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@@ -0,0 +1,1823 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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
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1749
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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
+
796
+ 1.00
797
+ 2x104
798
+ [np]
799
+ 1.5x104
800
+ 104
801
+ 5000
802
+
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
+
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
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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
+
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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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
+ References
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+ Environmental Research and Public Health, vol. 19, no. 6, p. 3687, 2022.
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+ [3] A. Choudhary, T. Mian, and S. Fatima, “Convolutional neural network based bearing fault diagnosis of
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+ [4] H. Shahraki, S. Aalaei, and S. Moradi, “Infrared small target detection based on the dynamic particle
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+ [5] A. Raza, J. Liu, Y. Liu, J. Liu, Z. Li, X. Chen, H. Huo, and T. Fang, “Ir-msdnet: Infrared and visible
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+ survey,” IEEE Geoscience and Remote Sensing Magazine, 2022.
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+ [12] S. D. Deshpande, M. H. Er, R. Venkateswarlu, and P. Chan, “Max-mean and max-median filters for
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