A Generalized Correlation-Based Model for Out-of-Plane Motion Estimation in Freehand Ultrasound

A big challenge in sensorless image-based ultrasound tracking is in the out-of-plane motion estimation. The correlation value of a specific model of speckle known as fully developed speckle (FDS) can be used to estimate the out-of-plane displacement. In real tissue, this kind of pattern is rare and the deviation of speckle pattern from the ideal FDS model diminishes the accuracy of the out-of-plane motion estimation. In this paper a new method for estimation of the out-of-plane motion is proposed. Firstly a closed-form mathematical derivation is provided for the correlation of two RF echo signal patches at different positions. A linear regression model of the ultrasound beam profile is proposed to account for the spatial variability of the ultrasound beam and enhance the accuracy of out-of-plane motion estimation in real tissue. The statistical model of speckle used here is based on the Rician-Inverse Gaussian (RiIG) stochastic process of the speckle formation, which can be considered as a generalized form of the K-distribution with richer parametrization. In this work, for the first time the second-order statistics of the RIG model is used for speckle tracking. This statistical model allows for derivation of a closed-form formulation for the correlation coefficient based on the statistical parameters of every patch. Since the effect of coherency is considered in the RiIG model, it increases the reliability of the out-of-plane motion estimation. The flexibility of the proposed method enables almost any patch through the whole image to be used for the purpose of displacement estimation. The method has been evaluated both on ex vivo and in vivo tissues in various experiments including out-of-plane rotation (tilt, yaw) and free-hand imaging. The overall outcome demonstrates the potential of the proposed method for in vivo tissues.