
lution of a single simulation and the resulting outflow
|
200 |
+
for more than 100,000 years as the central star grows
|
201 |
+
from an small initial mass of m∗ = 1 M⊙. The setup of
|
202 |
+
the simulation is described below.
|
203 |
+
2.1. Simulation domain and boundary conditions
|
204 |
+
We use a Cartesian grid with 168 × 280 × 280 cells
|
205 |
+
in the x1, x2, and x3 directions, respectively, for our
|
206 |
+
“medium” resolution simulation.
|
207 |
+
A “high” resolution
|
208 |
+
simulation is also run for the earlier phases of the evolu-
|
209 |
+
tion with 336 × 560 × 560 cells. A logarithmic grid (“ra-
|
210 |
+
tioed” in ZEUS terminology) is employed, where cells
|
211 |
+
become larger in each direction in a regular fashion as
|
212 |
+
the distance from the origin increases. This allows us to
|
213 |
+
cover a fairly large spatial region, while maintaining a
|
214 |
+
reasonably high resolution in the central region. The x1
|
215 |
+
direction (perpendicular to the disk and parallel to the
|
216 |
+
outflow) extends from 100 au above the disk midplane
|
217 |
+
to 26, 500 au, while the x2 and x3 directions (parallel to
|
218 |
+
the disk plane) extend out to ±16, 000 au. Compared
|
219 |
+
to the Paper I simulations, this domain is about twice
|
220 |
+
as long in the x1 direction, and slightly larger in the x2
|
221 |
+
and x3 directions.
|
222 |
+
All boundaries, except for the inner x1 boundary, are
|
223 |
+
outflow boundaries.
|
224 |
+
The inner x1 boundary is more
|
225 |
+
complicated, as the outflow is injected through it, and
|
226 |
+
mass can “accrete” onto the disk through it. The fastest
|
227 |
+
part of the disk wind is injected in a circular region with
|
228 |
+
radius ri centered on the origin. Following Paper I, ri
|
229 |
+
is related to the size of the disk around the protostar,
|
230 |
+
rd (see eq. 2 of Paper I). Just outside of the injection
|
231 |
+
region is a smoothing region, through which material
|
232 |
+
is also injected.
|
233 |
+
The role of this smoothing region is
|
234 |
+
to gradually transition from the density and velocity of
|
235 |
+
the injected disk wind profile to that of the surround-
|
236 |
+
ing environment. The smoothing region has a radius of
|
237 |
+
ro = 1.8ri, somewhat larger than the value of ro = 1.3ri
|
238 |
+
used in Paper I to ensure that it contains several cells
|
239 |
+
in the x2 and x3 directions at all times. Going further
|
240 |
+
out is the accretion region, extending from ro to racc,
|
241 |
+
through which material is removed to join the accretion
|
242 |
+
disk at a controlled rate. Beyond this, we use reflecting
|
243 |
+
boundaries, to prevent any additional mass from flowing
|
244 |
+
off the grid.
|
245 |
+
The value of ri (and thus also ro) increases during the
|
246 |
+
evolution as the star grows in mass, since rd ∝ m2/3
|
247 |
+
∗
|
248 |
+
in
|
249 |
+
the fiducial model of Zhang et al. (2014b) in the limit
|
250 |
+
of constant star formation efficiency, a fixed disk to star
|
251 |
+
mass ratio, and a constant profile of rotational energy
|
252 |
+
to gravitational energy ratio of material in the initial
|
253 |
+
core. The radius of the accretion region, racc, adjusts
|
254 |
+
over time so that the integrated mass flow rate through
|
255 |
+
the annulus given by racc−ro that has outflow boundary
|
256 |
+
conditions is ˙msim = 1
|
257 |
+
2 ˙m∗(1 + 1/3 + 1/10) ≃ 0.72 ˙m∗.
|
258 |
+
Note, the term 1/3 accounts for the growth of the accre-
|
259 |
+
tion disk, which is assumed to have a mass md = m∗/3).
|
260 |
+
The term 1/10 is present to account for the injected
|
261 |
+
mass flux of the disk wind that is immediately returned
|
262 |
+
to the simulation grid through the injection region. The
|
263 |
+
factor 1/2 is present since we simulate only one hemi-
|
264 |
+
sphere. The outer radius of the accretion region, racc,
|
265 |
+
is adjusted so that the desired accretion rate is achieved
|
266 |
+
via this region of outflow boundary condition.
|
267 |
+
2.2. Initial core
|
268 |
+
We initialize the simulation with a 1 M⊙ protostar
|
269 |
+
located at the origin of our coordinate system, which is
|
270 |
+
100 au below the inner x1 boundary. On the grid, we
|
271 |
+
include one hemisphere of a 60 M⊙ core, with a radius
|
272 |
+
of 12,000 au, which is the size expected for such a core
|
273 |
+
embedded in a clump with mass surface density of Σcl =
|
274 |
+
1 g cm−2.
|
275 |
+
In the TCA model, the fiducial initial density struc-
|
276 |
+
ture of the prestellar core is assumed to be spherical,
|
277 |
+
with a power-law of the form ρ ∝ r−kρ with kρ = 3/2.
|
278 |
+
Thus our density structure is given by
|
279 |
+
ρ(t = 0) = ρs (r/Rc)−3/2 ,
|
280 |
+
(1)
|
281 |
+
|
282 |
+
4
|
283 |
+
where ρs is the density at the surface of the core. Note,
|
284 |
+
in Paper I, which was mainly considering snapshots of
|
285 |
+
later phases of the evolution, we adopted kρ = 1 as an
|
286 |
+
approximation of the expected structure that develops
|
287 |
+
in the expansion wave of the collapse solution. For our
|
288 |
+
core with kρ = 3/2, we have ρs = 2.5 × 10−18 g cm−3,
|
289 |
+
i.e., nH = 1.1 × 106 cm−3 assuming a mass per H of
|
290 |
+
2.34 × 10−24 g cm−3. Beyond Rc we adopt a constant
|
291 |
+
ambient density of 0.1ρs. The material in the core and
|
292 |
+
its surroundings is initialized to be at rest.
|
293 |
+
Following Paper I, the initial magnetic field configura-
|
294 |
+
tion is the canonical Blandford & Payne (“BP”) config-
|
295 |
+
uration (Blandford & Payne 1982), with a constant field
|
296 |
+
added to it to ensure that the core flux is ∼ 1 mG × R2
|
297 |
+
c.
|
298 |
+
The BP configuration is a force-free, hour-glass shaped,
|
299 |
+
purely poloidal magnetic field configuration.
|
300 |
+
At the
|
301 |
+
mid-plane, the BP field varies as Bp ∝ r−1.25.
|
302 |
+
The 1D velocity dispersion of the fiducial 60 M⊙
|
303 |
+
prestellar core, i.e., assuming virial equilibrium, is
|
304 |
+
1.09(Mc/60 M⊙)1/4(Σcl/1 g cm−2)1/4 km s−1.
|
305 |
+
In our
|
306 |
+
simulation we adopt an isothermal equation of state
|
307 |
+
with an effective sound speed, i.e., signal speed, of
|
308 |
+
cs = 0.90 km s−1. This choice is made so that the core
|
309 |
+
is moderately sub-virial and will undergo gravitational
|
310 |
+
contraction.
|
311 |
+
The gravitational field is treated with a simple ap-
|
312 |
+
proximation in which the mass of the star and the disk,
|
313 |
+
residing outside of the simulation domain, are treated
|
314 |
+
as a point mass. For the contribution of the potential
|
315 |
+
of the envelope material, we assume a simple model of
|
316 |
+
a fixed core envelope size, i.e., of radius Rc, and a fixed
|
317 |
+
power law index describing the radial distribution, i.e.,
|
318 |
+
ρ ∝ r−3/2, but with the normalization of the profile
|
319 |
+
adjusted to match the mass that is remaining in the
|
320 |
+
envelope.
|
321 |
+
When the simulation starts, the core immediately be-
|
322 |
+
gins to contract as the initial setup is unstable to grav-
|
323 |
+
itational collapse.
|
324 |
+
Initially, the plasma-β (i.e., where
|
325 |
+
β ≡ Pgas/Pmag) is slightly above unity in the core.
|
326 |
+
However, as the envelope collapses, the plasma-β drops
|
327 |
+
below unity, meaning that the magnetic field starts to
|
328 |
+
dominate. The collapse will therefore not be spherically-
|
329 |
+
symmetric towards the protostar, but instead be guided
|
330 |
+
along the field lines towards the mid-plane.
|
331 |
+
2.3. Injection of the disk wind
|
332 |
+
We launch the disk wind through the injection re-
|
333 |
+
gion on the inner x1 boundary, with ˙minj = 1
|
334 |
+
2
|
335 |
+
1
|
336 |
+
10 ˙m∗ =
|
337 |
+
0.05 ˙m∗. We also enforce that the injected outflow has
|
338 |
+
the same momentum rate in the x1 direction as in Zhang
|
339 |
+
et al. (2014b). Together, this can be used to constrain
|
340 |
+
the injected density and velocity in the x1 direction (per-
|
341 |
+
pendicular to the injection boundary). As in Paper I,
|
342 |
+
we then have an injected density:
|
343 |
+
ρinj =
|
344 |
+
�
|
345 |
+
�
|
346 |
+
�
|
347 |
+
�
|
348 |
+
�
|
349 |
+
exp (0.0289 rcyl/r∗)φρρ0
|
350 |
+
rcyl < x0
|
351 |
+
2.77
|
352 |
+
�rcyl
|
353 |
+
x0
|
354 |
+
�−1
|
355 |
+
φρρ0
|
356 |
+
rcyl ≥ x0
|
357 |
+
(2)
|
358 |
+
and an injected v1 velocity:
|
359 |
+
vinj = (rcyl/r∗)−1/2φinjvK∗,
|
360 |
+
(3)
|
361 |
+
where r∗ is the stellar radius, x0 = 35.3r∗, rcyl is the
|
362 |
+
distance from the x1 axis, ρ0 is the injection density
|
363 |
+
at the axis, vK∗ is the Keplerian speed on the stellar
|
364 |
+
surface. φρ and φinj are time dependent dimensionless
|
365 |
+
factors that are needed in order to obtain the desired
|
366 |
+
mass flow and momentum rates of the inflowing wind,
|
367 |
+
as is discussed in Paper I.
|
368 |
+
The velocity components of the injected flow in the
|
369 |
+
2- and 3-directions are set so that the flow is along the
|
370 |
+
direction of the initial magnetic field lines. The injected
|
371 |
+
flow is also given an additional toroidal velocity compo-
|
372 |
+
nent:
|
373 |
+
vφ,inj = 0.23
|
374 |
+
� rcyl
|
375 |
+
22.4r∗
|
376 |
+
�−1/2
|
377 |
+
vK∗.
|
378 |
+
(4)
|
379 |
+
The values employed for ri, ρ0, ˙m∗, ˙minj, and ˙pinj are
|
380 |
+
given for protostellar masses of 1, 2, 4, 8, 16, and 24 M⊙
|
381 |
+
in Table 1.
|
382 |
+
In the smoothing region, at ri < r < ro, the veloc-
|
383 |
+
ity is gradually reduced by multiplying it by a factor
|
384 |
+
w = cos2[ π
|
385 |
+
2 (r − ri)/(ro − ri)]. The initial density of the
|
386 |
+
surrounding envelope is gradually joined with the den-
|
387 |
+
sity in the injection region by dividing the core density
|
388 |
+
by 1+w(fjump−1), where fjump is the ratio of the initial
|
389 |
+
core density to the density in the injection region.
|
390 |
+
We note that, especially in the outflow cavity, if the
|
391 |
+
density in a cell drops too low, the Alfv´en time step
|
392 |
+
drops to such a low value that the simulation effectively
|
393 |
+
grinds to a halt. To avoid this, it is common practice in
|
394 |
+
outflow simulations to implement a density floor, which
|
395 |
+
prevents the Alfv´en time step from becoming extremely
|
396 |
+
small. However, including such a density floor means
|
397 |
+
mass is being artificially added to the grid. In this work,
|
398 |
+
we have used a density floor that depends on height x1
|
399 |
+
above the disk: nH,floor = (x1/105 au)−1 cm−3. The rea-
|
400 |
+
son for this choice is that near the inner x1 boundary
|
401 |
+
where mass is accreting, we need a fairly large density
|
402 |
+
floor to maintain a reasonable Alfv´en time step as the
|
403 |
+
magnetic fields are stronger. High above the disk, the
|
404 |
+
density in the outflow cavity drops to values much below
|
405 |
+
what the floor needs to be near the inner x1 boundary,
|
406 |
+
and hence the density floor in the outer part of the sim-
|
407 |
+
ulation box can be lower than in the inner part. We
|
408 |
+
|
409 |
+
5
|
410 |
+
Table 1. Values of the radius of the injection region ri, the injected density along the axis ρ0, the desired accretion rate ˙macc,
|
411 |
+
the desired injected mass flow rate ˙minj, and the desired injected momentum rate ˙pinj employed at the lower x1 boundary for
|
412 |
+
protostellar masses m∗.
|
413 |
+
m∗
|
414 |
+
ri
|
415 |
+
ρ0
|
416 |
+
˙macc
|
417 |
+
˙minj
|
418 |
+
˙pinj
|
419 |
+
[M⊙]
|
420 |
+
[au]
|
421 |
+
[10−17 g cm−3]
|
422 |
+
[10−4M⊙ yr−1]
|
423 |
+
[10−5M⊙ yr−1]
|
424 |
+
[10−3M⊙ km s−1 yr−1]
|
425 |
+
1
|
426 |
+
92
|
427 |
+
5.8
|
428 |
+
1.0
|
429 |
+
1.0
|
430 |
+
5.9
|
431 |
+
2
|
432 |
+
106
|
433 |
+
4.4
|
434 |
+
1.4
|
435 |
+
1.4
|
436 |
+
9.9
|
437 |
+
4
|
438 |
+
124
|
439 |
+
1.1
|
440 |
+
2.0
|
441 |
+
2.0
|
442 |
+
9.4
|
443 |
+
8
|
444 |
+
150
|
445 |
+
0.6
|
446 |
+
2.7
|
447 |
+
2.7
|
448 |
+
14.2
|
449 |
+
16
|
450 |
+
196
|
451 |
+
1.5
|
452 |
+
3.2
|
453 |
+
3.2
|
454 |
+
41.2
|
455 |
+
24
|
456 |
+
282
|
457 |
+
1.0
|
458 |
+
3.3
|
459 |
+
3.3
|
460 |
+
49.5
|
461 |
+
note that when mass is added to a cell in the simula-
|
462 |
+
tion, we do not adjust the velocity of that cell, and as a
|
463 |
+
consequence momentum is also added to the simulation.
|
464 |
+
3. RESULTS
|
465 |
+
3.1. Density, velocity and magnetic field structures
|
466 |
+
We have simulated the evolution of the protostellar
|
467 |
+
core for ∼ 105yr as the protostar grows from m∗ = 1M⊙
|
468 |
+
to about 26 M⊙. In Fig. 1 we show slices of the density
|
469 |
+
structure in the x1 − x2 plane at x3 = 0. These images
|
470 |
+
show the general structure of the disk-wind outflow cav-
|
471 |
+
ity as it gradually carves open a larger and larger vol-
|
472 |
+
ume from the initial core infall envelope.
|
473 |
+
Concurrent
|
474 |
+
with this evolution of the outflow cavity, we also see the
|
475 |
+
collapse of the infall envelope down towards the central
|
476 |
+
midplane base of the core. A movie showing the evo-
|
477 |
+
lution of this structure is shown in Fig. 2. During the
|
478 |
+
course of the evolution the range of densities present in
|
479 |
+
the simulation extends from nH ∼ 4cm−3 (in the outflow
|
480 |
+
cavity) to ≳ 108 cm−3 (in the inner infall envelope).
|
481 |
+
Figure 3 shows the magnitude of the outflowing ve-
|
482 |
+
locity along the x1 direction, i.e., v1 > 0.9 km s−1, for
|
483 |
+
the same slices through the simulation domain shown
|
484 |
+
in Fig. 1. At any given evolutionary stage, the highest
|
485 |
+
velocities are found close to the central axis of the out-
|
486 |
+
flow cavity. At the earliest stages shown in Fig. 3, i.e.,
|
487 |
+
m∗ = 2M⊙, these velocities are already ∼ 2, 000km s−1.
|
488 |
+
By the later stages with m∗ = 24 M⊙, these velocities
|
489 |
+
have risen to ∼ 5, 000 km s−1.
|
490 |
+
Figure 4 shows the magnitude of the total magnetic
|
491 |
+
field strength for the same slices through the simula-
|
492 |
+
tion domain shown in Fig. 1. The largest magnetic field
|
493 |
+
strengths are ∼ 100mG near the base of the outflow and
|
494 |
+
inner infall envelope. In the outflow cavity, the magnetic
|
495 |
+
field strength is much lower than in the infall envelope,
|
496 |
+
with values at low as ∼ 0.01 mG.
|
497 |
+
3.2. Evolution of the outflow cavity opening angle
|
498 |
+
To evaluate the opening angle of the outflow cavity
|
499 |
+
at a given height x1, we first calculate the area A in
|
500 |
+
the x2-x3 plane of the outflowing matter that has v1 >
|
501 |
+
0.9 km s−1. We then approximate the outflow as having
|
502 |
+
a conical shape with a circular cross section of area A =
|
503 |
+
πr2, giving r =
|
504 |
+
�
|
505 |
+
A/π, and then find the half opening
|
506 |
+
angle of that cone, tan(θoutflow) = r/x1. In Fig. 5 we
|
507 |
+
show the evolution of the calculated opening angle over
|
508 |
+
time for several different heights above the disk. These
|
509 |
+
direct estimates of the opening angles are stopped when
|
510 |
+
the outflow cavity region approaches the lateral edges
|
511 |
+
of the simulation domain.
|
512 |
+
Beyond this point, shown
|
513 |
+
with dashed lines, we make an approximate estimate for
|
514 |
+
opening angle at a given height via linear extrapolation
|
515 |
+
from the closest lower height where the geometry of the
|
516 |
+
outflow is still contained within the domain.
|
517 |
+
From our results we see that the outflow cavity open-
|
518 |
+
ing angle is larger at lower heights (e.g., at 5,000 au),
|
519 |
+
and is smaller at larger heights due to collimation of the
|
520 |
+
outflow. In other words, the outflow cavity is not truly
|
521 |
+
conical (as is evidenced in Figs. 1 and 2). Considering
|
522 |
+
a fidcuial height equal to the initial radius of the core,
|
523 |
+
i.e., 12,000 au, we see that the outflow cavity opening
|
524 |
+
angle has achieved a value of about 10◦ at the earliest
|
525 |
+
stages of the simulation, i.e., when m∗ = 2 M⊙. It then
|
526 |
+
rises slowly until m∗ ∼ 4 M⊙. After this it increases
|
527 |
+
at a slightly faster rate, reach about 42◦ by the time
|
528 |
+
m∗ = 18M⊙, i.e., the last stage where it can be directly
|
529 |
+
evaluated in the simulation domain. An extrapolation
|
530 |
+
based estimate at m∗ = 24 M⊙ yields θoutflow ≃ 50◦.
|
531 |
+
In Figure 5 we also compare our results to those of
|
532 |
+
Paper I (without pre-clearing), which were calculated
|
533 |
+
at the top of the grid in those simulations, i.e., at a
|
534 |
+
height of about 12, 000 au. Recall that in Paper I, with
|
535 |
+
models run at fixed m∗, it was somewhat uncertain at
|
536 |
+
which time to evaluate the results for the opening angle.
|
537 |
+
Paper I also considered a case “with pre-clearing” that
|
538 |
+
attempted to allow for the earlier stages of evolution and
|
539 |
+
these yielded larger opening angles at the later stages,
|
540 |
+
i.e., about 50◦ at m∗ = 16M⊙ and 78◦ at 24M⊙. We find
|
541 |
+
that our new simulations with a continuous evolution
|
542 |
+
followed from low to high values of m∗ yield moderately
|
543 |
+
|
544 |
+
6
|
545 |
+
Figure 1. Slices of simulation results for density in the x1 − x2 plane at x3 = 0, with x1 corresponding to the outflow axis.
|
546 |
+
The top, middle and bottom rows show m∗ = 2 M⊙ and 4 M⊙, 8 M⊙ and 12 M⊙, and 16 M⊙ and 24 M⊙, respectively.
|
547 |
+
smaller cavity opening angles than the results of Paper I,
|
548 |
+
with the biggest differences being at the highest masses.
|
549 |
+
We also compare our results to the opening angles
|
550 |
+
predicted by the semi-analytic model of Zhang et al.
|
551 |
+
(2014b), following the method of Matzner & McKee
|
552 |
+
(2000), which is based on the condition of whether the
|
553 |
+
material in a given direction can be accelerated to the es-
|
554 |
+
cape speed. We find that our numerical results predict a
|
555 |
+
moderately narrower outflow cavity geometry than this
|
556 |
+
semi-analytic model, with the difference being about 20◦
|
557 |
+
by the end of the simulation.
|
558 |
+
3.3. Mass and momentum fluxes of the outflow
|
559 |
+
We evaluate the rate at which mass flows out of the
|
560 |
+
top of the simulation box at the x2 − x3 boundary face
|
561 |
+
via (1/2) ˙moutflow =
|
562 |
+
�
|
563 |
+
ρv1dA, i.e., performing the sum-
|
564 |
+
mation over the actual area of the outflow with no as-
|
565 |
+
sumption of it being circular and equating this to half
|
566 |
+
the total mass flux in a bipolar protostellar outflow. The
|
567 |
+
evolution of this outflowing mass flux is shown in Fig-
|
568 |
+
ure 6a.
|
569 |
+
Initially, there is a transient phase with a fairly high
|
570 |
+
mass flux out of the simulation box of ∼ 4×10−5M⊙yr−1
|
571 |
+
|
572 |
+
8.20
|
573 |
+
2.5×104
|
574 |
+
2×104
|
575 |
+
[np]
|
576 |
+
1.5x104
|
577 |
+
6.93
|
578 |
+
x
|
579 |
+
104
|
580 |
+
5000
|
581 |
+
5.67
|
582 |
+
2.5x104
|
583 |
+
2x104
|
584 |
+
[np]
|
585 |
+
4.40
|
586 |
+
104
|
587 |
+
5000
|
588 |
+
3.13
|
589 |
+
2.5x104
|
590 |
+
t=94,000 yrs, M=24
|
591 |
+
2x104
|
592 |
+
[np]
|
593 |
+
1.5x104
|
594 |
+
1.87
|
595 |
+
x
|
596 |
+
104
|
597 |
+
5000
|
598 |
+
0.60
|
599 |
+
-10000
|
600 |
+
0
|
601 |
+
10000
|
602 |
+
-10000
|
603 |
+
0
|
604 |
+
10000
|
605 |
+
Log(n/[cm-3])
|
606 |
+
Lnp]
|
607 |
+
X2
|
608 |
+
npl7
|
609 |
+
Figure 2. Movie showing the temporal evolution of the x1 − x2 at x3 = 0 density slices, i.e., same as the examples shown in
|
610 |
+
Fig. 1.
|
611 |
+
while the outflow cavity is being cleared out. After this
|
612 |
+
the mass flow rate grows from about 2 × 10−5 M⊙ yr−1
|
613 |
+
to ∼ 1 × 10−4 M⊙ yr−1 by the time the star has reached
|
614 |
+
∼ 10M⊙. We note that the mass flux exhibits moderate,
|
615 |
+
∼ 30%, fluctuations during this evolution. After this the
|
616 |
+
mass flux stops increasing and exhibits more dramatic
|
617 |
+
fluctuations during the evolution to m∗ = 16 M⊙. After
|
618 |
+
this, it shows a more steady, smooth decline, which is
|
619 |
+
mostly caused by the outflow cavity expanding beyond
|
620 |
+
the size of the top face of the simulation domain. For
|
621 |
+
this reason, we do not calculate the mass flow rate out
|
622 |
+
of the grid for masses beyond ∼ 20 M⊙: i.e., at this
|
623 |
+
stage a significant amount of mass is now leaving across
|
624 |
+
the side boundaries (as can be observed in the movie in
|
625 |
+
Fig. 2 and in Fig. 3).
