The Mehler-Fock Transform and some Applications in Texture Analysis and Color Processing
This work provides a theoretical framework for signal processing on non-Euclidean spaces, which is incremental as it extends existing harmonic analysis methods to specific groups like SU(1,1).
The paper tackles the problem of analyzing probability density functions on geometric objects like spheres and cones by applying harmonic analysis tools, specifically the Mehler-Fock transform on the Lorentz group and unit disk, and demonstrates its utility in signal processing with examples from texture analysis and color image processing.
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.