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.
