Multiscale Geometric Analysis for Higher Dimensional Signal Processing

William Marsh Rice University, Houston TX

Investigators

Abstract

Over the past twenty years multiscale methods like the discrete wavelet transform (DWT) have evolutionized signal processing; for example, wavelets lie at the core of the FBI fingerprint data base and the new JPEG 2000 image compression standard for digital photos. While wavelets are arguably ideal or near-ideal for analyzing and processing 1-dimensional (1-D) signals (price fluctuations of interest rates, for example), a surprising realization of the past few years is their inability to capitalize in a similar way on 2-D, 3-D, and higher-D signals containing "singularities" (edges and ridges in 2-D digital photos or 3-D videos, for example). The new, confounding aspect is "geometry": the singularities are typically localized along smooth, lower-dimensional manifolds. There is a great need for new theory and tools to exploit these geometric structures. Other than a few scattered promising results, however, such theory and tools do not exist today. This project aims toward a unified theory and practical toolset for the analysis and processing of higher-dimensional piecewise smooth signals that feature geometric manifold structures. In particular,the research involves (1) new multiscale signal representations based on local, directional atoms having a clear geometric meaning with respect to the underlying manifolds; (2) new statistical models for these representations; (3) new multiscale processing algorithms that exploit the new representations and models; and (4) educational outreach to engage undergraduates, graduate students, and other researchers in this timely research area. Given recent successful preliminary results in 2-D and 3-D, there is significant potential for progress in these directions to eventually yield orders of magnitude better compression, approximation, modeling, and denoising performance in real applications in higher dimensions.