Multiselective Pyramidal Decompositions of Images: How to exploit wavelets with adaptive angular selectivity

Many techniques have been devised these last ten years to add an appropriate directionality concept in decompositions of images from the (affine) transformations of a small set of atomic functions (e.g. directional wavelets, steerable filters, curvelets, wave atoms). Generally features which are best represented are straight lines (as those defining contours of objects), smooth curves (e.g. curvelets processing) or oriented textures (e.g. wave atoms). However, real images present also a set of details less oriented and more isotropic (like corners, spots, texture components, . . . ). This paper aims at developing one possible adaptive representation for all these image elements ranging from highly directional ones to fully isotropic ones. This new tool can indeed be tuned relatively to these image features by decomposing them into a (linear) frame of directional wavelets with variable angular selectivity. Inside such a decomposition, wavelets inherit some particularities of the (biorthogonal) circular multiresolution framework. This simple link qualifies our method of multiselectivity analysis. Two applications of the proposed method are given at the end of the paper, namely, in the fields of image denoising and N -term nonlinear approximation.