Learning sparse representations on the sphere

¹ Laboratoire AIM, CEA, CNRS, Université Paris-Saclay, Université Paris Diderot, Sorbonne Paris Cité, 91191 Gif-sur-Yvette, France
e-mail: florent.sureau@cea.fr
² Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany
³ Lehrstuhl für Wissenschaftliches Rechnen, Katholische Universität Eichstätt-Ingolstadt, Ostenstraße 26, 85072 Eichstätt, Germany

Received: 8 August 2018
Accepted: 27 October 2018

Abstract

Many representation systems on the sphere have been proposed in the past, such as spherical harmonics, wavelets, or curvelets. Each of these data representations is designed to extract a specific set of features, and choosing the best fixed representation system for a given scientific application is challenging. In this paper, we show that one can directly learn a representation system from given data on the sphere. We propose two new adaptive approaches: the first is a (potentially multiscale) patch-based dictionary learning approach, and the second consists in selecting a representation from among a parametrized family of representations, the α-shearlets. We investigate their relative performance to represent and denoise complex structures on different astrophysical data sets on the sphere.