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

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