Representation and compression of multidimensional piecewise functions using surflets
Author(s)Chandrasekaran, Venkat; Wakin, Michael B.; Baron, Dror; Baraniuk, Richard G.
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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.
DepartmentMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
IEEE Transactions on Information Theory
Institute of Electrical and Electronics Engineers
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
Final published version
wavelets, surflets, sparse representations, rate–distortion, nonlinear approximation, multiscale representations, multidimensional signals, metric entropy, discontinuities, compression