Wavelet-domain approximation and compression of piecewise smooth images

• Published in: IEEE transactions on image processing 2006-05, Vol.15 (5), p.1071-1087
• Main Authors: Wakin, M.B., Romberg, J.K., Hyeokho Choi, Baraniuk, R.G.
• Format: Article
• Language: English
• Subjects: Algorithms; Application software; Applied sciences; Approximation; Asymptotic properties; Atomic structure; Coding, codes; Computer Graphics; Data Compression - methods; Detection, estimation, filtering, equalization, prediction; Edges; Exact sciences and technology; Geometry; Image coding; image compression; Image Enhancement - methods; Image Interpretation, Computer-Assisted - methods; Image processing; Image segmentation; Information, signal and communications theory; Longitudinal waves; Mathematical analysis; Noise reduction; nonlinear approximation; Numerical Analysis, Computer-Assisted; Prototypes; Rate-distortion; Representations; Signal and communications theory; Signal processing; Signal Processing, Computer-Assisted; Signal representation. Spectral analysis; Signal, noise; Solid modeling; Sparsity; Studies; Telecommunications and information theory; Wavelet; Wavelet coefficients; Wavelet transforms; wavelets; wedgelets; wedgeprints
• ISSN: 1057-7149; 1941-0042
• DOI: 10.1109/TIP.2005.864175

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 C/sup 2/ regions separated by edges along smooth C/sup 2/ contours, for example, the asymptotic rate-distortion (R-D) performance of zerotree-based wavelet coding is limited to D(R) /spl lsim/1/R, well below the optimal rate of 1/R/sup 2/. 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)/spl lsim/(logR)/sup 2//R/sup 2/ for piecewise smooth C/sup 2//C/sup 2/ 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.