SIAM Journal on Imaging Sciences, Volume 14, Issue 1, Page 1-25, January 2021.
This paper considers the image recovery problem by taking group sparsity into account as the prior knowledge. This problem is formulated as a group sparse optimization over the intersection of a polyhedron and a possibly degenerate ellipsoid. It is a convexly constrained optimization problem with a group cardinality objective function. We use a capped folded concave function to approximate the group cardinality function and show that the solution set of the continuous approximation problem and the set of group sparse solutions are the same. Moreover, we use a penalty method to replace the constraints in the approximation problem by adding a convex nonsmooth penalty function in the objective function. We show the existence of positive penalty parameters such that the solution sets of the unconstrained penalty problem and the group sparse problem are the same. We propose a smoothing penalty algorithm and show that any accumulation point of the sequence generated by the algorithm is a directional stationary point of the continuous approximation problem. Numerical experiments for recovery of group sparse image are presented to illustrate the efficiency of the smoothing penalty algorithm with adaptive capped folded concave functions.

中文翻译：

---

使用封顶折叠凹函数进行图像稀疏的组稀疏优化

SIAM影像科学杂志，第14卷，第1期，第1-25页，2021年1月。
本文通过将组稀疏性作为先验知识来考虑图像恢复问题。这个问题被表述为在多面体和可能退化的椭球的交点上的群稀疏优化。这是具有基数基函数的凸约束优化问题。我们使用一个带帽的折叠凹函数来近似组基数函数，并证明连续逼近问题的解集和组稀疏解的集是相同的。此外，我们通过在目标函数中添加凸的非光滑罚函数来使用罚函数方法来替代逼近问题中的约束。我们证明了正惩罚参数的存在，使得无约束惩罚问题和群稀疏问题的解集相同。我们提出了一种平滑惩罚算法，并表明该算法生成的序列的任何累加点都是连续逼近问题的有向静止点。提出了用于恢复组稀疏图像的数值实验，以说明具有自适应带帽折叠凹函数的平滑惩罚算法的效率。