|
626 |
+
Figure 6b shows the ratio of the mass flux leaving
|
627 |
+
the top of the simulation domain to the mass injected
|
628 |
+
at the base of the outflow. After the initial peak asso-
|
629 |
+
ciated with first breakout of the outflow, this ratio is
|
630 |
+
about 2, but then rises up to a peak just below 10 when
|
631 |
+
m∗ = 10M⊙. At higher masses it generally declines, but
|
632 |
+
with large fluctuations, eventually reaching values near
|
633 |
+
2 again.
|
634 |
+
Figure 6c shows the time evolution of the total mass
|
635 |
+
that has left the top of the simulation domain. We find
|
636 |
+
that more than 4 M⊙ has left the grid as part of the
|
637 |
+
outflow by the time the protostar reaches 20 M⊙.
|
638 |
+
Figure 7a shows the momentum flux passing through
|
639 |
+
the top of the simulation domain, evaluated as ˙p =
|
640 |
+
�
|
641 |
+
ρv2
|
642 |
+
1dA.
|
643 |
+
As in Fig. 6, we cut off the measurements
|
644 |
+
when substantial mass and momentum start to leave
|
645 |
+
the domain through the side boundaries. We find that
|
646 |
+
the momentum flux leaving the domain stays approxi-
|
647 |
+
mately constant at about 0.005 M⊙ km s−1 yr−1, until
|
648 |
+
the star reaches ∼ 7 M⊙.
|
649 |
+
Then it increases to reach
|
650 |
+
nearly 0.02 M⊙ km s−1 yr−1 when the star is ∼ 16 M⊙.
|
651 |
+
It then continues to increase, but at a slower rate. How-
|
652 |
+
ever, at this stage we begin to lose track of mass that is
|
653 |
+
leaving through the sides of the domain.
|
654 |
+
Figure 7a also shows the injected momentum flux at
|
655 |
+
the base of the outflow. In general, as expected, we see
|
656 |
+
a very good agreement between the injected and ejected
|
657 |
+
momentum fluxes, with the largest deviation occurring
|
658 |
+
|
659 |
+
8.20
|
660 |
+
2.5x104
|
661 |
+
6.93
|
662 |
+
2x104
|
663 |
+
5.67
|
664 |
+
[αu]
|
665 |
+
1.5x104
|
666 |
+
4.40
|
667 |
+
+2
|
668 |
+
104
|
669 |
+
3.13
|
670 |
+
5000
|
671 |
+
1.87
|
672 |
+
1.5x104-1x104-5000
|
673 |
+
0
|
674 |
+
5000
|
675 |
+
104
|
676 |
+
1.5×104
|
677 |
+
0.60
|
678 |
+
x, [αu]
|
679 |
+
28
|
680 |
+
Figure 3.
|
681 |
+
Slices in the x1 − x2 plane at x3 = 0 of simulation results for total velocity, v, but only showing cells with
|
682 |
+
v1 > 0.9 km s−1 to highlight outflowing gas. The top, middle and bottom rows show m∗ = 2 M⊙ and 4 M⊙, 8 M⊙ and 12 M⊙,
|
683 |
+
and 16 M⊙ and 24 M⊙, respectively.
|
684 |
+
at late times due to some outflow material leaving via
|
685 |
+
the sides of the domain. The ratio of these momentum
|
686 |
+
fluxes is shown explicitly in Figure 7b.
|
687 |
+
Figure 7c shows the total momentum that has left via
|
688 |
+
the top of the simulation domain. This grows steadily
|
689 |
+
to reach ∼ 800M⊙ km s−1 by the time the protostar has
|
690 |
+
reached ∼ 20 M⊙.
|
691 |
+
3.4. Star formation efficiency
|
692 |
+
Here we evaluate the star formation efficiency (SFE),
|
693 |
+
i.e., the ratio of the final stellar mass to the initial core
|
694 |
+
mass, that is implied by our simulation results. After
|
695 |
+
100,000 years, the protostar has grown to m∗ ≃ 26 M⊙.
|
696 |
+
Thus we estimate that ¯ϵ∗f ≥ 0.43.
|
697 |
+
This is a lower
|
698 |
+
limit since in our model the disk has a mass of mdisk =
|
699 |
+
(1/3)m∗ ≃ 9 M⊙ and a significant portion of this ma-
|
700 |
+
terial is expected to be able to accrete to the star. If
|
701 |
+
the only process diverting material from the accretion
|
702 |
+
disk is injection into the disk wind with ˙mw = 0.1 ˙m∗,
|
703 |
+
then the final stellar mass would be at least 34 M⊙, i.e.,
|
704 |
+
¯ϵ∗f ≥ 0.56. It is possible that a larger fraction of ma-
|
705 |
+
terial could be diverted from the accretion disk if other
|
706 |
+
forms of feedback, especially disk photoevaporation, are
|
707 |
+
significant.
|
708 |
+
However, Tanaka et al. (2017) considered
|
709 |
+
such models and found that disk photoevaporation was
|
710 |
+
relatively unimportant compared to the disk wind mass
|
711 |
+
flux for this mass and accretion rate regime.
|
712 |
+
The above estimates are likely to still be lower limits,
|
713 |
+
since there is still 12M⊙ (3M⊙ from the initial core and
|
714 |
+
9 M⊙ from the surrounding clump) remaining in the
|
715 |
+
|
716 |
+
2.5×10
|
717 |
+
t=9,000 yrs.
|
718 |
+
t=21,000
|
719 |
+
3.70
|
720 |
+
[np]
|
721 |
+
yrs.
|
722 |
+
M=2
|
723 |
+
M= 4
|
724 |
+
2.95
|
725 |
+
5000
|
726 |
+
t=39,000
|
727 |
+
t=54,000
|
728 |
+
2x104
|
729 |
+
2.20
|
730 |
+
[no
|
731 |
+
1.5x104
|
732 |
+
yrs,
|
733 |
+
yrs,
|
734 |
+
M=8
|
735 |
+
M=12
|
736 |
+
5000
|
737 |
+
W
|
738 |
+
1.45
|
739 |
+
2.5x104
|
740 |
+
t=68,000 yrs.
|
741 |
+
t=94,000
|
742 |
+
2x104
|
743 |
+
1.5x104
|
744 |
+
yrs,
|
745 |
+
0.70
|
746 |
+
104
|
747 |
+
M=16
|
748 |
+
M=24
|
749 |
+
5000
|
750 |
+
@
|
751 |
+
10000
|
752 |
+
0.05
|
753 |
+
-10000
|
754 |
+
0
|
755 |
+
-10000
|
756 |
+
0
|
757 |
+
10000
|
758 |
+
X2
|
759 |
+
[np]
|
760 |
+
X2
|
761 |
+
[au]
|
762 |
+
log(v/[km s-1])9
|
763 |
+
Figure 4. Slices of simulation results for magnetic field strength, B, in the x1 − x2 plane at x3 = 0. The top, middle and
|
764 |
+
bottom rows show m∗ = 2 M⊙ and 4 M⊙, 8 M⊙ and 12 M⊙, and 16 M⊙ and 24 M⊙, respectively.
|
765 |
+
simulation domain, i.e., 24 M⊙ in the global, mirrored
|
766 |
+
domain. One expects that a significant fraction of this
|
767 |
+
material would be accreted to the central protostar. In
|
768 |
+
the case that all of the remaining initial core mass is
|
769 |
+
accreted, i.e., 6 M⊙, then this would thus result in a
|
770 |
+
SFE of ¯ϵ∗f ≃ 0.67.
|
771 |
+
Comparing the semi-analytic model of Zhang et al.
|
772 |
+
(2014b), they also reached a final value of m∗ = 26 M⊙.
|
773 |
+
Thus, with the same considerations of residual disk ac-
|
774 |
+
cretion, they expect to reach ¯ϵ∗f ≥ 0.56. However, their
|
775 |
+
model at this point would be exhausted of gas and so
|
776 |
+
this would be the final estimate of SFE. Thus we con-
|
777 |
+
clude that the expected SFE from our numerical model
|
778 |
+
is moderately (∼ 20%) larger than that predicted by
|
779 |
+
the semi-analytic model.
|
780 |
+
This is consistent with the
|
781 |
+
generally smaller outflow opening angles found during
|
782 |
+
the course of the evolution in the numerical model com-
|
783 |
+
pared the Zhang et al. (2014b) semi-analytic model (see
|
784 |
+
Fig. 5).
|
785 |
+
However, we note that in the fiducial TCA model of
|
786 |
+
McKee & Tan (2003), the initial core is expected to in-
|
787 |
+
teract with significant surrounding clump gas during its
|
788 |
+
collapse to a protostar, so with this consideration the
|
789 |
+
results of Zhang et al. (2014b) for the final stellar mass,
|
790 |
+
m∗f, are also lower limits. If SFE is defined with respect
|
791 |
+
to the initial core mass, then the values of ¯ϵ∗f would also
|
792 |
+
be lower limits.
|
793 |
+
|
794 |
+
2.5x104
|
795 |
+
一
|
796 |
+
1.00
|
797 |
+
2x104
|
798 |
+
[np]
|
799 |
+
1.5x104
|
800 |
+
104
|
801 |
+
5000
|
802 |
+
一
|
803 |
+
-2.00
|
804 |
+
2.5x10*
|
805 |
+
2×104
|
806 |
+
[np]
|
807 |
+
1.5×104
|
808 |
+
-3.00
|
809 |
+
104
|
810 |
+
5000
|
811 |
+
2.5x104
|
812 |
+
4.00
|
813 |
+
[np]
|
814 |
+
1.5x104
|
815 |
+
x
|
816 |
+
104
|
817 |
+
5000
|
818 |
+
-5.00
|
819 |
+
-10000
|
820 |
+
0
|
821 |
+
10000
|
822 |
+
-10000
|
823 |
+
0
|
824 |
+
10000
|
825 |
+
X2
|
826 |
+
[np]
|
827 |
+
×2 [αu]
|
828 |
+
Log(B/[G])10
|
829 |
+
0
|
830 |
+
10
|
831 |
+
20
|
832 |
+
30
|
833 |
+
40
|
834 |
+
50
|
835 |
+
60
|
836 |
+
70
|
837 |
+
80
|
838 |
+
0
|
839 |
+
5
|
840 |
+
10
|
841 |
+
15
|
842 |
+
20
|
843 |
+
25
|
844 |
+
θoutflow [degrees]
|
845 |
+
m* [M⊙]
|
846 |
+
Staff et al. (2019)
|
847 |
+
Zhang et al. (2014)
|
848 |
+
height 5,000 au
|
849 |
+
height 12,000 au
|
850 |
+
height 20,000 au
|
851 |
+
height 25,000 au
|
852 |
+
12,000 au extrapolated
|
853 |
+
20,000 au extrapolated
|
854 |
+
25,000 au extrapolated
|
855 |
+
Figure 5. Outflow cavity opening angle measured at different heights above the disk (solid lines). Extrapolated estimates
|
856 |
+
(dashed lines) are needed once the cavity nears the simulation boundary at a given height (see text). Also shown are the outflow
|
857 |
+
cavity opening angles found in the numerical models of Paper I (squares) and the semi-analytic models of Zhang et al. (2014b)
|
858 |
+
(crosses).
|
859 |
+
3.5. Outflow mass spectra
|
860 |
+
One method of comparing our model results with ob-
|
861 |
+
served systems is via the distribution of outflowing gas
|
862 |
+
mass with line of sight velocity velocity, i.e., “mass spec-
|
863 |
+
tra”, since this can be inferred from observations of CO
|
864 |
+
emission lines.
|
865 |
+
Note, in this paper we will not make
|
866 |
+
synthetic CO spectra of our models, deferring this step
|
867 |
+
to a future work. To produce the distribution of mass
|
868 |
+
with line of sight velocity, we need to produce a “global”
|
869 |
+
simulation domain, which is achieved by mirroring our
|
870 |
+
simulation grid about the x1 = 0 boundary, i.e., the
|
871 |
+
disk plane. In this way we produce a symmetric bipolar
|
872 |
+
outflow structure, which we then view at various angles,
|
873 |
+
θview, to the outflow axis. Note, θview = 0◦ is defined as
|
874 |
+
a line of sight that is parallel to the outflow axis.
|
875 |
+
Figure 8 shows the mass spectra within the global do-
|
876 |
+
main at various evolutionary stages. Note, these spectra
|
877 |
+
include all gas, i.e., both outflowing and infalling mate-
|
878 |
+
rial. We have chosen three values of θview that are part
|
879 |
+
of the grid of uniformly sampled grid of cos θview values
|
880 |
+
in the radiative transfer models of Zhang & Tan (2018).
|
881 |
+
The mass spectra show a sharp peak at low velocities,
|
882 |
+
and, except for θview values close to 90◦, long tails to
|
883 |
+
larger velocities. As the protostellar mass increases, we
|
884 |
+
find more mass at larger velocities. For m∗ > 16 M⊙,
|
885 |
+
the largest velocities are > 3000km s−1 when the system
|
886 |
+
is viewed close to the outflow axis. One point to note is
|
887 |
+
that between 2 M⊙ and 4 M⊙, the maximum velocities
|
888 |
+
decrease somewhat. This is due to the protostellar ra-
|
889 |
+
dius (which also sets the inner disk radius) growing from
|
890 |
+
3.45 R⊙ at 2 M⊙ to 20.5 R⊙ at 4 M⊙. The injection ve-
|
891 |
+
locity of the outflow is proportional to the Keplerian
|
892 |
+
speed at the launching point (vKep ∝ m1/2
|
893 |
+
∗
|
894 |
+
r−1/2; Eq.
|
895 |
+
3). Hence, the highest velocity outflow is launched from
|
896 |
+
the inner disk and, as the inner disk radius expands, the
|
897 |
+
velocity of the material launched from the inner disc de-
|
898 |
+
creases, even though the central mass is growing. We
|
899 |
+
use these mass spectra in the next subsection to make
|
900 |
+
detailed comparisons to some observed massive proto-
|
901 |
+
stars.
|
902 |
+
3.6. Comparison with observed outflow mass spectra
|
903 |
+
In Figures 9 and 10 we compare the simulation out-
|
904 |
+
flow mass spectra to equivalent outflow mass spectra
|
905 |
+
of G35.20-0.74N and G339.88-1.26 (hereafter G35.2 and
|
906 |
+
G339) as derived from ALMA observations of CO(2-
|
907 |
+
1) line emission by Zhang et al. (2022) and Zhang
|
908 |
+
et al. (2019), respectively. Note, the observed line emis-
|
909 |
+
sion from these sources was extracted from regions of
|
910 |
+
|
911 |
+
11
|
912 |
+
0
|
913 |
+
0.2
|
914 |
+
0.4
|
915 |
+
0.6
|
916 |
+
0.8
|
917 |
+
1
|
918 |
+
1.2
|
919 |
+
1.4
|
920 |
+
0
|
921 |
+
5
|
922 |
+
10
|
923 |
+
15
|
924 |
+
20
|
925 |
+
1/2 m
|
926 |
+
.
|
927 |
+
outflow [10-4 M☉ yr-1]
|
928 |
+
m* [M⊙]
|
929 |
+
0
|
930 |
+
1
|
931 |
+
2
|
932 |
+
3
|
933 |
+
4
|
934 |
+
5
|
935 |
+
6
|
936 |
+
7
|
937 |
+
8
|
938 |
+
9
|
939 |
+
10
|
940 |
+
0
|
941 |
+
5
|
942 |
+
10
|
943 |
+
15
|
944 |
+
20
|
945 |
+
m
|
946 |
+
.
|
947 |
+
outflow/m
|
948 |
+
.
|
949 |
+
inj
|
950 |
+
m* [M⊙]
|
951 |
+
0
|
952 |
+
0.5
|
953 |
+
1
|
954 |
+
1.5
|
955 |
+
2
|
956 |
+
2.5
|
957 |
+
3
|
958 |
+
3.5
|
959 |
+
4
|
960 |
+
4.5
|
961 |
+
0
|
962 |
+
5
|
963 |
+
10
|
964 |
+
15
|
965 |
+
20
|
966 |
+
∫ 1/2 m
|
967 |
+
.
|
968 |
+
outflow dt [M☉]
|
969 |
+
m* [M⊙]
|
970 |
+
Figure 6. (a) Top: Evolution of outflow mass flux through
|
971 |
+
the top of the simulation domain (x2 − x3 face at x1 =
|
972 |
+
25, 000 au) (purple solid line).
|
973 |
+
The red dashed line shows
|
974 |
+
the injected mass flow rate of the outflow. (b) Middle: Ratio
|
975 |
+
of the mass flow rate out of the top of the simulation box to
|
976 |
+
the injected mass flow rate at base of the outflow. (c) Top:
|
977 |
+
Evolution of total mass that has left the top of the simulation
|
978 |
+
domain by being swept-up by the outflow.
|
979 |
+
∼25,000 au in radial size centered on the protostars,
|
980 |
+
similar to the size of our simulation box. We consider
|
981 |
+
a velocity range of ±50 km s−1 and exclude the inner
|
982 |
+
±10 km s−1, which is affected by the presence of ambi-
|
983 |
+
ent clump gas.
|
984 |
+
To quantify the differences between the models and
|
985 |
+
observations, we calculate the reduced χ2 between the
|
986 |
+
two, following the method of Zhang & Tan (2018) (de-
|
987 |
+
veloped for spectral energy distribution fitting), as:
|
988 |
+
χ2 = 1
|
989 |
+
N
|
990 |
+
�
|
991 |
+
i
|
992 |
+
�mi,data − mi,sim
|
993 |
+
σ
|
994 |
+
�2
|
995 |
+
,
|
996 |
+
(5)
|
997 |
+
0
|
998 |
+
0.005
|
999 |
+
0.01
|
1000 |
+
0.015
|
1001 |
+
0.02
|
1002 |
+
0.025
|
1003 |
+
0
|
1004 |
+
5
|
1005 |
+
10
|
1006 |
+
15
|
1007 |
+
20
|
1008 |
+
1/2 p
|
1009 |
+
.
|
1010 |
+
outflow [M☉ km s-1 yr-1]
|
1011 |
+
m* [M⊙]
|
1012 |
+
0
|
1013 |
+
0.2
|
1014 |
+
0.4
|
1015 |
+
0.6
|
1016 |
+
0.8
|
1017 |
+
1
|
1018 |
+
1.2
|
1019 |
+
1.4
|
1020 |
+
1.6
|
1021 |
+
1.8
|
1022 |
+
0
|
1023 |
+
5
|
1024 |
+
10
|
1025 |
+
15
|
1026 |
+
20
|
1027 |
+
p
|
1028 |
+
.
|
1029 |
+
outflow/p
|
1030 |
+
.
|
1031 |
+
inj
|
1032 |
+
m* [M⊙]
|
1033 |
+
0
|
1034 |
+
100
|
1035 |
+
200
|
1036 |
+
300
|
1037 |
+
400
|
1038 |
+
500
|
1039 |
+
600
|
1040 |
+
700
|
1041 |
+
800
|
1042 |
+
900
|
1043 |
+
0
|
1044 |
+
5
|
1045 |
+
10
|
1046 |
+
15
|
1047 |
+
20
|
1048 |
+
∫ 1/2 p
|
1049 |
+
.
|
1050 |
+
outflow dt [M☉ km s-1]
|
1051 |
+
m* [M⊙]
|
1052 |
+
Figure 7. (a) Top: Evolution of outflow momentum flux
|
1053 |
+
through the top of the simulation domain (x2 − x3 face at
|
1054 |
+
x1 = 25, 000 au) (purple solid line).
|
1055 |
+
The red dashed line
|
1056 |
+
shows the injected momentum flux at the base of the outflow.
|
1057 |
+
The green solid line shows the momentum flux injected in
|
1058 |
+
the semi-analytic model of Zhang et al. (2014b). (b) Middle:
|
1059 |
+
Evolution of the ratio of the momentum flux through the top
|
1060 |
+
of the simulation domain to the injected momentum flux at
|
1061 |
+
the base of the outflow. (c) Bottom: Evolution of the total
|
1062 |
+
momentum that has left the top of the simulation domain.
|
1063 |
+
where N is the number of data points, mi,data and mi,sim
|
1064 |
+
are the mass in the i’th velocity bin in the observed data
|
1065 |
+
and in the simulation, and σ is the uncertainty on the
|
1066 |
+
observed data. The uncertainty in the data is assumed
|
1067 |
+
to be comprised of a systematic uncertainty of 40% and
|
1068 |
+
a noise level that is ∼ 6 × 10−5 M⊙/(km s−1) (for both
|
1069 |
+
G35.2 and G339). Note that while the mass spectra are
|
1070 |
+
shown in log space, we perform the χ2 fitting in linear
|
1071 |
+
space.
|
1072 |
+
|
1073 |
+
12
|
1074 |
+
Figure 8. Distribution of outflow mass with line of sight velocity for material within a global (i.e., mirrored) simulation domain
|
1075 |
+
at various evolutionary stages (i.e., protostellar masses) and as viewed at different inclination angles, θview = 12.8◦, 61.4◦, 88.6◦.
|
1076 |
+
As seen in Figure 9, G35.2’s outflow mass spectrum at
|
1077 |
+
negative velocities is affected by a significant absorption
|
1078 |
+
feature at −20km s−1, which may be due to other molec-
|
1079 |
+
ular cloud components along the line of sight. Thus, for
|
1080 |
+
this source we restrict fitting to only the positive veloc-
|
1081 |
+
ity range. Figure 10 shows that G339’s mass spectrum
|
1082 |
+
at positive velocities is similarly affected by absorption
|
1083 |
+
features and so here we only fit to the negative velocity
|
1084 |
+
range.
|
1085 |
+
Each of the panels in Figures 9 and 10 shows the mod-
|
1086 |
+
els at a particular evolutionary stage as seen over the full
|
1087 |
+
range of viewing angles, i.e., uniformly sampling cosθview
|
1088 |
+
from 0.025 to 0.975 in steps of 0.05. We can see that at
|
1089 |
+
small values of m∗ the models generally fail to to match
|
1090 |
+
the observational data. In particular, they underpredict
|
1091 |
+
the amount of outflowing gas at low and intermediate
|
1092 |
+
velocities. For G35.2, there is a better agreement in the
|
1093 |
+
shape of the mass spectrum when m∗ ∼ 16M⊙ to 24M⊙,
|
1094 |
+
although the model is systematically low by a factor of
|
1095 |
+
about 3. For G339, the shape of the mass spectrum has
|
1096 |
+
a best match when m∗ ∼ 20 M⊙, but is again low be
|
1097 |
+
about a factor of 3. We note that such systematic off-
|
1098 |
+
sets could be explained, at least in part, by uncertainties
|
1099 |
+
in the conversion of CO(2-1) line flux to mass. The dif-
|
1100 |
+
ference could also simply be due to the observed systems
|
1101 |
+
being more massive protostellar cores, i.e., involving an
|
1102 |
+
initial core mass that is > 60 M⊙. Within the context
|
1103 |
+
of the Turbulent Core Accretion model, there is also the
|
1104 |
+
additional parameter of Σcl, which could be varied from
|
1105 |
+
the fiducial value of 1 g cm−2 assumed here.
|
1106 |
+
Given the above considerations, we do not attempt
|
1107 |
+
to adjust our models further to find a better match to
|
1108 |
+
the data, since such a step will likely require running a
|
1109 |
+
much larger grid of simulations to explore the Mc and
|
1110 |
+
Σcl parameter space. Nevertheless, with the context of
|
1111 |
+
the models we have presented, there is formally a best
|
1112 |
+
fitting model for each of G35.2 and G339. To illustrate
|
1113 |
+
these and the dependence of χ2 on model parameters, in
|
1114 |
+
Figure 11 we plot χ2 versus cos θview for all the consid-
|
1115 |
+
ered models at various evolutionary stages. Again, we
|
1116 |
+
can see that the observations are more consistent with
|
1117 |
+
higher protostellar masses.
|
1118 |
+
However, in these higher
|
1119 |
+
mass cases, we note that the goodness of fit does not
|
1120 |
+
depend very sensitively on the viewing angle.
