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In this paper, we present a stochastic deconvolution method for a class of inverse problems that arc naturally formulated as group convolutions. Examples of such problems include Radon transform inversion for tomography, radar and sonar imaging, as well as channel estimation in communications. Key components of our approach are group representation theory and the concept of group stationarity. We formulate a minimum mean square solution to the deconvolution problem in the presence of nonstationary measurement noise. Our approach incorporates a priori information about the noise and the unknown signal into the inversion problem, which leads to a natural regularized solution.