  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.
