Nonlinear Wavelet Methods for Recovery of Signals, Densities, and Spectra from Indirect and Noisy Da

Noise reduction by constrained reconstructions in the wavelet-transform domain. Department of Mathematics, Dart-NONLINEAR WAVELET METHODS 203 Andrew Bruce and Carl Taswell for many discussions about wavelet software. The NMR datasets were provided by Chris Raphael (Figure 1) and Jee Hoch (Figure 11), the ESCA dataset by Jean-Paul Bib erian, the image dataset by Ingrid Daubechies, the seismic dataset by Paul Donoho. Many thanks to Tina Sharp for intense last-minute editorial work. Consider now a simple recursive nonlinear multiresolution scheme based on decimating by factors of 3. The ne-to-coarse mapping is obtained by grouping the signal in triplets of successive points, and replacing each group of three by a single number { the median of the group of 3. (This is a sort of nonlinear Haar analysis, since dyadic Haar wavelets correspond to grouping data in pairs and keeping only the mean of each pair.) This triadic nonlinear coarsening operator gives rise in an obvious way to a telescoping nonlinear multiresolution decomposition. Figure 28 shows the result of setting to zero the ne scale coeecients of this nonlinear triadic transform applied to the noisy data in Figure 26(b). This clearly performs much better than the linear recovery in Figure 27. Theoretical work to date on nonlinear multiresolution analysis has been done by Ron DeVore (S. Carolina) and Bradley Lucier (Purdue). Interesting applied work with mammograms has been done by Rich Richardson (Univ. of Texas at along with the author, have developed a variety of algorithms based on ideas from robust statistics. Acknowledgements. It is a pleasure to thank Iain Johnstone, G erard Kerkyacharian, and Dominique Picard, with whom much of the theory described here has been developed , and to thank Yves Meyer, Ronald Coifman, and Ingrid Daubechies for encouragement at key moments. Speciic inspirations provided by the work of Ronald DeVore and Bjj orn Jawerth are also gratefully acknowledged, as well as stimulating conversations with Albert Cohen and Bradley Lucier. Thanks to NONLINEAR WAVELET METHODS 201 Of course, in Figure 25 we are showing what happens when the wavelet transform is segmented exactly at the point of discontinuity. How are we to obtain, in analyzing noisy data, information about the proper segmentation point? Viewing the collection of segmented wavelet transforms with diierent values of t as a collection of bases B (t) , this is really a problem of selecting a best basis. Therefore we propose a best-basis segmentation SURE(y; …