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SCIENTIA SINICA Informationis, Volume 48, Issue 7: 903-918(2018) https://doi.org/10.1360/N112017-00292

Radio tomographic imaging based on cluster Bayesian compressive sensing

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  • ReceivedDec 29, 2017
  • AcceptedFeb 14, 2018
  • PublishedJul 16, 2018

Abstract

In narrowband radio tomographic imaging, the crucial challenge is to detect multipath interference effectively and obtain a better estimate of shallow fading. Based on structural cluster Bayesian compressive-sensing theory, an analysis is presented of the possible shallow fading status, and more spatial distribution information of the shallow fading is explored. As a result, a more accurate prior model combining the sparsity and cluster property of shallow fading and a better computational imaging mechanism is proposed. The experimental results show that this cluster sparsity Bayesian compressive-sensing model restrains the artifacts in the image by improving the link discrimination ability between shallow fading and multipath interference so that better recovered images can be obtained, and device-free localization performance is improved.


Funded by

国家自然科学基金(61772574,61375080)

大型科学仪器设备共享专项(2015B030304001)

广东省自然科学基金重点项目(2015A030311049)


Supplement

Appendix

阴影衰落聚集稀疏Bayesian压缩感知模型中的参数学习规则

采用变分Bayesian方法[20]推导阴影衰落聚集稀疏Bayesian压缩感知模型的总模型(7)中阴影衰落相关的目标变量${\boldsymbol~w}$, ${\boldsymbol~z}$与隐变量$b$, $\boldsymbol{\alpha}$, $\boldsymbol{\pi}$, 其中$\langle~\cdot~\rangle$表示期望算子.

更新稀疏向量${w}$

稀疏向量${\boldsymbol~w}$的近似后验概率为 begindisplaymath q(bm w∝ rm expłeft(łeftłangle ∑_i=1^nłn p(w_iα_i)rightrangle_α_iright)rm expłangle łn p(bm ybm wbm zb)rangle_bm z β∼ mathcalNμ,Σ), enddisplaymath 其中, \begin{align}& \boldsymbol{\Sigma} = (\tilde{\boldsymbol A}+\tilde{\beta}\langle {\boldsymbol Z}{\boldsymbol \Phi}^{\rm T}{\boldsymbol \Phi} {\boldsymbol Z}\rangle)^{-1}, \tag{10} \\ & \boldsymbol{\mu} = \tilde{\beta}{\boldsymbol \Sigma}\tilde{\boldsymbol Z}{\boldsymbol \Phi}^{\rm T}{\boldsymbol y}, \tag{11} \end{align} 其中${\boldsymbol~A}={\rm~diag}({\boldsymbol~\alpha})$, ${\boldsymbol~Z}={\rm~diag}({\boldsymbol~z})$, 且当聚集向量的估计值${\boldsymbol~{\tilde~z}}$给定, 则有 \begin{equation}\langle {\boldsymbol Z}{\boldsymbol \Phi}^{\rm T}{\boldsymbol \Phi} {\boldsymbol Z}\rangle = ({\boldsymbol \Phi}^{\rm T}{\boldsymbol \Phi})\cdot({\boldsymbol {\tilde z}}{\boldsymbol {\tilde z}}^{\rm T}+{\rm diag}({\boldsymbol {\tilde z}}\cdot(1-{\boldsymbol {\tilde z}}))), \tag{12}\end{equation} 因而稀疏向量${\boldsymbol~w}$的更新为 \begin{equation}{\boldsymbol {\tilde w}}=\boldsymbol{\mu}. \tag{13}\end{equation} 同时此向量整体更新问题可以再转化为分量单独更新进一步提升速度 1).

He L H, Carin L. Exploiting structure in wavelet-based bayesian compressive sensing. IEEE Trans Signal Process, 2009, 57: 3488–3497.

