Wavelet-domain approximation and compression of piecewise smooth images

The wavelet transform provides a sparse representation for smooth images, enabling efficient approximation and compression using techniques such as zerotrees. Unfortunately, this sparsity does not extend to piecewise smooth images, where edge discontinuities separating smooth regions persist along smooth contours. This lack of sparsity hampers the efficiency of wavelet-based approximation and compression. On the class of images containing smooth C² regions separated by edges along smooth C² contours, for example, the asymptotic rate-distortion (R-D) performance of zerotree-based wavelet coding is limited to D(R) ≲1/R, well below the optimal rate of 1/R². In this paper, we develop a geometric modeling framework for wavelets that addresses this shortcoming. The framework can be interpreted either as 1) an extension to the "zerotree model" for wavelet coefficients that explicitly accounts for edge structure at fine scales, or as 2) a new atomic representation that synthesizes images using a sparse combination of wavelets and wedgeprints-anisotropic atoms that are adapted to edge singularities. Our approach enables a new type of quadtree pruning for piecewise smooth images, using zerotrees in uniformly smooth regions and wedgeprints in regions containing geometry. Using this framework, we develop a prototype image coder that has near-optimal asymptotic R-D performance D(R)≲(logR)²/R² for piecewise smooth C²/C² images. In addition, we extend the algorithm to compress natural images, exploring the practical problems that arise and attaining promising results in terms of mean-square error and visual quality.