Vector flow estimator isomorphism and wall filter requirements

Ultrasound vector flow estimation can be complicated by the spatial anisotropy of the ultrasound pulse, a limitation most commonly characterized in terms of the different spatial resolutions achieved along the three axes of a typical imaging system. For vector Doppler or spatial quadrature flow estimation techniques, this anisotropy limits, relative to the axial modulation, the maximum achievable frequency of the modulation associated with lateral motion. This anisotropy has one advantage in establishing higher aliasing thresholds in the non-axial dimensions than are typically encountered axially. However, it is likely that one of the obstacles to practical implementation of vector velocity estimation will be the attenuation of non-axial modulation components by conventional wall and clutter filters. As part of our investigation of this topic, we describe a commonality among several vector flow estimators: the vector Doppler described by Overbeck, et. al, the estimator described by Jensen and Munk, and the heterodyning spatial quadrature estimator we have previously described. When implemented with the same transducer array, these estimators produce very similar spatial impulse responses despite the use of radically different complex apodization schemes. In two cases this common response is not formed during beamforming but rather after post-processing has been applied, We describe this isomorphism and discuss the differences among the estimators that we expect to become evident under realistic imaging conditions. We present an analysis of the multi-dimensional spatial frequency responses of these vector velocity estimators in comparison with typical wall filter responses. The axial/lateral anisotropy for all three estimators was found to be on the order of 10 in a pulse-echo regime. This implies that detection of lateral flow at a rate of, for example, 10 m/s would require a wall filter setting providing an axial cut-off velocity of less than 1 m/s. The practical implications of these findings and alternative approaches are discussed.