Image Compression with Geometrical Wavelets

We introduce a sparse image representation that takes advantage 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 reconstruct singularities. Foveal wavelets are extended in two dimensions, 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 image coding algorithm is described. Edges are coded with quantized bandelet coefficients, and a smooth residual image is coded in a standard two-dimensional wavelet basis. 1. GEOMETRICAL COMPRESSION Currently, the most efficient image transform codes are obtained in orthonormal wavelet bases. For a given distortion associated to a quantizer, at high compression rates the bit budget is proportional to the number of non-zero quantized coefficients [1]. 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 advantage 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 optimize the distortion rate. In this paper, we construct “bandelet” orthonormal bases that carry all the edge information 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 singularities 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 Space Industries grant and a DARPAFastVDO grant 25-74100-F0945 of a signal. We define two mother wavelets Ψ(t) and Ψ(t), which are respectively antisymmetric and symmetric with respect to t = 0, and such that ∫ Ψ(t)dt = 0 for k = {1, 2}. For any location u we denote Ψj,u(t) = 2 −j/2 Ψ(2(t− u)) for k = 1, 2. There exists such mother wavelets, which are C and such that for any u ∈ R and J ∈ Z, the family {Ψj,u(t)}−∞<j≤J , k∈{1,2} is orthonormal [7]. These wavelets zoom on a single position u and are thus called foveal wavelets, by analogy with the foveal vision. To reconstruct discontinuities, we insure that left and right indicator functions, 1[u,+∞) and 1(−∞,u] can be written as linear combinations of foveal wavelets. This is the case for the mother wavelets shown in Figure 1. Foveal wavelets of larger support, which also reconstruct discontinuities of higher derivatives are constructed in [7], but will not be used here. Foveal wavelet families are easily discretized while retaining their orthogonality properties. The scale parameter 2 is then limited by the resolution of the signal measurement. −2 −1 0 1 2 −1 −0.5 0 0.5 1 −2 −1 0 1 2 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 Fig. 1. Foveal mother wavelets Ψ and Ψ Let Vu be the space generated by the foveal family located at u. The orthogonal projection of f in Vu is