|
1121 |
+
|
1122 |
+
c0s(0)=0.025 (0=88.69
|
1123 |
+
cos(8)=0.475 (8=61.4))
|
1124 |
+
cos(9)-0.975 (0-12.80)
|
1125 |
+
2
|
1126 |
+
M=2 M
|
1127 |
+
M=1E M
|
1128 |
+
4
|
1129 |
+
_ )
|
1130 |
+
[
|
1131 |
+
2
|
1132 |
+
4
|
1133 |
+
M=8
|
1134 |
+
8
|
1135 |
+
2000
|
1136 |
+
0
|
1137 |
+
2000
|
1138 |
+
2000
|
1139 |
+
0
|
1140 |
+
2000
|
1141 |
+
v[km s
|
1142 |
+
v[krm s-]]13
|
1143 |
+
Figure 9. The mass velocity spectra from the simulation compared to that from observations of G35.20-0.74N (Zhang et al.
|
1144 |
+
2022) for velocities less than ±50 km s−1.
|
1145 |
+
3.7. Comparison to other observational metrics of
|
1146 |
+
massive protostars
|
1147 |
+
The mass flow rate out of the simulation box (see
|
1148 |
+
Fig. 6) starts out at a few ×10−5 M⊙ yr−1 for the first
|
1149 |
+
∼ 50, 000 years until the star reaches ∼ 10 M⊙, before
|
1150 |
+
increasing to more than 10−4 M⊙ yr−1 and becoming
|
1151 |
+
quite variable during the latter parts of the simulation.
|
1152 |
+
The momentum flux out of the simulation box (Fig. 7)
|
1153 |
+
is, meanwhile, about 5 × 10−3 M⊙ km s−1 yr−1 for the
|
1154 |
+
first ∼ 40, 000 years until the star reaches ∼ 8 M⊙,
|
1155 |
+
after which the momentum rate grows steadily to ∼
|
1156 |
+
2 × 10−2 M⊙ km s−1 yr−1, and also shows time-variable
|
1157 |
+
behaviour. Such values are in general agreement with
|
1158 |
+
observations of outflows from massive protostars (Wu
|
1159 |
+
et al. 2004; Maud et al. 2015; Fedriani et al. 2019), al-
|
1160 |
+
though it should be noted that there are significant un-
|
1161 |
+
certainties associated with the observational derivation
|
1162 |
+
of these mass and momentum fluxes.
|
1163 |
+
There have been a few measurements of magnetic field
|
1164 |
+
strengths in the outflows of massive protostars. In Orion
|
1165 |
+
Source I, which is thought to be 10 − 20 M⊙ protostar
|
1166 |
+
(e.g., see discussion in Hirota et al. 2020), the magnetic
|
1167 |
+
field strength was estimated to be 30 mG on a scale of
|
1168 |
+
a few hundred au. This is in reasonable agreement with
|
1169 |
+
our simulations on similar scales (Fig. 4).
|
1170 |
+
4. DISCUSSION
|
1171 |
+
4.1. Comparison with previous simulation studies
|
1172 |
+
Here we discuss how our simulation results to those of
|
1173 |
+
other relevant studies of massive star formation, mostly
|
1174 |
+
restricting our consideration to those including pro-
|
1175 |
+
tostellar outflow feedback with magnetohydrodynamic
|
1176 |
+
(MHD) simulations. The simulation we have presented,
|
1177 |
+
in addition to its initial core, has a well defined boundary
|
1178 |
+
condition during the evolution for the input protostel-
|
1179 |
+
lar outflow, which is tied to the evolution of the fidu-
|
1180 |
+
|
1181 |
+
M=2
|
1182 |
+
M
|
1183 |
+
M=16
|
1184 |
+
M
|
1185 |
+
2
|
1186 |
+
4
|
1187 |
+
1
|
1188 |
+
M=20
|
1189 |
+
2
|
1190 |
+
4
|
1191 |
+
M=24
|
1192 |
+
2
|
1193 |
+
4
|
1194 |
+
M=12
|
1195 |
+
Mo
|
1196 |
+
cos(0)
|
1197 |
+
了
|
1198 |
+
4
|
1199 |
+
5
|
1200 |
+
0.00
|
1201 |
+
0.17
|
1202 |
+
0.33
|
1203 |
+
0.50
|
1204 |
+
0.67
|
1205 |
+
0.83
|
1206 |
+
1.00
|
1207 |
+
-40
|
1208 |
+
-20
|
1209 |
+
0
|
1210 |
+
20
|
1211 |
+
40
|
1212 |
+
Ikm s14
|
1213 |
+
Figure 10. The mass velocity spectrum from the simulation compared to that from observations of G339.88-1.25 (Zhang et al.
|
1214 |
+
2019), for velocities less than ±50 km s−1.
|
1215 |
+
cial massive protostar in the Turbulent Core Accretion
|
1216 |
+
model (McKee & Tan 2003; Zhang et al. 2014b). One
|
1217 |
+
comparable non-MHD simulation is that of Kuiper &
|
1218 |
+
Hosokawa (2018), who presented a simulation of a mas-
|
1219 |
+
sive protostar forming from a surrounding mass reservoir
|
1220 |
+
from 100 M⊙ to 1000 M⊙. The simulation code Pluto
|
1221 |
+
was utilized with a logarithmically spaced spherical co-
|
1222 |
+
ordinate grid assuming axial and midplane symmetry
|
1223 |
+
of the system. Feedback from radiation pressure, ion-
|
1224 |
+
ization and injected protostellar outflows was included.
|
1225 |
+
However, the simulation did not include magnetic fields.
|
1226 |
+
In contrast, the following simulation studies generally
|
1227 |
+
present collapse of a fully 3D gas structure to a sink
|
1228 |
+
particle representing a protostellar source. For example,
|
1229 |
+
Rosen & Krumholz (2020) performed radiation MHD
|
1230 |
+
simulations of a collapsing 150 M⊙ core (significantly
|
1231 |
+
more massive than the 60 M⊙ core we consider in this
|
1232 |
+
study), and followed the evolution until the star reached
|
1233 |
+
a mass of 33.64 M⊙. They found that once the stellar
|
1234 |
+
mass reached about 30 M⊙, radiation pressure created
|
1235 |
+
by the central star starts driving an expanding bubble.
|
1236 |
+
Radiative effects like this could potentially be relevant
|
1237 |
+
in our case if we continued the simulation beyond 30M⊙
|
1238 |
+
(see also Tanaka et al. 2017).
|
1239 |
+
Commer¸con et al. (2021) compared collapse simula-
|
1240 |
+
tions of a 100 M⊙ core in several scenarios: without
|
1241 |
+
magnetic fields, with ideal MHD, and with ambipolar
|
1242 |
+
diffusion. In the case of the non-magnetized simulation,
|
1243 |
+
they found a very weak outflow dominated by episodes
|
1244 |
+
of accretion bursts. In their ideal MHD simulation, they
|
1245 |
+
found that an increased pressure in the central region,
|
1246 |
+
due to increased stellar luminosity and build-up of mag-
|
1247 |
+
netic field, causes the outflow to almost disappear when
|
1248 |
+
the protostar reaches ∼ 10M⊙. However, this behaviour
|
1249 |
+
is not observed in their non-ideal MHD simulation.
|
1250 |
+
|
1251 |
+
1
|
1252 |
+
M=2
|
1253 |
+
M= 16 M
|
1254 |
+
2
|
1255 |
+
4
|
1256 |
+
1
|
1257 |
+
M=4
|
1258 |
+
M
|
1259 |
+
M=20 M
|
1260 |
+
+
|
1261 |
+
4
|
1262 |
+
1
|
1263 |
+
M=8
|
1264 |
+
M
|
1265 |
+
M=24
|
1266 |
+
3
|
1267 |
+
4
|
1268 |
+
1
|
1269 |
+
2
|
1270 |
+
cos(0)
|
1271 |
+
4
|
1272 |
+
L
|
1273 |
+
0.00
|
1274 |
+
0.17
|
1275 |
+
0.33
|
1276 |
+
0.50
|
1277 |
+
0.67
|
1278 |
+
0.83
|
1279 |
+
1.00
|
1280 |
+
-40
|
1281 |
+
-20
|
1282 |
+
0
|
1283 |
+
20
|
1284 |
+
40
|
1285 |
+
[km s15
|
1286 |
+
Figure 11. Dependence of χ2 derived from fitting our sim-
|
1287 |
+
ulated mass spectra for different evolutionary stages (i.e.,
|
1288 |
+
various values of m∗) to the observational data of massive
|
1289 |
+
protostars G35.2 (top) and G339 (bottom) as a function of
|
1290 |
+
the cosine of the viewing angle.
|
1291 |
+
Mignon-Risse et al. (2021b,a) performed radiation
|
1292 |
+
MHD collapse simulations also of a 100 M⊙ core.
|
1293 |
+
Mignon-Risse
|
1294 |
+
et
|
1295 |
+
al.
|
1296 |
+
(2021a)
|
1297 |
+
focused
|
1298 |
+
on
|
1299 |
+
the
|
1300 |
+
out-
|
1301 |
+
flow.
|
1302 |
+
They found mass outflow rates of ∼ 10−5 −
|
1303 |
+
10−4 M⊙ yr−1. The momentum rate that they found
|
1304 |
+
was ∼ 10−4M⊙km s−1 yr−1, which is much smaller than
|
1305 |
+
the ∼ 10−3 − 10−2 M⊙ km s−1 yr−1 that we measure in
|
1306 |
+
our simulation. We also note that our model involves
|
1307 |
+
the momentum rate growing as the protostellar mass
|
1308 |
+
grows, while they found a roughly constant momentum
|
1309 |
+
rate with time. Also, contrary to our work, the opening
|
1310 |
+
angle in their simulations for the most part decreased
|
1311 |
+
with time.
|
1312 |
+
4.2. The role of the magnetic field
|
1313 |
+
In ideal MHD, the gas is forced to follow the field lines.
|
1314 |
+
This therefore creates a natural separation between the
|
1315 |
+
outflowing gas and the collapsing envelope, because the
|
1316 |
+
field lines found in the outflow are anchored in the in-
|
1317 |
+
jection region. To demonstrate this we performed a test
|
1318 |
+
simulation with the same set up, but without magnetic
|
1319 |
+
field. In Fig. 12, we show slices of the density structures
|
1320 |
+
and velocity fields of the outflowing gas for simulations
|
1321 |
+
with and without magnetic field after 39,000 years (i.e.,
|
1322 |
+
when the protostar has reached 8 M⊙). A consequence
|
1323 |
+
of the lack of magnetic field is less collimated, slower
|
1324 |
+
outflow, which interacts with much more envelope ma-
|
1325 |
+
terial, causing a larger mass flow rate out of the simu-
|
1326 |
+
lation box as more envelope material is entrained in the
|
1327 |
+
outflow. We also find that the outflow cavity is much
|
1328 |
+
less distinct, i.e., in its density contrast with the infall
|
1329 |
+
envelope, in the simulation without magnetic field. Be-
|
1330 |
+
cause of this, there is no high-velocity outflow, and the
|
1331 |
+
momentum flow rate at a height of 25,000 au is smaller
|
1332 |
+
than in the simulation with magnetic field. Interestingly,
|
1333 |
+
the outflow pushes more material sideways when there is
|
1334 |
+
no magnetic field to confine it, forcing envelope material
|
1335 |
+
farther away from the protostar where the gravitational
|
1336 |
+
force is weaker, causing the envelope to collapse more
|
1337 |
+
slowly. As a consequence, the envelope “puffs up” side-
|
1338 |
+
ways in the no-magnetic field simulation, and at 39,000
|
1339 |
+
years it extends beyond the side boundaries (see density
|
1340 |
+
panels in Fig. 12).
|
1341 |
+
4.3. Effect of numerical resolution
|
1342 |
+
To examine the dependence on numerical resolution,
|
1343 |
+
we ran the same simulation set up with twice as many
|
1344 |
+
cells in each direction (i.e., 336×560×560 cells; see §2.1),
|
1345 |
+
but keeping other parameters the same. In this higher
|
1346 |
+
resolution simulation, the smallest cells are now roughly
|
1347 |
+
6 au on each side, compared to roughly 12 au in our
|
1348 |
+
primary “medium” resolution simulation. This higher
|
1349 |
+
resolution simulation is much more computationally ex-
|
1350 |
+
pensive, and it was not feasible to run it for the entire
|
1351 |
+
evolution (i.e., up to ∼ 24M⊙). Instead, we compare the
|
1352 |
+
results between the two resolutions at t = 39, 000 years,
|
1353 |
+
when the star has reached 8M⊙. In Fig. 13, we compare
|
1354 |
+
the logarithm of the number density, and the velocity
|
1355 |
+
field of the outflowing gas (where v1 > 0.9 km s−1), in
|
1356 |
+
a slice through the middle of the grid (x3 = 0).
|
1357 |
+
The medium and high resolution simulations are qual-
|
1358 |
+
itatively and quantitatively similar. For example, the
|
1359 |
+
opening angle of the outflow in the high resolution simu-
|
1360 |
+
lation measured at 12,000 au is 17.0◦, compared to 20.0◦
|
1361 |
+
in the medium resolution simulation. Note, while the
|
1362 |
+
low density part of the outflow cavity appears slightly
|
1363 |
+
larger in the slice of the high resolution simulation shown
|
1364 |
+
in Fig. 13, the cavity defined by the outflowing gas is in
|
1365 |
+
fact slightly smaller. At 39,000 years, in the high resolu-
|
1366 |
+
tion simulation we find that 1.5 M⊙ has left the simula-
|
1367 |
+
tion box with the outflow through the outer x1 bound-
|
1368 |
+
ary, while in the medium resolution simulation 1.2 M⊙
|
1369 |
+
has left the box. These example diagnostics indicates a
|
1370 |
+
fairly good agreement between the higher and medium
|
1371 |
+
resolution simulations.
|
1372 |
+
|
1373 |
+
6
|
1374 |
+
M:
|
1375 |
+
M
|
1376 |
+
M=
|
1377 |
+
5
|
1378 |
+
M=
|
1379 |
+
M:
|
1380 |
+
20
|
1381 |
+
24
|
1382 |
+
4
|
1383 |
+
2.
|
1384 |
+
3
|
1385 |
+
2
|
1386 |
+
0
|
1387 |
+
0.2
|
1388 |
+
0.4
|
1389 |
+
0.6
|
1390 |
+
0.8
|
1391 |
+
1
|
1392 |
+
cos(0)6
|
1393 |
+
M:
|
1394 |
+
248
|
1395 |
+
M:
|
1396 |
+
M=
|
1397 |
+
M=
|
1398 |
+
5
|
1399 |
+
M=
|
1400 |
+
M:
|
1401 |
+
20
|
1402 |
+
24
|
1403 |
+
4
|
1404 |
+
2.
|
1405 |
+
3
|
1406 |
+
2
|
1407 |
+
0
|
1408 |
+
0.2
|
1409 |
+
0.4
|
1410 |
+
0.6
|
1411 |
+
0.8
|
1412 |
+
1
|
1413 |
+
cos(0)16
|
1414 |
+
Figure 12. The effect of magnetic fields on the outflow structure is illustrated by a comparison of the number density in the
|
1415 |
+
x1 − x2 slice at x3 = 0 and time 39,000 years, when the protostar is 8 M⊙ for a case without magnetic field (|B| = 0) (left
|
1416 |
+
panels) and with a magnetic field (i.e., our fiduical model) (right panels). The upper panels show density structure; the lower
|
1417 |
+
panels show the velocity field of the outflowing gas.
|
1418 |
+
|
1419 |
+
Log(n/[cm-3])
|
1420 |
+
0.60
|
1421 |
+
1.87
|
1422 |
+
3.13
|
1423 |
+
4.40
|
1424 |
+
5.67
|
1425 |
+
6.93
|
1426 |
+
8.20
|
1427 |
+
time=39,000 yeurs
|
1428 |
+
Without B-field
|
1429 |
+
Medium resolution
|
1430 |
+
2.5x104
|
1431 |
+
2x104
|
1432 |
+
[np]
|
1433 |
+
1.5x104
|
1434 |
+
x
|
1435 |
+
104
|
1436 |
+
5000
|
1437 |
+
-10000
|
1438 |
+
0
|
1439 |
+
10000
|
1440 |
+
-10000
|
1441 |
+
0
|
1442 |
+
10000
|
1443 |
+
X
|
1444 |
+
[au]
|
1445 |
+
X
|
1446 |
+
Inplog(v/[km s-1])
|
1447 |
+
-0.05
|
1448 |
+
0.63
|
1449 |
+
1.31
|
1450 |
+
1.99
|
1451 |
+
2.67
|
1452 |
+
3.35
|
1453 |
+
time=39,000 ye0rs
|
1454 |
+
Without B-field
|
1455 |
+
with B-field
|
1456 |
+
2.5x10
|
1457 |
+
2x104
|
1458 |
+
x
|
1459 |
+
104
|
1460 |
+
5000
|
1461 |
+
10000
|
1462 |
+
0
|
1463 |
+
10000
|
1464 |
+
-10000
|
1465 |
+
0
|
1466 |
+
10000
|
1467 |
+
X>
|
1468 |
+
[nD17
|
1469 |
+
Figure 13. Effect of numerical resolution is illustrated by a comparison of the density structure in the x1 − x2 plane at x3 = 0
|
1470 |
+
at 39,000 years (m∗ = 8 M⊙) for the high resolution simulation (left panels) and fiducial medium resolution simulation (right
|
1471 |
+
panels). The upper panels show density structure; the lower panels show the velocity field of the outflowing gas.
|
1472 |
+
5. CONCLUSIONS
|
1473 |
+
We have presented a 3D-MHD simulation of a
|
1474 |
+
magnetically-powered disk wind outflow from a massive
|
1475 |
+
protostar located at the center of a core with initial mass
|
1476 |
+
of 60 M⊙ and radius of 12,000 au. Such a core is the
|
1477 |
+
fiducial case of the Turbulent Core Accretion model of
|
1478 |
+
McKee & Tan (2003), which involves the core being pres-
|
1479 |
+
sure confined by an ambient clump medium with mass
|
1480 |
+
surface density of Σcl = 1 g cm−2. We have followed the
|
1481 |
+
evolution for 100,000 years as the protostar grows from
|
1482 |
+
m∗ = 1 M⊙ to about 26 M⊙, following the protostellar
|
1483 |
+
evolutionary track of Zhang et al. (2014b), which sets
|
1484 |
+
both the accretion rate to the star and the mass and
|
1485 |
+
momentum injection rate to the disk wind outflow.
|
1486 |
+
We find that the protostar drives a powerful, colli-
|
1487 |
+
mated outflow that breaks out of the core at relatively
|
1488 |
+
early times, i.e., within ∼ 1, 000 yr of the start of the
|
1489 |
+
simulation. At the scale of the initial core, the outflow
|
1490 |
+
has an opening angle (from outflow axis to cavity edge)
|
1491 |
+
of just over 10◦ until m∗ = 4 M⊙ at 21,000 yr. There-
|
1492 |
+
after, as the protostar grows in mass and contracts to-
|
1493 |
+
|
1494 |
+
Log(n/[cm-3])
|
1495 |
+
0.60
|
1496 |
+
1.87
|
1497 |
+
3.13
|
1498 |
+
4.40
|
1499 |
+
5.67
|
1500 |
+
6.93
|
1501 |
+
8.20
|
1502 |
+
time=39,000 yeurs
|
1503 |
+
High resolution
|
1504 |
+
Medium resolution
|
1505 |
+
2.5x104
|
1506 |
+
2×104
|
1507 |
+
[nD
|
1508 |
+
1.5x104
|
1509 |
+
x
|
1510 |
+
104
|
1511 |
+
5000
|
1512 |
+
-10000
|
1513 |
+
0
|
1514 |
+
10000
|
1515 |
+
-10000
|
1516 |
+
0
|
1517 |
+
10000
|
1518 |
+
Lau]
|
1519 |
+
X2
|
1520 |
+
Inplog(v/[km s-1])
|
1521 |
+
-0.05
|
1522 |
+
0.63
|
1523 |
+
1.31
|
1524 |
+
1.99
|
1525 |
+
2.67
|
1526 |
+
3.35
|
1527 |
+
time=39,000 yeurs
|
1528 |
+
High resolution
|
1529 |
+
Medium resolution
|
1530 |
+
2.5x10
|
1531 |
+
2x104
|
1532 |
+
x
|
1533 |
+
104
|
1534 |
+
5000
|
1535 |
+
10000
|
1536 |
+
0
|
1537 |
+
10000
|
1538 |
+
-10000
|
1539 |
+
0
|
1540 |
+
10000
|
1541 |
+
Lau]
|
1542 |
+
X>
|
1543 |
+
au18
|
1544 |
+
wards the zero age main sequence, the outflow becomes
|
1545 |
+
more powerful causing the cavity to open up gradually,
|
1546 |
+
reaching opening angles of about 50◦ by the end of the
|
1547 |
+
simulation. This disk wind outflow feedback thus dra-
|
1548 |
+
matically affects the density structure and morphology
|
1549 |
+
of the protostar. While we have not performed radia-
|
1550 |
+
tive transfer (RT) calculations on these simulations (de-
|
1551 |
+
ferring this step for a future work), the RT models of
|
1552 |
+
Zhang et al. (2014b) based on a semi-analytic core and
|
1553 |
+
outflow structure already illustrate the importance of
|
1554 |
+
such cavities for determining the infrared images and
|
1555 |
+
SEDs of the protostars.
|
1556 |
+
The outflow also is the main factor determining the
|
1557 |
+
star formation efficiency (SFE) from the core. We find
|
1558 |
+
a lower limit to this SFE of ¯ϵ∗f = 0.43, but, considering
|
1559 |
+
the presence of a massive accretion disk and residual
|
1560 |
+
infall envelope, we estimate that the final value could
|
1561 |
+
reach as high as ¯ϵ∗f ≃ 0.7. Such values are moderately
|
1562 |
+
higher than the efficiencies assumed of 0.5 in the fiducial
|
1563 |
+
TCA model of McKee & Tan (2003).
|
1564 |
+
Inside the outflow cavity we find that the magnetic
|
1565 |
+
field is relatively weak, ∼ 10−4−10−5 G, while it retains
|
1566 |
+
its initial core value ∼ 10−3 G just outside the outflow
|
1567 |
+
cavity. Near the base of the outflow, however, we find
|
1568 |
+
magnetic field strengths of ∼ 0.1 G. The magnetic field
|
1569 |
+
structure we have implemented acts to help separate the
|
1570 |
+
outflow from the collapsing core, limiting the amount of
|
1571 |
+
the envelope material being entrained in the outflow.
|
1572 |
+
The mass flow and momentum rates of our simu-
|
1573 |
+
lation are ∼ 2 × 10−5 − 2 × 10−4 M⊙ yr−1 and ∼
|
1574 |
+
2 × 10−3 − 2 × 10−2 M⊙ km s−1 yr−1 respectively, with
|
1575 |
+
these values controlled by the boundary conditions we
|
1576 |
+
have implemented, but also comparable to rates mea-
|
1577 |
+
sured from observed massive protostars. We have also
|
1578 |
+
compared the distribution of outflow mass with veloc-
|
1579 |
+
ity, i.e., outflow mass spectra, of our simulations out
|
1580 |
+
to velocities of ±50 km s−1 with two example massive
|
1581 |
+
protostars G35.2 and G339 observed by ALMA. This
|
1582 |
+
comparison indicates that such observations have di-
|
1583 |
+
agnostic power to constrain model parameters related
|
1584 |
+
to evolutionary stage, i.e., m∗, and viewing angle, i.e.,
|
1585 |
+
θview. While precise agreement between model and ob-
|
1586 |
+
servation is not found (and is not expected given po-
|
1587 |
+
tential systematic uncertainties in measure mass from
|
1588 |
+
CO line emission and from the limited range of TCA
|
1589 |
+
model parameters explored in our simulation), we do
|
1590 |
+
find quite striking agreement in the shape of the out-
|
1591 |
+
flow mass spectra for some models. Further diagnostic
|
1592 |
+
tests involving full synthetic position-position-velocity
|
1593 |
+
cubes of synthetic CO line emission will be presented in
|
1594 |
+
a follow-up paper.
