Summary: We speed up the screening of a computer image compressed as an iterated function system (IFS) of affine transformations \(w_1,\dots, w_N\). The idea is to avoid applying a given map to a given point if nothing new would be obtained. Let \(\underline w(x, y)\) be the result of applying transformation \(w\) to pixel coordinates \((x, y)\) then rounding to the nearest integer pair, say \((u, v)\). Then the \(w\)-jigsaw pieces \(\underline w^{-1}(u, v)\) tile the plane. We develop a foundation of results about such jigsaws based on the coefficients defining the transformations \(w\), and apply them to a collection of 50 IFSs. Much time is indeed saved and the method is to be pursued further.