Modeling Very Oscillating Signals. Application to Image Processing

This article is a companion paper of a previous work where we havedeveloped the numerical analysis of a variational model first introduced by Rudin et al. and revisited by Meyer forremoving the noise and capturing textures in an image. The basic idea in this model isto decompose an image f into two components (u + v) and then to search for (u,v) asa minimizer of an energy functional. The first component u belongs to BV andcontains geometrical information, while the second one v is sought in a space Gwhich contains signals with large oscillations, i.e. noise and textures. In Meyer carried out his study in the whole ² and his approach is rather built on harmonic analysis tools. We place ourselves in the case of a bounded set of ² which is the proper setting for image processing and our approach isbased upon functional analysis arguments. We define in this context the space G, givesome of its properties, and then study in this continuous setting the energy functionalwhich allows us to recover the components u and v. We present some numericalexperiments to show the relevance of the model for image decomposition and for imagedenoising.