|
1595 |
+
JES, JPR and JCT acknowledge support from Collab-
|
1596 |
+
orative NSF grant AST-1910675. JES also acknowledges
|
1597 |
+
support from NASA through grant HST-AR-15053 from
|
1598 |
+
the Space Telescope Science Institute, which is operated
|
1599 |
+
by AURA, Inc., under NASA contract NAS 5-26555.
|
1600 |
+
JPR also acknowledges support from the Virginia Ini-
|
1601 |
+
tiative on Cosmic Origins (VICO). JCT also acknowl-
|
1602 |
+
edges support from ERC Advanced Grant MSTAR.
|
1603 |
+
We acknowledge the use of NASA High-End Comput-
|
1604 |
+
ing (HEC) resources through the NASA Advanced Su-
|
1605 |
+
percomputing (NAS) division at Ames Research Cen-
|
1606 |
+
ter to support this work.
|
1607 |
+
The analysis and the fig-
|
1608 |
+
ures have been made using GDL (Coulais et al. 2010),
|
1609 |
+
VisIt:
|
1610 |
+
https://visit-dav.github.io/visit-website/ , and
|
1611 |
+
Gnuplot: http://www.gnuplot.info/ .
|
1612 |
+
1
|
1613 |
+
2
|
1614 |
+
3
|
1615 |
+
4
|
1616 |
+
5
|
1617 |
+
6
|
1618 |
+
7
|
1619 |
+
8
|
1620 |
+
9
|
1621 |
+
10
|
1622 |
+
11
|
1623 |
+
12
|
1624 |
+
13
|
1625 |
+
14
|
1626 |
+
15
|
1627 |
+
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|
1628 |
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D9E0T4oBgHgl3EQfggHn/content/tmp_files/2301.02420v1.pdf.txt
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|
1 |
+
Noname manuscript No.
|
2 |
+
(will be inserted by the editor)
|
3 |
+
The shape of gold
|
4 |
+
B. Bally1,2,a, G. Giacalone3,b, M. Bender4,c
|
5 |
+
1 ESNT, IRFU, CEA, Universit´e Paris-Saclay, 91191 Gif-sur-Yvette, France
|
6 |
+
2 Departamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid, E-28049 Madrid, Spain
|
7 |
+
3 Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany
|
8 |
+
4 Universit´e de Lyon, Universit´e Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822, 4 rue Enrico Fermi, F-69622,
|
9 |
+
Villeurbanne, France
|
10 |
+
Received: January 9, 2023 / Revised version: date
|
11 |
+
Abstract Having a detailed theoretical knowledge of
|
12 |
+
the low-energy structure of the heavy odd-mass nucleus
|
13 |
+
197Au is of prime interest as the structure of this isotope
|
14 |
+
represents an important input to theoretical simulations
|
15 |
+
of collider experiments involving gold ions performed
|
16 |
+
worldwide at relativistic energies. In the present article,
|
17 |
+
therefore, we report on new results on the structure
|
18 |
+
of 197Au obtained from state-of-the-art multi-reference
|
19 |
+
energy density functional (MR-EDF) calculations. Our
|
20 |
+
MR-EDF calculations were realized using the Skyrme-
|
21 |
+
type pseudo-potential SLyMR1, and include beyond
|
22 |
+
mean-field correlations through the mixing, in the spirit
|
23 |
+
of the Generator Coordinate Method (GCM), of particle-
|
24 |
+
number and angular-momentum projected triaxially de-
|
25 |
+
formed Bogoliubov quasi-particle states. Comparison
|
26 |
+
with experimental data shows that the model gives a
|
27 |
+
reasonable description of 197Au with in particular a
|
28 |
+
good agreement for most of the spectroscopic proper-
|
29 |
+
ties of the 3/2+
|
30 |
+
1 ground state. From the collective wave
|
31 |
+
function of the correlated state, we compute an average
|
32 |
+
deformation ¯β(3/2+
|
33 |
+
1 ) = 0.13 and ¯γ(3/2+
|
34 |
+
1 ) = 40◦ for the
|
35 |
+
ground state. We use this result to construct an intrinsic
|
36 |
+
shape of 197Au representing a microscopically-motivated
|
37 |
+
input for precision simulations of the associated collider
|
38 |
+
processes. We discuss, in particular, how the triaxial-
|
39 |
+
ity of this nucleus is expected to impact 197Au+197Au
|
40 |
+
collision experiments at ultrarelativistic energy.
|
41 |
+
1 Introduction
|
42 |
+
For millennia, gold has held a prominent role in human
|
43 |
+
societies, whether it be as a symbol of wealth, a stan-
|
44 |
+
aE-mail: benjamin.bally@cea.fr
|
45 |
+
bE-mail: giacalone@thphys.uni-heidelberg.de
|
46 |
+
cE-mail: bender@ip2i.in2p3.fr
|
47 |
+
dard in international economic trades or because of its
|
48 |
+
medicinal and industrial applications. Interestingly, all
|
49 |
+
the gold of the world, whether it is used as jewelry, in
|
50 |
+
computer chips or kept in secured bank vaults, shares
|
51 |
+
one important feature: it is made of a single isotope.
|
52 |
+
Indeed, zooming in on the structure of this special el-
|
53 |
+
ement at the nuclear scale, one discovers that there is
|
54 |
+
only one stable gold isotope known to exist, namely
|
55 |
+
197Au.
|
56 |
+
As a matter of fact, nuclear physics essentially began
|
57 |
+
with the 197Au nucleus, which has been the first to be
|
58 |
+
discovered in 1909 by Rutherford, Geiger and Mardsen
|
59 |
+
from the scattering of α particles off a gold foil [1,2]. Over
|
60 |
+
100 years later, we have now a wealth of data available
|
61 |
+
on the structure of 197Au [3–10]. The low-energy spec-
|
62 |
+
trum of the nucleus is well known and electromagnetic
|
63 |
+
moments were measured for the ground state as well as
|
64 |
+
for several excited states [11,12]. Within a simple single-
|
65 |
+
particle picture, the 3/2+
|
66 |
+
1 ground state of 197Au can be
|
67 |
+
interpreted as a proton 2d3/2 particle (hole) coupled to
|
68 |
+
a 196Pt (198Hg) core. Considering the naive picture of
|
69 |
+
a many-body state built as the product of independent
|
70 |
+
harmonic oscillator single-particles (holes) on top of a
|
71 |
+
suitably chosen core, oblate deformations are favoured
|
72 |
+
for nuclei close to the end of a major shell [13,14]. Given
|
73 |
+
the proximity of the Z = 82 and N = 126 shell closures,
|
74 |
+
we can thus expect 197Au to adopt a small oblate-like
|
75 |
+
deformation. Actually, axially-symmetric mean-field cal-
|
76 |
+
culations based on the Gogny D1S functional [15, 16]
|
77 |
+
reported in the AMEDEE database [17] do find an oblate
|
78 |
+
minimum with a magnitude of β ≈ 0.12.
|
79 |
+
Starting from the early 2000’s, 197Au has played a cen-
|
80 |
+
tral role as well in high-energy nuclear physics. Indeed,
|
81 |
+
gold ions are employed in various scattering experiments
|
82 |
+
ranging from fixed-target experiments at a nucleon-
|
83 |
+
arXiv:2301.02420v1 [nucl-th] 6 Jan 2023
|
84 |
+
|
85 |
+
2
|
86 |
+
nucleon center-of-mass energy of 2-3 GeV performed
|
87 |
+
at GSI, Darmstadt, to ultrarelativsitic collisions at a
|
88 |
+
nucleon-nucleon center-of-mass energy of 200 GeV per-
|
89 |
+
formed at the at the BNL Relativsitic Heavy Ion Collider
|
90 |
+
(RHIC). Gold is, in particular, the prime species used at
|
91 |
+
the BNL RHIC, and the first conclusive evidence of the
|
92 |
+
formation of quark-gluon plasma in a laboratory has
|
93 |
+
been indeed obtained in ultrarelativistic 197Au+197Au
|
94 |
+
collisions [18–21].
|
95 |
+
The theoretical interpretation of the results of high-
|
96 |
+
energy scattering experiments starts with an input from
|
97 |
+
nuclear structure theory [22]. The great success of the
|
98 |
+
hydrodynamic modeling of the quark-gluon plasma [23]
|
99 |
+
combined with the availability of data from collisions of
|
100 |
+
several ion species has recently lead to a precise identifi-
|
101 |
+
cation of the impact of the structural properties of the
|
102 |
+
collided nuclei on several experimental observables. In
|
103 |
+
particular, the azimuthal distributions of particles pro-
|
104 |
+
duced in relativistic collision experiments are observed
|
105 |
+
to present a strong sensitivity to spatial correlations of
|
106 |
+
nucleons (i.e. deformations) in the ground-state many-
|
107 |
+
body wave function of the colliding species [24–29]. For
|
108 |
+
example, in a recent article [30], we argued that we
|
109 |
+
could identify fingerprints of the triaxiality of 129Xe
|
110 |
+
in collisions performed at the CERN Large Hadron
|
111 |
+
Collider (LHC). The picture of a triaxial 129Xe drawn
|
112 |
+
from the analysis of high-energy data [31] is in excel-
|
113 |
+
lent agreement with results obtained from low-energy
|
114 |
+
Coulomb excitation experiments performed on the ad-
|
115 |
+
jacent isotopes, 128,130Xe [32, 33], as well as with our
|
116 |
+
recent theoretical calculations dedicated to these three
|
117 |
+
xenon isotopes [34]. Our goal for this manuscript is, in
|
118 |
+
a sense, to perform a similar analysis focused on 197Au,
|
119 |
+
to assess and potentially improve the current structure
|
120 |
+
input to high-energy 197Au+197Au collisions.
|
121 |
+
To this aim, we first investigate the low-energy structure
|
122 |
+
of 197Au on microscopic grounds using the MR-EDF
|
123 |
+
formalism [35,36]. More precisely, we present new results
|
124 |
+
obtained from state-of-the-art calculations based on the
|
125 |
+
configuration mixing of symmetry-projected triaxially
|
126 |
+
deformed Bogoliubov quasi-particle states [37–42] and
|
127 |
+
the use of the Skyrme-type pseudo-potential SLyMR1
|
128 |
+
[43,44]. Secondly, we employ these results to construct a
|
129 |
+
point-nucleon density for 197Au, which we subsequently
|
130 |
+
employ in state-of-the-art simulations of the initial states
|
131 |
+
of high-energy 197Au+197Au collisions. We point out,
|
132 |
+
thus, the expected consequences of implementing our
|
133 |
+
newly-derived nucleon density in future hydrodynamic
|
134 |
+
simulations of such processes, with a focus on the role
|
135 |
+
played by the presence of a slight triaxiality in the
|
136 |
+
colliding ions.
|
137 |
+
This article is organized as follows: In Sec. 2, we re-
|
138 |
+
port on MR-EDF calculations dedicated to the study of
|
139 |
+
the structure of 197Au. Then, in Sec. 3 we analyze the
|
140 |
+
consequences of our results on the modeling and the ob-
|
141 |
+
servables of relativistic heavy-ion collisions. Finally, our
|
142 |
+
conclusions and prospects are reported in Sec. 4.
|
143 |
+
2 Nuclear structure
|
144 |
+
2.1 Method
|
145 |
+
In the present study, we use the same theoretical frame-
|
146 |
+
work as the one that was presented in Ref. [34] and
|
147 |
+
refer to that article for more details on our method
|
148 |
+
such as the definitions of the usual operators or the
|
149 |
+
symmetries used in our calculations. Nevertheless, to
|
150 |
+
deal with the heavy-mass 197Au nucleus we changed a
|
151 |
+
few numerical parameters compared to the ones used
|
152 |
+
in Ref. [34]. Firstly, the Bogoliubov reference states
|
153 |
+
were represented on a three-dimensional Cartesian La-
|
154 |
+
grange mesh [45] in a box of 32 points in each direction.
|
155 |
+
Secondly, when exploring the triaxial deformations, we
|
156 |
+
used a mesh with a spacing1 ∆q1 = ∆q2 = 375 fm2
|
157 |
+
starting from (q1, q2) = (0, 0) and restricting ourselves
|
158 |
+
to positive values of q1 and q2, which maps the first
|
159 |
+
sextant of the β-γ plane. Finally, concerning the cutoffs
|
160 |
+
applied during the mixing of reference states: before the
|
161 |
+
mixing, we remove the projected components that in
|
162 |
+
the decomposition of the original reference states have
|
163 |
+
a weight that is lower than 10−3, whereas during the
|
164 |
+
mixing of K-components (performed individually for
|
165 |
+
each reference state) we remove the norm eigenstates
|
166 |
+
with an eigenvalue smaller than 10−2, and during the
|
167 |
+
final diagonalization mixing projected states originating
|
168 |
+
from different Bogoliubov vacua, we remove the norm
|
169 |
+
eigenstates with an eigenvalue smaller than 10−4 for all
|
170 |
+
nuclei. The values for the the cutoffs are more restric-
|
171 |
+
tive than the ones used when tackling the 128,129,130Xe
|
172 |
+
isotopes because the configuration mixing performed in
|
173 |
+
the present calculations for 197Au proved to be more
|
174 |
+
sensitive to the inclusion of components with a small
|
175 |
+
weight that are probably not well represented on our
|
176 |
+
cartesian mesh and, therefore, have to be discarded. Un-
|
177 |
+
fortunately, the improvement of the numerical accuracy
|
178 |
+
of our lattice, by increasing the number of mesh points
|
179 |
+
and/or reducing the spacing between them, implies a
|
180 |
+
substantial increase of the computational cost of the
|
181 |
+
MR-EDF calculations that is at present out of reach for
|
182 |
+
us.
|
183 |
+
1When considering axial deformations, this corresponds to a
|
184 |
+
step of ∆β ≈ 0.05.
|
185 |
+
|
186 |
+
3
|
187 |
+
Fig. 1: Particle-number restored total energy surfaces for
|
188 |
+
197Au and π = +1 (top panel) or π = −1 (bottom panel).
|
189 |
+
Black lines are separated by 1 MeV. The minimum for
|
190 |
+
positive (negative) parity, indicated by a silver star,
|
191 |
+
is located at a deformation of β = 0.12 and γ = 38◦
|
192 |
+
(β = 0.12 and γ = 19◦)
|
193 |
+
2.2 Structure of 197Au
|
194 |
+
2.2.1 Energy surfaces
|
195 |
+
The first step in our approach is the generation of a set
|
196 |
+
of one-quasi-particle states that will be used as reference
|
197 |
+
states in the final configuration mixing calculations. To
|
198 |
+
generate and select the reference states, we follow the
|
199 |
+
strategy detailed in Ref. [34]. We briefly recall here
|
200 |
+
that this implies: i) the self-consistent blocking of four
|
201 |
+
different one-quasi-particle states at each point of the
|
202 |
+
deformation mesh, ii) the projection onto good particle
|
203 |
+
numbers and good angular momentum of all (converged)
|
204 |
+
one-quasi-particle states, and iii) the selection of the ones
|
205 |
+
having a projected energy lower than a given threshold
|
206 |
+
above the projected minimum of same parity. In this
|
207 |
+
work, we use a threshold of 5 MeV for both positive and
|
208 |
+
negative parity states.
|
209 |
+
But before discussing the final results obtained after con-
|
210 |
+
figuration mixing, let us first analyze the intermediate
|
211 |
+
steps in our method. Figure 1 displays the particle-
|
212 |
+
number restored (PNR) total energy surface for the
|
213 |
+
positive and negative parity states of 197Au. As can be
|
214 |
+
seen, the two energy surfaces exhibit a γ-soft topography
|
215 |
+
with a slightly deformed minimum2 located at β = 0.12.
|
216 |
+
Also, we notice that the surface for positive parity is
|
217 |
+
softer at small deformation than the surface for negative
|
218 |
+
parity. Finally, the minimum for positive parity states is
|
219 |
+
approximately 200 keV lower than the one for negative
|
220 |
+
parity states.
|
221 |
+
Performing the full symmetry restoration, we display
|
222 |
+
in Fig. 2 the angular-momentum and particle-number
|
223 |
+
restored (AMPNR) total energy surfaces for the lowest
|
224 |
+
Jπ = 1/2+, 3/2+ and 11/2− projected states, which
|
225 |
+
are the three values of Jπ giving the lowest projected
|
226 |
+
energies. A first remark is that the energy surfaces are
|
227 |
+
much more rigid with now a well pronounced triaxial
|
228 |
+
minimum with β = 0.13. Compared to the PNR case,
|
229 |
+
the minima of the AMPNR surfaces gain rouhgly 5 MeV
|
230 |
+
in binding energy and the absolute minimum is obtained
|
231 |
+
for Jπ = 3/2+. It is also worth mentioning that the one-
|
232 |
+
quasi-particle state giving the lowest projected state is
|
233 |
+
obtained by blocking a quasi-particle that is dominated
|
234 |
+
by a single-particle state originating from the spherical
|
235 |
+
2d3/2 shell. The latter observations are consistent with
|
236 |
+
the experimental spin-parity assignment 3/2+
|
237 |
+
1 for the
|
238 |
+
ground state of 197Au as well as its naive single-particle
|
239 |
+
interpretation. However, we notice that the minimum
|
240 |
+
for the Jπ = 3/2+ surface is located at a deformation
|
241 |
+
with an angle γ = 24◦, which seems to be at variance
|
242 |
+
with the oblate-like shape expected from simple argu-
|
243 |
+
ments as mentioned above. Nevertheless, it is important
|
244 |
+
to remark that the configuration mixing may change
|
245 |
+
this picture. In addition, we displayed in Fig. 2 only
|
246 |
+
the surface for the lowest Jπ = 3/2+ projected states,
|
247 |
+
but given the fact that we explore triaxial deformations,
|
248 |
+
all the reference states with a non-zero average value
|
249 |
+
of γ will generate after angular-momentum restoration
|
250 |
+
two projected states with Jπ = 3/2+ that will enter the
|
251 |
+
configuration mixing. Ultimately, given the fact that the
|
252 |
+
AMPNR is only an intermediate step in our approach,
|
253 |
+
it is neither possible nor desirable to definitively charac-
|
254 |
+
terize the structure of the final correlated state at this
|
255 |
+
level of approximation.
|
256 |
+
Additionally, we note that the angular-momentum pro-
|
257 |
+
jection does not shift the energy minimum towards larger
|
258 |
+
values of β compared to the plain PNR case, which is
|
259 |
+
2Note that all the extrema discussed in this article are com-
|
260 |
+
puted from an interpolation based on the results obtained at
|
261 |
+
the points on the discretized deformation mesh.
|
262 |
+
|
263 |
+
4
|
264 |
+
Jπ = 1/2+
|
265 |
+
Jπ = 3/2+
|
266 |
+
Jπ = 11/2−
|
267 |
+
Fig. 2: Angular-momentum and particle-number re-
|
268 |
+
stored total energy surface for 197Au and for the lowest
|
269 |
+
Jπ = 1/2+ (top panel), the lowest Jπ = 3/2+ (mid-
|
270 |
+
dle panel) and lowest Jπ = 11/2− (bottom panel).
|
271 |
+
Black lines are separated by 1 MeV. The minima for
|
272 |
+
Jπ = 1/2+, 3/2+ and 11/2−, indicated by silver stars,
|
273 |
+
are located at deformations of β = 0.13 and γ = 39◦,
|
274 |
+
β = 0.13 and γ = 24◦ and β = 0.13 and γ = 22◦,
|
275 |
+
respectively
|
276 |
+
contrary to what is often observed in MR-EDF calcula-
|
277 |
+
tions [30,37–40].
|
278 |
+
Fig. 3: Low-energy spectrum for 197Au. Experimental
|
279 |
+
data are taken from [46], which are based on the evalu-
|
280 |
+
ation [3]
|
281 |
+
2.2.2 Low-energy spectroscopy
|
282 |
+
Finally, we perform the full configuration mixing of sym-
|
283 |
+
metry projected reference states considering for positive
|
284 |
+
(negative) parity a set containing 24 (19) one-quasi-
|
285 |
+
particle states. In Fig. 3 we compare the theoretical
|
286 |
+
results to the available experimental data for the low-
|
287 |
+
lying states up to 1 MeV of excitation energies. First
|
288 |
+
of all, we remark that the theory is able to reproduce
|
289 |
+
the spin-parity assignment for the ground state (3/2+
|
290 |
+
1 )
|
291 |
+
as well as for the the first (1/2+
|
292 |
+
1 ), second (3/2+
|
293 |
+
2 ) and
|
294 |
+
third (5/2+
|
295 |
+
1 ) excited states. As in experimental data,
|
296 |
+
the theory predicts a staggering between the two fomer
|
297 |
+
and two latter states but the relative spacing between
|
298 |
+
the two pairs of levels, as well as the spacing between
|
299 |
+
the levels within a pair, are too large. The 5/2+
|
300 |
+
2 and
|
301 |
+
7/2+
|
302 |
+
1 states also appear in our calculations but at too
|
303 |
+
high excitation energy.
|
304 |
+
The low-lying spectrum of positive parity states of 197Au
|
305 |
+
has been interpreted with De-Shalit’s core-excitation
|
306 |
+
model of odd-mass nuclei [47], within which an odd-
|
307 |
+
even nucleus is treated as a single nucleon coupled to
|
308 |
+
an even-even core. Whenever the excitation of the core
|
309 |
+
is energetically favored compared to the promotion of
|
310 |
+
the single nucleon to a higher orbital, in this model the
|
311 |
+
lowest lying excited states of the odd-even nucleus can
|
312 |
+
be interpreted as the single-particle configuration of the
|
313 |
+
|
314 |
+
5
|
315 |
+
ground state coupled in different ways to the lowest
|
316 |
+
excitation of the even-even core. When applied to 197Au
|
317 |
+
[6,48–51], the ground state of the nucleus is constructed
|
318 |
+
as a proton 2d3/2 particle (hole) coupled to a 196Pt
|
319 |
+
(198Hg) core with Jπ = 0+.3 Then, the weak coupling of
|
320 |
+
the same 2d3/2 particle, or hole, to the Jπ = 2+ excited
|
321 |
+
state of the (appropriate) core generates a quartet of
|
322 |
+
state with Jπ = 1/2+, 3/2+, 5/2+, 7/2+ whose energy
|
323 |
+
centroid Ec = {�
|
324 |
+
J(2J + 1)E(Jπ)} / {�
|
325 |
+
J(2J + 1)} is
|
326 |
+
equal to the energy E(2+) of the excited core [52]. Using
|
327 |
+
the experimental excitation energies, we obtain Ec =
|
328 |
+
364.2 keV that is close to the values of E(2+) = 355.7
|
329 |
+
keV and for 411.8 keV for 196Pt and 198Hg, respectively.
|
330 |
+
Computing the energy centroid within our approach,
|
331 |
+
we obtain the value Ec = 630.7 keV that is obviously
|
332 |
+
too large compared to the experimental one but should
|
333 |
+
be compared the theoretical values of E(2+) for the
|
334 |
+
neighboring even-even nuclei calculated within the same
|
335 |
+
theoretical framework, which is technically possible but
|
336 |
+
falls outside the scope of the present article.
|
337 |
+
The MR-EDF theory also correctly predicts the 11/2−
|
338 |
+
1
|
339 |
+
state to be lowest state of negative parity but with an
|
340 |
+
excitation energy of 792 keV, about 400 keV too high
|
341 |
+
compared to the experimental value of 409 keV. This
|
342 |
+
is especially surprising given that the energy difference
|
343 |
+
between the AMPNR minima for Jπ = 3/2+ and 11/2−
|
344 |
+
has the correct order of magnitude as can be seen Figs. 2.
|
345 |
+
As a matter of fact, the minimum for Jπ = 11/2− has
|
346 |
+
roughly the same energy as the one for Jπ = 1/2+.
