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Wavelet-domain approximation and compression of piecewise smooth images

Wakin, Michael B. and Romberg, Justin K. and Choi, Hyeokho and Baraniuk, Richard G. (2006) Wavelet-domain approximation and compression of piecewise smooth images. IEEE Transactions on Image Processing, 15 (5). pp. 1071-1087. ISSN 1057-7149. https://resolver.caltech.edu/CaltechAUTHORS:WAKieeetip06

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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.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1109/TIP.2005.864175DOIUNSPECIFIED
ORCID:
AuthorORCID
Baraniuk, Richard G.0000-0002-0721-8999
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.
Funders:
Funding AgencyGrant Number
NSF Graduate Research FellowshipUNSPECIFIED
NSFCCR-9973188
Office of Naval Research (ONR)N00014-02-1-0353
Air Force Office of Scientific Research (AFOSR)F49620-01-1-0378
Texas InstrumentsUNSPECIFIED
Subject Keywords:Edges, image compression, nonlinear approximation, rate-distortion, wavelets, wedgelets, wedgeprints
Issue or Number:5
Record Number:CaltechAUTHORS:WAKieeetip06
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:WAKieeetip06
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:4005
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:24 Jul 2006
Last Modified:02 Oct 2019 23:08

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