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>Bayesian Speckle Tracking. Part I: An Implementable Perturbation to the Likelihood Function for Ultrasound Displacement Estimation
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Bayesian Speckle Tracking. Part I: An Implementable Perturbation to the Likelihood Function for Ultrasound Displacement Estimation
A hallmark of clinical ultrasound has been accurate and precise displacement estimation. Displacement estimation accuracy has largely been considered to be limited by the Cramer-Rao lower bound (CRLB). However, the CRLB only describes the minimum variance obtainable from unbiased estimators. Unbiased estimators are generally implemented using Bayes’ theorem, which requires a likelihood function. The classic likelihood function for the displacement estimation problem is not discriminative and hard to implement for clinically relevant ultrasound with diffuse scattering. Since the classic likelihood function is not effective a perturbation is proposed.The proposed likelihood function was evaluated and compared against the classic likelihood function by converting both to posterior probability density functions (PDFs) using a non-informative prior. Example results are reported for bulk motion simulations using a 6λ tracking kernel and 30 dB SNR for 1000 data realizations. The canonical likelihood function assigned the true displacement a mean probability of only 0.070±0.020, while the new likelihood function assigned the true displacement a much higher probability of 0.22±0.16. The new likelihood function shows improvements at least for bulk motion, acoustic radiation force induced motion and compressive motion, and at least for SNRs greater than 10 dB and kernel lengths between 1.5 and 12λ.
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机译：临床超声的特点是精确而精确的位移估计。位移估计的准确性在很大程度上被认为受Cramer-Rao下界（CRLB）的限制。但是，CRLB仅描述了可从无偏估计量获得的最小方差。通常使用贝叶斯定理来实现无偏估计量，这需要似然函数。位移估计问题的经典似然函数没有区别，并且对于具有弥散散射的临床相关超声很难实现。由于经典似然函数无效，因此提出了一种摄动。通过使用非信息先验将这两者都转换为后验概率密度函数（PDF），对提出的似然函数进行了评估并与经典似然函数进行了比较。报告了使用6λ跟踪内核和30 dB SNR进行1000次数据实现的批量运动仿真的示例结果。规范似然函数将真实位移分配给的平均概率仅为0.070±0.020，而新的似然函数将真实位移分配给的概率高得多，为0.22±0.16。新的似然函数显示出至少对于整体运动，声辐射力诱导的运动和压缩运动以及至少对于大于10 dB的SNR和1.5至12λ之间的内核长度的改进。
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