|
347 |
+
What happens is that during the configuration mixing,
|
348 |
+
the 11/2−
|
349 |
+
1 state does not gain nearly as much correla-
|
350 |
+
tion energy as the positive parity states and, therefore,
|
351 |
+
ends up at a too high excitation energy. It is not en-
|
352 |
+
tirely clear why the mixing is less important in this
|
353 |
+
case. It might be due to the deficiency of the effective
|
354 |
+
interaction but we can not exclude the possibility that
|
355 |
+
other factors may play a role. For example, we had to
|
356 |
+
use more restrictive values for the cutoffs before and
|
357 |
+
after K-mixing to remove states not well represented on
|
358 |
+
our Cartesian mesh. Therefore, it is possible that some
|
359 |
+
important components or non-diagonal matrix elements
|
360 |
+
suffer from numerical inaccuracy. Another possibility
|
361 |
+
is that our selection strategy for the one-quasi-particle
|
362 |
+
to self-consistent block at the mean-field level misses
|
363 |
+
some configurations with negative parity relevant in the
|
364 |
+
subsequent shape mixing.
|
365 |
+
In general, the theoretical spectrum is too spread in
|
366 |
+
energy, which is an often-encountered deficiency of MR-
|
367 |
+
EDF calculations based on reference states generated by
|
368 |
+
3We mention in passing that some authors argue that using a
|
369 |
+
198Hg core provides a better global description of experimental
|
370 |
+
data [6].
|
371 |
+
Quantity
|
372 |
+
Experiment
|
373 |
+
Theory
|
374 |
+
E(3/2+
|
375 |
+
1 )
|
376 |
+
-1559.384
|
377 |
+
-1556.044
|
378 |
+
rrms(3/2+
|
379 |
+
1 )
|
380 |
+
5.4371(38)
|
381 |
+
5.389
|
382 |
+
µ(1/2+
|
383 |
+
1 )
|
384 |
+
+0.416(3)
|
385 |
+
+0.01
|
386 |
+
µ(3/2+
|
387 |
+
1 )
|
388 |
+
+0.1452(2)
|
389 |
+
-0.38
|
390 |
+
µ(5/2+
|
391 |
+
1 )
|
392 |
+
+0.74(6)
|
393 |
+
+0.15
|
394 |
+
µ(5/2+
|
395 |
+
2 )
|
396 |
+
+3.0(5)
|
397 |
+
+0.14
|
398 |
+
µ(7/2+
|
399 |
+
1 )
|
400 |
+
+0.84(7)
|
401 |
+
+0.51
|
402 |
+
µ(9/2+
|
403 |
+
1 )
|
404 |
+
+1.5(5)
|
405 |
+
+0.81
|
406 |
+
µ(11/2−
|
407 |
+
1 )
|
408 |
+
(+)5.96(9)
|
409 |
+
+6.87
|
410 |
+
Qs(3/2+
|
411 |
+
1 )
|
412 |
+
+0.547(16)
|
413 |
+
+0.65
|
414 |
+
Qs(11/2−
|
415 |
+
1 )
|
416 |
+
+1.68(5)
|
417 |
+
+2.05
|
418 |
+
Table 1: Spectroscopic quantities for the low-lying states
|
419 |
+
of 197Au: total energy E (MeV), root-mean-square (rms)
|
420 |
+
charge radius rrms (fm), magnetic dipole moments µ
|
421 |
+
(µN), and spectroscopic quadrupole moments Qs (eb).
|
422 |
+
Experimental data are taken from [11,12,55–57]. The
|
423 |
+
experimental error on the binding energy is much smaller
|
424 |
+
than the rounded value given here
|
425 |
+
a variation of the total energy without consideration for
|
426 |
+
the angular momentum of the trial states. Indeed, such
|
427 |
+
a variation tend to energetically favor the ground state.
|
428 |
+
This deficiency can be in principle corrected by adding
|
429 |
+
a constraint on the average angular momentum of the
|
430 |
+
trial states during the minimization and using the value
|
431 |
+
of the constraint as an additional generator coordinate.
|
432 |
+
Unfortunately, such calculations are computationally
|
433 |
+
expensive and very few practical applications exist [53,
|
434 |
+
54].
|
435 |
+
In Table 1, we report spectroscopic quantities for some
|
436 |
+
of the low-lying states. First, we see that the calcu-
|
437 |
+
lations reproduce fairly well the binding energy and
|
438 |
+
root-mean-square charge radius of the ground state,
|
439 |
+
with a relative accuracy below 1%. The spectroscopic
|
440 |
+
quadrupole moments for the 3/2+
|
441 |
+
1 and 11/2−
|
442 |
+
1 states
|
443 |
+
are also reasonably well described in spite of being
|
444 |
+
slightly too large. While we indicate in Table 1 the value
|
445 |
+
for spectroscopic quadrupole moment of the ground
|
446 |
+
state, Qs(3/2+
|
447 |
+
1 ) = 0.547(16) eb, currently taken as
|
448 |
+
the accepted value in the compilation of Ref. [58], and
|
449 |
+
which was determined using muonic hyperfine measur-
|
450 |
+
ments [51], we remark that other values appear in the
|
451 |
+
literature that are slightly larger, i.e. 0.60 eb and 0.64
|
452 |
+
eb in Ref. [59] and 0.59 eb in Ref. [60], and in better
|
453 |
+
agreement with the value of 0.65 eb obtained in our
|
454 |
+
calculations.
|
455 |
+
Concerning the magnetic moments, they are, overall,
|
456 |
+
poorly described. The values of most of them are sig-
|
457 |
+
nificantly underestimated in our calculations and the
|
458 |
+
|
459 |
+
6
|
460 |
+
Transition
|
461 |
+
Type
|
462 |
+
Experiment
|
463 |
+
Theory
|
464 |
+
1/2+
|
465 |
+
1 → 3/2+
|
466 |
+
1
|
467 |
+
E2
|
468 |
+
35(3)
|
469 |
+
45
|
470 |
+
M1
|
471 |
+
0.004
|
472 |
+
0.019
|
473 |
+
3/2+
|
474 |
+
2 → 1/2+
|
475 |
+
1
|
476 |
+
E2
|
477 |
+
18(3)
|
478 |
+
6
|
479 |
+
M1
|
480 |
+
0.089(9)
|
481 |
+
0.048
|
482 |
+
3/2+
|
483 |
+
3 → 1/2+
|
484 |
+
1
|
485 |
+
E2
|
486 |
+
9
|
487 |
+
M1
|
488 |
+
0.34
|
489 |
+
3/2+
|
490 |
+
2 → 3/2+
|
491 |
+
1
|
492 |
+
E2
|
493 |
+
18.5(19)
|
494 |
+
0.4
|
495 |
+
M1
|
496 |
+
< 0.001
|
497 |
+
0.002
|
498 |
+
3/2+
|
499 |
+
3 → 3/2+
|
500 |
+
1
|
501 |
+
E2
|
502 |
+
4
|
503 |
+
M1
|
504 |
+
0.02
|
505 |
+
5/2+
|
506 |
+
1 → 1/2+
|
507 |
+
1
|
508 |
+
E2
|
509 |
+
14.4(17)
|
510 |
+
12
|
511 |
+
5/2+
|
512 |
+
1 → 3/2+
|
513 |
+
1
|
514 |
+
E2
|
515 |
+
26(6)
|
516 |
+
30
|
517 |
+
M1
|
518 |
+
0.034(4)
|
519 |
+
0.065
|
520 |
+
5/2+
|
521 |
+
2 → 1/2+
|
522 |
+
1
|
523 |
+
E2
|
524 |
+
7.6(23)
|
525 |
+
8
|
526 |
+
5/2+
|
527 |
+
2 → 3/2+
|
528 |
+
1
|
529 |
+
E2
|
530 |
+
7(6)
|
531 |
+
0.4
|
532 |
+
M1
|
533 |
+
0.083(10)
|
534 |
+
< 0.001
|
535 |
+
7/2+
|
536 |
+
1 → 5/2+
|
537 |
+
1
|
538 |
+
E2
|
539 |
+
0.18(7)
|
540 |
+
1
|
541 |
+
M1
|
542 |
+
0.012(1)
|
543 |
+
0.106
|
544 |
+
7/2+
|
545 |
+
1 → 3/2+
|
546 |
+
1
|
547 |
+
E2
|
548 |
+
33(3)
|
549 |
+
38
|
550 |
+
7/2+
|
551 |
+
1 → 3/2+
|
552 |
+
2
|
553 |
+
E2
|
554 |
+
6.8(20)
|
555 |
+
0.3
|
556 |
+
7/2+
|
557 |
+
1 → 3/2+
|
558 |
+
3
|
559 |
+
E2
|
560 |
+
3
|
561 |
+
7/2+
|
562 |
+
2 → 3/2+
|
563 |
+
2
|
564 |
+
E2
|
565 |
+
6(4)
|
566 |
+
22
|
567 |
+
7/2+
|
568 |
+
2 → 3/2+
|
569 |
+
3
|
570 |
+
E2
|
571 |
+
2
|
572 |
+
7/2+
|
573 |
+
2 → 5/2+
|
574 |
+
1
|
575 |
+
E2
|
576 |
+
21(6)
|
577 |
+
13
|
578 |
+
M1
|
579 |
+
0.175(23)
|
580 |
+
0.010
|
581 |
+
9/2+
|
582 |
+
1 → 7/2+
|
583 |
+
1
|
584 |
+
E2
|
585 |
+
10(7)
|
586 |
+
10
|
587 |
+
M1
|
588 |
+
0.028(10)
|
589 |
+
0.047
|
590 |
+
9/2+
|
591 |
+
1 → 5/2+
|
592 |
+
1
|
593 |
+
E2
|
594 |
+
41(5)
|
595 |
+
43
|
596 |
+
Table 2: Reduced transition probabilities among the
|
597 |
+
low-lying state of 197Au given in Weisskopf units. Ex-
|
598 |
+
perimental data are taken from [46], which are based on
|
599 |
+
the evaluation [3]
|
600 |
+
moment of the ground state has even the wrong sign.
|
601 |
+
Surprisingly, the best (relative) agreement with exper-
|
602 |
+
imental data is obtained for the magnetic moment of
|
603 |
+
the 11/2−
|
604 |
+
1 state. A similar mediocre description of the
|
605 |
+
magnetic moments was already observed in our study of
|
606 |
+
the 128,129,130Xe nuclei and we refer to this article [34]
|
607 |
+
for a discussion of the large spectrum of possible reasons
|
608 |
+
for this problem that is faced by the vast majority of
|
609 |
+
EDF calculations of magnetic properties.
|
610 |
+
In Table 2, we compare the theoretical values for the
|
611 |
+
reduced transition probabilities B(E2) and B(M1) to
|
612 |
+
available experimental data. Concerning the E2 transi-
|
613 |
+
tions, the theory gives reasonable estimates for most of
|
614 |
+
the decays. In particular, all of the strong transitions,
|
615 |
+
i.e. 1/2+
|
616 |
+
1 → 3/2+
|
617 |
+
1 , 5/2+
|
618 |
+
1 → 3/2+
|
619 |
+
1 , 7/2+
|
620 |
+
1 → 3/2+
|
621 |
+
1 and
|
622 |
+
9/2+
|
623 |
+
1 → 5/2+
|
624 |
+
1 , are well described. More generally, the
|
625 |
+
hierarchy between the transitions seems to be respected,
|
626 |
+
i.e. strong (weak) experimental transitions tend to be
|
627 |
+
strong (weak) in our calculations. One notable excep-
|
628 |
+
tion are the transitions towards/from the 3/2+
|
629 |
+
2 state
|
630 |
+
that are largely underestimated in our calculations. A
|
631 |
+
possible interpretation is that the 3/2+
|
632 |
+
2 and 3/2+
|
633 |
+
3 states
|
634 |
+
are inverted in our calculation compared to the experi-
|
635 |
+
mental spectrum. Indeed, in Table 2 we also report the
|
636 |
+
calculated transitions towards/from the 3/2+
|
637 |
+
3 state that
|
638 |
+
are in better agreement with the experimental data for
|
639 |
+
the transitions towards/from 3/2+
|
640 |
+
2 state. In the limit
|
641 |
+
case of the core-excitation model discussed above, the
|
642 |
+
reduced transition probabilities from the states of quar-
|
643 |
+
tet 1/2+
|
644 |
+
1 , 3/2+
|
645 |
+
2 , 5/2+
|
646 |
+
1 , 7/2+
|
647 |
+
1 to the 3/2+
|
648 |
+
1 ground state
|
649 |
+
are supposed to be equal among each other and with
|
650 |
+
the B(E2 : 2+
|
651 |
+
1 → 0+
|
652 |
+
1 ) of the even-even core. Obviously,
|
653 |
+
these equalities are not verified exactly for experimen-
|
654 |
+
tal data but the values remain somewhat close.4 In
|
655 |
+
particular, within the same model, the electromagnetic
|
656 |
+
transition probabilities are very sensitive to the mixing
|
657 |
+
of the Jπ = 3/2+ intrinsic states [48], a problem that
|
658 |
+
might also be present in our approach.
|
659 |
+
Concerning the M1 transitions, the model performs
|
660 |
+
poorly and most of the probabilities are either widely
|
661 |
+
underestimated or widely overestimated. These lacking
|
662 |
+
results are consistent with the observation made above
|
663 |
+
on the magnetic moments. Again, this characteristic is
|
664 |
+
a deficiency found in many nuclear EDF calculations.
|
665 |
+
While the projection techniques used here are crucial for
|
666 |
+
the reliable and unambiguous comparison of calculated
|
667 |
+
and experimental data for magnetic properties, they do
|
668 |
+
by themselves not lead to a satisfying description of
|
669 |
+
data. We refer again to [34] for further discussion of this
|
670 |
+
issue.
|
671 |
+
2.2.3 Collective wave functions
|
672 |
+
We now turn our attention towards the analysis of the
|
673 |
+
collective wave functions gJπ
|
674 |
+
σ (β, γ) of the correlated
|
675 |
+
states as defined in Ref. [34]. We recall here that the
|
676 |
+
squared collective wave functions (scwf) is a quantity
|
677 |
+
that can be used to gauge the importance of a given
|
678 |
+
deformation in the correlated wave function obtained
|
679 |
+
in the final step of the MR-EDF calculations, with the
|
680 |
+
caveat that, strictly speaking, it cannot be interpreted as
|
681 |
+
a probability distribution due to the non-orthogonality
|
682 |
+
of the reference states in the set.
|
683 |
+
In Fig. 4, we display the scwf for several low-lying states
|
684 |
+
of 197Au. Interestingly, in all cases, the distribution
|
685 |
+
of the scwf squared is dominated by triaxial shapes,
|
686 |
+
4We mention that the B(E2 : 2+
|
687 |
+
1 → 0+
|
688 |
+
1 ) values are 40.6(20)
|
689 |
+
and 28.8(4) W.u. for 196Pt and 198Hg, respectively [46,61,62].
|
690 |
+
|
691 |
+
7
|
692 |
+
Jπσ = 1/2+1
|
693 |
+
Jπσ = 3/2+1
|
694 |
+
Jπσ = 3/2+2
|
695 |
+
Jπσ = 3/2+3
|
696 |
+
Jπσ = 5/2+1
|
697 |
+
Jπσ = 5/2+2
|
698 |
+
Jπσ = 7/2+1
|
699 |
+
Jπσ = 7/2+2
|
700 |
+
Jπσ = 9/2+1
|
701 |
+
Jπσ = 11/2−
|
702 |
+
1
|
703 |
+
Fig. 4: Collective wave function squared for several low-lying states of 197Au with different values of Jπ
|
704 |
+
σ . Black lines
|
705 |
+
are separated by 10% of the (respective) maximum value indicated by a silver star
|
706 |
+
with a sharply peaked maximum that has a quadrupole
|
707 |
+
deformation of β ≃ 0.14. But depending on the value of
|
708 |
+
Jπ
|
709 |
+
σ , the maximum is either located at angle γ ≈ 40◦ or
|
710 |
+
γ ≈ 20◦. In particular, we remark that the scwf of the
|
711 |
+
3/2+
|
712 |
+
1 and 3/2+
|
713 |
+
2 states exhibit different behaviour with
|
714 |
+
the former being located closer to the oblate axis whereas
|
715 |
+
the latter favours the prolate side of the sextant. Also,
|
716 |
+
the scwf of the 3/2+
|
717 |
+
3 state is very similar to the one of
|
718 |
+
the 3/2+
|
719 |
+
1 state. This is interesting for two reasons. First,
|
720 |
+
it is contrary to what could have been expected looking
|
721 |
+
at the AMPNR energy surface for Jπ = 3/2+ in Fig. 2.
|
722 |
+
Second, this is consistent with the oblate-like behavior
|
723 |
+
expected in an independent-particle model for a nucleus
|
724 |
+
close to the Z = 82 and N = 126 shell closures. Still, it
|
725 |
+
is important to stress that non-axial deformations carry
|
726 |
+
a substantial percentage of the scwf.
|
727 |
+
Looking more closely at the scwfs of the positive parity
|
728 |
+
states, we can arrange them into three groups of similar
|
729 |
+
appearance: a) the states 1/2+
|
730 |
+
1 , 3/2+
|
731 |
+
1 , 3/2+
|
732 |
+
3 and 7/2+
|
733 |
+
1
|
734 |
+
that have a scwf mostly located in the range 30◦ ≤
|
735 |
+
γ ≤ 60◦ b) the states 3/2+
|
736 |
+
2 , 5/2+
|
737 |
+
2 and 7/2+
|
738 |
+
2 that have a
|
739 |
+
scwf mostly located in the range 0◦ ≤ γ ≤ 30◦ and c)
|
740 |
+
the states 5/2+
|
741 |
+
1 and 9/2+
|
742 |
+
1 whose scwf are more evenly
|
743 |
+
distributed as a function of γ and seem a combination
|
744 |
+
of the cases a) and b). This is in good agreement with
|
745 |
+
the data for the reduced transition probabilities given
|
746 |
+
in Table 2. Indeed, the transitions between the states of
|
747 |
+
a given group have very large B(E2) values whereas the
|
748 |
+
transitions between the states belonging to a different
|
749 |
+
group are less likely. This is not a perfect rule, however,
|
750 |
+
because the 5/2+
|
751 |
+
1 state has also an strong transition
|
752 |
+
towards the 3/2+
|
753 |
+
1 ground state but not towards the 1/2+
|
754 |
+
1
|
755 |
+
excited state even if the scwfs of the two latter states
|
756 |
+
have similarities. To come back to the core-excitation
|
757 |
+
model analysis, the fact that the scwf of the 1/2+
|
758 |
+
1 , 3/2+
|
759 |
+
3 ,5
|
760 |
+
5/2+
|
761 |
+
1 and 7/2+
|
762 |
+
1 excited states have a large overlap with
|
763 |
+
5Provided that we interpret the 3/2+
|
764 |
+
2 and 3/2+
|
765 |
+
3 states as being
|
766 |
+
inverted in our calculations compared to experimental data.
|
767 |
+
|
768 |
+
8
|
769 |
+
the scwf of the 3/2+
|
770 |
+
1 ground state is consistent with the
|
771 |
+
interpretation of the quartet of positive parity state as
|
772 |
+
being weak coupling of the same single-particle state to
|
773 |
+
a collective even-even core with an angular momentum
|
774 |
+
of either Jπ = 0+ or 2+.
|
775 |
+
As a last comment, we remark that the scwf of the
|
776 |
+
11/2−
|
777 |
+
1 state has a narrower distribution than the other
|
778 |
+
ones displayed, which is consistent with the fact that
|
779 |
+
this state does not mix as much when diagonalizing the
|
780 |
+
Hamiltonian within the space spanned by the symmetry-
|
781 |
+
projected reference states.
|
782 |
+
2.2.4 Average deformation
|
783 |
+
Finally, following the strategy presented in our previous
|
784 |
+
article on xenon isotopes [34], we use the scwf to compute
|
785 |
+
deformation parameters for the 3/2+
|
786 |
+
1 ground state of
|
787 |
+
197Au and obtain: an average elongation of ¯β(3/2+
|
788 |
+
1 ) =
|
789 |
+
0.13, with a standard deviation of ∆β(3/2+
|
790 |
+
1 ) = 0.03,
|
791 |
+
and an average angle of ¯γ(3/2+
|
792 |
+
1 ) = 40◦, with a standard
|
793 |
+
deviation of ∆γ(3/2+
|
794 |
+
1 ) = 15◦. This average deformation
|
795 |
+
is consistent with the distribution displayed in Fig. 4
|
796 |
+
as the maximum is located at a deformation of β =
|
797 |
+
0.14 and γ = 41◦ but the distribution extends towards
|
798 |
+
smaller values of β and is more or less equally distributed
|
799 |
+
with respect to the γ = 40◦ axis.
|
800 |
+
Within the rigid rotor model, it is also possible to com-
|
801 |
+
pute a deformation βr for the 0+
|
802 |
+
1 ground state of an
|
803 |
+
even-even nucleus using the experimental B(E2) values,
|
804 |
+
for more details see for example Refs. [34,63]. Comput-
|
805 |
+
ing βr for the even-even nuclei adjacent to 197Au one
|
806 |
+
obtains the value 0.13 for 196Pt and 0.11 198Hg. Our
|
807 |
+
average deformation ¯β(3/2+
|
808 |
+
1 ) = 0.12 fits nicely between
|
809 |
+
these two values, although we have to mention that
|
810 |
+
the definitions of the two elongations are model depen-
|
811 |
+
dent such that this excellent agreement may be partly
|
812 |
+
accidental.
|
813 |
+
The results of axially-symmetric EDF calculations based
|
814 |
+
on the Gogny D1S parametrization [15,16] reported in
|
815 |
+
the AMEDEE database [17] indicate a sharp minimum
|
816 |
+
at a deformation of about β ≈ 0.12 for 197Au. This is
|
817 |
+
perfectly consistent with our estimate. The AMEDEE
|
818 |
+
database also reports average deformations obtained
|
819 |
+
from large-scale five-dimensional collective Hamiltonian
|
820 |
+
(5DCH) calculations of even-even nuclei throughout the
|
821 |
+
nuclear chart [64]. We recall here that the 5DCH can be
|
822 |
+
derived as an approximation to the full GCM performed
|
823 |
+
here [63]. While their definition for the average deforma-
|
824 |
+
tion differs from ours, we mention that for 196Pt (198Hg),
|
825 |
+
they obtain an average elongation of 0.135 (0.110), with
|
826 |
+
a standard deviation of 0.032 (0.030), and average angle
|
827 |
+
of 32◦ (31◦), with a standard deviation of 12◦ (12◦). If
|
828 |
+
the values for the elongation are consistent with our
|
829 |
+
result, the average angles differ slightly with the 5DCH
|
830 |
+
result indicating a deformation right at the center of
|
831 |
+
the triaxial plane, although the fluctuations are large
|
832 |
+
enough such that the results are compatible.
|
833 |
+
3 Heavy-ion collisions
|
834 |
+
As previously mentioned, knowing the structure of 197Au
|
835 |
+
is of particular relevance in the context of high-energy
|
836 |
+
nuclear experiments, as gold is the primary species col-
|
837 |
+
lided at the BNL RHIC. This section analyzes the con-
|
838 |
+
sequences of our results for model simulations of ultra-
|
839 |
+
relativistic 197Au+197Au collisions.
|
840 |
+
3.1 Woods-Saxon parameterization of the ground
|
841 |
+
state
|
842 |
+
Traditionally, simulations of high-energy nuclear col-
|
843 |
+
lisions take as input from nuclear structure a point-
|
844 |
+
nucleon density which is used to sample nucleon coordi-
|
845 |
+
nates and define an interaction region between two ions
|
846 |
+
on a collision-by-collision basis.6 The standard choice for
|
847 |
+
the nucleon density is that of a deformed Woods-Saxon
|
848 |
+
(WS) profile:
|
849 |
+
ρ(r, θ, φ) =
|
850 |
+
ρ0
|
851 |
+
1 + e[r−R(θ,φ)]/a ,
|
852 |
+
(1)
|
853 |
+
where r, θ, φ are the usual spherical coordinates, ρ0 is
|
854 |
+
the saturation density, a is the surface diffuseness and
|
855 |
+
R(θ, φ) is the nuclear radius parameterized as
|
856 |
+
R(θ, φ) = R0
|
857 |
+
�
|
858 |
+
1 + βWS
|
859 |
+
2
|
860 |
+
�
|
861 |
+
cos(γWS)Y20(θ, φ)
|
862 |
+
(2)
|
863 |
+
+
|
864 |
+
√
|
865 |
+
2 sin(γWS)Re
|
866 |
+
�
|
867 |
+
Y22(θ, φ)
|
868 |
+
��
|
869 |
+
+ βWS
|
870 |
+
4
|
871 |
+
Y40(θ, φ)
|
872 |
+
�
|
873 |
+
,
|
874 |
+
where the spherical harmonics Ylm(θ, φ) are in complex
|
875 |
+
form. Note that the shape parameters βWS
|
876 |
+
2
|
877 |
+
, γWS and
|
878 |
+
βWS
|
879 |
+
4
|
880 |
+
represent surface deformations that differ from
|
881 |
+
the volume deformation reported in the analysis of the
|
882 |
+
previous sections [69].
