Wavelet-domain approximation and compression of piecewise smooth images

Michael B. Wakin, Justin K. Romberg, Hyeokho Choi, Richard G. Baraniuk

Research output: Contribution to journalArticlepeer-review

64 Scopus citations


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)lesssim ≲ 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)≲(log R)2/R2 for piecewise smooth C2/ C2 algorithm to compress natural images, exploring the practical problems that arise and attaining promising results in terms of mean-square error and visual quality.

Original languageEnglish (US)
Pages (from-to)1071-1087
Number of pages17
JournalIEEE Transactions on Image Processing
Issue number5
StatePublished - May 2006


  • Edges
  • Image compression
  • Nonlinear approximation
  • Rate-distortion
  • Wavelets
  • Wedgelets
  • Wedgeprints

ASJC Scopus subject areas

  • Electrical and Electronic Engineering
  • Computer Graphics and Computer-Aided Design
  • Software
  • Theoretical Computer Science
  • Computational Theory and Mathematics
  • Computer Vision and Pattern Recognition


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