Sparsity-inducing nonconvex nonseparable regularization for convex image processing

Alessandro Lanza, Serena Morigi, Ivan Selesnick, Fiorella Sgallari

Research output: Contribution to journalArticle

Abstract

A popular strategy for determining solutions to linear least-squares problems relies on using sparsitypromoting regularizers and is widely exploited in image processing applications such as, e.g., image denoising, deblurring, and inpainting. It is well known that, in general, nonconvex regularizers hold the potential for promoting sparsity more effectively than convex regularizers such as, e.g., those involving the \ell 1 norm. To avoid the intrinsic difficulties related to non-convex optimization, the convex nonconvex (CNC) strategy has been proposed, which allows the use of nonconvex regularization while maintaining convexity of the total objective function. In this paper, a new CNC variational model is proposed, based on a more general parametric nonconvex nonseparable regularizer. The proposed model is applicable to a greater variety of image processing problems than prior CNC methods. We derive the convexity conditions and related theoretical properties of the presented CNC model, and we analyze existence and uniqueness of its solutions. A primal-dual forward-backward splitting algorithm is proposed for solving the related saddle-point problem. The convergence of the algorithm is demonstrated theoretically and validated empirically. Several numerical experiments are presented which prove the effectiveness of the proposed approach.

Original languageEnglish (US)
Pages (from-to)1099-1134
Number of pages36
JournalSIAM Journal on Imaging Sciences
Volume12
Issue number2
DOIs
StatePublished - Jan 1 2019

Fingerprint

Nonseparable
Sparsity
Image Processing
Regularization
Image processing
Image denoising
Convexity
Variational Model
Deblurring
Inpainting
Saddle Point Problems
Linear Least Squares
Nonconvex Optimization
Image Denoising
Primal-dual
Least Squares Problem
Existence and Uniqueness of Solutions
Experiments
Objective function
Numerical Experiment

Keywords

  • Convex nonconvex strategy
  • Sparsity-promoting regularization
  • Variational method

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Sparsity-inducing nonconvex nonseparable regularization for convex image processing. / Lanza, Alessandro; Morigi, Serena; Selesnick, Ivan; Sgallari, Fiorella.

In: SIAM Journal on Imaging Sciences, Vol. 12, No. 2, 01.01.2019, p. 1099-1134.

Research output: Contribution to journalArticle

Lanza, Alessandro ; Morigi, Serena ; Selesnick, Ivan ; Sgallari, Fiorella. / Sparsity-inducing nonconvex nonseparable regularization for convex image processing. In: SIAM Journal on Imaging Sciences. 2019 ; Vol. 12, No. 2. pp. 1099-1134.
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