Deautoconvolution in the two-dimensional case

TL;DR

This paper extends the mathematical understanding of the deautoconvolution inverse problem from one dimension to two dimensions, providing theoretical results and numerical methods for reconstructing functions from their autoconvolution data.

Contribution

It offers the first analytical and numerical study of 2D deautoconvolution, including uniqueness, ill-posedness, and regularization techniques.

Findings

Proven twofoldness and uniqueness in 2D deautoconvolution.

Characterized the ill-posedness of the 2D problem.

Numerical experiments demonstrate the effectiveness of regularization methods.

Abstract

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.