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SCIENTIA SINICA Informationis
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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

Jiaju TAN 1,2, Le QIN 3,5, Xin ZHAO 1,2, Xuemei GUO 4,5, Guoli WANG 4,5,*
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  • ReceivedDec 29, 2017
  • AcceptedFeb 14, 2018
  • PublishedJul 16, 2018
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Abstract


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.


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    Figure 1

    (Color online) The shadow fading in RTI

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    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

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    Figure 3

    Bayes compressive sensing model

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    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

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    Figure 5

    Bayes compressive sensing model of the cluster-sparse shadow fading

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    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

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    Figure 7

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

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    Figure 8

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

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    Figure 9

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

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    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

    a) The bold one is the best in comparison of the row.

  •   

    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

    a) The bold one is the best in comparison of the row.

  • 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

    a) The bold one is the best in comparison of the row.