Reflections on the 2006 IEEE Joint Communications / Information Theory Paper Award

Suppose the binary images shown in Figure 1 were transmitted over a binary symmetric channel (BSC). What could be done to clean up or denoise the resulting noisy images? A quick scan of the image processing literature suggests the use of such algorithms as median filtering and morphological filtering. In the binary case a median filter replaces each pixel in the noisy image by the color of the majority of pixels in a neighborhood about that pixel, while a morphological filter carries out dilation and erosion operations to denoise a pixel, again based on the values of nearby pixels. Indeed, many of the simplest and most practical denoising algorithms in the image processing literature are of this form. They denoise any given pixel by applying a function to the pixel values in a nearby neighborhood. The denoising scenario above, however, raises an important issue with this class of algorithms, namely that a neighborhood function that works well for denoising one image may be disastrous for another. For example, a median filter would be an excellent choice for denoising the text image side of Figure 1, but would obliterate the half-toning in the Einstein image, and thereby actually amplify the distortion. Thus, in general, we are faced with the following problem: Given a noisy binary image and a neighborhood size of k, which of the 22 k binary valued denoising functions should be used? If a genie were to reveal the underlying clean image we could determine the “best” function that would lead to the smallest distortion between the denoised and clean images. Can we hope to even come close to this level of performance with access only to the noisy image? It turns out that with the additional knowledge of the BSC cross-over probability, the answer is yes in a very strong sense, and, further, that it can be done using an eminently practical algorithm, which we call DUDE, as defined and analyzed in our paper [1].