|
883 |
+
We consider now the intrinsic shape of 197Au computed
|
884 |
+
from a single Hartree-Fock-Bogoliubov (HFB) calcula-
|
885 |
+
tion with the SLyMR1 interaction in which the expecta-
|
886 |
+
tion value of the quadrupole operators are constrained
|
887 |
+
6More sophisticated calculations based on nuclear configura-
|
888 |
+
tions obtained from ab initio nuclear theory have also been
|
889 |
+
recently performed [65–68]. For the moment, they are limited
|
890 |
+
to the description of collisions of 16O ions.
|
891 |
+
|
892 |
+
9
|
893 |
+
Parameter
|
894 |
+
Proton
|
895 |
+
Neutron
|
896 |
+
Nucleon
|
897 |
+
ρ0
|
898 |
+
0.067
|
899 |
+
0.090
|
900 |
+
0.157
|
901 |
+
R0
|
902 |
+
6.44
|
903 |
+
6.65
|
904 |
+
6.56
|
905 |
+
a
|
906 |
+
0.46
|
907 |
+
0.49
|
908 |
+
0.48
|
909 |
+
βWS
|
910 |
+
2
|
911 |
+
0.134
|
912 |
+
0.137
|
913 |
+
0.135
|
914 |
+
γWS
|
915 |
+
43◦
|
916 |
+
43◦
|
917 |
+
43◦
|
918 |
+
βWS
|
919 |
+
4
|
920 |
+
-0.024
|
921 |
+
-0.023
|
922 |
+
-0.023
|
923 |
+
Table 3: Parameters for the point-proton, point-neutron
|
924 |
+
and point-nucleon densities defined as in Eq. (1) and
|
925 |
+
fitted to reproduce the one-body densities of a quasi-
|
926 |
+
particle state constrained to have, on average, β = 0.13
|
927 |
+
and γ = 40◦; see the body of the text for more details.
|
928 |
+
The parameters R0 and a are given in units of fm,
|
929 |
+
whereas ρ0 is given in units of fm−3
|
930 |
+
Fig. 5: Schematic illustration of the shape of 197Au based
|
931 |
+
on the surface parametrization of the matter density of
|
932 |
+
Eq. (2), and using the parameters reported in Tab. 3
|
933 |
+
such that the one-body density of the trial one-quasi-
|
934 |
+
particle state7 verifies, on average, β = ¯β(3/2+
|
935 |
+
1 ) = 0.13
|
936 |
+
and γ = ¯γ(3/2+
|
937 |
+
1 ) = 40◦.8 We fit the resulting one-
|
938 |
+
body nucleon density with the Woods-Saxon profile
|
939 |
+
given in Eq. (1). The fit parameters are reported in
|
940 |
+
Tab. 3. We obtain, thus, a new microscopically mo-
|
941 |
+
tivated parametrization for the Woods-Saxon profile
|
942 |
+
representing the nucleon density of the ground state of
|
943 |
+
197Au which can be employed in simulations of high-
|
944 |
+
energy collisions. This profile corresponds to a triaxial
|
945 |
+
ellipsoid with radii 6.02 fm, 6.68 fm, and 6.97 fm, as
|
946 |
+
illustrated in Fig. 5.
|
947 |
+
7The trial one-quasi-particle state is built by blocking a single-
|
948 |
+
particle state originating from the spherical 2d3/2 shell.
|
949 |
+
8All other non-vanishing multipole moments authorized by
|
950 |
+
the symmetries of our calculations are let free to adopt a value
|
951 |
+
that minimizes the total energy of the trial quasi-particle
|
952 |
+
state.
|
953 |
+
For completeness, we evaluate as well the neutron skin
|
954 |
+
of the intrinsic shape, as defined by the difference of rms
|
955 |
+
radii, ∆rnp = ⟨r2⟩1/2
|
956 |
+
n
|
957 |
+
−⟨r2⟩1/2
|
958 |
+
p
|
959 |
+
. For the density returned
|
960 |
+
by the constrained HFB calculation, we find
|
961 |
+
∆rnp[HFB(¯β, ¯γ)] = 0.17 fm,
|
962 |
+
(3)
|
963 |
+
which is in perfect agreement with the result obtained
|
964 |
+
from the full MREDF calculation
|
965 |
+
∆rnp[MREDF] = 0.17 fm.
|
966 |
+
(4)
|
967 |
+
On the other hand, the fitted Woods-Saxon profile gives
|
968 |
+
a neutron skin
|
969 |
+
∆rnp[WS fit] = 0.19 fm,
|
970 |
+
(5)
|
971 |
+
meaning that, even for a large nucleus such as 197Au, the
|
972 |
+
Woods-Saxon parametrization does not fully capture
|
973 |
+
skin differences of order 0.1 fm between neutrons and
|
974 |
+
protons. We note that both the above estimates agree
|
975 |
+
with a recent measurement of the STAR collaboration
|
976 |
+
obtained via diffractive photo-production of ρ0 mesons
|
977 |
+
in ultra-peripheral 197Au+197Au collisions [70],
|
978 |
+
∆rnp[STAR] = 0.17 ± 0.03 (stat.) ± 0.08 (syst.) fm. (6)
|
979 |
+
We note, in addition, that the half-width radius obtained
|
980 |
+
for 197Au by the STAR collaboration, R0[STAR] =
|
981 |
+
6.53 ± 0.06 fm, is fully consistent with that exhibited by
|
982 |
+
our nucleon density, R0[WS fit] = 6.56 fm. This suggests
|
983 |
+
that the density of gluons relevant for scattering at
|
984 |
+
these beam energies is in fact very close to the rest-
|
985 |
+
frame point-nucleon density. This potentially adds to
|
986 |
+
the circumstantial evidence of a small nucleon width in
|
987 |
+
high-energy collisions mediated by gluons [71–75].
|
988 |
+
We discuss now the observational consequences of our
|
989 |
+
newly-derived nucleon density for relativistic 197Au+197Au
|
990 |
+
collisions. Model calculations of such processes (see e.g.
|
991 |
+
Ref. [76] for a state-of-the-art Bayesian analysis) have
|
992 |
+
so far employed the charge density of the nucleus, as
|
993 |
+
inferred from low-energy electron-nucleus scattering ex-
|
994 |
+
periments [77], as a proxy for the nucleon density. The
|
995 |
+
corresponding radial profiles are R0 = 6.38 fm, and
|
996 |
+
a = 0.53 fm. Nuclear quadrupole deformation has been
|
997 |
+
instead included by simply implementing βWS
|
998 |
+
2
|
999 |
+
= −0.13,
|
1000 |
+
as reported by finite-range liquid drop model evalua-
|
1001 |
+
tions [78]. In terms of radial profiles, there are, thus,
|
1002 |
+
minor differences between the WS parametrization that
|
1003 |
+
we show in Tab. 3 and that implemented in the lit-
|
1004 |
+
erature. We only note a reduction by 0.05 fm in the
|
1005 |
+
diffuseness parameter, a, which is due to the inclusion
|
1006 |
+
of the neutron density. This will have a mild, though
|
1007 |
+
visible impact on the initial eccentricities, εn, of the sys-
|
1008 |
+
tem [79–81]. A new feature of our calculation is instead
|
1009 |
+
the fact that 197Au is not fully oblate, but presents
|
1010 |
+
γWS = 43◦. We investigate now the impact of such a
|
1011 |
+
feature on high-energy collisions.
|
1012 |
+
|
1013 |
+
10 cm
|
1014 |
+
100%
|
1015 |
+
x1013
|
1016 |
+
6.02 fm
|
1017 |
+
6.68 fm
|
1018 |
+
6.97 fm10
|
1019 |
+
3.2 Impact of the triaxiality
|
1020 |
+
In the context of multi-particle correlation measure-
|
1021 |
+
ments in the soft sector of high-energy nuclear colli-
|
1022 |
+
sions, the strongest sensitivity to the triaxial structure
|
1023 |
+
of the colliding nuclei is carried by the mean momentum-
|
1024 |
+
elliptic flow correlation [82–84],
|
1025 |
+
ρ2 ≡ ρ(⟨pt⟩, v2
|
1026 |
+
2) = ⟨⟨pt⟩v2
|
1027 |
+
2⟩ − ⟨⟨pt⟩⟩⟨v2
|
1028 |
+
2⟩
|
1029 |
+
σ(⟨pt⟩)σ(⟨v2
|
1030 |
+
2⟩)
|
1031 |
+
,
|
1032 |
+
(7)
|
1033 |
+
where outer brackets denote a statistical average over
|
1034 |
+
events, and σ(o) is the standard deviation of observable
|
1035 |
+
o. This quantity can be evaluated in the final states
|
1036 |
+
as a three-particle correlation [85], and it measures
|
1037 |
+
the strength of the statistical correlation between the
|
1038 |
+
charged-particle average transverse momentum, ⟨pt⟩,
|
1039 |
+
and the charged-particle elliptic flow, v2, at a given
|
1040 |
+
collision multiplicity.
|
1041 |
+
To assess the impact of γWS = 43◦ on the ρ2 correlator of
|
1042 |
+
197Au+197Au collisions, we follow Ref. [30] and provide
|
1043 |
+
an estimate of the measured ρ2 from high-statistics simu-
|
1044 |
+
lations of the initial condition of these processes. For the
|
1045 |
+
details of such simulations, we refer to the exhaustive de-
|
1046 |
+
scriptions given in Ref. [30]. Briefly, we assume that the
|
1047 |
+
distribution of final-state multiplicities is proportional
|
1048 |
+
to the distribution of initial-state entropy, S, which we
|
1049 |
+
calculate event-to-event following the original TRENTo
|
1050 |
+
parametrization [86] (s(x, τ0) ∝ √TATB, S =
|
1051 |
+
�
|
1052 |
+
d2x s)
|
1053 |
+
with a nucleon size w = 0.5 fm, and a fluctuation pa-
|
1054 |
+
rameter, k, tuned to reproduce measured multiplicity
|
1055 |
+
histograms in 208Pb+208Pb collisions at CERN LHC
|
1056 |
+
energy. We consider that i) the mean transverse momen-
|
1057 |
+
tum is, at a given entropy, proportional to the initial
|
1058 |
+
E/S, where E is the total energy of the system [87,88],
|
1059 |
+
obtained upon application of the equation of state of
|
1060 |
+
high-temperature QCD (e(x) ∝ s(x)4/3, E =
|
1061 |
+
�
|
1062 |
+
d2x e),
|
1063 |
+
and ii) that the elliptic flow is proportional to the initial
|
1064 |
+
eccentricity of the system, ε2. The Pearson correlation
|
1065 |
+
coefficient of Eq. (7) can then be estimated by replac-
|
1066 |
+
ing v2
|
1067 |
+
2 and ⟨pt⟩ with, respectively, ε2
|
1068 |
+
2 and E/S. Note
|
1069 |
+
that the resulting estimator should not be compared
|
1070 |
+
directly to the experimental measurements, as it misses
|
1071 |
+
effects related to the cuts in transverse momentum, pt,
|
1072 |
+
implemented in the experimental analysis, which have
|
1073 |
+
been shown to be sizable for the magnitude of this
|
1074 |
+
observable [31,89,90]. That said, it is the initial-state
|
1075 |
+
estimator that carries the dependence on the deforma-
|
1076 |
+
tion parameters, such that the relative impact of the
|
1077 |
+
value of γWS on the final-state result can be assessed
|
1078 |
+
from it [30,91].
|
1079 |
+
We perform 20 × 106 minimum bias simulations of
|
1080 |
+
197Au+197Au collisions for three structure scenarios,
|
1081 |
+
300
|
1082 |
+
350
|
1083 |
+
400
|
1084 |
+
450
|
1085 |
+
500
|
1086 |
+
550
|
1087 |
+
600
|
1088 |
+
Nrec
|
1089 |
+
ch (|η| < 0.5)
|
1090 |
+
0.00
|
1091 |
+
0.05
|
1092 |
+
0.10
|
1093 |
+
0.15
|
1094 |
+
0.20
|
1095 |
+
0.25
|
1096 |
+
ρ
|
1097 |
+
�
|
1098 |
+
⟨pt⟩, v2
|
1099 |
+
2
|
1100 |
+
�
|
1101 |
+
← uncertainty on STAR data at Nrec
|
1102 |
+
ch ≈ 550
|
1103 |
+
TRENTo, 200 GeV Au+Au
|
1104 |
+
oblate gold (βWS
|
1105 |
+
2
|
1106 |
+
= 0.135, γWS = 60◦)
|
1107 |
+
triaxial gold (βWS
|
1108 |
+
2
|
1109 |
+
= 0.135, γWS = 43◦)
|
1110 |
+
prolate gold (βWS
|
1111 |
+
2
|
1112 |
+
= 0.135, γWS = 0)
|
1113 |
+
16
|
1114 |
+
9
|
1115 |
+
3
|
1116 |
+
1
|
1117 |
+
centrality (%)
|
1118 |
+
Fig. 6: Initial-state estimates of ρ(⟨pt⟩, v2
|
1119 |
+
2) in 200 GeV
|
1120 |
+
197Au+197Au collisions for prolate ions (dot-dashed
|
1121 |
+
line), oblate ions (dotted line) and triaxial ions (dashed
|
1122 |
+
line) presenting γWS = 43◦, as a function of the number
|
1123 |
+
of reconstructed charged tracks in the STAR detector.
|
1124 |
+
Shaded bands (of the same width as the lines) are statis-
|
1125 |
+
tical uncertainties. The figure reports as well the total
|
1126 |
+
uncertainty on preliminary STAR measurements for this
|
1127 |
+
observable at high multiplicities.
|
1128 |
+
namely, we set βWS
|
1129 |
+
2
|
1130 |
+
= 0.135, and consider γWS = 0◦,
|
1131 |
+
43◦, and 60◦.9 Rescaling the TRENTo entropy to match
|
1132 |
+
the observed mutliplicity of reconstructed charged tracks
|
1133 |
+
in the STAR detector, N rec
|
1134 |
+
ch , at midrapidity (|η| < 0.5),
|
1135 |
+
our results for ρ2 are reported in Fig. 6. Qualitatively,
|
1136 |
+
the impact of γWS follows the generic parametric expec-
|
1137 |
+
tation ρ2 ∝ c0 − c1(βWS
|
1138 |
+
2
|
1139 |
+
)3 cos(3γWS), where c0 and c1
|
1140 |
+
are positive coefficients [30,91]. We conclude that a 17◦
|
1141 |
+
deviation from oblateness in 197Au leads to a correction
|
1142 |
+
of order 10-15% to ρ2 for collisions in the 0-2% centrality
|
1143 |
+
range. We reiterate that, while our results for the magni-
|
1144 |
+
tude of the Pearson coefficient should not be compared
|
1145 |
+
directly to data, we expect the correction induced by
|
1146 |
+
the triaxiality, relative to the oblate scenario, to be ro-
|
1147 |
+
bustly captured by our initial-state evaluation. In Fig. 6
|
1148 |
+
we report as well the size of the experimental error on
|
1149 |
+
preliminary ρ2 data at high multiplicity from the STAR
|
1150 |
+
collaboration [92]. The error bar turns out to be signifi-
|
1151 |
+
cantly smaller than the splitting that we find between
|
1152 |
+
the triaxial scenario (red dashed line) and the oblate
|
1153 |
+
scenario (dotted blue line). Therefore, according to our
|
1154 |
+
results the impact of the triaxiality has been already iso-
|
1155 |
+
lated in the preliminary data, and it will be possible to
|
1156 |
+
9We safely neglect the effect of the very small hexadecapolarity
|
1157 |
+
of the nucleus, βWS
|
1158 |
+
4
|
1159 |
+
= −0.023, in these simulations.
|
1160 |
+
|
1161 |
+
11
|
1162 |
+
quantify it in the future via high-precision hydrodynamic
|
1163 |
+
simulations. We stress, though, that the most effective
|
1164 |
+
way to access the value of γWS is by studying the ρ2
|
1165 |
+
correlator of 197Au+197Au collisions normalized with
|
1166 |
+
that of 238U+238U collisions, as done in Refs. [30, 31]
|
1167 |
+
to extract such an information in the comparisons of
|
1168 |
+
129Xe+129Xe and 208Pb+208Pb collisions, which allows
|
1169 |
+
one to fully cancel theoretical and experimental system-
|
1170 |
+
atical uncertainties and isolate transparent information
|
1171 |
+
about the nuclear structure. The current mismatch be-
|
1172 |
+
tween hydrodynamic results and experimental data for
|
1173 |
+
238U+238U collisions [93] prevents us, for the moment,
|
1174 |
+
from performing such an analysis, which will be thus
|
1175 |
+
reported in future work.
|
1176 |
+
4 Conclusions
|
1177 |
+
In the present article, we first reported on new re-
|
1178 |
+
sults on the low-energy structure of the heavy odd-
|
1179 |
+
mass nucleus 197Au obtained by performing state-of-
|
1180 |
+
the-art MR-EDF calculations that include the mixing
|
1181 |
+
of angular-momentum and particle-number projected
|
1182 |
+
Bogoliubov quasi-particle states with different average
|
1183 |
+
triaxial shapes. All the calculations were realized using
|
1184 |
+
the parametrization SLyMR1 of a Skyrme-type pseudo-
|
1185 |
+
potential [44,94].
|
1186 |
+
Although odd-mass nuclei represent half of the existing
|
1187 |
+
nuclei in the nuclear chart, their calculations within
|
1188 |
+
the full-fledged MR-EDF framework are still scarce, ex-
|
1189 |
+
ceptions being [34,40,41,95]. In this work, to generate
|
1190 |
+
reference states adapted to the modeling of odd-mass
|
1191 |
+
nuclei, we performed self-consistent blocking of Bogoli-
|
1192 |
+
ubov one-quasi-particle states and considered exactly
|
1193 |
+
all the time-odd terms of the functional.
|
1194 |
+
The results obtained on the low-energy spectroscopy
|
1195 |
+
of 197Au are reasonable. The spin-parity assignments
|
1196 |
+
for the 3/2+
|
1197 |
+
1 ground state and for the first few excited
|
1198 |
+
states are correct even if the levels are too spread out,
|
1199 |
+
a well-known deficiency of usual MR-EDF calculations
|
1200 |
+
that can be corrected by adding a supplemental con-
|
1201 |
+
straint on the average angular momentum of the trial
|
1202 |
+
wave functions when generating the set of reference
|
1203 |
+
states to be projected and mixed [53,54]. The binding
|
1204 |
+
energy, root-mean-square charge radius and spectro-
|
1205 |
+
scopic quadrupole moment of the of the ground state
|
1206 |
+
are also well reproduced. By contrast, the calculations
|
1207 |
+
fail to reproduce the known magnetic moments for the
|
1208 |
+
ground and excited states. Concerning the electromag-
|
1209 |
+
netic transitions, the values for the reduced transition
|
1210 |
+
probabilities B(E2) are, overall, well described whereas
|
1211 |
+
the values for the B(M1) are off, sometimes by more
|
1212 |
+
than one order of magnitude.
|
1213 |
+
Starting from the collective wave function of the ground
|
1214 |
+
state, we computed average triaxial deformation param-
|
1215 |
+
eters ¯β(3/2+
|
1216 |
+
1 ) = 0.13 and ¯γ(3/2+
|
1217 |
+
1 ) = 40◦. Following the
|
1218 |
+
the strategy of Ref. [30], we then fitted the parameters
|
1219 |
+
of a deformed Woods-Saxon density profile, to obtain
|
1220 |
+
a new state-of-the-art microscopically-motivated input
|
1221 |
+
for the simulation of high-energy 197Au+197Au colli-
|
1222 |
+
sions. In terms of radial profile parameters, our result
|
1223 |
+
corrects to some extent the widely- and incorrectly-
|
1224 |
+
employed charge-density parametrization, which has in
|
1225 |
+
particular a too large skin thickness. For future precision
|
1226 |
+
phenomenological studies of 197Au+197Au collisions, es-
|
1227 |
+
pecially in view of the upcoming sPHENIX program
|
1228 |
+
at the BNL RHIC, it will be crucial to implement real-
|
1229 |
+
istic properties of the point-nucleon density in Monte
|
1230 |
+
Carlo simulations. This includes as well implementing
|
1231 |
+
an appropriate triaxiality, of order 45◦, for gold ions.
|
1232 |
+
Our estimates indicate that this magnitude of the tri-
|
1233 |
+
axiality does impact the final state in a significant way,
|
1234 |
+
and we expect future theoretical work to be able to
|
1235 |
+
cleanly isolate such a contribution from the data. As
|
1236 |
+
an outlook, we emphasize that measurements of the
|
1237 |
+
third centered moment (skewness) of the distribution of
|
1238 |
+
⟨pt⟩ [96] provide additional and independent information
|
1239 |
+
about γWS [91], and can be used in conjunction with
|
1240 |
+
hydrodynamic simulations to further test our prediction
|
1241 |
+
for this parameter.
|
1242 |
+
Acknowledgements We thank Chunjian Zhang for help with
|
1243 |
+
the entropy-to-multiplicity conversion used in Fig. 6, and
|
1244 |
+
Wouter Ryssens for useful discussions. This project has re-
|
1245 |
+
ceived funding from the European Union’s Horizon 2020 re-
|
1246 |
+
search and innovation programme under the Marie Sk�lodowska-
|
1247 |
+
Curie grant agreement No. 839847. M.B. acknowledges sup-
|
1248 |
+
port by the Agence Nationale de la Recherche, France, un-
|
1249 |
+
der grant No. 19-CE31-0015-01 (NEWFUN). G.G. is funded
|
1250 |
+
by the Deutsche Forschungsgemeinschaft (DFG, German Re-
|
1251 |
+
search Foundation) under Germany’s Excellence Strategy
|
1252 |
+
EXC2181/1-390900948 (the Heidelberg STRUCTURES Ex-
|
1253 |
+
cellence Cluster), within the Collaborative Research Center
|
1254 |
+
SFB1225 (ISOQUANT, Project-ID 273811115). The calcula-
|
1255 |
+
tions were performed by using HPC resources from CIEMAT
|
1256 |
+
(Turgalium), Spain (FI-2021-3-0004, FI-2022-1-0004).
|
1257 |
+
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|
1258 |
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|
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1302 |
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|
1303 |
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|
1304 |
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|
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|
1 |
+
Assessing the applicability of common performance
|
2 |
+
metrics for real-world infrared small-target detection
|
3 |
+
Saed Moradi1, Alireza Memarmoghadam1, Payman Moallem∗1, and Mohamad Farzan
|
4 |
+
Sabahi1
|
5 |
+
1Department of Electrical Engineering, Faculty of Engineering, University of Isfahan, Isfahan, Iran
|
6 |
+
Abstract
|
7 |
+
Infrared small target detection (IRSTD) is a challenging task in computer vision. During the last
|
8 |
+
two decades, researchers’ efforts are devoted to improving detection ability of IRSTDs. Despite the huge
|
9 |
+
improvement in designing new algorithms, lack of extensive investigation of the evaluation metrics are
|
10 |
+
evident. Therefore, in this paper, a systematic approach is utilized to: First, investigate the evaluation
|
11 |
+
ability of current metrics; Second, propose new evaluation metrics to address shortcoming of common
|
12 |
+
metrics. To this end, after carefully reviewing the problem, the required conditions to have a successful de-
|
13 |
+
tection are analyzed. Then, the shortcomings of current evaluation metrics which include pre-thresholding
|
14 |
+
as well as post-thresholding metrics are determined. Based on the requirements of real-world systems,
|
15 |
+
new metrics are proposed. Finally, the proposed metrics are used to compare and evaluate four well-
|
16 |
+
known small infrared target detection algorithms. The results show that new metrics are consistent with
|
17 |
+
qualitative results.
