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IMAGE COMPRESSION WITH GEOMETRICAL WAVELETS Erwan Le Pennec Ecole Polytechnique Centre de MathCmatiques AppliquCes 91 128 Palaiseau Cedex France ABSTRACT We introduce a sparse image representation that takes ad- vantage of the geometrical regularity of edges in images. A new class of one-dimensional wavelet orthonormal bases, called foveal wavelets, are introduced to detect and recon- struct singularities. Foveal wavelets are extended in two di- mensions, to follow the geometry of arbitrary curves. The resulting two dimensional “bandelets” define orthonormal families that can restore close approximations of regular edges with few non-zero coefficients. A double layer im- age coding algorithm is described. Edges are coded with quantized bandelet coefficients, and a smooth residual im- age is coded in a standard two-dimensional wavelet basis. 1. GEOMETRICAL COMPRESSION Currently, the most efficient image transform codes are ob- tained in orthonormal wavelet bases. For a given distor- tion associated to a quantizer, at high compression rates the bit budget is proportional to the number of non-zero quan- tized coefficients [ 11. For images decomposed in wavelet orthonormal bases, these non-zero coefficients are created by singularities and contours. When the contours are along regular curves, this bit budget can be reduced by taking ad- vantage of this regularity [2]. Many image compression with edge coding have already been proposed [3,4,5,6], but they rely on ad-hoc algorithms to represent the edge information, which makes it difficult to compute and opti- mize the distortion rate. In this paper, we construct “ban- delet” orthonormal bases that carry all the edge informa- tion and take advantage of their regularity by concentrating their energy over few coefficients. An application to image compression is studied. 2. FOVEAL WAVELET BASES Contours are considered here as one-dimensional singular- ities that move in the image plane. We first construct a new family of orthonormal wavelets, all centered as the same location, which can “absorb” the singular behavior ‘Support in parts by an Alcatel-Espace grant and a DARPA-Fastvideo grant 25-741OO-Fo945 Stkphane Mallat* Ecole Polytechnique Centre de MathCmatiques AppliquCes New York University Courant Institute of Mathematical Sciences of a signal. We define two mother wavelets Q’(t) and Q2 (t), which are respectively antisymmetric and symmet- ric with respect to t = 0, and such that J @(t)dt = 0 for IC = { 1,2}. For any location U we denote Q? 39u (t) = 2-j/’ qk(2-j(t - U)) for L = 1,2. There exists such mother wavelets, which are C’ and such that for any U E R and J E Z, the family I is orthonormal [7]. These wavelets zoom on a single posi- tion U and are thus called foveal wavelets, by analogy with the foveal vision. To reconstruct discontinuities, we insure that left and right indicator functions, l[u,+m) and l(-m,u~ can be written as linear combinations of foveal wavelets.
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This note was uploaded on 05/28/2010 for the course EE EE564 taught by Professor Runyiyu during the Spring '10 term at Eastern Mediterranean University.

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