张小群论文获选Inverse problems杂志 2013 Highlights

近期,上海交大自然科学研究院及数学系特别研究员张小群及合作者陈培军(上海交通大学数学系博士生),黄建国(上海交大数学系教授)题为“A primal–dual fixed point algorithm for convex separable minimization with applications to image restoration”的论文(2013年29期2号,Inverse problems) 被选为该杂志2013年20篇Highlights文章中的一篇 (http://iopscience.iop.org/0266-5611/page/Highlights-of-2013)。Inverse problems 是应用数学领域重要学术杂志之一。 该篇论文主要研究了图像处理领域一类凸分离问题的对偶不动点算法框架,在不动点理论框架下证明了算法的收敛性与收敛率,并研究了与其他已有算法的联系。该类算法形式简单,参数选取简单,能够很好的扩展到很多图像应用领域。

参考文献:“ A primal–dual fixed point algorithm for convex separable minimization with applications to image restoration,Peijun Chen, Jianguo Huang and Xiaoqun Zhang, Inverse Problems 29 025011” http://iopscience.iop.org/0266-5611/29/2/025011/article


Recently, the minimization of a sum of two convex functions has received considerable interest in a variational image restoration model. In this paper, we propose a general algorithmic framework for solving a separable convex minimization problem from the point of view of fixed point algorithms based on proximity operators (Moreau 1962 C. R. Acad. Sci., Paris I 255 2897–99). Motivated by proximal forward–backward splitting proposed in Combettes and Wajs (2005 Multiscale Model. Simul. 4 1168–200) and fixed point algorithms based on the proximity operator (FP2O) for image denoising (Micchelli et al 2011 Inverse Problems 27 45009–38), we design a primal–dual fixed point algorithm based on the proximity operator (PDFP2Oκ for κ ∈ [0, 1)) and obtain a scheme with a closed-form solution for each iteration. Using the firmly nonexpansive properties of the proximity operator and with the help of a special norm over a product space, we achieve the convergence of the proposed PDFP2Oκ algorithm. Moreover, under some stronger assumptions, we can prove the global linear convergence of the proposed algorithm. We also give the connection of the proposed algorithm with other existing first-order methods. Finally, we illustrate the efficiency of PDFP2Oκ through some numerical examples on image supper-resolution, computerized tomographic reconstruction and parallel magnetic resonance imaging. Generally speaking, our method PDFP2O (κ = 0) is comparable with other state-of-the-art methods in numerical performance, while it has some advantages on parameter selection in real applications.