Estimating Uncorrupted-Signal Markov Statistics

A Bayesian denoising framework implicitly assumes the existence of a prior statistical model of the uncorrupted signal. We can, potentially, derive such priors from a suitable database of high-SNR brain MR images (e.g., different images of the same modality and anatomy). This effectively amounts to training the denoising system. Effective training data, however, are not easily available for many applications. Alternatively, we can infer the uncorrupted signal statistics from the observed data by making suitable assumptions. Let us assume a fixed, but unknown, Markov model $P ({\bf Z})$ for the uncorrupted signal that generates all uncorrupted data. These data, subsequently, get corrupted by Rician noise. What we observe is only the corrupted data--the prior remains unknown. However, the following analysis provides a way of inferring the prior.

Given sufficiently many corrupted observations, we can infer the Markov statistics of the corrupted signal accurately [9,5]. With this knowledge of the corrupted-signal Markov statistics and knowing the properties of the corruption process, we can accurately estimate the uncorrupted-signal Markov statistics. In this way, we can empirically estimate the unknown prior PDF. This essentially amounts to solving an inverse problem, which we discuss in detail in the next section.