Phase-coupled two-dimensional speckle tracking algorithm

A new two-dimensional (2-D) speckle tracking method for displacement estimation based on the gradients of the magnitude and phase of 2-D complex correlation in a search region is presented. The novelty of this approach is that it couples the phase and magnitude gradients near the correlation peak to determine its coordinates with subsample accuracy in both axial and lateral directions. This is achieved with a minimum level of lateral interpolation determined from the angles between the magnitude and phase gradient vectors on the sampled (laterally interpolated) 2-D cross-correlation grid. The key result behind this algorithm is that the magnitude gradient vectors´ final approach to the true peak is orthogonal to the zero-phase contour. This leads to a 2-D robust projection on the zero-phase contour that results in subsample accuracy at interpolation levels well below those needed using previously proposed methods. A full description of the 2-D, phase-coupled approach is given, including two implementations based on a geometric projection and constrained optimization. In addition, a robust fast search algorithm that allows the localization of the true peak without the need for exhaustive search is given. Experimental validation on three data sets from speckle-generating phantoms undergoing uniform diagonal motion, uniform axial deformation, and nonuniform lateral flow is given. It is shown that estimated 2-D displacement fields obtained using the phase-coupled technique display a full range of values covering the dynamic range without evidence of quantization. In comparison, a previously published method using 1-D phase-projection after lateral interpolation produces severely quantized lateral displacement fields (at the same levels of interpolation as the 2-D, phase-coupled method)