﻿ QR分解与特征值优化观测矩阵的算法研究
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QR分解与特征值优化观测矩阵的算法研究

1. 上海海事大学 信息工程学院, 上海 201306;
2. 西安理工大学 自动化与信息工程学院, 陕西 西安 710000

An algorithm for measurement matrix based on QR decomposition and eigenvalue optimizatio
ZHENG Xiao1 , BO Hua1, SUN Qiang2
1. College of Information Engineering, Shanghai Maritime University, Shanghai 201306, China;
2. Automation and Information Engineering College, Xi'an University of Technology, Xi'an, 710000, China
Abstract: Measurement matrix is a core part of compressed sensing. The column independence of measurement matrix and the incoherence between measurement matrix and sparse basis have a major impact on the quality of a reconstructed image. This paper proposes a new algorithm of measurement matrix based on QR decomposition and eigenvalue. The column independence of the measurement matrix is increased by QR decomposition and at the same time the Gram matrix is optimized. Therefore, the eigenvalue of the normalized Gram matrix approximates to N/M so as to increases the incoherence between measurement matrix and sparse basis. The simulation results showed that the proposed method has excellent results on the aspects of increasing the quality of reconstructed image. In addition, the stability of the reconstructed results had more apparent advantages than other algorithms in the case of less number of observed values.
Key words: compressed sensing     sparse basis     measurement matrix     reconstruction algorithm     QR decomposition     eigenvalue     column independence     incoherenc

1 压缩感知

1)寻求稀疏基ΨRN×N,并将信号XRN进行稀疏表示,由式(1)可以得到稀疏向量ΘRN:

2) 设计一个平稳的、与稀疏基ψ不相关的观测矩阵ΦRM×N,保证稀疏向量Θ能够从N维降到M维,同时不丢失重要信息,进而得到观测集合YRM,如式(2)所示:

3) 设计快速重构算法,利用观测集合Y并根据式(3)恢复原信号:

2 文中优化算法

2.1 QR分解增大列独立性

QR分解优化不但能够增大矩阵的最小奇异值,同时能够保持矩阵的性质基本不变[16]。QR分解优化主要原理为:首先对观测矩阵ΦRM×N进行QR分解,得到正交矩阵QRM×N和上三角矩阵ARN×N。然后对A分析,看出其主对角线元素远远大于非对角线元素,利用该发现,把A的非对角线元素全置零得A1,再根据A1得优化后的观测矩阵Φ1RM×N。通过QR分解优化,Φ的最小奇异值小于Φ1的最小奇异值[12]

2.2 基于特征值的优化

2.3 算法流程及具体实现

 图 1 QT算法流程图Fig. 1 The flow of QT algorithm

1) 构造DRM×N,并且令D=ΦΨ,对其进行归一化处理,得到归一化矩阵D1;

2) 由构造出Gram矩阵G,然后对G进行奇异值分解,即G=USVT,其中U,V均为酉矩阵,S为对角矩阵,并且URN×N,VRN×N,SRN×N;

3) 将S中对角线上的非零元素置为N/M,得到S1,接着得到G1=US1VT;

4)根据S1=LTL,对S1进行分解,得到矩阵L。其中LRM×N,并且它的对角线元素为,其他元素为零;

5)设D2RM×N,并且令D2=LVT,然后通过Φ1=D2Ψ-1,求得Φ1;

6)对Φ1进行近似QR分解的优化:即首先对Φ1进行QR分解,得到Φ1=EF。其中,正交阵ERM×N,而上三角矩阵FRM×N;然后将F中的非对角线元素设置为零,得到F1;最后根据Φ2=EF1,求得进一步更新的观测矩阵Φ2;

7)利用D2=Φ2Ψ求出D2,接着对D2进行归一化处理,得到D3,然后继续利用得到新的Gram矩阵G2,求出G2中除主对角线元素外的所有值的平方和(sum),当(sum)-((N/M)2-N)| < 0.1时,输出此时的观测矩阵Φ2,否则跳转到②继续新的循环。

3 实验结果与分析

3.1 不同观测长度的比较

 dB 观测 矩阵 高斯 矩阵 QR分解优化 特征值优化 QT 算法 m=80 18.395 18.607 21.644 29.205 m=100 26.197 26.572 27.317 31.086 m=120 28.190 28.581 29.444 32.902 m=140 29.805 30.334 31.640 34.151 m=160 31.384 32.077 33.375 35.115

 dB 观测 矩阵 高斯 矩阵 QR分解优化 特征值优化 QT 算法 m=80 16.942 16.994 19.547 25.070 m=100 22.210 22.551 23.322 26.423 m=120 23.398 24.063 24.809 27.760 m=140 25.629 25.629 26.401 28.661 m=160 27.111 27.112 28.275 29.441

3.2 算法的鲁棒性比较

 图 2 M=80时的PNSR图(lena)Fig. 2 The graph of PNSR when the value of M is 80(lena)
 图 3 M=80时的PNSR图(cameraman)Fig. 3 The graph of PNSR when the value of M is 80(cameraman)

 图 4 不同方法优化的重构图(lena)Fig. 4 The reconstructed images by different optimization methods(lena)
 图 5 不同方法优化的重构图(cameraman)Fig. 5 The reconstructed images by different optimization methods(cameraman)

4 结束语

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DOI: 10.3969/j.issn.2013-0934.201309034

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#### 文章信息

ZHENG Xiao, BO Hua, SUN Qiang
QR分解与特征值优化观测矩阵的算法研究
An algorithm for measurement matrix based on QR decomposition and eigenvalue optimizatio

CAAI Transactions on Intelligent Systems, 2015, 10(01): 149-155.
DOI: 10.3969/j.issn.2013-0934.201309034