Sparse Approximation of Images Inspired from the Functional Architecture of the Primary Visual Areas
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Research Article Sparse Approximation of Images Inspired from the Functional Architecture of the Primary Visual Areas 1 ´ Sylvain Fischer,1, 2 Rafael Redondo,1 Laurent Perrinet,2 and Gabriel Cristobal
´ de Optica - CSIC, Serrano 121, 28006 Madrid, Spain UMR 6193, CNRS and Aix-Marseille University, 31 chemin Joseph Aiguier, 13402 Marseille Cedex 20, France
1 Instituto 2 INCM,
Received 1 December 2005; Revised 7 September 2006; Accepted 18 September 2006 Recommended by Javier Portilla Several drawbacks of critically sampled wavelets can be solved by overcomplete multiresolution transforms and sparse approximation algorithms. Facing the difficulty to optimize such nonorthogonal and nonlinear transforms, we implement a sparse approximation scheme inspired from the functional architecture of the primary visual cortex. The scheme models simple and complex cell receptive fields through log-Gabor wavelets. The model also incorporates inhibition and facilitation interactions between neighboring cells. Functionally these interactions allow to extract edges and ridges, providing an edge-based approximation of the visual information. The edge coefficients are shown sufficient for closely reconstructing the images, while contour representations by means of chains of edges reduce the information redundancy for approaching image compression. Additionally, the ability to segregate the edges from the noise is employed for image restoration. Copyright © 2007 Sylvain Fischer et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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INTRODUCTION
Recent works on multiresolution transforms showed the necessity of using overcomplete transformations to solve drawbacks of (bi-)orthogonal wavelets, namely their lack of shift invariance, the aliasing between subbands, their poor resolution in orientation and their insufficient match with image features [1–4]. Nevertheless the representations from linear overcomplete transforms are highly redundant and consequently inefficient for such tasks needing sparseness as, for example, for image compression. Several sparse approximation algorithms have been proposed to address this problem by approximating the images through a reduced number of decomposition functions chosen in an overcomplete set called dictionary [5–8] (see reviews in [6, 9]). In some very particular cases there exist algorithms achieving the optimal solutions. In the general case, two main classes of algorithms are available: matching pursuit (MP) [5, 10] which recursively chooses the most relevant coefficients in all the dictionary and basis pursuit (BP) [6] which minimizes a penalizing function corresponding to the sum of the amplitude of all coefficients. Both these algorithms perform iteratively and globally through all the dictionary. They are computationally costly algorithms which generally only achieve approximations of the optimal solutions.
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