Journal PaperWe 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 demonstrate that surflets
provide superior compression performance when compared to other
state-of-the-art approximation schemes
Is data on this page outdated, violates copyrights or anything else? Report the problem now and we will take corresponding actions after reviewing your request.