We consider the problem of registering two observations on an arbitrary object, where the two are related by a geometric affine transformation of their coordinate systems, and by a non-linear mapping of their intensities. More generally, the framework is that of jointly estimating the geometric and radiometric deformations relating two observations on the same object. We show that the original high-dimensional, non-linear, non-convex search problem of simultaneously recovering the geometric and radiometric deformations can be represented by an equivalent sequence of two linear systems. A solution of this sequence yields an exact, explicit, and efficient solution to the joint estimation problem.