更新聚集向量${z}$

聚集向量${\boldsymbol~z}$中每个分量$z_i$的近似后验概率为 begindisplaymath q(z_i)∝ rm expłangle rm ln p(z_iuppi_i)rangle_uppi_irm expłanglerm ln p(bm y_-iz_i,w_i,β)rangle_w_i, β, enddisplaymath 其中${\boldsymbol~y}_{-i}={\boldsymbol~y}-\sum_{k\neq~i}z_k~w_k\boldsymbol{\phi}_k$ 表示第$i$个像素点的阴影衰落情况对信号测量${\boldsymbol~y}$的影响, 则事件$z_i=1$发生的概率为 \begin{equation}p(z_i=1)\propto {\rm exp}\langle {\rm ln} \pi_i\rangle{\rm exp}\left(-\frac{\tilde{\beta}}{2} \left(\langle w_i^2 \rangle \boldsymbol{\phi}_i^{\rm T}\boldsymbol{\phi}_i-2\tilde{w}_i\boldsymbol{\phi}_i^{\rm T}{\boldsymbol y}_{-i}\right)\right), \tag{14}\end{equation} 其中$\langle~w_i^2~\rangle~=~\tilde{w}_i^2+\Sigma_{ii}$, $\Sigma_{ii}$为(10)中协方差矩阵$\boldsymbol{\Sigma}$中第$i$个对角元, 而事件$z_i=0$发生的概率为 \begin{equation}p(z_i=0)\propto {\rm exp}(\langle {\rm ln}(1-\pi_i)\rangle), \tag{15}\end{equation} 因而聚集向量${\boldsymbol~z}$的更新为 \begin{equation}\tilde{z}_i = \frac{p(z_i=1)}{p(z_i=1)+p(z_i=0)}. \tag{16}\end{equation}

更新稀疏向量的精度$\alpha$

结合${\rm~Gamma}$先验分布为正态分布的共轭先验, 稀疏向量的精度$\boldsymbol{\alpha}$中各分量$\alpha_i$的近似后验概率为 \begin{equation}q(\alpha_i) \propto p(\alpha_i|a_i, b_i){\rm exp}(\langle p(w_i|\alpha_i) \rangle_{w_i}) \propto \Gamma \left(\alpha \Big|a+\frac{1}{2}, b+\frac{\langle w_i^2 \rangle}{2}\right), \tag{17}\end{equation} 从而$\alpha_i$的更新为 \begin{equation}\tilde\alpha_i = \frac{a+\frac{1}{2}}{b+\frac{\langle w_i^2 \rangle}{2}}. \tag{18}\end{equation}

更新信号测量的精度$\beta$

根据${\rm~Gamma}$先验分布为正态分布的共轭先验, 信号测量的精度$\beta$的近似后验概率为 \begin{equation}q(\beta) \propto p(\beta|a, b){\rm exp}(\langle p({\boldsymbol y}| {\boldsymbol w}, {\boldsymbol z}, \beta) \rangle_{{\boldsymbol w}, {\boldsymbol z}}) \propto \Gamma \left(\beta \Big|c+\frac{m}{2}, d+\frac{\langle \lVert {\boldsymbol y} - {\boldsymbol \Phi}({\boldsymbol w}\cdot{\boldsymbol z}) \rVert^2 \rangle_{{\boldsymbol w}, {\boldsymbol z}}}{2}\right), \tag{19}\end{equation} 而且当稀疏向量和聚集向量的估计值${\boldsymbol~{\tilde~w}}$和${\boldsymbol~{\tilde~z}}$给定, 则上式中 \begin{equation}\langle \lVert {\boldsymbol y} - {\boldsymbol \Phi}({\boldsymbol w}\cdot{\boldsymbol z}) \rVert^2 \rangle_{{\boldsymbol w}, {\boldsymbol z}} = {\boldsymbol y}^{\rm T}{\boldsymbol y} - 2({\boldsymbol {\tilde w}}\cdot {\boldsymbol {\tilde z}}) {\boldsymbol \Phi}^{\rm T}{\boldsymbol y} +{\boldsymbol I}^{\rm T}[({\boldsymbol {\tilde z}}{\boldsymbol {\tilde z}}^{\rm T}+{\rm diag}({\boldsymbol {\tilde z}}\cdot(1-{\boldsymbol {\tilde z}}))) \cdot({\boldsymbol {\tilde w}}{\boldsymbol {\tilde w}}^{\rm T}+{\rm diag}({\boldsymbol \Sigma})) \cdot({\boldsymbol \Phi}^{\rm T}{\boldsymbol \Phi})]{\boldsymbol I}, \tag{20}\end{equation} 从而$\beta$的更新为 \begin{equation}\tilde\beta =\frac{2c+m}{2d+\langle \lVert {\boldsymbol y} - {\boldsymbol \Phi}({\boldsymbol w}\cdot{\boldsymbol z}) \rVert^2 \rangle_{{\boldsymbol w}, {\boldsymbol z}}}. \tag{21}\end{equation}

