Reconstruction of noisy signals by minimization of non-convex functionals

Non-convex functionals have shown sharper results in signal reconstruction as compared to convex ones, although the existence of a minimum has not been established in general. This paper addresses the study of a general class of either convex or non-convex functionals for denoising signals which combines two general terms for fitting and smoothing purposes, respectively. The first one measures how close a signal is to the original noisy signal. The second term aims at removing noise while preserving some expected characteristics in the true signal such as edges and fine details. A theoretical proof of the existence of a minimum for functionals of this class is presented. The main merit of this result is to show the existence of minimizer for a large family of non-convex functionals.The rst author gratefully acknowledges many helpful discussion with Professor H. Frid from IMPA. Also thanks the Promeps Project that support this work. The second author is grateful to the Spanish Ministry of Economy and Competitiveness for the grant TIN2013-46522-P, and to the Generalitat Valenciana for the grant PROMETEOII/2014/062