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)
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 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∼ mathcalN 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 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 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 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, 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: 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:
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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
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 | ||
Figure | 2$\sim$3 | 1.32 | 3.13 | 1.18 | 3.55 | 1.17 | 3.05 | ||
Outdoor | 1 | 0.21 | 2.22 | 0.16 | 0.32 | 0.22 | 1.56 | ||
Figure | 2$\sim$5 | 1.07 | 3.54 | 1.28 | 3.19 | 1.09 | 3.14 |
a) The bold one is the best in comparison of the row.
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( |
更新目标变量根据式( |
计算损失函数根据式( |
衰落值的估计:${\tilde x} = {\tilde w}\cdot{\tilde z}$. |
Scene | Number of targets | BCS | GHBCS | BSBL2 | Cluster |
Indoor | 1 | 59 | 33 | 9 | |
Figure | 2$\sim$3 | 70 | 45 | 13 | |
Outdoor | 1 | 98 | 40 | 15 | |
Figure | 2$\sim$5 | 117 | 72 | 28 |
a) The bold one is the best in comparison of the row.
Scene (s) | BCS | GHBCS | BSBL2 | Cluster |
Indoor Figure | 1.01 | 0.48 | 3.02 | |
Outdoor Figure | 13.45 | 1.30 | 21.78 |
a) The bold one is the best in comparison of the row.