  There is extensive mathematical literature on the inverse problem of
deautoconvolution for a function with support in the unit interval $[0,1]
\subset \mathbb R$, 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 $[0,2]^2 \subset \mathbb
R^2$ (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.
