Robust Tracking of Small Displacements With a Bayesian Estimator

Abstract

Radiation-force-based elasticity imaging describes a group of techniques that use acoustic radiation force (ARF) to displace tissue in order to obtain qualitative or quantitative measurements of tissue properties. Because ARF-induced displacements are on the order of micrometers, tracking these displacements in vivo can be challenging. Previously, it has been shown that Bayesian-based estimation can overcome some of the limitations of a traditional displacement estimator like normalized cross-correlation (NCC). In this work, we describe a Bayesian framework that combines a generalized Gaussian-Markov random field (GGMRF) prior with an automated method for selecting the prior’s width. We then evaluate its performance in the context of tracking the micrometer-order displacements encountered in an ARF-based method like acoustic radiation force impulse (ARFI) imaging. The results show that bias, variance, and mean-square error performance vary with prior shape and width, and that an almost one order-of-magnitude reduction in mean-square error can be achieved by the estimator at the automatically-selected prior width. Lesion simulations show that the proposed estimator has a higher contrast-to-noise ratio but lower contrast than NCC, median-filtered NCC, and the previous Bayesian estimator, with a non-Gaussian prior shape having better lesion-edge resolution than a Gaussian prior. In vivo results from a cardiac, radiofrequency ablation ARFI imaging dataset show quantitative improvements in lesion contrast-to-noise ratio over NCC as well as the previous Bayesian estimator.

Improving Displacement Signal-to-Noise Ratio for Low-Signal Radiation Force Elasticity Imaging Using Bayesian Techniques

“…The tracking of small displacements using Bayesian techniques has been described previously (refer to our earlier work for more detail: Byram et al 2013a, 2013b; Dumont and Byram 2016). Briefly, the Bayesian estimator maximizes the global log-posterior probability between a given set of displacement estimates and the corresponding RF reference-track paired data.…”

“…The log-posterior probability P k ( τ k |x ) is computed as the following summation: $$lnfalse(P_{k}false({\tau }_{k}false|xfalse)false)\propto lnfalse(P_{k}false(xfalse|{\tau }_{k}false)false)+lnfalse(P_{k}false({\tau }_{k}false)false),$$which gives the log-posterior probability of the displacement estimate τ k for kernel k given the log-likelihood, ln ( P k ( x|τ k )) and the log-prior ln ( P k ( τ k )) (Dumont and Byram 2016). The log-likelihood is given by: $$lnfalse(P_{k}false(xfalse|{\tau }_{k}false)false)=-\frac{1}{4{\sigma }_n^2}true\sum _{n=0}^{M-1}{\left(s_1[n]-s_2[n;-{\tau }_k]\right)}^2,$$which evaluates the likelihood of obtaining data x , taken here to be some reference RF signal s 1 (containing n samples indexed over kernel-length M ) after undelaying the tracked RF signal s 2 by − τ k (Carlson and Sjoberg 2004; Dumont and Byram 2016). Parameter σ n characterizes the quality of the tracked RF data and is derived from a peak correlation coefficient-derived estimation of the SNR, which captures both thermal noise and tissue decorrelation based signal distortions (Byram et al 2013b; Dumont and Byram 2016).…”

“…The log-likelihood is given by: $$lnfalse(P_{k}false(xfalse|{\tau }_{k}false)false)=-\frac{1}{4{\sigma }_n^2}true\sum _{n=0}^{M-1}{\left(s_1[n]-s_2[n;-{\tau }_k]\right)}^2,$$which evaluates the likelihood of obtaining data x , taken here to be some reference RF signal s 1 (containing n samples indexed over kernel-length M ) after undelaying the tracked RF signal s 2 by − τ k (Carlson and Sjoberg 2004; Dumont and Byram 2016). Parameter σ n characterizes the quality of the tracked RF data and is derived from a peak correlation coefficient-derived estimation of the SNR, which captures both thermal noise and tissue decorrelation based signal distortions (Byram et al 2013b; Dumont and Byram 2016). Additional information is incorporated into the problem in the form of a generalized Gaussian Markov random field prior: $$lnfalse(P_{k}false({\tau }_{k}false)false)=-\frac{1}{p{\lambda }^p}true\sum _{k,j\in B}{w}_j{false|{\tau }_k-{\tau }_{j}false|}^p$$where τ j are displacement estimates located within a neighborhood that is spatially adjacent to the current estimate τ k .…”

“…Previously, we proposed using Bayesian-based displacement estimation to improve the tracking of small displacements, and found that the estimator could reduce the mean-square error of the displacement estimate by approximately an order of magnitude over a traditional correlation-based estimator (Byram et al 2013a, 2013b; Dumont and Byram 2016). Our prior investigation quantified the performance in a mean-square error sense in the context of thermal noise and shearing-induced decorrelation solely for ARFI imaging, whereas here we investigate the recovery of data quality in low displacement scenarios at the initial on-axis excitation site and after attenuation due to shear wave propagation (Dumont and Byram 2016).…”

“…Our prior investigation quantified the performance in a mean-square error sense in the context of thermal noise and shearing-induced decorrelation solely for ARFI imaging, whereas here we investigate the recovery of data quality in low displacement scenarios at the initial on-axis excitation site and after attenuation due to shear wave propagation (Dumont and Byram 2016). …”