Infimal post-composition approach for composite convex optimization applied to image restoration

• Published in: Signal processing Vol. 223; p. 109549
• Main Authors: Briceño-Arias, Luis M., Pustelnik, Nelly
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-10-2024
• Subjects: Composite convex optimization problems; Douglas–Rachford algorithm; Image restoration; Infimal post-composition; Splitting algorithms; Strong convexity; Douglas–Rachford algorithm; Image restoration; Composite convex optimization problems; Splitting algorithms; Strong convexity; Infimal post-composition
• ISSN: 0165-1684; 1872-7557
• DOI: 10.1016/j.sigpro.2024.109549

In this paper we introduce a new approach for solving image restoration problems by using the infimal postcomposition of a convex function by a linear operator. We derive this formulation for general linear composite convex problems in Hilbert spaces and we provide globally weakly convergent algorithms based on the Douglas–Rachford splitting. We apply our algorithms to the image restoration problem, giving an explicit closed expression for the proximity operator of the infimal postcomposition. Comprehensive numerical experiments are performed in order to serve two key objectives: first, to highlight the advantages of the proposed procedure over a wide array of state-of-the-art methods, considering diverse levels of image degradations; and second, to assess the impact of TV-l2 penalization, which introduces strong convexity while maintaining high performance, contrary to conventional beliefs.
•Use of infimal postcomposition of a convex function by a linear operator for solving inverse problems.•Globally weakly convergent algorithms based on the Douglas–Rachford splitting.•Explicit closed expression for the proximity operator of the infimal postcomposition.•Application to image restoration.