Décomposition du Potentiel de l’Opérateur de Schrödinger Semi-Classique Bidimensionnel : Application à la Représentation d’Images

Abstract

This work emphasizes on the generalization to two dimensions of a new approach of function decomposition based on a semi-classical quantification namely Semi-Classical Signal Analysis. The SCSA approach consists in considering the function, to be analyzed, as a potential of a semi-classical Schrödinger operator. The resolution of the spectral problem of the Schrödinger operator allows approximating the function using the resulting negative eigenvalues and its corresponding eigenfunctions. Precisely, the approximation is expressed as the truncated sum of the squared L2-normalized eigenfunctions weighted by the associated negative eigenvalues. The novelty of this approach is the fact that the underlying basis functions are dependent on the function itself. The relation between the approximation accuracy and the semi-classical parameter together with the eigenfunctions' interesting features are studied. Furthermore, the extension of the method to higher dimensions, using tensor product, is investigated. Numerical results on both academic functions of two variables and standard gray test images illustrate the performance of the proposed approach. Through the promising results, the SCSA opens great perspectives on theoretical developments that can handle several image processing questions including denoising, compression and analysis of singularities.