更新聚集向量的概率$\pi$

当聚集向量的估计值${\boldsymbol~{\tilde~z}}$给定, 则可以得到每种聚集模式对${\boldsymbol~\pi}$的影响比例, \begin{equation}\begin{aligned} p(s^{[0]}) = & (1-\tilde{z}_{i-M})(1-\tilde{z}_{i-1}) (1-\tilde{z}_{i+1})(1-\tilde{z}_{i+M}) (1-\tilde{z}_{i-M-1})(1-\tilde{z}_{i-M+1}) (1-\tilde{z}_{i+M-1})(1-\tilde{z}_{i+M+1}), \\ p(s^{[1]}) = & \sum_{k_1=0}^{1}\sum_{k_1=0}^{1}\sum_{k_2=0}^{1}\sum_{k_3=0}^{1} \{(1-\tilde{z}_{i-M-1})(1-\tilde{z}_{i-M+1}) (1-\tilde{z}_{i+M-1})(1-\tilde{z}_{i+M+1}) \\ & \cdot[(k_1)+(-1)^{k_1}\tilde{z}_{i-M}] [(k_2)+(-1)^{k_2}\tilde{z}_{i-1}] [(k_3)+(-1)^{k_3}\tilde{z}_{i+1}] [(k_4)+(-1)^{k_4}\tilde{z}_{i+M}]\} \\ & -(1-\tilde{z}_{i-M-1})(1-\tilde{z}_{i-M+1}) (1-\tilde{z}_{i+M-1})(1-\tilde{z}_{i+M+1}) (1-\tilde{z}_{i-M})(1-\tilde{z}_{i-1}) (1-\tilde{z}_{i+1})(1-\tilde{z}_{i+M}) \\ & -(1-\tilde{z}_{i-M-1})(1-\tilde{z}_{i-M+1}) (1-\tilde{z}_{i+M-1})(1-\tilde{z}_{i+M+1}) \tilde{z}_{i-M}\tilde{z}_{i-1} \tilde{z}_{i+1}\tilde{z}_{i+M}, \\ p(s^{[2]}) = & (1-\tilde{z}_{i-M-1})(1-\tilde{z}_{i-M+1}) (1-\tilde{z}_{i+M-1})(1-\tilde{z}_{i+M+1}) \tilde{z}_{i-M}\tilde{z}_{i-1} \tilde{z}_{i+1}\tilde{z}_{i+M}, \\ p(s^{[3]}) = & \sum_{k_5=0}^{1}\sum_{k_6=0}^{1}\sum_{k_7=0}^{1}\sum_{k_8=0}^{1} \{\tilde{z}_{i-M}\tilde{z}_{i-1} \tilde{z}_{i+1}\tilde{z}_{i+M} \\ & \cdot[(k_5)+(-1)^{k_5}\tilde{z}_{i-M-1}] [(k_6)+(-1)^{k_6}\tilde{z}_{i-M+1}] [(k_7)+(-1)^{k_7}\tilde{z}_{i+M-1}] [(k_8)+(-1)^{k_8}\tilde{z}_{i+M+1}]\} \\ & -\tilde{z}_{i-M}\tilde{z}_{i-1} \tilde{z}_{i+1}\tilde{z}_{i+M} (1-\tilde{z}_{i-M-1})(1-\tilde{z}_{i-M+1}) (1-\tilde{z}_{i+M-1})(1-\tilde{z}_{i+M+1}) \\ & \cdot \tilde{z}_{i-M}\tilde{z}_{i-1} \tilde{z}_{i+1}\tilde{z}_{i+M} \tilde{z}_{i-M-1}\tilde{z}_{i-M+1} \tilde{z}_{i+M-1}\tilde{z}_{i+M+1}, \\ p(s^{[4]}) = & \tilde{z}_{i-M}\tilde{z}_{i-1} \tilde{z}_{i+1}\tilde{z}_{i+M} \tilde{z}_{i-M-1}\tilde{z}_{i-M+1} \tilde{z}_{i+M-1}\tilde{z}_{i+M+1}, \end{aligned} \tag{22}\end{equation} 其中通过系数$k_1$, $k_2$, $k_3$, $k_4$的遍历使近邻渐变衰落$s^{[1]}$包含$2^4-1-1=14$种渐变模式, 通过系数$k_5$, $k_6$, $k_7$, $k_8$的遍历使对角渐变衰落$s^{[3]}$包含$2^4-1-1=14$种渐变方式.

