Life Becomes More Colorful When You Know EM, Bayes, and Wavelets ...

Abstract

Two common tasks in signal and image processing are denoising and dealing with missing values (e.g., missing pixels). The literature on either of them is substantial, especially the former and especially in the engineering literature. Dealing with both problems simultaneously is much less commonly done, especially when using wavelets, for it require much more brain power and computing power to do well. For example, concatenating the interpolation and denoising methods sequentially typically yields unsatisfactory results. On the other hand, there are many real-life problems where both problems arise, such as with digital color pictures and picture in-painting. In this talk we attempt to build a unified framework by combining several statistical and engineering methods to address this problem. We first use Bayesian hierarchical modelings to regulate wavelet reconstructions, and we then invoke a partial empirical Bayesian (PEB) approach to avoid excessive modelling and computational complexity. Because the EM algorithm provides a natural way of going back and forth between denoising and dealing with missing values, we adopt the EM algorithm for carrying out the maximization needed by PEB, which also provides, via its E-step, useful reconstructions as a by-product. This statistically principled approach also helps us to seek practical engineering solution by constructing various sensible "short cuts" to reduce computation. While we will show that the problem is somewhat surprisingly difficult even with all the tools we have, we also demonstrate, in the context of simultaneously demosaicing and denoising, the power of principled statistical modelling and computational approach via a set of colorful pictures ... (This is a join work with Keigo Hirakawa, a Ph. D. in Electrical and Computing Engineering from Cornell University and an MM in Jazz Performance from New England Conservatories.)