图像恢复问题中的非凸变分模型及其原始-对偶算法研究

在变分图像处理领域中，越来越多的实验证实了非凸模型对于刻画实际问题的有效性。因此，如何设计快速收敛的优化方法求解这些非凸变分模型成为亟待解决的问题。尽管前期对非凸优化方法已经有一些理论结果，但假设条件要求严格，还需进一步研究。本项目拟开展图像恢复问题中非凸变分模型及其原始-对偶算法的研究，主要包括三类典型的非凸变分模型和对应的凸函数差方法、交替方向乘子法和原始-对偶混合梯度法。利用邻近点、连续化策略与自适应参数技巧，提高收敛速度。借助凸分析、对偶理论及Kurdyka-Lojasiewicz性质，减弱算法收敛的条件。对新方法在噪音去除和少量数据医学图像重建问题上进行数值实验，编写实用高效的数值程序包。该项目的实施不仅能为图像恢复问题提供新方法，也可促进非凸优化理论的发展，具有重要的科学意义和实用价值。

In the field of variational image processing, more and more experiments have confirmed the effectiveness of nonconvex models for characterizing practical problems. Therefore, it is urgent to design the fast optimization method with convergence guarantee for solving the nonconvex variational models. Although there have been some theoretical results in the nonconvex optimization method, the assumptions are strict and further research is needed. This project aims to study the nonconvex variational model and primal-dual algorithm in image restoration problems, including three typical nonconvex variational models and difference of convex functions algorithm, alternating direction method of multipliers, primal-dual hybrid gradient method. The convergence speed is improved by using the techniques of proximal points, continuity strategy, adaptive parameters and the convergence of the new method under weaker conditions is proved by the convex analysis, dual theory and Kurdyka-Lojasiewicz properties. Noise remove problems and medical reconstruction problems with a small amount of data are tested. Practical and efficient mathematical software will be developed. This project is scientific significant and extremely valuable, not only for providing new methods for image restoration problems, but also for promoting the development of nonconvex optimization theory.