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Shift invariant restoration-an overcomplete maxent MAP framework

Translation-invariant denoising was introduced by Coifman and Donoho (1995) to overcome Gibbs-type phenomena produced by transform-domain shrinkage estimators in the vicinity of signal discontinuities. Shrinkage estimators are in general not shift-invariant. Shift-invariant denoising consists of a simple averaging of the shrinkage estimates over a family of cyclic spatial-shifts of the image. Shift-invariant denoising is denoising in an overcomplete basis, and work in this area has been devoted towards finding a best basis in the overcomplete family. This paper presents a maximum a posteriori (MAP) framework for shift-invariant restoration of images using the maximum-entropy prior consistent with moment constraints on the transform coefficients in different subbands. The simple averaging of estimates in the classical shift-invariant denoising can then be shown to be a certain limiting case within this framework.