Deautoconvolution in the two-dimensional case
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There is extensive mathematical literature on the inverse problem of deautoconvolution for a function with support in the unit interval [0,1] ⊂ℝ, 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 ⊂ℝ^2 (full data case) or on [0,1]^2 (limited data case). In an L^2-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.
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