Quantitative image recovery theorems (Q2451917)

Let \(H\) be a real Hilbert space and let \(C_1,C_2,\dots,C_r\) be nonempty closed convex subsets of \(H\). The problem of image recovery in this context is to find the original (unknown) image \(z\) which is a priori known to belong to \(C_0=\cap_{i=1}^{r} C_i\), the so-called feasibility set for the problem (or the class of all images that satisfy the recovery criterion). By an appropriate iterative scheme, involving the metric projections \(P_i: H\rightarrow C_i\), in which some initial estimate is sequentially projected onto the individual sets according to a periodic schedule, one can recover some \(z_0\in C_0\). In the context of the image recovery problem, a family of (nonexpansive) mappings is defined as the averaged map of \(P_i\): \[ T_i:=I+\lambda_i (P_i-I),\;i=1,2,\dots,r, \] where \(I\) is the identity map. It is known that \(\cap_{i=1}^{r} C_i=\cap_{i=1}^{r} \mathrm{Fix}\,(T_i)\), where \(\mathrm{Fix}\,(T_i)\) stands for the set of fixed points of \(T_i\). Starting from the fact that in the classical image recovery problem there is no information on how a \(\delta\)-fixed point \(\cap_{i=1}^{r} C_i\) relates to being in the intersection \(C_{0,\varepsilon}\) of \(\varepsilon\)-neighborhoods \(C_{i,\varepsilon}\) of \(C_i\), the authors introduce an \(\varepsilon\)-version of the classical image recovery problem. This approach thus provides an approximate solution of the problem even in the case of inconsistency.