Stochastic Deconvolution Over Groups

Abstract

Abstract-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 roposed 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.

Decompounding on Compact Lie Groups

“…Our presentation of characteristic functions is adapted from [5], [12]. Harmonic analysis on compact Lie groups is presented in more detail in recent papers [8], [7]. More thorough classical references thereon include [13], [14].…”

Stationary Random Fields Arising From Second-Order Partial Differential Equations on Compact Lie Groups

“…If is a so-called type I separable topological group, then a generalization of Bochner's theorem gives rise to an operator-valued measure which plays the role of spectral power distribution. This is the main fact used by Yaglom and it is also central in the signal-processing techniques proposed in [5].…”

“…WSS random fields on Lie groups, or even topological groups in general, were studied in [5]. They were shown to appear in a variety of applications and to admit useful generalizations of common signal-processing techniques.…”

An algebraic approach to multiresolution analysis

“…Other papers that mention or pursue some representationtheoretic aspects of multiresolution analysis include [3], [6], [8], [9], [11], [12], [13], [21], [22], [26], [27], but these complement rather than overlap substantially with this paper. The work of Chirikjian and his collaborators, [5], [15], uses the tools of harmonic analysis on the affine group and motion groups in various pattern recognition problems.…”