DEAUTOCONVOLUTION IN THE TWO-DIMENSIONAL CASE

There is extensive mathematical literature on the inverse problem of deautoconvolution for a function with support in the unit interval [0, 1] subset of I8, but little is known about the multidimensional situation. This article tries to fill this gap with analytical and numerical studies on the reconstruction of a real function of two real variables over the unit square from observations of its autoconvolution on [0, 2]2 subset of I82 (full data case) or on [0, 1]2 (limited data case). In an L2-setting, twofoldness and uniqueness assertions are proven for the deautoconvolution problem in 2D. Moreover, its ill-posedness is characterized and illustrated. Extensive numerical case studies give an overview of the behaviour of stable approximate solutions to the two-dimensional deautoconvolution problem obtained by Tikhonov-type regularization with different penalties and the iteratively regularized Gauss-Newton method.