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We consider the scenario where additive, independent, and identically distributed (i.i.d) noise in an image is removed using an overcomplete set of linear transforms and thresholding. Rather than the standard approach, where one obtains the denoised signal by ad hoc averaging of the denoised estimates provided by denoising with each of the transforms, we formulate the optimal combination as a conditional linear estimation problem and solve it for optimal estimates. Our approach is independent of the utilized transforms and the thresholding scheme, and as we illustrate using oracle-based denoisers, it extends established work by exploiting a separate degree of freedom that is, in general, not reachable using previous techniques. Our derivation of the optimal estimates specifically relies on the assumption that the utilized transforms provide sparse decompositions. At the same time, our work is robust as it does not require any assumptions about image statistics beyond sparsity. Unlike existing work, which tries to devise ever more sophisticated transforms and thresholding algorithms to deal with the myriad types of image singularities, our work uses basic tools to obtain very high performance on singularities by taking better advantage of the sparsity that surrounds them. With well-established transforms, we obtain results that are competitive with state-of-the-art methods.