Patch complexity, finite pixel correlations and optimal denoising
Image restoration tasks are ill-posed problems, typically solved withpriors. Since the optimal prior is the exact unknown density of natural images,actual priors are only approximate and typically restricted to small patches. Thisraises several questions: How much may we hope to improve current restorationresults with future sophisticated algorithms? And more fundamentally, even withperfect knowledge of natural image statistics, what is the inherent ambiguity ofthe problem? In addition, since most current methods are limited to finite supportpatches or kernels, what is the relation between the patch complexity of naturalimages, patch size, and restoration errors? Focusing on image denoising, we makeseveral contributions. First, in light of computational constraints, we study the relation between denoising gain and sample size requirements in a non parametricapproach. We present a law of diminishing return, namely that with increasingpatch size, rare patches not only require a much larger dataset, but also gain littlefrom it. This result suggests novel adaptive variable-sized patch schemes for denoising. Second, we study absolute denoising limits, regardless of the algorithmused, and the converge rate to them as a function of patch size. Scale invarianceof natural images plays a key role here and implies both a strictly positive lowerbound on denoising and a power law convergence. Extrapolating this parametriclaw gives a ballpark estimate of the best achievable denoising, suggesting thatsome improvement, although modest, is still possible.