记$t$表示模型中的稀疏聚集模式的类别, 则$t\in~\{0,~1,~2,~3,~4\}$, 当通过上式得到各个稀疏聚集模式$t$在$\pi_i$上的影响比例后, 根据$\rm~Beta$分布为$\rm~Bernouli$分布的共轭先验, 可计算各个模型中各个模式$\pi_{i}^{[t]}$的后验概率: \begin{equation}q\left(\pi_{i}^{[t]}\right) \propto p \left(\pi_{i}^{[t]}|e_{i}^{[t]}, f_{i}^{[t]}\right) {\rm exp}\left( \left\langle p \left(z_i|\pi_{i}^{[t]}\right) \right\rangle_{z_i}\right) {\rm exp}\left(p \left(s^{[t]}\right)\right) \propto {\rm Beta}\left(p \left(\pi_{i}^{[t]}|e_{i}^{\prime[t]}, f_{i}^{\prime[t]}\right)\right), \tag{23}\end{equation} 其中根据$\rm~Beta$分布为$\rm~Bernouli$分布的共轭先验, 有 \begin{equation}\begin{aligned} e_{i}^{\prime[t]} = e^{[t]}+p \left(s^{[t]}\right)\tilde{z_i}, f_{i}^{\prime[t]} = f^{[t]}+p \left(s^{[t]}\right)(1-\tilde{z}_i), \end{aligned} \tag{24}\end{equation} 从而有 \begin{equation}\begin{aligned} \left \langle {\rm ln} \pi_{i}^{[t]} \right \rangle & = \psi \left(e_{i}^{\prime[t]}\right) - \psi \left(e_{i}^{\prime[t]}+f_{i}^{\prime[t]}\right), \left \langle {\rm ln} \left(1-\pi_{l, i}^{[t]}\right) \right \rangle = \psi \left(f_{i}^{\prime[t]}\right) - \psi \left(e_{i}^{\prime[t]}+f_{i}^{\prime[t]}\right), \end{aligned} \tag{25}\end{equation} 其中$\psi(x)=\frac{\rm~d}{{\rm~d}x}{\rm~ln}\Gamma(x)$为双Gamma (Digamma)函数, 从而${\boldsymbol~\pi}$的更新为 \begin{equation}\begin{aligned} \langle {\rm ln} \tilde\pi_{i} \rangle & = \sum_{t=0}^{4}p(s^{[t]}) \left\langle {\rm ln} \pi_{i}^{[t]} \right\rangle , \langle {\rm ln} (1-\tilde\pi_{i}) \rangle = \sum_{t=0}^{4}p(s^{[t]}) \left\langle {\rm ln} \left(1-\pi_{i}^{[t]}\right) \right\rangle . \end{aligned} \tag{26}\end{equation}

重构终止准则

考虑测量向量${\boldsymbol~y}$关于参数的边际似然函数: \begin{equation}p({\boldsymbol y}|{\boldsymbol w}, {\boldsymbol z}, \boldsymbol\alpha, \beta)=\int p({\boldsymbol y}|{\boldsymbol w}, {\boldsymbol z}, \beta)p({\boldsymbol w}|\boldsymbol\alpha){\rm d}{\boldsymbol w} \sim \mathcal{N}(\textbf{0}, {\boldsymbol \Sigma}_0), \tag{27}\end{equation} 其中${\boldsymbol~\Sigma}_0=\beta~{\boldsymbol~I}_m+~{\boldsymbol~\Phi}~{\boldsymbol~Z}{\boldsymbol~A}^{-1}{\boldsymbol~Z}{\boldsymbol~\Phi}^{\rm~T}$, 从而重构算法的损失函数可定义为此边际似然函数的对数函数: \begin{equation}Q = {\rm ln}\lvert {\boldsymbol \Sigma}_0 \rvert + {\boldsymbol y}^{\rm T}{\boldsymbol \Sigma}_0^{-1}{\boldsymbol y}, \tag{28}\end{equation} 从而重构算法的收敛判别函数可定义为损失函数在第$l$次迭代的相对误差: \begin{equation}D(l) = \frac{\lvert Q(l)-Q(l-1)\rvert}{Q(l-1)}, \tag{29}\end{equation} 即给定一个阈值$\delta$, 当$D(l)<\delta$时重构算法收敛终止迭代.


