Bayesian speckle tracking. Part II: biased ultrasound displacement estimation

Abstract

Ultrasonic displacement estimates have numerous clinical uses including blood-flow, elastography, therapeutic guidance and ARFI imaging. These clinical tasks could be improved with better ultrasonic displacement estimates. Traditional ultrasonic displacement estimates are limited by the Cramer-Rao lower bound (CRLB). The CRLB can be surpassed using biased estimates. In this paper a framework for biased estimation using Bayes’ theorem is described. The Bayesian displacement estimation method is tested against simulations of several common types of motion: bulk, step, compression, and acoustic radiation force induced motion. Bayesian estimation is also applied to in vivo acoustic radiation force imaging of cardiac ablation lesions. The Bayesian estimators are compared to the unbiased estimator, normalized cross-correlation. As an example, the peak displacement of the simulated acoustic radiation force response is reported since this position results in the noisiest estimates. Estimates were made with a 1.5 λ kernel and 20 dB SNR on 100 data realizations. Estimates using normalized cross-correlation and the Bayes’ estimator had mean-square errors of 17 and 7.6 μm2, respectively, and contextualized by the true displacement magnitude, 10.9 μm. Biases for normalized cross-correlation and the Bayes’ estimator are −.12 and −.28 μm, respectively. In vivo results show qualitative improvements. The results show that with small amounts of additional information significantly improved performance can be realized.

Robust Tracking of Small Displacements With a Bayesian Estimator

“…These methods—termed Bayesian speckle tracking or Bayesian regularization—use prior knowledge of the estimation task in order to improve the current estimate. Byram et al showed that Bayesian speckle tracking could produce displacement estimates with a lower mean-square error relative to a CRLB-limited estimator for bulk displacement, strain-based elastography, and radiation-force based elasticity imaging [13]. Mc-Cormick et al proposed an iterative, Bayesian regularization method and showed that significant improvements in estimate quality could be achieved in very few iterations for ultrasound strain images with strains greater than 5% [14].…”

“…While both approaches demonstrate the reduction in estimate error that can be realized with these techniques, there is room for improvement. For example, Byram et al proposed a directionally-dependent prior scheme, and hypothesized that the falsely-imposed causality limited its performance [13]. McCormick et al reported a reduction in image quality for strain-fields smaller than 1% [14], making it unclear how suitable their approach will be for radiation-force based techniques such as Acoustic Radiation Force Impulse (ARFI) imaging or Shear Wave Elasticity Imaging (SWEI) [8], [10].…”

“…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]. …”

Acute and chronic myocardial infarct differentiation using atrial kick induced strain (AKIS) imaging

“…In order to create AKIS images, ventricular motion during atrial contraction is estimated in 3D using Bayesian speckle tracking [17], [18]. Bayesian speckle tracking appropriately weights prior information about the expected motion against a likelihood function scaled based on local data quality.…”

New phase-matching method for ultrasonic tissue displacement and strain measurements: Application of maximum likelihood and maximum a posteriori estimations to multidimensional autocorrelation method

“…ML and MAP estimations are also performed in the medical US field, for instance, in Refs. [43][44][45], in which a likelihood function is derived for the 1D cross-correlation of local rf-echo data for axial displacement measurement, or a prior is derived for neighboring axial displacements. In this report, a new likelihood function that is derived for the phase difference between a pair of rf-echo signals generated by tissue displacement (vector) and a prior that is derived straightforwardly for the target displacement (vector) are used.…”