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摘要:
压缩感知（CS）理论是在充分利用信号稀疏性或可压缩性的情况下，对信号进行少量采样即可实现信号的精确重建。本文尝试将CS理论应用于电容层析成像（ECT）图像重建中，首先，使用快速傅里叶变换（FFT）基将原始图像灰度信号进行稀疏化处理；其次，将ECT灵敏度矩阵的各行按随机顺序进行排列，得到ECT系统随机观测矩阵；最后，选取当前普遍使用的基于内点法、梯度投影（GPSR）算法以及贪婪算法的CS图像重建算法进行ECT图像重建，并与线性反投影及Landweber迭代算法进行了对比。仿真实验结果表明：基于CS图像理论的ECT图像重建算法，其重建精度有所提高。本文同时分析了3种CS图像重建算法的优缺点及适用范围。
Abstract:Based on the sparsity or compressibility of the signal, compressed sensing (CS) theory can achieve high-accuracy reconstruction of the signal by sampling a small amount of data. In this paper, CS theory was used for the image reconstruction of electrical capacitance tomography (ECT). First, using the fast Fourier transformation (FFT) basis, the gray signals of original images can be transformed into the sparse signals. Then, the random observation matrix of ECT system was designed by rearranging the rows of the sensitivity matrix of ECT in a random order. Finally, interior point method, gradient projection for sparse reconstruction (GPSR) algorithm and greedy algorithm which are the three commonly used reconstruction algorithms of CS were used for ECT image reconstruction and the comparison was made with linear back projection algorithm and Landweber iterative algorithm. Simulation results indicate that reconstructed images with higher accuracy can be obtained using the ECT image reconstruction algorithm based on CS theory. Meanwhile, the advantages and disadvantages of the three CS image reconstruction algorithms were analyzed. The advice of selecting which type of image reconstruction algorithm was given.
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图 2 基于最优阈值处理的重建图像
Figure 2. Reconstructed images based on optimal threshold processing
表 1 初始重建图像相对误差
Table 1. Relative error of initial reconstructed image
流型 Er LBP Landweber 内点法 GPSR OMP 流型1 1.405 3 0.793 3 0.807 8 0.590 7 0.798 8 流型2 1.238 3 0.807 0 0.689 1 0.675 2 0.689 3 流型3 1.570 9 0.882 6 0.718 9 0.774 0 0.863 9 流型4 2.002 7 1.020 4 0.892 5 0.883 7 0.938 1 流型5 2.052 8 1.114 7 0.904 1 1.091 0 0.971 5 
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表 2 初始重建图像相关系数
Table 2. Correlation coefficient of initial reconstructed image
流型 Cc LBP Landweber 内点法 GPSR OMP 流型1 0.616 9 0.769 6 0.813 8 0.874 2 0.737 5 流型2 0.569 3 0.742 3 0.843 1 0.839 0 0.855 8 流型3 0.445 8 0.685 0 0.730 3 0.812 9 0.575 2 流型4 0.357 8 0.603 0 0.662 7 0.728 8 0.551 6 流型5 0.322 8 0.536 6 0.675 2 0.632 2 0.525 6
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表 3 后处理重建图像相对误差
Table 3. Relative error of reconstructed image after processing
流型 Er 后处理
LBP后处理
Landweber后处理内点法 后处理
GPSR后处理
OMP流型1 0.854 9 0.339 7 0.392 2 0.277 4 0.425 1 流型2 0.784 5 0.392 2 0.240 2 0.219 3 0.325 2 流型3 0.872 0 0.692 2 0.489 5 0.102 1 0.747 8 流型4 1.258 3 0.640 3 0.517 4 0.408 2 0.822 9 流型5 1.030 8 0.698 2 0.497 6 0.559 0 0.702 0
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表 4 后处理重建图像相关系数
Table 4. Correlation coefficient of reconstructed image after processing
流型 Cc 后处理
LBP后处理
Landweber后处理内点法 后处理
GPSR后处理
OMP流型1 0.630 6 0.943 1 0.813 8 0.958 4 0.738 8 流型2 0.696 6 0.920 4 0.966 6 0.972 2 0.938 3 流型3 0.681 9 0.700 6 0.870 6 0.994 1 0.507 8 流型4 0.552 2 0.846 1 0.797 4 0.915 4 0.542 7 流型5 0.500 2 0.716 9 0.819 3 0.799 4 0.663 0
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表 5 重建图像所用时间
Table 5. Consumed time of image reconstruction
流型 重建图像所用时间/s LBP Landweber 内点法 GPSR OMP 流型1 0.013 43 4.799 04 8.609 49 2.779 11 0.060 46 流型2 0.014 73 4.585 68 8.791 92 2.623 62 0.043 96 流型3 0.011 82 5.560 35 9.095 21 3.539 22 0.054 18 流型4 0.013 81 5.262 73 9.862 51 4.610 25 0.087 01 流型5 0.015 43 5.040 01 9.932 23 4.735 56 0.119 94
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