Convex Non-convex Variational Models

Alessandro Lanza (), Serena Morigi (), Ivan W. Selesnick () and Fiorella Sgallari ()
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Alessandro Lanza: University of Bologna, Department of Mathematics
Serena Morigi: University of Bologna, Department of Mathematics
Ivan W. Selesnick: New York University, Department of Electrical and Computer Engineering
Fiorella Sgallari: University of Bologna, Department of Mathematics

Chapter 1 in Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging, 2023, pp 3-59 from Springer

Abstract: Abstract 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 sparsity-characterized images corrupted by blur and noise.

Keywords: Convex non-convex optimization; Sparsity regularization; Image restoration; Alternating direction method of multipliers; Forward backward algorithm (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/978-3-030-98661-2_61

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