<title>Hierarchical parallel matching pursuit</title>

Abstract

The by now well-known matching pursuit method of S. Mallat as well as the recently proposed orthogonal matching pursuit work purely sequential and are based on the idea of choosing only one single atom at a given time. Pursuing ideas which are related to modifications of the POCS method, we suggest a new type of orthogonalization procedure which allows to operate in parallel at different 'levels'. More precisely, we assume that we have a dictionary which consists of similar 'pages', i.e. those pages are collection of functions, generated from a single function by translations along a subgroup which is the same for all such pages. Based on a certain hierarchical structure (preference of pages) we apply an appropriate Gram-Schmidt type orthogonalization procedure which allows to deal with the matching pursuit problem in parallel at different levels. After carrying out appropriate approximations at the different, now orthogonal levels, one comes back to a representation based on the given family of atoms, in a straight-forward way.