Weighted overcomplete denoising

We consider the familiar scenario where independent and identically distributed (i.i.d) noise in an image is removed using a set of overcomplete linear transforms and thresholding. Rather than the standard approach where one obtains the denoised signal by ad hoc averaging of the denoised estimates (corresponding to each transform), we formulate the optimal combination as a linear estimation problem for each pixel and solve it for optimal estimates. Our approach is independent of the utilized transforms and the thresholding scheme, and extends established work by exploiting a separate degree of freedom that is in general not reachable using previous techniques. Surprisingly, our derivation of the optimal estimates does not require explicit image statistics but relies solely on the assumption that the utilized transforms provide sparse decompositions. Yet it can be seen that our adaptive estimates utilize implicit conditional statistics and they make the biggest impact around edges and singularities where standard sparsity assumptions fail.