Stochastic deconvolution over groups

In this paper, we address a class of inverse problems that are formulated as group convolutions. This is a rich area of research with applications to Radon transform inversion for tomography, wide-band and narrow-band signal processing, inverse rendering in computer graphics, and channel estimation in communications, as well as robotics and polymer science. We present a group-theoretic framework for signal modeling and analysis for such problems and propose a minimum mean-square error (MMSE) deconvolution method in a probabilistic setting. Key components of our approach are group representation theory and the concept of group stationarity. The proposed deconvolution method incorporates a priori information and noise statistics into the inversion process, which leads to a natural regularized solution. We present recovery of self-similar processes that are "blurred" and embedded in noise as a demonstration example. The method is applicable to a wide range of inverse problems involving both commutative and noncommutative groups including finite, compact, and majority of well-behaved locally compact groups.