Résumé:
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.
