摘要压缩感知理论突破了乃奎斯特定律对采样频率的限制，允许仅用少量的测量值就能 准确恢复原始信号。压缩感知利用了自然信号的稀疏性，将采样和压缩同时进行，是目 前重要的信号处理方式。目前，进一步利用组稀疏或结构稀疏提出的重构算法都取得了 不错的重构效果，利用相似块组形成的矩阵具有低秩特性来对图像块进行重构，因此秩 最小化的问题是至关重要的。针对利用图像非局部低秩特性的重构算法，本文使用 Schatten p-范数代替logdet作为秩的替代函数来求解矩阵秩最小化的问题，利用交替方向 乘子法对优化模型求解，实验结果验证了改进后算法的优越性。74309

毕业论文关键词 压缩感知 图像重构 非局部相似模型 低秩近似 Schatten p-范数

Title Implementation and verification of compressive sensing image reconstruction algorithm

Abstract The compressive sensing theory breaks through the limitation of Nyquist's law on the sampling frequency, and allows to recover the original signal with only a small amount of measurements。 Compressive sensing utilizes the sparsity of natural signals, and it is an important signal processing method。 Recent advances have suggested that structured or group sparsity often leads to more powerful signal reconstruction techniques in various compressed sensing studies。 Exploiting the low-rank property of the matrix which is formed by the similar patches can lead to a better result。 For the reconstruction algorithm which exploits the non-local low rank property, we use the Schatten p-norm instead of logdet as the surrogate function for the rank to solve the matrix rank minimization problem, and exploit the alternating direction multiplier method to solve the optimization model。

Experimental results has verified the superiority of the proposed algorithm。

Keywords compressive sensing image reconstruction non-local similarity model low-rank approximation Schatten p-norm

目 次

1 绪论 1

2 压缩感知的主要内容 3

2。1 信号的稀疏表示 3

2。2 测量矩阵 3

2。3 图像重构算法 4

3 基于非局部低秩正则化的压缩感知重构算法 6

3。1 非局部相似模型 6

3。2 图像重构中的非局部低秩正则化 7

3。3 交替方向乘子法实现图像重构 9

4 算法的实现与实验验证 13

4。1 对无噪声图像进行实验 15

4。2 对含噪声图像进行实验 19

4。3 参数 p 对实验结果的影响 22

4。4 算法的收敛性 23

结论