Feasibility of using a generalized-Gaussian Markov random field prior for Bayesian speckle tracking of small displacements

Abstract

Accurate displacement estimation can be a challenging task in acoustic radiation force elastography, where signal decorrelation can degrade the ability of a normalized cross-correlation (NCC) estimator to characterize the tissue response. In this work, we describe a Bayesian estimation scheme which models both signal decorrelation and thermal noise, and uses an edge-preserving, generalized Gaussian Markov random field prior. The performance of the estimator was evaluated in FEM simulations modeling the acoustic radiation force impulse response in a linearly-isotropic material. Bias, variance, and mean-square error were calculated over a range of estimator parameters, and compared to NCC. The results demonstrate that a significant reduction in mean-square error can be achieved with the proposed estimator. Finally, in vivo data of an radio-frequency ablation in a canine model are shown, demonstrating the in vivo feasibility of the proposed method.

Robust Tracking of Small Displacements With a Bayesian Estimator

“…To overcome these limitations, we proposed an iterative approach that combines an edge-preserving generalized Gaussian-Markov random field (GGMRF) prior with a reformulated likelihood function [15]. We review the algorithm here.…”

“…First, the likelihood P m ( x |τ 0 ) is rewritten explicitly as a function of the two time-shifted RF signals instead of the normalized cross-correlation function, $$P_{m}false(xfalse|{\tau }_{0}false)=\prod _{n=0}^{N-1}{false(2\pi {\sigma }_n^{2}false)}^{-\frac{1}{2}}\text{ exp }true[-\frac{1}{{\sigma }_n^2}{false(\zeta false)}^{2}true]={false(2\pi {\sigma }_n^{2}false)}^{-\frac{N}{2}}\text{ exp }true[-\frac{1}{{\sigma }_n^2}\sum _{n=0}^{N-1}{false(\zeta false)}^{2}true]$$ $$\zeta =s_{1}false[nfalse]-s_{2}false[n;-{\tau }_{0}false]$$ where the data x is not taken as the cross-correlation function, but rather the RF signal s 1 [ n ]. N is the kernel length, τ 0 is the displacement estimate for a kernel m containing n samples, and ${\sigma }_n^2$ is a noise term (described later) that quantifies the uncertainty in the probability distribution of the data [15], [18]. 1 The function described in (5) expresses the likelihood of observing s 1 [ n ] given delayed signal s 2 [ n ;−τ 0 ] that has been un-delayed by −τ 0 .…”

“…The term λ scales the influence of the prior on the final posterior PDF. Z is a normalizing constant, τ 0 is the current estimate for kernel m , and w j weights the influence of adjacent estimates τ j within the neighborhood B [15]. …”

“…To address these limitations, we recently proposed a Bayesian estimator that uses a generalized-Gaussian Markov Random Field (GGMRF) prior, one that has no direction-dependence and allows for adjustments in prior shape [15]. Our initial results showed improvement in mean-square error compared to normalized cross-correlation, but the error analysis was limited to tracking ARF-induced displacements in simulated, homogeneous data with little noise.…”

“…Our initial results showed improvement in mean-square error compared to normalized cross-correlation, but the error analysis was limited to tracking ARF-induced displacements in simulated, homogeneous data with little noise. It was unclear in our preliminary work on how well the estimator performed in visualizing non-homogeneous structures in the presence of increased noise, how to reliably select the prior width from the data or what advantages the new prior offered in comparison to the previous approach [12], [13], [15]. …”