Signal-Dependent Noise Characterization in Haar Filterbank Representation

Owing to the properties of joint time-frequency analysis that compress energy and approximately decorrelate temporal redundancies in sequential data, filterbank and wavelets are popular and convenient platforms for statistical signal modeling. Motivated by the prior knowledge and empirical studies, much of the emphasis in signal processing has been placed on the choice of the prior distribution for these transform coefficients. In this paradigm however, the issues pertaining to the loss of information due to measurement noise are difficult to reconcile because the effects of point-wise signal-dependent noise permeate across scale and through multiple coefficients. In this work, we show how a general class of signal-dependent noise can be characterized to an arbitrary precision in a Haar filterbank representation, and the corresponding maximum a posteriori estimate for the underlying signal is developed. Moreover, the structure of noise in the transform domain admits a variant of Stein’s unbiased estimate of risk conducive to processing the corrupted signal in the transform domain. We discuss estimators involving Poisson process, a situation that arises often in real-world applications such as communication, signal processing, and imaging.