Convex-NonConvex (CNC) Variational Models and Algorithms for Inverse Imaging Problems

  • Lanza, Alessandro (University of Bologna)
  • Sgallari, Fiorella (University of Bologna)
  • Morigi, Serena (University of Bologna)
  • Huska, Martin (University of Bologna)
  • Selesnick, Ivan (New York University)

Please login to view abstract download link

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, namely 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 a lot of research towards 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 recently proposed - see, e.g., [1-3] - which allows the use of non-convex (sparsity-promoting) regularization while maintaining convexity of the total cost function. In this talk, after outlining the key ideas at the basis of the CNC strategy, a general class of parameterized non-convex sparsity-inducing separable and non-separable regularizers and their associated CNC variational models are presented [4]. Convexity conditions for the total cost functions together with suitable algorithms for their minimization based on a general forward-backward splitting strategy are discussed. Numerical experiments on the two classes of considered separable and non-separable CNC variational models are presented which show how they outperform the purely convex counterparts when applied to some inverse imaging problems.