Representation and compression of multidimensional piecewise functions using surflets
Name
Chandrasekaran_Representation-and-Compression.pdf
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1.13 MB
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Adobe PDF
Checksum (MD5)
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Author(s)
Chandrasekaran, Venkat
Wakin, Michael B.
Baron, Dror
Baraniuk, Richard G.
Date Issued
December 2008
Journal
IEEE Transactions on Information Theory
Publisher
Institute of Electrical and Electronics Engineers
Citation
Chandrasekaran, V. et al. “Representation and Compression of Multidimensional Piecewise Functions Using Surflets.” Information Theory, IEEE Transactions on 55.1 (2009): 374-400. © 2008 Institute of Electrical and Electronics Engineers
Version
Final published version
Abstract
We study the representation, approximation, and compression of functions in M dimensions that consist of constant or smooth regions separated by smooth (M-1)-dimensional discontinuities. Examples include images containing edges, video sequences of moving objects, and seismic data containing geological horizons. For both function classes, we derive the optimal asymptotic approximation and compression rates based on Kolmogorov metric entropy. For piecewise constant functions, we develop a multiresolution predictive coder that achieves the optimal rate-distortion performance; for piecewise smooth functions, our coder has near-optimal rate-distortion performance. Our coder for piecewise constant functions employs surflets, a new multiscale geometric tiling consisting of M-dimensional piecewise constant atoms containing polynomial discontinuities. Our coder for piecewise smooth functions uses surfprints, which wed surflets to wavelets for piecewise smooth approximation. Both of these schemes achieve the optimal asymptotic approximation performance. Key features of our algorithms are that they carefully control the potential growth in surflet parameters at higher smoothness and do not require explicit estimation of the discontinuity. We also extend our results to the corresponding discrete function spaces for sampled data. We provide asymptotic performance results for both discrete function spaces and relate this asymptotic performance to the sampling rate and smoothness orders of the underlying functions and discontinuities. For approximation of discrete data, we propose a new scale-adaptive dictionary that contains few elements at coarse and fine scales, but many elements at medium scales. Simulation results on synthetic signals provide a comparison between surflet-based coders and previously studied approximation schemes based on wedgelets and wavelets.
Subjects
wavelets
surflets
sparse representations
rate–distortion
nonlinear approximation
multiscale representations
multidimensional signals
metric entropy
discontinuities
compression
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
http://dx.doi.org/10.1109/TIT.2008.2008153
