Signal recovery by proximal forward-backward splitting. (English) Zbl 1179.94031

Combettes, Patrick L.; Wajs, Valérie R.

Multiscale Model. Simul. 4, No. 4, 1168-1200 (2005).

Summary: We show that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. This formulation makes it possible to derive existence, uniqueness, characterization, and stability results in a unified and standardized fashion for a large class of apparently disparate problems. Recent results on monotone operator splitting methods are applied to establish the convergence of a forward-backward algorithm to solve the generic problem. In turn, we recover, extend, and provide a simplified analysis for a variety of existing iterative methods. Applications to geometry/texture image decomposition schemes are also discussed. A novelty of our framework is to use extensively the notion of a proximity operator, which was introduced by Moreau in the 1960s.

MSC:

94A12	Signal theory (characterization, reconstruction, filtering, etc.)
65K10	Numerical optimization and variational techniques
94A08	Image processing (compression, reconstruction, etc.) in information and communication theory
90C25	Convex programming

Keywords:

denoising; proximal Landweber method; forward-backward algorithm; image decomposition; image restoration; multiresolution analysis; inverse problem; signal recovery; iterative soft-thresholding; proximity operator