Convex Non-convex Variational Models

Alessandro Lanza, Serena Morigi, Ivan W. Selesnick, Fiorella Sgallari

Research output: Chapter in Book/Report/Conference proceedingChapter


An important class of computational techniques to solve inverse problems in image processing relies on a variational approach: the optimal output is obtained by finding a minimizer of an energy function or “model” composed of two terms, the data-fidelity term, and the regularization term. Much research has focused on models where both terms are convex, which leads to convex optimization problems. However, there is evidence that non-convex regularization can improve significantly the output quality for images characterized by some sparsity property. This fostered recent research toward the investigation of optimization problems with non-convex terms. Non-convex models are notoriously difficult to handle as classical optimization algorithms can get trapped at unwanted local minimizers. To avoid the intrinsic difficulties related to non-convex optimization, the convex non-convex (CNC) strategy has been proposed, which allows the use of non-convex regularization while maintaining convexity of the total cost function. This work focuses on a general class of parameterized non-convex sparsity-inducing separable and non-separable regularizers and their associated CNC variational models. Convexity conditions for the total cost functions and related theoretical properties are discussed, together with suitable algorithms for their minimization based on a general forward-backward (FB) splitting strategy. Experiments on the two classes of considered separable and non-separable CNC variational models show their superior performance than the purely convex counterparts when applied to the discrete inverse problem of restoring sparsitycharacterized images corrupted by blur and noise.

Original languageEnglish (US)
Title of host publicationHandbook of Mathematical Models and Algorithms in Computer Vision and Imaging
Subtitle of host publicationMathematical Imaging and Vision
PublisherSpringer International Publishing
Number of pages57
ISBN (Electronic)9783030986612
ISBN (Print)9783030986605
StatePublished - Jan 1 2023


  • Alternating direction method of multipliers
  • Convex non-convex optimization
  • Forward backward algorithm
  • Image restoration
  • Sparsity regularization

ASJC Scopus subject areas

  • General Mathematics
  • General Computer Science


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