非凸优化及稀疏相似性与图像恢复问题研究

Restoration of an image is the process of removing or minimizing degradations in the observed image. Image restoration techniques frequently arise in a variety of applications in different fields of scientific computing and engineering such as computer vision, medical image, astronomy and microscopy. The design of fast and numerically reliable algorithms for image restoration problems has become an increasingly important activity, especially in recent years, driven by the ever-increasing complexity of practical applications. The major challenge in this area is to develop algorithms that blend speed and numerical accuracy. These two requirements often have been regarded as competitive, so that the design of fast and numerically reliable algorithms for image restoration problems has remained a significant open issue in many instances. This project is aimed at studying non-convex optimization and sparse similarity technology, especially their applications to image restoration. The main contents include: 1. We study a fast and numerically effective non-convex optimization algorithm for image restoration problems and analyze convergence properties for the algorithm. 2. We focus on establishing sparse similarity image model for image restoration problems and giving a fast and numerically effective algorithm for the model. 3. According to the characteristics of imaging system, we propose effective algorithms for the image restoration problems from SAR and PET images, discuss the choice of the regulaization parameter. 4. We develop a package of Matlab software for large-scale image restoration problems based on the non-convex optimization and the sparse similarity method...

图像复原是去除或者减轻观察图像中的退化的过程. 图像复原技术在计算机视觉、医学图像处理、天文学、显微镜学等等不同科学计算和工程领域都有广泛的应用. 特别是在实际应用中日益增长的复杂性的驱动下, 设计快速的数值可靠的算法求解图像复原问题已变得日益重要. 这方面最主要的挑战是需要发展速度快与数值精度高的算法, 但这两个要求常常不可兼得, 以至于在许多情况下, 怎样设计快速的数值可靠的算法求解图像复原问题依然是一个关键问题. 本项目旨在研究非凸优化和稀疏相似性技术, 特别是在图像恢复问题中的应用, 主要包括：1.研究求解大规模图像恢复问题的非凸优化模型及相关高效求解算法; 2. 研究基于稀疏相似性的数字图像恢复模型及相应计算大规模问题的高效稀疏算法; 3. 根据系统成像特点，研究SAR图像和PET图像恢复问题的高效算法, 讨论正则参数的选取; 4. 开发适用于大规模稀疏优化图像复原问题的程序包.

图像复原技术在计算机视觉、医学图像处理、天文学、显微镜学等等不同科学计算和工程领域都有广泛的应用。本项目主要研究凸优化和非凸优化以及稀疏相似性技术，特别是在图像恢复问题中的应用。具体研究成果如下：对于不同类型噪声的图像复原问题，为了抑制阶梯效应，建立了基于全变分重叠组稀疏正则化模型，采用交替方向乘子算法和优化最小化方法进行求解；为了抑制阶梯效应和减少Gibbs现象，建立了基于紧框架和全变分的非凸优化模型，通过变量替换把原问题转化成约束问题，用交替方向乘子算法进行求解；基于加权核范数最小化思想，提出了高效的解耦迭代算法求解图像复原问题；建立非凸优化模型求解矩阵完成问题，提出了全局收敛的紧阈值迭代算法；利用逆滤波方法求解图像盲复原问题，给出了高效的原始对偶求解算法。该项目研究取得了一系列有意义的研究成果，开发了适用于大规模稀疏优化图像复原问题的程序包，共发表了高水平SCI论文10篇，完成了项目的预期考核指标。