Adapted Wavelet Algorithms

Washington University, Saint Louis MO

Investigators

Abstract

ADAPTED WAVELET ALGORITHMS NSF DMS 0072234 Mladen Victor Wickerhauser Department of Mathematics Washington University in St. Louis Using wavelet packets and other signal-adapted waveforms, we build cheap and efficient `best basis' representations for complicated digital signals and images. Used on very large datasets, these reduce the complexity of processing, transmitting, and storing data. Some near-term applications suggested by our prior work are: - fast approximate singular value decomposition, and fast approximate discrete Hilbert transforms; - signal segmentation through local spectrum change detection; - fast discrete atomic decomposition for feature detection and adapted data compression. The long-term goals of this research program are: - understanding, controlling, and constructing signal-adapted waveforms; - high-performance computing with adapted waveforms. We can achieve these goals through a deeper understanding of some discoveries from our prior supported research. We must know, using certain signal-adapted representations, - can we measure information content more accurately, and thus reduce the storage size of a signal or image? -can we lower the fundamental limit of automatic detection of signal features? - what algorithm modifications are needed to obtain adapted waveforms with additional desirable properties? Our work is used to automate or improve tasks such as: segmentation of continuous speech into syllables for speech recognition; analysis and compression of seismic petroleum exploration data; fingerprint image compression that preserves features needed for automatic identification; de-noising and preconditioning of radar signals before automatic feature detection and classification; and speedups of certain basic matrix computations. Our methods include: devising new high-performance algorithms to compute certain quantities; running numerical experiments and simulations to evaluate their range of usefulness; and proving mathematical guarantees for these algorithms such as maximums for running times with minimums for accuracy.