Wavelets and their associated transforms are highly efficient when approximating and analyzing onedimensional signals. However, multivariate signals such as images or videos typically exhibit curvilinear singularities, which wavelets are provably deficient of sparsely approximating and also of analyzing in the sense of, for instance, detecting their direction. Shearlets are a directional repres...

This paper introduces a Parseval frame of shearlets for the representation of 3D data, which is especially designed to handle geometric features such as discontinuous boundaries with very high efficiency. This system of 3D shearlets forms a multiscale pyramid of well-localized waveforms at various locations and orientations, which become increasingly thin and elongated at fine scales. We prove ...

Synthetic Aperture Radar (SAR) images suffer of multiplicative speckle noise, which damages the radiometric resolution of SAR images and makes the data interpretation difficult. Bayesian shrinkage in a transformed domain is a well-known method based on finding threshold value to suppress the speckle noise. This paper present a new approach to obtain the optimum threshold values for Bayesian shr...

Convolutional Neural Networks (CNNs) are a type of deep neural network which have performed well at image and audio classification. One approach to understanding the success of CNNs is Mallat’s scattering transform, which formalizes the observation that the filters learned by a CNN have wavelet-like structure. The resulting transform generates a representation that is approximately translation ...

Driven by an overwhelming amount of applications numerical approximation of partial differential equations was established as one of the core areas in applied mathematics. During the last decades a trend for the solution of PDEs emerged, that focuses on employing systems from applied harmonic analysis for the adaptive solution of these equations. Most notably wavelet systems have been used and ...

Shearlets emerged in recent years among the most successful frameworks for the efficient representation of multidimensional data. Indeed, after it was recognized that traditional multiscale methods are not very efficient at capturing edges and other anisotropic features which frequently dominate multidimensional phenomena, several methods were introduced to overcome their limitations. The shear...