The Mehler-Fock Transform and some Applications in Texture Analysis and Color Processing

TL;DR

This paper explores the Mehler-Fock transform's mathematical properties and demonstrates its applications in texture analysis and color processing within the context of harmonic analysis on hyperbolic geometries.

Contribution

It introduces the Mehler-Fock transform in the context of harmonic analysis on Lie groups and applies it to signal processing tasks in vision and color image analysis.

Findings

The Mehler-Fock transform relates to group convolutions and Laplace-Beltrami operator.

It provides an isometric mapping between signal and transform spaces.

Applications demonstrated in low-level vision and color image processing.

Abstract

Many stochastic processes are defined on special geometrical objects like spheres and cones. We describe how tools from harmonic analysis, i.e. Fourier analysis on groups, can be used to investigate probability density functions (pdfs) on groups and homogeneous spaces. We consider the special case of the Lorentz group SU(1,1) and the unit disk with its hyperbolic geometry, but the procedure can be generalized to a much wider class of Lie-groups. We mainly concentrate on the Mehler-Fock transform which is the radial part of the Fourier transform on the disk. Some of the characteristic features of this transform are the relation to group-convolutions, the isometry between signal and transform space, the relation to the Laplace-Beltrami operator and the relation to group representation theory. We will give an overview over these properties and their applications in signal processing. We will illustrate the theory with two examples from low-level vision and color image processing.