ITS TECHNICAL REPORT NO. 31/2004 1 A Matching Pursuit Full Search Algorithm for Image Approximations Rosa M. Figueras i Ventura, Oscar Divorra Escoda and Pierre Vandergheynst Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL) Signal Processing Institute (ITS) CH-1015 Lausanne, Switzerland e-mail: {rosa.figueras,oscar.divorra,pierre.vandergheynst}@epfl.ch Technical Report No. 31/2004 Abstract There is a growing interest on adapted signal expansions for efficient sparse approximations. For this purpose, signal expansions on over-complete bases are of high interest. Several strategies exist in order to get sparse approximations of a signal as a superposition of functions from a redundant dictionary. One of these strategies is the well known Matching Pursuit (MP). MP is an algorithm where complexity depends on the accuracy of the desired approximation. This is due to its greedy iterative nature. For very large dictionaries, however, complexity depends in great manner on the size of this, i.e. at each iteration the whole dictionary has to be browsed. Sometimes, heuristic procedures need to be adopted due to the overwhelming complexity that this may represent. However, these reduce the search space and, consequently, a poorer signal approximation is retrieved. In this work, we propose a feasible approach for Full Search Matching Pursuit (FSMP) for the particular case of natural image approximations with an-isotropically refined oriented atoms (which have the purpose of exploiting image geometry). Thanks to the structure of the dictionary and its spatio-temporal localisation, several enhancements are possible to speed-up the calculation of the most critical step: the scalar product of the signal with all the functions from the dictionary. Index Terms Matching Pursuit, Full Search, Fast Algorithm, FFT, Sparse Approximations, Redundant Dictionaries, Anisotropic Refinement Atoms, Edges, Image Geometry, Memory Compression, Steerability, Complexity Optimization. Contents I Introduction 2 I-A Image Model ................................................ 2 I-B Dictionary of basis functions ....................................... 2 I-C Purpose of this Work ........................................... 3 II Matching Pursuit 3 III Genetic Algorithm based MP: a Weak Matching Pursuit Implementation 4 IV Full Search MP 5 IV-A “Brute Force” Full Search MP ...................................... 5 IV-B Spatial Invariance in Scalar Product Computations and Boundary Renormalization ........ 6 IV-C FFT Based Full Search MP: From Scalar Products to Spatial Convolution ............. 7 IV-D Results: Full Search vs Genetic Algorithm Search ........................... 8 V Exploiting the dictionary features 10 V-A Exploiting Spatio-Temporal Energy Localization: Compact Support and Atoms Approximation . 10 V-A.1 Memory Compression ..................................... 12 V-A.2 Examples ............................................ 12 V-B Steerability of Atoms and Complexity Benefits ............................. 12 VI Conclusions 14 Web page: http://lts2www.epfl.ch This report concerns the work done during automn 2002 for the developement of a full search Matching Pursuit algorithm using AR atoms for image approximations. The work of Oscar Divorra Escoda was sponsored by Swiss Federal Office for Education and Technology grant number 6044.1 KTS.