2D Empirical Transforms. Wavelets, Ridgelets, and Curvelets Revisited

A recently developed approach, called empirical wavelet transform, aims to build one-dimensional (1D) adaptive wavelet frames accordingly to the analyzed signal. In this paper, we present several extensions of this approach to two-dimensional (2D) signals (images). We revisit some well-known transforms (tensor wavelets, Littlewood-Paley wavelets, ridgelets, and curvelets) and show that it is possible to build their empirical counterparts. We prove that such constructions lead to different adaptive frames which show some promising properties for image analysis and processing.