References

[1] Wilson J, Patwari N. Radio tomographic imaging with wireless networks. IEEE Trans Mobile Comput, 2010, 9: 621-632 CrossRef Google Scholar

[2] Patwari N, Wilson J. RF sensor networks for device-free localization: measurements, models, and algorithms. Proc IEEE, 2010, 98: 1961-1973 CrossRef Google Scholar

[3] Yang Z Y, Huang K D, Guo X M. A real-time device-free localization system using correlated RSS measurements. J Wirel Commun Netw, 2013, 2013: 186 CrossRef Google Scholar

[4] Huang K D, Luo Y B, Guo X M, et al. Data-efficient radio tomographic imaging with adaptive Bayesian compressive sensing. In: Proceedings of IEEE International Conference on Information and Automation, Lijiang, 2015. 1859--1864. Google Scholar

[5] Huang K D, Guo Y, Guo X M. Heterogeneous Bayesian compressive sensing for sparse signal recovery. IET Signal Proc, 2014, 8: 1009-1017 CrossRef Google Scholar

[6] Shang B L, Tan J J, Guo X M, et al. Dual radio tomographic imaging with bayesian compressive sensing. In: Proceedings of the 36th Chinese Intelligent Systems Conference, Dalian, 2017. 443--452. Google Scholar

[7] Wang J, Gao Q H, Wang H Y. Device-free localization with multidimensional wireless link information. IEEE Trans Veh Technol, 2015, 64: 356-366 CrossRef Google Scholar

[8] Wang J X, Gao Q H, Cheng P. Lightweight robust device-free localization in wireless networks. IEEE Trans Ind Electron, 2014, 61: 5681-5689 CrossRef Google Scholar

[9] Guo Y, Huang K D, Jiang N Y. An exponential-rayleigh model for RSS-based device-free localization and tracking. IEEE Trans Mobile Comput, 2015, 14: 484-494 CrossRef Google Scholar

[10] Wang J, Gao Q H, Pan M. Toward accurate device-free wireless localization with a saddle surface model. IEEE Trans Veh Technol, 2016, 65: 6665-6677 CrossRef Google Scholar

[11] Wang J, Gao Q, Wang H Y. Device-free localisation with wireless networks based on compressive sensing. IET Commun, 2012, 6: 2395-2403 CrossRef Google Scholar

[12] Wang J, Gao Q H, Wang H Y. Time-of-flight-based radio tomography for device free localization. IEEE Trans Wirel Commun, 2013, 12: 2355-2365 CrossRef Google Scholar

[13] Huang K D, Guo Y, Yang L W, et al. Optimal information based adaptive compressed radio tomographic imaging. In: Proceedings of the 32th Chinese Intelligent Systems Conference, Xi'an, 2013. 7438--7444. Google Scholar

[14] Zhao Y, Patwari N. Robust estimators for variance-based device-free localization and tracking. IEEE Trans Mobile Comput, 2015, 14: 2116-2129 CrossRef Google Scholar

[15] Wang J, Gao Q H, Yu Y. Time and energy efficient TOF-based device-free wireless localization. IEEE Trans Ind Inf, 2017, 12: 158-168 CrossRef Google Scholar

[16] Huang K D, Tan S B, Luo Y B. Enhanced radio tomographic imaging with heterogeneous Bayesian compressive sensing. Pervasive Mobile Comput, 2017, 40: 450-463 CrossRef Google Scholar

[17] Shang B L, Tan J J, Guo X M, et al. A RF-based spatiotemporal RTI localization algorithm using sparse Bayesian learning. In: Proceedings of the 2nd IEEE International Conference on Agents, Beijing, 2017. 151--153. Google Scholar

[18] Chang L Q, Chen X J, Wang Y. FitLoc: fine-grained and low-cost device-free localization for multiple targets over various areas. IEEE/ACM Trans Netw, 2017, 25: 1994-2007 CrossRef Google Scholar

[19] Zhang Z L, Rao B D. Extension of SBL algorithms for the recovery of block sparse signals with intra-block correlation. IEEE Trans Signal Process, 2013, 61: 2009-2015 CrossRef ADS arXiv Google Scholar

[20] Yu L, Wei C, Jia J Y. Compressive sensing for cluster structured sparse signals: variational Bayes approach. IET Signal Proc, 2011, 10: 770-779 CrossRef Google Scholar

[21] Mohammad-Djafari A, Dumitru M. Bayesian sparse solutions to linear inverse problems with non-stationary noise with Student-t priors. Digital Signal Process, 2015, 47: 128-156 CrossRef Google Scholar

[22] Schuhmacher D, Vo B T, Vo B N. A consistent metric for performance evaluation of multi-object filters. IEEE Trans Signal Process, 2008, 56: 3447-3457 CrossRef ADS Google Scholar

