Isotropic Multiresolution Analysis and Rotational Invariance in Image Analysis

In this thesis, we study the theory of Isotropic Multiresolution Analysis (IMRA) and its application to image analysis problems that require rotational invariance. Multiresolution Analysis (MRA) is a mathematical tool that gives us the ability to process a signal or an image at multiple levels of resolution and detail. IMRA is a new type of MRA for which the core resolution sub-space is invariant under all rotations. We give a characterization of IMRAs using the Lax-Wiener theorem, which shows that all the resolution and detail spaces of an IMRA are invariant under all rigid motions. We further develop examples of isotropic wavelet frames associated to IMRA via the Extension Principles. This facilitates the fast implementation of isotropic wavelet decomposition and reconstruction algorithms. We derive an IMRA-based explicit scheme for the numerical solution of the acoustic wave equation in the context of seismic migration. The multiscale structure of the IMRA offers the possibility of improving the computational efficiency of the standard explicit schemes used in seismic imaging. We develop a novel rotationally invariant three-dimensional texture classification scheme using Gaussian Markov Random Fields on Z3 to model textures sampled on a discrete lattice. These are considered to be sampled versions of continuous textures, which are viewed as realizations of stationary Gaussian random fields on R3. IMRA is used to bridge the gap between the discrete and the continuous domains, where the rotation invariance of the resolution spaces plays a key role.