SIGGRAPH '96 Proceedings of the 23rd annual conference on Computer graphics and interactive techniques
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
Data compression and harmonic analysis
IEEE Transactions on Information Theory
On denoising and best signal representation
IEEE Transactions on Information Theory
On the importance of combining wavelet-based nonlinear approximation with coding strategies
IEEE Transactions on Information Theory
De-noising by soft-thresholding
IEEE Transactions on Information Theory
The context-tree weighting method: basic properties
IEEE Transactions on Information Theory
Space-frequency quantization for wavelet image coding
IEEE Transactions on Image Processing
Wavelet-domain approximation and compression of piecewise smooth images
IEEE Transactions on Image Processing
Multidimensional Directional Filter Banks and Surfacelets
IEEE Transactions on Image Processing
3-D Data Denoising and Inpainting with the Low-Redundancy Fast Curvelet Transform
Journal of Mathematical Imaging and Vision
| Hi-index | 754.87 |
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.