Wavelet-domain approximation and compression of piecewise smooth images
Abstract
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.
Additional Information
© 2006 IEEE. Reprinted with permission. Manuscript received November 22, 2004; revised April 6, 2005. [Posted online: 2006-04-18] This work was supported in part by a National Science Foundation (NSF) Graduate Research Fellowship, NSF Grant CCR-9973188, Office of Naval Research Grant N00014-02-1-0353, Air Force Office of Scientific Research Grant F49620-01-1-0378, and the Texas Instruments Leadership University Program. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Amir Said. While this paper was in press, Hyeokho Choi passed away. We will forever remember his broad vision, his keen insights, and our lively discussions. His legacy will live on through his many contributions to the signal processing community. The authors would like to thank A. Cohen, R. DeVore, and M. Orchard for many helpful discussions and the anonymous reviewers for helpful comments.Attached Files
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Additional details
- Eprint ID
- 4005
- Resolver ID
- CaltechAUTHORS:WAKieeetip06
- NSF Graduate Research Fellowship
- CCR-9973188
- NSF
- N00014-02-1-0353
- Office of Naval Research (ONR)
- F49620-01-1-0378
- Air Force Office of Scientific Research (AFOSR)
- Texas Instruments
- Created
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2006-07-24Created from EPrint's datestamp field
- Updated
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2021-11-08Created from EPrint's last_modified field