The authors consider a signal, i.e. a function \(\varphi \in L^2(\mathbb R)\), and assume that only a foveated version of this signal is known, i.e.a discretized version with a very high resolution in a small region and a much lower resolution elsewhere. This situation occurs, for example, in many optical systems. The foveated image \(\psi_n^m\) is supposed to be of the form \(\psi_n^m = P_n Q_m T Q_m P_n \phi\), where \(P_n\) is a projection onto a finite dimensional subspace of \(L^2(\mathbb R)\) and \(Q_m f(x) = f(x)\) for \(| x| \leq m\) while \(Q_m f(x) = 0\) else. Finally the integral operator \(T\) is chosen to be of the form \(T f(x) = \int_{-\infty}^\infty f(t) (\beta | x| )^{-1} g((t-x)/(\beta | x| )) dt\) with a certain \(\beta > 0\) and a suitable function \(g\).
The main purpose of the paper is to give sufficient conditions on the functions and parameters involved such that it is possible to reconstruct an approximate uniform pre-image \(Q_m P_n \varphi\) of \(\varphi\) from the given foveated image \(\psi_n^m\). Moreover the authors investigate bounds for the difference between this pre-image and the original function \(\varphi\) itself.