压缩感知与矩阵填充及其在图像处理中的应用
Compressed sensing，matrix completion and their application in image processing

压缩感知和矩阵填充是当前的两个研究热点，压缩感知的性能取决于3个要素：信号的稀疏性、压缩感知矩阵的非相干性和重构算法的快速有效性。相应地，矩阵填充性能也取决于3个要素：矩阵的低秩性、矩阵的不相关性和重构算法的快速有效性。文中首先论述了压缩感知和矩阵填充的应用背景，阐述了两者的数学模型，分析了信号的稀疏性和观测矩阵的不相关性对压缩感知性能的影响，研究了矩阵的低秩和不相关性在矩阵填充中的作用，进而对压缩感知和矩阵填充的稀疏性和非相干性进行了对比，总结了压缩感知和矩阵填充的重构算法，介绍了压缩感知和矩阵填充在图像处理中的应用。
The compressed sensing and the matrix completion are the two research focus currently. The performances of the compressed sensing depend on three elements: sparsity of signal, incoherence of compressed sensing matrix and efficient recontruction algorithm. Accordingly, the performances of matrix completion depend on three elements: low rank matrix, incoherence of matrix and efficient reconstruction algorithm. Firstly the background of the compressed sensing and matrix completion is presented, the mathematics model for them is described. Then, the influences of sparsity and incoherence on compressed sensing, and the effect of low rank and incoherence on matrix completion are analyzed, and the sparsity and the incoherence on compressed sensing and matrix completion are compared. Finally, we summarize a reconstruction algorithm for compressed sensing and matrix completion, and introduce their applications in image processing