Two-dimensional least-squares estimation for motion tracking in ultrasound elastography

This paper proposes a method of 2-D translations estimation using an a priori signal model. Two analytical signals defined with multidimensional Hilbert transform are considered and shown to have linear phases with respect to the translations to estimate. A least squares estimator (LSE) is then developed to adjust the measured phases of the complex signals to their theoretical forms. Moreover, the LSE provides an analytical solution to the 2-D translation estimation problem. The estimator is then included in a block matching method for motion tracking with ultrasound images. We compared our results with those obtained with a classical sum of absolute differences (SAD) cost function. We show that with our method there is no need of interpolating the images. Thus, for images at the original resolution level, the results obtained with the proposed estimator are largely more accurate than with SAD. Moreover, we show that using SAD on images with resolution five times higher provide roughly the same results as with our method, but the processing time is ten times higher in this case.