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This paper considers the problem of registering two observations of the same object, where the observations differ due to a combined effect of an affine geometric transformation and nonuniform illumination changes. The problem of deriving new representations of the observations that are both invariant to geometric transformations and linear in the illumination model is analyzed. In this framework, we present a novel method for linear estimation of illumination changes in an affine invariant manner, thus, decoupling the original problem into two simpler ones. The computational complexity of the method is low as it requires no more than solving a linear set of equations. The prior step of illumination estimation is shown to improve the accuracy of state-of-the-art registration techniques by a factor of two.