Image registration, or equivalently motion estimation, plays a central role in a broad range of ultrasound applications
including elastography, estimation of blood or tissue motion, radiation force imaging, and extended field of view
imaging. Because of its central significance, motion estimation accuracy, precision, and computational cost are of critical
importance. Furthermore, since motion estimation is typically performed on sampled signals, while estimates are usually
desired over a continuous domain, performance should be considered in conjunction with associated interpolation.
We have previously presented a highly accurate, spline-based time delay estimator that directly determines sub-sample
time delay estimates from sampled data. The algorithm uses cubic splines to produce a continuous time representation of
a reference signal and then computes an analytical matching function between this reference and a delayed signal. The
location of the minima of this function yields estimates of the time delay. In this paper we describe a MUlti-dimensional
Spline-based Estimator (MUSE) that allows accurate and precise estimation of multi-dimensional displacements/strain
components from multi-dimensional data sets.
In this paper we describe the mathematical formulation for three-dimensional (3D) motion/strain estimation and present
simulation results to assess the intrinsic bias and standard deviation of this algorithm and compare it to currently
available multi-dimensional estimators. In 1,000 noise-free simulations we found that 2D MUSE exhibits maximum bias
errors of 4.8nm and 297nm in range and azimuth respectively. The maximum simulated standard deviation of estimates
in both dimensions was comparable at 0.0026 samples (corresponding to 54nm axially and 378nm laterally). These
results are two to three orders of magnitude lower than currently used 2D tracking methods. Simulation of performance
in 3D yielded similar results to those observed in 2D. We also performed experiments using 2D MUSE on an Ultrasonix
Sonix RP imaging system with an L14-5/38 linear array transducer operating at 6.6MHz. With this experimental data we
found that bias errors were significantly smaller than geometric errors induced by machining of the transducer mount.