  • Figure 1

    (Color online) The shadow fading in RTI

  • Figure 2

    (Color online) The multi-path interfere in RTI. (a) The original link status; (b) the original recovered image; (c) 5 low SNR links have been added; (d) the recovered image after link addition

  • Figure 3

    Bayes compressive sensing model

  • Figure 4

    Shallow fading cluster pattern. (a) Isolated shallow fading; (b) adjacent transition shallow fading; (c) adjacent covered shallow fading; (d) diagonal transition shallow fading; (e) diagonal covered shallow fading

  • Figure 5

    Bayes compressive sensing model of the cluster-sparse shadow fading

  • Figure 6

    (Color online) The experimental scene. (a) The indoor graph; (b) the indoor scene; (c) the outdoor graph;protect łinebreak (d) the outdoor scene

  • Figure 7

    (Color online) The recovered RTI image in the indoor experiment. (a) BCS; (b) GHBCS; (c) BSBL2; (d) cluster

  • Figure 8

    (Color online) The recovered RTI image in the outdoor experiment. (a) BCS; (b) GHBCS; (c) BSBL2;protect łinebreak (d) Cluster

  • Figure 9

    (Color online) The cumulative distribution curve of the localization error. (a) The indoor scene; (b) the outdoor scene

  • Figure 10

    (Color online) The average localization error via measurement. (a) The indoor scene; (b) the outdoor scene

  • Table 1   Localization error comparison$^{\rm~a)}$
    Scene (m) Number of targets BCS GHBCS BSBL2 Cluster
    Average Max Average Max Average Max Average Max
    Indoor 1 0.49 3.3 0.35 1.22 0.33 1.22 0.16 0.70
    Figure 6(a)(b) 2$\sim$3 1.32 3.13 1.18 3.55 1.17 3.05 0.42 0.81
    Outdoor 1 0.21 2.22 0.16 0.32 0.22 1.56 0.13 0.32
    Figure 6(c)(d) 2$\sim$5 1.07 3.54 1.28 3.19 1.09 3.14 0.40 2.76
  •   

    Algorithm 1 聚集阴影衰落稀疏模型重构算法

    超参数初始化: $a=10^{-6}$, $b=10^{-6}$, $c=10^{-6}$, $d=10^{-6}$, $\delta=10^{-4}$, $T=100$ $e^{[0]}=\frac{1}{9}$, $f^{[0]}=\frac{8}{9}$, $e^{[1]}=\frac{1}{9}$, $f^{[1]}=\frac{1}{9}$, $e^{[2]}=\frac{1}{9}$, $f^{[2]}=\frac{1}{9}$, $e^{[3]}=\frac{1}{9}$, $f^{[3]}=\frac{1}{9}$, $e^{[4]}=\frac{8}{9}$, $f^{[4]}=\frac{1}{9}$;

    测量行归一化: $\Phi_{i,~:}=\frac{\Phi_{i,~:}}{\lVert~\Phi_{i,~:}\rVert_2}$, ${\boldsymbol~y}_i=\frac{{\boldsymbol~y}_i}{\lVert~\Phi_{i,~:}\rVert_2}$;

    更新隐变量根据式(22)更新聚集模型各衰落模式的概率$p(s^{[t]})$;根据式(24)$\sim$(26)更新阴影衰落聚集向量的概率$\boldsymbol{\pi}$;根据式(20), (21)更新测量模型噪声向量的精度$\beta$;根据式(18)更新阴影衰落稀疏向量的精度$\boldsymbol{\alpha}$;

    更新目标变量根据式(10)$\sim$(13)更新阴影衰落稀疏向量${\boldsymbol~w}$;根据式(14)$\sim$(16)更新阴影衰落聚集向量${\boldsymbol~z}$;

    计算损失函数根据式(28)更新损失函数$Q(l)$;根据式(29)更新判别函数$D(l)$;

    衰落值的估计:${\tilde x} = {\tilde w}\cdot{\tilde z}$.

  • Table 2   Average sparsity comparison$^{\rm~a)}$
    Scene Number of targets BCS GHBCS BSBL2 Cluster
    Indoor 1 59 33 9 2
    Figure 6(a)(b) 2$\sim$3 70 45 13 4
    Outdoor 1 98 40 15 3
    Figure 6(c)(d) 2$\sim$5 117 72 28 8
  • Table 3   Average localization time comparison$^{\rm~a)}$
    Scene (s) BCS GHBCS BSBL2 Cluster
    Indoor Figure 6(a)(b) 1.01 0.48 3.02 0.36
    Outdoor Figure 6(c)(d) 13.45 1.30 21.78 1.23

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