We propose two methods for sparse approximation of images under l₂ error metric. First one performs an approximation error minimization given a l_p-norm of the representation through alternated orthogonal projections onto two sets. We study the cases p = 0 (sub-optimal) and p = 1 (optimal), and find that the l₀-AP method is neatly superior, for typical images and overcomplete oriented pyramids. Given that l₁-AP is optimal, this shows that it is not equivalent in practical image processing conditions to minimize one or the other norm, contrarily to what is often assumed. The second method is more powerful, and it performs gradient descent onto decreasingly smoothed versions of the sparse approximation cost function, yielding a method previously proposed as a heuristic. We adapt these techniques for being applied to image restoration, with very positive results.

Date Published: 20 September 2007
PDF: 15 pages
Proc. SPIE 6701, Wavelets XII, 67011Z (20 September 2007); doi: 10.1117/12.736231

Published in SPIE Proceedings Vol. 6701:
Wavelets XII
Dimitri Van De Ville; Vivek K. Goyal; Manos Papadakis, Editor(s)