Multiscale Analysis and Diffusion Semigroups with Applications

Abstract

Multiscale (or multiresolution) analysis is used to represent signals or functions at increasingly high resolution. In this thesis, we develop multiresolution representa- tions based on frames, which are overcomplete sets of vectors or functions that span an inner product space.

First, we explore composite frames, which generalize certain representations capable of capturing directionality in data. We show that we can obtain composite frames for L^2(R^n) given two main ingredients: 1) dilation operators based on matrices from admissible subgroups G_A and G, and 2) a generating function that is refinable with respect to G_A and G.

We also construct frame multiresolution analyses (MRA) for L^2-functions of spaces of homogeneous type. In this instance, dilations are represented by operators that come from the discretization of a compact symmetric diffusion semigroup. The eigenvectors shared by elements of the compact symmetric diffusion semigroup can be used to define an orthonormal MRA for L^2. We introduce several frame systems that yield an equivalent MRA, notably composite diffusion frames, which are built with the composition of two "similar" compact symmetric diffusion semigroups.

The last part of this thesis is an application of Laplacian Eigenmaps (LE) to a biomedical problem: Age-Related Macular Degeneration. LE, a tool in the family of diffusion methods, uses similarities at local scales to provide global analysis of data sets. We propose a novel approach with two steps. First, we apply LE to retinal images, provided by the National Institute of Health, for feature enhancement and dimensionality reduction. Then, using an original Vectorized Matched Filtering technique, we detect retinal anomalies in eigenimages produced by the LE algorithm.