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Optimal Denoising in Redundant Bases

Image denoising methods are often based on estimators chosen to minimize mean squared error (MSE) within the sub-bands of a multi-scale decomposition. But this does not guarantee optimal MSE performance in the image domain, unless the decomposition is orthonormal. We prove that despite this suboptimality, the expected image-domain MSE resulting from a representation that is made redundant through spatial replication of basis functions (e.g., cycle-spinning) is less than or equal to that resulting from the original non-redundant representation. We also develop an extension of Stein's unbiased risk estimator (SURE) that allows minimization of the image-domain MSE for estimators that operate on subbands of a redundant decomposition. We implement an example, jointly optimizing the parameters of scalar estimators applied to each subband of an overcomplete representation, and demonstrate substantial MSE improvement over the sub-optimal application of SURE within individual subbands.