Solving inverse problems for the Hammerstein integral equation and its random analog using the "collage method" for fixed points

Many inverse problems in applied mathematics can be formulated as the approximation of a target element $u$ in a complete metric space $(X,d_X)$ by the fixed point $\bar x$ of an appropriate contraction mapping $T : X \mapsto X$. The method of collage coding seeks to solve this problem by finding a contraction mapping $T$ that minimizes the so-called collage distance $d(x,Tx)$. In this paper, we develop a collage coding framework for inverse problems involving deterministic or random Hammerstein integral operators. Such operators are used to model image blurring. We illustrate the method with examples