|
18 |
+
Keywords: Infrared small target detection; thresholding; pre-thresholding metrics; post-thresholding
|
19 |
+
metrics
|
20 |
+
1
|
21 |
+
Introduction
|
22 |
+
Nowadays, infrared (IR) imaging has a wide range of application from medical [1, 2] and industrial diagnosis
|
23 |
+
[3] to defense [4] and remote sensing [5]. Generally, processing IR images is a challenging task [6] due to the
|
24 |
+
specifications of IR imaging. Among all the aforementioned applications, IR small target detection (IRSTD)
|
25 |
+
is a highly challenging research field because:
|
26 |
+
• Since the IR small targets are far from the imaging device, the target has low local contrast and appears
|
27 |
+
as a dim spot in the image plane [7].
|
28 |
+
• The small target in IR images typically occupies handful of pixels [8]. Thus, the region of interest
|
29 |
+
(ROI) does not represent distinguished features.
|
30 |
+
• The edges of the small target are blurred due to atmospheric thermal fields [9]. Therefore, there are
|
31 |
+
not a clear boundary between background area and target pixels.
|
32 |
+
The block diagram of a typical IRSTD pipeline is illustrated in Fig. 1.
|
33 |
+
As shown in the figure, the
|
34 |
+
input IR image is first process by the IRSTD algorithm to create a saliency map. Note that, while the
|
35 |
+
IRSTD algorithm may refer to the end-to-end IR image processing pipeline, here, the process of construction
|
36 |
+
of saliency map from input IR image is called IRSTD algorithm. The goal is to suppress the background
|
37 |
+
area and enhance target pixels. An ideal saliency map should eliminate the background intensities and only
|
38 |
+
preserve the target area. After saliency map reconstruction, a thresholding strategy is chosen to be applied
|
39 |
+
on the saliency map. Then, true (logical one) pixels in the resulting binary image is considered as target-like
|
40 |
+
objects.
|
41 |
+
Considering the pipeline in the Fig. 1, specific attributes are defined for images in pipeline. In IRSTD
|
42 |
+
terminology, the input image can be represented by two attributes:
|
43 |
+
∗corresponding author: p moallem@eng.ui.ac.ir
|
44 |
+
1
|
45 |
+
arXiv:2301.03796v1 [cs.CV] 10 Jan 2023
|
46 |
+
|
47 |
+
Input infrared image
|
48 |
+
Saliency map
|
49 |
+
Target-like candidates
|
50 |
+
IRSTD algorithm
|
51 |
+
Thresholding
|
52 |
+
Input attributes
|
53 |
+
Pre-thresholding attributes
|
54 |
+
Post-thresholding attributes
|
55 |
+
SCRin , σb,in
|
56 |
+
SCRout , σb,out
|
57 |
+
Pfa , Pd
|
58 |
+
Figure 1: The block diagram of a typical IRSTD pipline
|
59 |
+
• σb,in which denotes the standard deviation of background pixels in input image. This parameter directly
|
60 |
+
related to the background complexity. Smaller σb,in represents smooth backgrounds, while larger σb,in
|
61 |
+
belongs to a complicated background.
|
62 |
+
• SCRin stands for signal to clutter ratio in the input image. SCR is defined as µt−µb
|
63 |
+
σb
|
64 |
+
. Where, µt, µb,
|
65 |
+
and σb are the mean value of target pixels, mean value of the local background pixels, and standard
|
66 |
+
deviation of the local background, respectively.
|
67 |
+
Same argument is valid for saliency map (The processed image by IRSTD algorithm). Thus, just like the
|
68 |
+
input attributes, σb,out and SCRout represents the background complexity and the signal to clutter ration in
|
69 |
+
the saliency map. It is clear that for a typical IRSTD pipeline:
|
70 |
+
σb,out < σb,in
|
71 |
+
and
|
72 |
+
SCRout > SCRin
|
73 |
+
(1)
|
74 |
+
According to Eq. 1, two performance metrics are defined for evaluation of IRSTD algorithms: Background
|
75 |
+
suppression factor (BSF) and signal to clutter ration gain (SCRG) which are defined as follows [10]:
|
76 |
+
BSF = σb,in
|
77 |
+
σb,out
|
78 |
+
,
|
79 |
+
SCRG = SCRout
|
80 |
+
SCRin
|
81 |
+
(2)
|
82 |
+
Based on Eq. 1 and Eq. 2, larger values for both SCRG and BSF are desired. Note that, in case of
|
83 |
+
evaluation of different IRSTD algorithms, since the input images are the same for all baseline algorithms,
|
84 |
+
SCRout and
|
85 |
+
1
|
86 |
+
σb,out can be used as performance metrics, as well.
|
87 |
+
The IRSTD algorithms are well-studied in the literature. Mainly, these algorithms can be categorized
|
88 |
+
based on filtering method, contrast measure calculation, and data structure decomposition [11].
|
89 |
+
The filtering based methods are divided into two sub-categories.
|
90 |
+
The first one is the spatial domain
|
91 |
+
filtering, in which, the input infrared image is processed using local kernels to enhance the target area. Max-
|
92 |
+
mean [12], max-median [12], bilateral filtering [13], morphological operators [14], two dimensional least mean
|
93 |
+
square [15] are some instances of this sub-category. The second one refers to processing in the transformation
|
94 |
+
domain.
|
95 |
+
In these techniques, the input image is transformed to a desired transformation space like as
|
96 |
+
frequency [16] and wavelet [17] domains. Then, after processing the transformed information, the inverse
|
97 |
+
transform is applied to recover true targets.
|
98 |
+
Methods based on human visual systems (HVS) which lead to local contrast-based mechanism has received
|
99 |
+
researchers’ attention during last few years. These methods outperform filter-based methods in terms of
|
100 |
+
SCRG and BSF. However, they usually have higher computational complexity compared to filter-based ones.
|
101 |
+
Generally, local contrast can be constructed in either difference or ratio forms. Difference local contrast like
|
102 |
+
as Laplacian of Gaussian (LoG) [18], difference of Gaussian (DoG) [19], improved difference of Gabor [20],
|
103 |
+
center-surround difference measure [21], and local difference adaptive measure [22]. Unlike the difference form
|
104 |
+
local measures, ratio-form local measures utilize enhancement factor which is the ration between the center
|
105 |
+
cell and surrounding ones. Local contrast measure (LCM) [23], improved local contrast measure (ILCM) [24],
|
106 |
+
2
|
107 |
+
|
108 |
+
relative local contrast measure (RLCM) [25], Tri-Layer local constrast method (TLLCM) [26], novel local
|
109 |
+
contrast descriptor (NLCD) [27], and weighted strengthened local contrast measure (WSLCM) [28] are the
|
110 |
+
most effective IRSTDs in the literature. There is also a combined local measure which benefits from both
|
111 |
+
difference and ratio from of local contrast measure [29].
|
112 |
+
Data structure decomposition-based methods are also a newly introduced class of IRSTDs. Sparse and
|
113 |
+
low-rank matrices decomposition is the principal of these class of IRSTDs. Infrared patch image (IPI) model
|
114 |
+
[30], weighted infrared patch image (WIPI) model [31], non-negative infrared patch image model based
|
115 |
+
on partial sum minimization of singular values (NIPPS) [32], nonconvex rank approximation minimization
|
116 |
+
(NRAM) [33], and nonconvex optimization with an Lp norm constraint (NOLC) [34] are the recent efforts of
|
117 |
+
IR image decomposition-based approach.
|
118 |
+
The goal of all aforementioned methods is to obtain larger BSF and SCRG values . However, having larger
|
119 |
+
BSF and SCRG does not guarantee a successful detection. A high performance IRSTD algorithm should
|
120 |
+
be followed by a proper thresholding strategy to detect real targets and eliminate false responses. This is
|
121 |
+
why there are two more performance metrics after applying the thresholding operation to the saliency map.
|
122 |
+
These two metrics which demonstrate the ability of detection true targets and eliminating false responses
|
123 |
+
are called probability of detection Pd and probability of false alarms Pfa, respectively. In contrast to BSF
|
124 |
+
and SCRG which are measurable before applying the threshold (This is why we call them pre-thresholding
|
125 |
+
attributes), these two metrics are measured on binary images and therefore we call them post-thresholding
|
126 |
+
attributes (Fig. 1).
|
127 |
+
As mentioned in the previous paragraph, for a successful detection, both high performance IRSTD al-
|
128 |
+
gorithm as well as the proper thresholding strategy are required. Regardless of effectiveness of the IRSTD
|
129 |
+
algorithm, improper thresholding will leads to missing true targets and having false responses which could
|
130 |
+
be disaster for a practical system. Hence, in this paper, after investigating various thresholding strategies,
|
131 |
+
the best methods for applying threshold to the saliency map is presented. Then, current pre-thresholding as
|
132 |
+
well as the post-thresholding metrics are investigated, and some new metrics which are aligned with practical
|
133 |
+
considerations are proposed. The rest of this paper is organized as follows: in the next section, the role of
|
134 |
+
thresholding in practical systems is deeply investigated. Then, in section 3, current pre-thresholding metrics
|
135 |
+
are reviewed. After demonstrating their shortages, modified metrics are proposed for IRSTD performance
|
136 |
+
evaluation. In section 4, same process is performed for post-thresholding metrics. In section 5, The newly
|
137 |
+
proposed metrics are used for performance comparison of common IRSTD algorithms. Finally, the paper is
|
138 |
+
concluded in section 6.
|
139 |
+
2
|
140 |
+
The onus of thresholding on the overall performance
|
141 |
+
After performing target enhancement and clutter suppression procedure (saliency map construction), the
|
142 |
+
filtered IR image should be converted to binary one using thresholding operation that can be applied in
|
143 |
+
different forms (i.e manual, automatic, local, global). Since the target detection problem only consists of two
|
144 |
+
different classes namely as target and background clutter, single-level thresholding is a satisfactory option
|
145 |
+
for this purpose. The simplest method to achieve the classification goal, is to apply a global threshold T:
|
146 |
+
g(x, y) =
|
147 |
+
�
|
148 |
+
1
|
149 |
+
f(x, y) > T
|
150 |
+
0
|
151 |
+
f(x, y) ≤ T
|
152 |
+
(3)
|
153 |
+
where, f(x, y) and g(x, y) stand for the saliency map and binary image,respectively. The most challenging part
|
154 |
+
of global thresholding operation is how to set an effective threshold value T. Since target detection systems
|
155 |
+
continuously scan the environment, human operator cannot be helpful to choose the optimum threshold
|
156 |
+
value. The most simplest way to do this is to choose a unique threshold value based on experiments for all
|
157 |
+
incoming image frames. However, when the dynamic range of the filtered image is not equal to the dynamic
|
158 |
+
range of the input images, the false-alarm rate or the miss-rate will increase drastically. Fig. 2 shows the
|
159 |
+
change in the dynamic range of filtered images (saliency maps) using Tophat and AAGD IRSTDs. As shown
|
160 |
+
in the figure; the output dynamic range directly depends on the applied IRSTD. Therefore, the thresholding
|
161 |
+
procedure should be performed in an automated manner. There are various automatic image thresholding
|
162 |
+
algorithms in the literature. The Otsu’s method is one of the widely used one [35]. In this method the global
|
163 |
+
threshold value is chosen in a way, to maximize inter-class variance. When both foreground (Target) and
|
164 |
+
3
|
165 |
+
|
166 |
+
40
|
167 |
+
60
|
168 |
+
80
|
169 |
+
100
|
170 |
+
120
|
171 |
+
140
|
172 |
+
160
|
173 |
+
180
|
174 |
+
200
|
175 |
+
(a)
|
176 |
+
0
|
177 |
+
10
|
178 |
+
20
|
179 |
+
30
|
180 |
+
40
|
181 |
+
50
|
182 |
+
60
|
183 |
+
70
|
184 |
+
80
|
185 |
+
90
|
186 |
+
100
|
187 |
+
(b)
|
188 |
+
0
|
189 |
+
500
|
190 |
+
1000
|
191 |
+
1500
|
192 |
+
2000
|
193 |
+
2500
|
194 |
+
(c)
|
195 |
+
Figure 2: Variable dynamic range in saliency map.
|
196 |
+
a) Original infrared image.
|
197 |
+
b) filtering result using
|
198 |
+
TopHat algorithm [14], c) filtering result using AAGD [37] algorithm. The dynamic range of the saliency
|
199 |
+
map might be different than input infrared image depending on the applied IRSTD.
|
200 |
+
background classes include considerable number of pixels, and the image histogram is bimodal (i.e. there
|
201 |
+
is a deep valley between two peaks in the image histogram), the Otsu’s method works very well in object
|
202 |
+
segmentation problems. However, when the target area is too small compared to the background area, which
|
203 |
+
is always occurred in incoming infrared target detection problems, the segmentation result of Otsu’s method
|
204 |
+
is inaccurate Fig. 3. Another widely used automatic thresholding is presented in [36], where the followings
|
205 |
+
are performed to obtain the desired threshold value:
|
206 |
+
i) The gray image is segmented into two classes using threshold value equal to global mean of the image
|
207 |
+
(T = µG).
|
208 |
+
ii) The average values of the background and target are calculated (µB, µT ).
|
209 |
+
iii) The new threshold level is calculated
|
210 |
+
�
|
211 |
+
Tnew = 1
|
212 |
+
2 (µT + µB)
|
213 |
+
�
|
214 |
+
.
|
215 |
+
iv) While (Tnew − Told > ϵ), steps (ii) and (iii) are recursively repeated.
|
216 |
+
When the background noise is not strong, this automatic thresholding operation shows good performance
|
217 |
+
for final target detection (Fig. 3d). However, in strong noisy scenarios, the performance of this algorithm
|
218 |
+
is degraded significantly (Fig. 4d), which in turn, increases the false responses. Moreover, when infrared
|
219 |
+
scenario does not contain any small target, these histogram-based automatic thresholding methods always
|
220 |
+
return incorrect responses in non-target areas (Fig. 5).
|
221 |
+
Statistics-based image thresholding is the most effective thresholding strategy for small target detection
|
222 |
+
which can be applied in both local and global manners. Statistics-based global and local thresholding are
|
223 |
+
expressed in Eq. 4 and Eq. 5, respectively.
|
224 |
+
T = µG + kG × σG
|
225 |
+
(4)
|
226 |
+
T(x, y) = µ(x, y) + kL × σ(x, y)
|
227 |
+
(5)
|
228 |
+
where, µG, σG, µ(x, y), σ(x, y), kG, and kL indicate global mean of the image, global standard deviation of the
|
229 |
+
image, local mean around (x, y) position, local standard deviation around (x, y) position, control parameter
|
230 |
+
of global thresholding and local thresholding, respectively.
|
231 |
+
Global thresholding is a simple operation with low computational complexity. However, in multi-target
|
232 |
+
scenarios, some targets may be missed. Local thresholding can detect all targets. Since local mean and
|
233 |
+
standard deviation should be calculated for each pixel in the gray image, the local statistics-based thresholding
|
234 |
+
has higher computational complexity compared to the global one. Generally speaking, using statistics-based
|
235 |
+
thresholding has the following advantages:
|
236 |
+
• It can work with any gray-level dynamic ranges.
|
237 |
+
• The control parameter (k) can be determined by experiments to achieve reasonable false-alarm rate.
|
238 |
+
• The last but not the least, it is very effective for scenarios with no targets.
|
239 |
+
4
|
240 |
+
|
241 |
+
(a)
|
242 |
+
(b)
|
243 |
+
(c)
|
244 |
+
(d)
|
245 |
+
(e)
|
246 |
+
Figure 3: The automatic thresholding results. a) Original infrared image. b) Top-hat filtering result [14].
|
247 |
+
c) Otsu’s thresholding result (T = 0.48). d) automatic thresholding using average values of background and
|
248 |
+
target classes (T = 19). e) Manual thresholding (T = 29).
|
249 |
+
(a)
|
250 |
+
(b)
|
251 |
+
(c)
|
252 |
+
(d)
|
253 |
+
(e)
|
254 |
+
Figure 4: The automatic thresholding results. a) Original noisy infrared image. b) Top-hat filtering result.
|
255 |
+
c) Otsu’s thresholding result (T = 0.5). d) automatic thresholding using average values of background and
|
256 |
+
target classes (T = 7). e) Manual thresholding (T = 10).
|
257 |
+
3
|
258 |
+
Pre-thresholding evaluation
|
259 |
+
A detection process is successful as long as a single pixel of target area is correctly recognized. In this case,
|
260 |
+
the exact boundary extraction of the target area is not important at all. Therefore, a proper evaluation
|
261 |
+
metric should support this argument.
|
262 |
+
Signal to clutter ratio (SCR) is one of pre-thresholding metrics which shows the target enhancement
|
263 |
+
5
|
264 |
+
|
265 |
+
(a)
|
266 |
+
(b)
|
267 |
+
(c)
|
268 |
+
Figure 5: Drawback of automatic thresholding in scenarios with no targets. a) original infrared image which
|
269 |
+
does not contain small target. b) the result of Top-Hat filtering. c) automatic thresholding results.
|
270 |
+
ability of an IRSTD, which is defined as:
|
271 |
+
SCR = µT − µb
|
272 |
+
σb
|
273 |
+
,
|
274 |
+
(6)
|
275 |
+
where, µT , µb, and σb denote average intensity of the target area, average intensity and standard deviation of
|
276 |
+
its local surrounding background, respectively. While this evaluation measure is generally accepted in the lit-
|
277 |
+
erature, it can not correctly reflect the target enhancement capability of an IRSTD. To better understanding,
|
278 |
+
a simple scenario is provided here (Fig. 6).
|
279 |
+
Two different saliency maps are demonstrated in Fig. 6a and Fig. 6b. Fig. 6a shows the result of applying
|
280 |
+
AAGD algorithm [37] with 9 × 9 internal window. As depicted in the figure; the target area is relatively
|
281 |
+
enhanced while there are some remaining background clutter. Compared to the Fig. 6a, the second IRSTD
|
282 |
+
which again is an AAGD Fig. 6b algorithm with 3 × 3 internal window followed by a morphological erosion
|
283 |
+
with a 3 × 3 square-shape structural element, shows better target enhancement and background suppression.
|
284 |
+
As shown in Fig. 6c and Fig. 6d, the signal amplitude for the IRSTD #2 is almost twice as the one in the
|
285 |
+
IRSTD #1, which means in higher threshold values the target will be detected corectly in the second one,
|
286 |
+
while in the first one the target will be missed. One dimensional (1D) cross-section of target area in both
|
287 |
+
saliency maps are shown in Fig. 6e and Fig. 6f, respectively. To simplify the scenario, let’s approximate 1D
|
288 |
+
cross-section of the target area with closest square signal. The result is shown in Fig. 6g. As shown in the
|
289 |
+
figure:
|
290 |
+
AT1 = AT2
|
291 |
+
2
|
292 |
+
,
|
293 |
+
WT1 = 3 × WT2
|
294 |
+
(7)
|
295 |
+
where, AT1, AT2 denote the target amplitude in the output of IRSTD #1 and #2. Also, WT1, WT2 show the
|
296 |
+
target width (extension) in the output of IRSTD #1 and #2, respectively.
|
297 |
+
Based on SCR formulation (Eq. 6), and simply considering zero-mean background signal, the following
|
298 |
+
relationship can be easily derived:
|
299 |
+
SCR1 = 3
|
300 |
+
2 × SCR2
|
301 |
+
(8)
|
302 |
+
which implies that the target detection ability of the IRSTD #1 is 50% more than that of IRSTD #1.
|
303 |
+
However, by applying a global threshold level at Tapp, the #1 algorithms does not detect the true target
|
304 |
+
(Fig. 6). It can be clearly seen that the #2 algorithm can detect the true target at the same threshold
|
305 |
+
level. In order to address this issue when global thresholding is final choice in practical system, the SCR
|
306 |
+
formulation should be modified as:
|
307 |
+
SCRglobal = maxT −µG
|
308 |
+
σG
|
309 |
+
,
|
310 |
+
(9)
|
311 |
+
where, maxT denotes the maximum gray value of the target area.
|
312 |
+
According to Eq. 4, the maximum
|
313 |
+
acceptable control parameter is equal to newly defined SCR metric:
|
314 |
+
kGmax = SCRglobal.
|
315 |
+
(10)
|
316 |
+
There are two important points regarding the Eq. 10:
|
317 |
+
6
|
318 |
+
|
319 |
+
0
|
320 |
+
50
|
321 |
+
100
|
322 |
+
150
|
323 |
+
200
|
324 |
+
250
|
325 |
+
(a)
|
326 |
+
0
|
327 |
+
50
|
328 |
+
100
|
329 |
+
150
|
330 |
+
200
|
331 |
+
250
|
332 |
+
300
|
333 |
+
350
|
334 |
+
400
|
335 |
+
450
|
336 |
+
(b)
|
337 |
+
0
|
338 |
+
50
|
339 |
+
100
|
340 |
+
150
|
341 |
+
200
|
342 |
+
250
|
343 |
+
(c)
|
344 |
+
0
|
345 |
+
50
|
346 |
+
100
|
347 |
+
150
|
348 |
+
200
|
349 |
+
250
|
350 |
+
300
|
351 |
+
350
|
352 |
+
400
|
353 |
+
450
|
354 |
+
(d)
|
355 |
+
0
|
356 |
+
5
|
357 |
+
10
|
358 |
+
15
|
359 |
+
0
|
360 |
+
50
|
361 |
+
100
|
362 |
+
150
|
363 |
+
200
|
364 |
+
250
|
365 |
+
(e)
|
366 |
+
0
|
367 |
+
5
|
368 |
+
10
|
369 |
+
15
|
370 |
+
0
|
371 |
+
50
|
372 |
+
100
|
373 |
+
150
|
374 |
+
200
|
375 |
+
250
|
376 |
+
300
|
377 |
+
350
|
378 |
+
400
|
379 |
+
450
|
380 |
+
500
|
381 |
+
(f)
|
382 |
+
Threshold Value
|
383 |
+
Tapp
|
384 |
+
Signal
|
385 |
+
Amplitude
|
386 |
+
AT1
|
387 |
+
WT1
|
388 |
+
Signal
|
389 |
+
width
|
390 |
+
Output of #1 detection algorithm
|
391 |
+
Threshold Value
|
392 |
+
Tapp
|
393 |
+
Signal
|
394 |
+
Amplitude
|
395 |
+
AT2
|
396 |
+
WT2
|
397 |
+
Signal
|
398 |
+
width
|
399 |
+
Output of #2 detection algorithm
|
400 |
+
(g)
|
401 |
+
Figure 6: A simple scenario to demonstrate the drawback of common SCR metric. a, b) Saliency map of the
|
402 |
+
IRSTD #1 and #2, c, d) target area in the saliency map of the IRSTD #1 and #2, e, f) 1D plot of target
|
403 |
+
cross-section in c and d, g) simplified 1D representation of target area in both IRSTD #1 and #2.
|
404 |
+
1. Common SCR metric is not able to correctly reflect the target detection ability. The pre-thresholding
|
405 |
+
evaluation should be performed in a global manner on the saliency maps.
|
406 |
+
2. The thresholding operation should be consistent with pre-thresholding evaluation metrics. For instance,
|
407 |
+
in our case, the global statistics-based thresholding is the right choice.
|
408 |
+
So far, it is demonstrated that the global thresholding is the right one to be applied on the saliency map.
|
409 |
+
In the next subsection, we demonstrate the drawback of the local statistics-based thresholding.
|
410 |
+
3.1
|
411 |
+
Drawback of common local thresholding
|
412 |
+
Now, let consider the case that local thresholding is supposed to be applied on the saliency map. According
|
413 |
+
to Fig. 6, the local mean around target region can be calculated as follows:
|
414 |
+
µ(x) = AW
|
415 |
+
n
|
416 |
+
(11)
|
417 |
+
where, A, W, x, and n stand for the target amplitude, width (spatial extension), the current index and
|
418 |
+
number of samples in local neighborhood (n > W).
