求解可分凸优化模型的Peaceman-Rachford型分裂算法及其在图像处理中的应用

Many important mathematical problems arising in such areas as signal processing, image processing, machine learning and data science can be modeled, or relaxed as convex programming models with some structures. How to exploit these structures favorably and design efficient algorithms to solve these models become crucial, especially for the case with rapidly growing dimension. Recently, various first-order methods based on the operator splitting idea have received wide attention from varying fields. Among them is the strictly contractive Peaceman-Rachford splitting method (PRSM) proposed in 2014 and its efficiency has been well verified. Based on our previous work along this line, we will study this method more deeply, with the aims of developing some more advanced application-driven PRSM-type algorithms and investigating their applications to some image processing problems. Specifically, we will: 1) propose a linearized version of the strictly contractive PRSM with adaptively-chosen step sizes, an accelerated inertial version of the strictly contractive PRSM, and some improved PRSM-type methods for multi-block separable convex programming models; 2) analyze some theoretical properties of the new algorithms such as the convergence study and the iteration complexity estimate; and 3) apply the new algorithms to some challenging image processing problems such as computational tomography, magnetic resonance imaging and background extraction from surveillance video.

很多信号处理、图像处理、机器学习、数据科学等领域里出现的关键数学问题都可以归结为（或松驰为）一些带结构的凸优化问题。尤其是随着问题维数的增长，如何充分利用问题的结构来设计高效率的算法求解这些凸优化模型就至关重要。近年来，一阶算子分裂方法受到学界的广泛关注。其中，一类严格收缩的Peaceman-Rachford分裂算法(PRSM)于2014年提出; 其算法的有效性也得到了检验。本项目以此为背景，基于申请人前期关于一阶算子分裂方法的工作，旨在进一步设计问题驱动的PRSM改进加速类算法并且重点关注这些算法在图像处理领域里的应用。具体而言，在本项目中，我们将1)设计自适应步长的线性化PRSM算法，inertial加速的PRSM算法，多块的分裂算法等; 2)分析并建立算法的收敛性和复杂度理论;3）将结果应用于计算机断层扫描成像、磁共振成像和视频背景提取等图像处理问题。