Top-Down and Bottom-Up Tree Search Algorithms for Selecting Bases in Wavelet Packet Transforms
Search algorithms for finding signal decompositions called near-best bases using decision criteria called non-additive information costs have recently been proposed by Taswell  for selecting bases in wavelet packet transforms represented as binary trees. These methods are extended here to distinguish between top-down and bottom-up tree searches. Other new non-additive information cost functions are also proposed. In particular, the near-best basis with the non-additive cost of the Shannon entropy on probabilities is compared against the best basis with the additive cost of the Coifman-Wickerhauser entropy on energies . All wavelet packet basis decompositions are also compared with the nonorthogonal matching pursuit decomposition of Mallat and Zhang  and the orthogonal matching pursuit decomposition of Patiet al . Monte Carlo experiments using a constant-bit-rate variable-distortion paradigm for lossy compression suggest that the statistical performance of top-down near-best bases with non-additive costs is superior to that of bottom-up best bases with additive costs. Top-down near-best bases provide a significant increase in computational efficiency with reductions in memory, flops, and time while nevertheless maintaining similar coding efficiency with comparable re-construction errors measured by ℓ p -norms. Finally, a new compression scheme called parameterized model coding is introduced and demonstrated with results showing better compression than standard scalar quantization coding at comparable levels of distortion.
KeywordsEntropy Expense Sorting Terrell
Unable to display preview. Download preview PDF.
- Joseph Berkson. Relative precision of minimum chi-square and maximum likelihood estimates of regression coefficients. In Jerzy Neyman, editor,Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pages 471–479. University of California Press, 1951.Google Scholar
- Y.C. Pati, R. Rezaiifar, and P. S. Krishnaprasad. Orthogonal matching pursuit: Recursive function approximation with application to wavelet decomposition. InProceedings of the 27th Annual Asilomar Conference on Signals Systems and Computers, pages 40–44, Pacific Grove, CA, November 1993.CrossRefGoogle Scholar
- David A. Ratkowsky.Handbook of Nonlinear Regression Models, volume 107 ofStatistics: Textbooks and Monographs. Marcel Dekker, Inc., New York, NY, 1990.Google Scholar
- David W. Scott.Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley Series in Applied Probability and Statistics. John Wiley & Sons, Inc., New York, 1992.Google Scholar
- Carl Taswell. Near-best basis selection algorithms with non-additive information cost functions. In Moeness G. Amin, editor,Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis, pages 13–16, Philadelphia, PA, October 1994. IEEE Press 94TH8007.Google Scholar
- Carl Taswell. WavBox 4: A software toolbox for wavelet transforms and adaptive wavelet packet decompositions. In Anestis Antoniadis and Georges Oppenheim, editors,Wavelets and Statistics, Lecture Notes in Statistics. Springer Verlag, 1995. Proceedings of the Villard de Lans Conference November 1994.Google Scholar
- Mladen Victor Wickerhauser . INRIA lectures on wavelet packet algorithms. Technical report, INRIA, Roquencourt, France, 1991. minicourse lecture notes.Google Scholar