|
419 |
+
7
|
420 |
+
|
421 |
+
2
|
422 |
+
3
|
423 |
+
4
|
424 |
+
5
|
425 |
+
6
|
426 |
+
7
|
427 |
+
8
|
428 |
+
9
|
429 |
+
1
|
430 |
+
2
|
431 |
+
3
|
432 |
+
4
|
433 |
+
kL max
|
434 |
+
n=17
|
435 |
+
n=25
|
436 |
+
n=33
|
437 |
+
Figure 7: kLmax versus target spatial extension W (target width in 1D case)
|
438 |
+
The local standard deviation can be calculated as:
|
439 |
+
σ(x) =
|
440 |
+
�
|
441 |
+
�
|
442 |
+
�
|
443 |
+
� 1
|
444 |
+
n
|
445 |
+
n
|
446 |
+
�
|
447 |
+
i=1
|
448 |
+
(y(i) − µ(x))2
|
449 |
+
(12)
|
450 |
+
Where y(i) and µ(x) denote the saliency map samples and local mean, respectively. Since the detection
|
451 |
+
algorithm is supposed to suppress background clutter, for the sake of simplicity, we can assume that the
|
452 |
+
saliency map samples out of the target region are equal to zero. Then:
|
453 |
+
σ(x) = A
|
454 |
+
n
|
455 |
+
��
|
456 |
+
nW − W 2
|
457 |
+
�
|
458 |
+
(13)
|
459 |
+
The local thresholding (Eq. 5), can be rewritten as:
|
460 |
+
T(x) = µ(x) + kL × σ(x) = AW
|
461 |
+
n
|
462 |
+
+ kLA
|
463 |
+
n
|
464 |
+
��
|
465 |
+
nW − W 2
|
466 |
+
�
|
467 |
+
(14)
|
468 |
+
The detection process is established correctly for the threshold values lower than target amplitude (T(x) <
|
469 |
+
A). Therefore, for a successful target detection the following condition should be met:
|
470 |
+
µ(x) + kL × σ(x) < A
|
471 |
+
⇒ AW
|
472 |
+
n
|
473 |
+
+ kLA
|
474 |
+
n
|
475 |
+
��
|
476 |
+
nW − W 2
|
477 |
+
�
|
478 |
+
< A
|
479 |
+
(15)
|
480 |
+
The upper bound for control parameter (kLmax) to detect the target accurately can be find as follows:
|
481 |
+
AW
|
482 |
+
n
|
483 |
+
+ kLmaxA
|
484 |
+
n
|
485 |
+
��
|
486 |
+
nW − W 2
|
487 |
+
�
|
488 |
+
= A
|
489 |
+
⇒ kLmax =
|
490 |
+
�
|
491 |
+
n − W
|
492 |
+
W
|
493 |
+
(16)
|
494 |
+
Fig. 7 shows the upper bound of control parameter versus target width. As shown in the figure, the maximum
|
495 |
+
control parameter to detect target correctly using local thresholding decreases as the target width increases.
|
496 |
+
kLmax takes its maximum value when the target width is equal to one pixel (kLmax = √n − 1 for W = 1).
|
497 |
+
This result is quite consistent with the fact that the most effective target detection algorithm should suppress
|
498 |
+
all background region and only returns a single pixel (Target centroid).
|
499 |
+
Fig. 8 shows the local threshold value which is normalized to the target amplitude ( T
|
500 |
+
A) versus different
|
501 |
+
control parameter. It is clear that, only when the ( T
|
502 |
+
A) fraction is less than one the target can be detected
|
503 |
+
correctly. Another finding which can be derived from the figure is that the maximum control parameter
|
504 |
+
decreases as the target width increases. Therefore, unlike the global case, the effective control parameter to
|
505 |
+
extract real targets and eliminate background clutter depends on the target area in the saliency map. Also,
|
506 |
+
the reasonable rang for control parameter is narrowed when the local neighborhood is decreased (Fig. 8b).
|
507 |
+
Moreover, using Eq. 5 to extract true target from saliency map leads to many false alarms. Fig. 9 shows the
|
508 |
+
8
|
509 |
+
|
510 |
+
0
|
511 |
+
0.5
|
512 |
+
1
|
513 |
+
1.5
|
514 |
+
2
|
515 |
+
2.5
|
516 |
+
3
|
517 |
+
Local Adjustment Parameter (KL)
|
518 |
+
0
|
519 |
+
0.5
|
520 |
+
1
|
521 |
+
1.5
|
522 |
+
W=3
|
523 |
+
W=5
|
524 |
+
W=7
|
525 |
+
W=9
|
526 |
+
The target
|
527 |
+
is missed
|
528 |
+
The target
|
529 |
+
is correctly
|
530 |
+
detected
|
531 |
+
(a)
|
532 |
+
0
|
533 |
+
0.5
|
534 |
+
1
|
535 |
+
1.5
|
536 |
+
2
|
537 |
+
2.5
|
538 |
+
3
|
539 |
+
Local Adjustment Parameter (KL)
|
540 |
+
0
|
541 |
+
0.5
|
542 |
+
1
|
543 |
+
1.5
|
544 |
+
2
|
545 |
+
W=3
|
546 |
+
W=5
|
547 |
+
W=7
|
548 |
+
W=9
|
549 |
+
The target
|
550 |
+
is missed
|
551 |
+
The target
|
552 |
+
is correctly
|
553 |
+
detected
|
554 |
+
(b)
|
555 |
+
Figure 8: Local threshold value normalized to target amplitude ( T
|
556 |
+
A) versus different control parameter. a)
|
557 |
+
n = 33, b) n = 17.
|
558 |
+
(a)
|
559 |
+
(b)
|
560 |
+
(c)
|
561 |
+
(d)
|
562 |
+
Figure 9: Shortcoming of local thresholding. a) Original input image, the target area is marked by red
|
563 |
+
ellipse, b) the result of target enhancement using multi-scale Laplacian of Gaussian (LoG) method, c) the
|
564 |
+
local thresholding applied on (b) with k = 4, c) the local thresholding applied on (b) with k = 5.
|
565 |
+
local thresholding on the saliency map of multi-scale Laplacian of Gaussian (LoG) method [18]. As shown in
|
566 |
+
the Fig. 9b, the target area is the most salient region in saliency map. However, after thresholding using local
|
567 |
+
method (Fig. 9c), there are too many false responses. The only way to limit false responses to an acceptable
|
568 |
+
range is to increase the control parameter. However, the true target is not extracted when the local threshold
|
569 |
+
is increased. Note that there are still too many false responses in Fig. 9d.
|
570 |
+
Based on the local thresholding results Fig. 9, this method (Eq. 5) is not a proper strategy to discriminate
|
571 |
+
target area from background clutter.
|
572 |
+
4
|
573 |
+
Post-thresholding evaluation
|
574 |
+
After applying a predefined threshold to the saliency map, a binary image is obtained. In this case, the
|
575 |
+
prevalent metrics to evaluate the performance of the detection algorithms are probability of false-alarm Pfa
|
576 |
+
and detection Pd. These two metrics are defined as [38]:
|
577 |
+
Pfa = Nf
|
578 |
+
Ntot
|
579 |
+
,
|
580 |
+
Pd = Nd
|
581 |
+
Nr
|
582 |
+
(17)
|
583 |
+
9
|
584 |
+
|
585 |
+
10
|
586 |
+
−10
|
587 |
+
10
|
588 |
+
−5
|
589 |
+
10
|
590 |
+
0
|
591 |
+
0
|
592 |
+
0.2
|
593 |
+
0.4
|
594 |
+
0.6
|
595 |
+
0.8
|
596 |
+
1
|
597 |
+
Pfa
|
598 |
+
Pd
|
599 |
+
|
600 |
+
|
601 |
+
Alg. 1
|
602 |
+
Alg. 2
|
603 |
+
Figure 10: ROC curve for two typical detectors
|
604 |
+
(a)
|
605 |
+
(b)
|
606 |
+
(c)
|
607 |
+
Figure 11: a) original image, b) the LCM filtering result, c) the Top-Hat filtering result.
|
608 |
+
0
|
609 |
+
0.1
|
610 |
+
0.2
|
611 |
+
0.3
|
612 |
+
0.4
|
613 |
+
0.5
|
614 |
+
0.6
|
615 |
+
0.7
|
616 |
+
0.8
|
617 |
+
0.9
|
618 |
+
1
|
619 |
+
Pfa
|
620 |
+
0.6
|
621 |
+
0.65
|
622 |
+
0.7
|
623 |
+
0.75
|
624 |
+
0.8
|
625 |
+
0.85
|
626 |
+
0.9
|
627 |
+
0.95
|
628 |
+
1
|
629 |
+
PD
|
630 |
+
LCM
|
631 |
+
TopHat
|
632 |
+
(a)
|
633 |
+
0
|
634 |
+
20
|
635 |
+
40
|
636 |
+
60
|
637 |
+
80
|
638 |
+
100
|
639 |
+
120
|
640 |
+
140
|
641 |
+
160
|
642 |
+
180
|
643 |
+
200
|
644 |
+
T
|
645 |
+
0
|
646 |
+
0.1
|
647 |
+
0.2
|
648 |
+
0.3
|
649 |
+
0.4
|
650 |
+
0.5
|
651 |
+
0.6
|
652 |
+
0.7
|
653 |
+
0.8
|
654 |
+
0.9
|
655 |
+
1
|
656 |
+
Pfa
|
657 |
+
LCM
|
658 |
+
TopHat
|
659 |
+
(b)
|
660 |
+
Figure 12: a) The ROC curve, b) false-alarm rate versus different threshold levels.
|
661 |
+
where Nf, Ntot, Nd, and Nr denote the number of wrongly detected pixels, the total number of pixels, the
|
662 |
+
number of pixels which are detected correctly, and the target pixels in the ground-truth, respectively. The
|
663 |
+
receiver operational characteristics (ROC) curve is constructed by considering each (Pfa, Pd) pair at different
|
664 |
+
threshold level.
|
665 |
+
Fig. 10 shows the ROC curve for two typical detectors. As shown in the figure, for a constant false-alarm
|
666 |
+
rate, the detector #1 has higher detection rate, and outperforms the algorithm #2. The ROC curve is a
|
667 |
+
satisfactory tool to evaluate the performance of different detectors. However, if the detection rate and false-
|
668 |
+
alarm rate are not defined accurately, the final ROC curve is not a reliable measure anymore. In order to
|
669 |
+
demonstrate the deficiency of the definitions of the Pd and Pfa (Eq. 17), let consider the target detection
|
670 |
+
ability of two well-known small infrared target detection algorithms; Local contrast method (LCM) [23]
|
671 |
+
and Top-hat algorithm [39]. Fig. 11 shows the detection results of these two algorithms. As shown in the
|
672 |
+
figure, the Top-hat filtering method clearly outperforms the LCM algorithm. however, the ROC curve gives
|
673 |
+
contradictory result against visual perception (Fig. 12a). Also, by constructing the curve of the false-alarm
|
674 |
+
rate versus different threshold levels (Fig. 12b), the low performance of the LCM algorithm is clearly seen.
|
675 |
+
Therefore, the former definition of the Pd (Eq. 17) is not appropriate for this crucial metric.
|
676 |
+
Another alternative definition for Pd is suggested in the literature ([25]):
|
677 |
+
Pd = ND
|
678 |
+
NR
|
679 |
+
(18)
|
680 |
+
where ND and NR are number of detected true targets, and total number of true targets. While this new
|
681 |
+
definition addresses the deficiency of the former one (Eq. 17), there are still some drawbacks regrading this
|
682 |
+
formula; The real infrared scenarios usually contain limited number of targets. To overcome this drawback,
|
683 |
+
10
|
684 |
+
|
685 |
+
(a)
|
686 |
+
(b)
|
687 |
+
(c)
|
688 |
+
(d)
|
689 |
+
Figure 13: a) Synthetic targets in homogeneous local background, b) low contrast targets in background
|
690 |
+
clutter edges, c) the character filter response to a and b).
|
691 |
+
(a)
|
692 |
+
(b)
|
693 |
+
(c)
|
694 |
+
(d)
|
695 |
+
Figure 14: a, c) real infrared scenario, b , d) the response of character filter [40] to a and c, respectively.
|
696 |
+
synthetic targets are usually created using Gaussian spatial distribution. However, spatial distribution-based
|
697 |
+
target detection algorithms directly benefit from synthetic data, so the final evaluation is not fair. An example
|
698 |
+
is provided here to better demonstration of this situation. The character filter [40] utilizes Gaussian spatial
|
699 |
+
distribution as a measure to distinguish between real target and background clutter. As shown in Fig. 13,
|
700 |
+
when the small target has exactly Gaussian distribution, the character filter effectively can enhance the small
|
701 |
+
targets and eliminate background clutter. However, in real infrared scenarios, which the spatial distribution
|
702 |
+
of small targets does not follow the Gaussian distribution [38], the detection results of character filter is
|
703 |
+
chaotic (Fig. 14).
|
704 |
+
According to aforementioned issues regarding the post-thresholding performance evaluation metrics,
|
705 |
+
herein, awe present a new approach capable of addressing all the shortcomings. Since in a successful de-
|
706 |
+
tection operation, at least one pixel is detected after thresholding operation, the following procedure is
|
707 |
+
introduced to obtain new post-thresholding performance measure:
|
708 |
+
i) The upper bound for control parameter (kmax) is calculated (Eq. 10).
|
709 |
+
ii) The [0 − kmax] interval is chosen as valid interval for performance evaluation.
|
710 |
+
iii) For each different control parameters, the false-alarm rate is calculated using Eq. 17.
|
711 |
+
11
|
712 |
+
|
713 |
+
Table 1: The baseline algorithms
|
714 |
+
Detection Algorithm
|
715 |
+
Details
|
716 |
+
Top-Hat [39]
|
717 |
+
7 × 7 structural element
|
718 |
+
LoG [18]
|
719 |
+
With [0.50, 0.60, 0.72, 0.86, 1.03, 1.24, 1.49, 1.79, 2.14, 2.57, 3.09, 3.71] scale parameters
|
720 |
+
PCM [41]
|
721 |
+
With [3 × 3, 5 × 5, 7 × 7, and 9 × 9] cell-sizes
|
722 |
+
AAGD [42]
|
723 |
+
With [3 × 3, 5 × 5, 7 × 7, and 9 × 9] cell-sizes
|
724 |
+
Table 2: The value of maximum control parameter kmax for different algorithms
|
725 |
+
the 1st test image
|
726 |
+
the 2nd test image
|
727 |
+
the 3rd test image
|
728 |
+
the 4th test image
|
729 |
+
the 5th test image
|
730 |
+
the 6th test image
|
731 |
+
AAGD
|
732 |
+
39.8553
|
733 |
+
61.9809
|
734 |
+
45.8804
|
735 |
+
57.9033
|
736 |
+
8.0117
|
737 |
+
166.4851
|
738 |
+
LoG
|
739 |
+
16.5312
|
740 |
+
28.8976
|
741 |
+
5.4743
|
742 |
+
7.2515
|
743 |
+
13.4517
|
744 |
+
37.4031
|
745 |
+
TopHat
|
746 |
+
15.8858
|
747 |
+
22.8994
|
748 |
+
2.5513
|
749 |
+
3.8742
|
750 |
+
14.4408
|
751 |
+
26.9160
|
752 |
+
PCM
|
753 |
+
13.7368
|
754 |
+
67.1190
|
755 |
+
25.5538
|
756 |
+
14.4819
|
757 |
+
12.4008
|
758 |
+
87.0127
|
759 |
+
iv) The false-alarm rate versus control parameter (Pfa – k) curve is constructed. In the next step, the [0
|
760 |
+
– k] interval is linearly mapped to [0 – 1] range. This normalization allows us to fairly compare and
|
761 |
+
evaluate different algorithms.
|
762 |
+
After constructing (Pfa – k) curve, the following measures can be extracted:
|
763 |
+
• The maximum control parameter (kmax) is the first inferred performance evaluation metric. The larger
|
764 |
+
kmax, the higher detection ability.
|
765 |
+
• The false-alarm rate at kmax, which is called Pfa,min here, is the second evaluation metric. It is obvious
|
766 |
+
that the false-alarm rate of the system can not be less than Pfa,min while the true target is detected.
|
767 |
+
After normalizing [0 – k] interval to [0 – 1] range, the false alarm rate of the detection algorithms can
|
768 |
+
be plotted in single figure. Then, the algorithm with satisfying detection performance can be chosen for the
|
769 |
+
practical application.
|
770 |
+
5
|
771 |
+
Detection ability evaluation using new metrics
|
772 |
+
In order to evaluate the detection ability using the proposed metrics, four well-known small infrared target
|
773 |
+
detection algorithms are chosen to conduct the experiments. Tab. 1 reports the baseline algorithms and their
|
774 |
+
implementation details. The pre-thresholding enhancement results of each algorithm are depicted in Fig. 15.
|
775 |
+
Visually speaking, the AAGD algorithm has better performance in background suppression (the background
|
776 |
+
region is mapped to zero value). However, the most part of gray area in PCM output have zero values (Since
|
777 |
+
there are also negative values in the saliency map, the zero values are depicted by gray color instead of black
|
778 |
+
one). LoG and TopHat filters are sensitive to noise and sharp edges, therefore, there are too many false
|
779 |
+
responses in their saliency maps.
|
780 |
+
The results of evaluation using new metrics are reported in Tab. 2 and Tab. 3. As reported in Tab. 2,
|
781 |
+
AAGD and PCM algorithms have better enhancement for target area. However, by taking the false-alarms
|
782 |
+
into account, the PCM algorithm shows better clutter rejection ability.
|
783 |
+
Finally, Fig. 16 shows the normalized (Pfa – k) curve to investigate the detection performance charac-
|
784 |
+
teristics of different baseline algorithms, and fairly compare them. As shown in the figure, the PCM and
|
785 |
+
AAGD algorithm has overall superiority compared to LoG and TopHat algorithms. The AAGD algorithm
|
786 |
+
Table 3: The value of minimum probability of false alarm Pfa,min for different algorithms
|
787 |
+
1st test image
|
788 |
+
2nd test image
|
789 |
+
3rd test image
|
790 |
+
4th test image
|
791 |
+
5th test image
|
792 |
+
6th test image
|
793 |
+
AAGD
|
794 |
+
0
|
795 |
+
0
|
796 |
+
7.4627e-5
|
797 |
+
1.3412e-5
|
798 |
+
0.0019
|
799 |
+
0
|
800 |
+
LoG
|
801 |
+
0
|
802 |
+
0
|
803 |
+
7.4627e-5
|
804 |
+
5.3648e-5
|
805 |
+
0
|
806 |
+
0
|
807 |
+
TopHat
|
808 |
+
0
|
809 |
+
0
|
810 |
+
0.0044
|
811 |
+
1.7436e-4
|
812 |
+
0
|
813 |
+
0
|
814 |
+
PCM
|
815 |
+
0
|
816 |
+
0
|
817 |
+
0
|
818 |
+
1.3412e-5
|
819 |
+
0
|
820 |
+
0
|
821 |
+
12
|
822 |
+
|
823 |
+
Figure 15: Pre-thresholding results of the algorithms under the test on real infrared images (Target region is
|
824 |
+
marked by yellow circle). From the left:
|
825 |
+
the first column: original images, the second column: filtering
|
826 |
+
results of AAGD algorithm, the third column: filtering results of Tophat transform, the fourth column:
|
827 |
+
filtering results of LoG algorithm, the fifth column: filtering results of PCM algorithm.
|
828 |
+
shows poor detection performance in the 5th test image (Fig. 15). As shown in Fig. 16e, the new metrics is
|
829 |
+
completely consistent with the visual and qualitative results (Fig. 15).
|
830 |
+
6
|
831 |
+
Conclusion
|
832 |
+
The development of new algorithms for infrared small target detection is attracted more attention during
|
833 |
+
the last decade. However, many of these recently developed algorithms do not meet the requirements of the
|
834 |
+
practical applications. Also, there are some disadvantage regarding the common evaluation metrics. In order
|
835 |
+
to completely understand the requirements of the effective evaluation metrics, the practical procedure of small
|
836 |
+
target detection should be revealed. The thresholding operation has a great role in this procedure. Without
|
837 |
+
13
|
838 |
+
|
839 |
+
10-1
|
840 |
+
100
|
841 |
+
Normalized Adjustment Parameter
|
842 |
+
10-6
|
843 |
+
10-5
|
844 |
+
10-4
|
845 |
+
10-3
|
846 |
+
10-2
|
847 |
+
10-1
|
848 |
+
100
|
849 |
+
False Alarm Rate (P
|
850 |
+
fa)
|
851 |
+
AAGD
|
852 |
+
LoG
|
853 |
+
TopHat
|
854 |
+
PCM
|
855 |
+
(a)
|
856 |
+
10-1
|
857 |
+
100
|
858 |
+
Normalized Adjustment Parameter
|
859 |
+
10-5
|
860 |
+
10-4
|
861 |
+
10-3
|
862 |
+
10-2
|
863 |
+
10-1
|
864 |
+
100
|
865 |
+
False Alarm Rate (P
|
866 |
+
fa)
|
867 |
+
AAGD
|
868 |
+
LoG
|
869 |
+
TopHat
|
870 |
+
PCM
|
871 |
+
(b)
|
872 |
+
10-1
|
873 |
+
100
|
874 |
+
Normalized Adjustment Parameter
|
875 |
+
10-5
|
876 |
+
10-4
|
877 |
+
10-3
|
878 |
+
10-2
|
879 |
+
10-1
|
880 |
+
100
|
881 |
+
False Alarm Rate (P
|
882 |
+
fa)
|
883 |
+
AAGD
|
884 |
+
LoG
|
885 |
+
TopHat
|
886 |
+
PCM
|
887 |
+
(c)
|
888 |
+
10-1
|
889 |
+
100
|
890 |
+
Normalized Adjustment Parameter
|
891 |
+
10-5
|
892 |
+
10-4
|
893 |
+
10-3
|
894 |
+
10-2
|
895 |
+
10-1
|
896 |
+
100
|
897 |
+
False Alarm Rate (P
|
898 |
+
fa)
|
899 |
+
AAGD
|
900 |
+
LoG
|
901 |
+
TopHat
|
902 |
+
PCM
|
903 |
+
(d)
|
904 |
+
10-1
|
905 |
+
100
|
906 |
+
Normalized Adjustment Parameter
|
907 |
+
10-5
|
908 |
+
10-4
|
909 |
+
10-3
|
910 |
+
10-2
|
911 |
+
10-1
|
912 |
+
100
|
913 |
+
False Alarm Rate (P
|
914 |
+
fa)
|
915 |
+
AAGD
|
916 |
+
LoG
|
917 |
+
TopHat
|
918 |
+
PCM
|
919 |
+
(e)
|
920 |
+
10-1
|
921 |
+
100
|
922 |
+
Normalized Adjustment Parameter
|
923 |
+
10-6
|
924 |
+
10-5
|
925 |
+
10-4
|
926 |
+
10-3
|
927 |
+
10-2
|
928 |
+
10-1
|
929 |
+
100
|
930 |
+
False Alarm Rate (P
|
931 |
+
fa)
|
932 |
+
AAGD
|
933 |
+
LoG
|
934 |
+
TopHat
|
935 |
+
PCM
|
936 |
+
(f)
|
937 |
+
Figure 16: The normalized (Pfa – k) curve in logarithmic scale. The detection performance characteristics
|
938 |
+
curve for: a) the 1st test image, b) the 2nd test image, c) the 3rd test image, d) the 4th test image, e) the 5th
|
939 |
+
test image, f) the 6th test image.
|
940 |
+
a proper thresholding strategy, the previous efforts in target enhancement algorithm development would
|
941 |
+
become obsolete. It has been demonstrated that the local statistics-based thresholding is not an appropriate
|
942 |
+
option for the segmentation of saliency map, and the the global statistics-based threshold operation is better
|
943 |
+
choice.
|
944 |
+
By considering the global thresholding as final step for the detection algorithm, the signal to clutter ratio
|
945 |
+
(SCR) metric is modified for better detection ability reflection. Also, three post-thresholding metrics are
|
946 |
+
proposed to complete performance evaluation of different algorithms.
|
947 |
+
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|
948 |
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|
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