On the Expectation-Maximization algorithm for Rice-Rayleigh mixtures with application to noise parameter estimation in magnitude MR datasets

Abstract

Magnitude magnetic resonance (MR) images are noise-contaminated measurements of the true signal, and it is important to assess the noise in many applications. A recently introduced approach models the magnitude MR datum at each voxel in terms of a mixture of up to one Rayleigh and an a priori unspecified number of Rice components, all with a common noise parameter. The Expectation-Maximization (EM) algorithm was developed for parameter estimation, with the mixing component membership of each voxel as the missing observation. This paper revisits the EM algorithm by introducing more missing observations into the estimation problem such that the complete (observed and missing parts) dataset can be modeled in terms of a regular exponential family. Both the EM algorithm and variance estimation are then fairly straightforward without any need for potentially unstable numerical optimization methods. Compared to local neighborhood- and wavelet-based noise-parameter estimation methods, the new EM-based approach is seen to perform well not only in simulation experiments but also on physical phantom and clinical imaging data.

Appendix: Conditional distribution of \(\boldsymbol{\gamma_i}\) given \(\boldsymbol{R_i}\)

We first state and prove the following

Theorem 1

Let $\boldsymbol{W}_i$ be a realization from the one-trial J -class multinomial distribution with class probability vector $\boldsymbol{\pi}$ . Conditional on W _i,j  = 1 for some j = 1,2,...,J, let $U_i\sim N(\nu_j, \sigma^2)$ be independent of $V_i\sim N(0, \sigma^2)$ . Write (U _i , V _i ) in its polar form i.e. (U _i , V _i ) = (R _i cosγ _i , R _i sinγ _i ). Then the conditional distribution of γ _i given R _i and the event W _i,j  = 1 is $\mathcal M(0, {R_i\nu_j}/{\sigma^2})$ , i.e. it is von-Mises-distributed with mean angular direction 0 and concentration parameter R _i ν _j /σ ² . Consequently, ${\textrm{I\!E}}(\cos\gamma_i\mid R_i, W_{i,j} = 1) = {{\textrm{I}}_1({R_i\nu_j}/{\sigma^2} )}/{{\textrm{I}}_0({R_i\nu_j}/{\sigma^2})}$ .

Proof

From the characterization of the Rice distribution in Rice (1944, 1945), we know that $R_i\sim \varrho(x;\sigma,\nu_j)$ given that W _i,j  = 1. From standard results on probability distributions of transformation of variables, we get that the conditional joint density of (R _i ,γ _i ) given that W _i,j  = 1 is $f_{R_i,\gamma_i \mid W_{i,j}=1}(x, \gamma) = x/{2\pi\sigma^2}\exp{[-(x^2-2x\nu_j\cos\gamma_i + \nu_j^2)/2\sigma^2]}$. Thus $f(\gamma_i | R_i=x, W_{i,j}=1) = \exp{[{x\nu_j}/{\sigma^2}\cos\gamma_i]} / {2\pi{\textrm{I}}_0({x\nu_j}/{\sigma^2})}$ for 0 < γ _i  < 2π, which is the density of $\mathcal M(0, {x\nu_j}/{\sigma^2})$. From results in Mardia and Jupp (2000) on the expectation of the cosine of a von-Mises-distributed random variable, we get that ${\textrm{I\!E}}(\cos\gamma_i\mid R_i, W_{i,j} = 1) = {{\textrm{I}}_1({R_i\nu_j}/{\sigma^2} )}/{{\textrm{I}}_0({R_i\nu_j}/{\sigma^2})}$. This proves Theorem 1.□