压缩感知与矩阵填充问题的参数化阈值算法研究

This project will studies the reconstruction problems and their algorithms of compressed sensing and matrix completion in signal processing, image restoration, and so on. Research content can be generalized as follows: . (1) For the basic reconstruction problem of compressed sensing, this project considers transforming equivalently this NP-hard nonconvex optimization problem into the convex optimization problem or the solvable nonconvex optimization, and designs the parameterized thresholding algorithms with high accuracy to solve the considered problem.. (2) For the reconstruction problem of matrix completion, from the point of view of algorithm accuracy, this project uses the matrix theory and the optimization theory to design the stable and precise algorithms.. (3) This project comprehensively studies the practical applications for the reconstruction problems of compressed sensing and matrix completion, including the compression and restoration applications of signal and image. This project makes validity and convergence analysis for the algorithms designed to solve the practical problems.

本项目致力于研究源于信号压缩与图像恢复等领域的压缩感知与矩阵填充的重构问题及其计算方法。项目研究内容如下：.(1)对压缩感知的基本重构问题进行研究，将此NP难的非凸优化问题等价转化为凸优化问题或较容易的非凸优化问题，并设计出高精度的参数化阈值算法来求解此可解优化问题。.(2)对矩阵填充问题的重构问题进行研究，从提高计算精度的角度出发，运用矩阵理论与最优化理论中的有关理论与方法设计出精度高、稳定性好的计算方法。.(3)深入系统地研究压缩感知与矩阵填充的重构问题的实际应用，包括解决信号与图像的压缩与恢复问题。并对求解实际问题所设计的计算方法进行有效性与收敛性分析。

本课题围绕压缩感知与矩阵填充的重构问题进行研究。对于第一个问题，通过对拟软阈值算法进行参数化，得到参数化拟软阈值算法。再对参数化拟软阈值算子的参数进行迭代更新，得到变参数拟软阈值算法。证明了这个算法的收敛性，并用数值实验表明变参数拟软阈值算法能有效提高信号重构的精度。对于第二个问题，提出了矩阵填充的参数化拟软阈值算法与变参数拟软阈值算法。证明了这些算法的收敛性，且由数值实验表明矩阵参数化拟软阈值算法与变参数拟软阈值算法能提高矩阵填充的精度。