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We investigate inexact proximity operators for weakly convex functions. To this aim, we derive sum rules for proximal {\epsilon}-subdifferentials, by incorporating the moduli of weak convexity of the functions into the respective formulas. This allows us to investigate inexact proximity operators for weakly convex functions in terms of proximal {\epsilon}-subdifferentials.

The core of many approaches for the resolution of variational inverse problems arising in signal and image processing consists of promoting the sought solution to have a sparse representation in a well-suited space. A crucial task in this context is the choice of a good sparsity prior that can ensure a good trade-off between the quality of the solution and the resulting computational cost. The recently introduced Convex-Non-Convex (CNC) strategy appears as a great compromise, as it combines the high qualitative performance of non-convex sparsity-promoting functions with the convenience of dealing with convex optimization problems. This work proposes a new variational formulation to implement CNC approach in the context of image denoising. By suitably exploiting duality properties, our formulation allows to encompass sophisticated directional total variation (DTV) priors. We additionally propose an efficient optimisation strategy for the resulting convex minimisation problem. We illustrate on numerical examples the good performance of the resulting CNC-DTV method, when compared to the standard convex total variation denoiser., Comment: Accepted for EUSIPCO 2022 - 30th European Signal Processing Conference, Aug 2022, Belgrade, Serbia

The joint problem of reconstruction / feature extraction is a challenging task in image processing. It consists in performing, in a joint manner, the restoration of an image and the extraction of its features. In this work, we firstly propose a novel nonsmooth and non-convex variational formulation of the problem. For this purpose, we introduce a versatile generalised Gaussian prior whose parameters, including its exponent, are space-variant. Secondly, we design an alternating proximal-based optimisation algorithm that efficiently exploits the structure of the proposed non-convex objective function. We also analyse the convergence of this algorithm. As shown in numerical experiments conducted on joint deblurring/segmentation tasks, the proposed method provides high-quality results.

We propose a non-convex variational model for the super-resolution of Optical Coherence Tomography (OCT) images of the murine eye, by enforcing sparsity with respect to suitable dictionaries learnt from high-resolution OCT data. The statistical characteristics of OCT images motivate the use of {\alpha}-stable distributions for learning dictionaries, by considering the non-Gaussian case, {\alpha}=1. The sparsity-promoting cost function relies on a non-convex penalty - Cauchy-based or Minimax Concave Penalty (MCP) - which makes the problem particularly challenging. We propose an efficient algorithm for minimizing the function based on the forward-backward splitting strategy which guarantees at each iteration the existence and uniqueness of the proximal point. Comparisons with standard convex L1-based reconstructions show the better performance of non-convex models, especially in view of further OCT image analysis, Comment: 4 pages, 2 figures, 1 table, 1 algorithm, submitted to ISBI2021