A Demonstration of Optimal Apodization Determination for Proper Lateral Modulation

Abstract

We have realized effective ultrasound (US) beamformings by the steering of plural beams and apodizations for B-mode imaging with a high lateral resolution and accurate measurement of tissue or blood displacement vector and/or strain tensor using the multidimensional cross-spectrum phase gradient method (MCSPGM), or multidimensional autocorrelation or Doppler methods (MAM and MDM) using multidimensional analytic signals. For instance, the coherent superposition of the steered beams performed in the lateral cosine modulation method (LCM) has a higher potential for realizing a more accurate measurement of a displacement vector than the synthesis of the displacement vector using the accurately measured axial displacements obtained by the multidimensional synthetic aperture method (MDSAM), multidirectional transmission method (MTM) or the use of plural US transducers. Originally, the apodization function to be used for realizing a designed point spread function (PSF) was obtained by the Fraunhofer approximation (FA). However, to obtain the best approximated, designed PSF in the least-squares sense, we proposed a linear optimization (LO) method. Furthermore, on the basis of the knowledge about the losts of US energy during the propagation, we have recently developed a nonlinear optimization (NLO) method, in which the feet of the main lobes in apodization function are properly truncated. Thus, NLO also allows the decrease in the number of channels or the confinement of the effective aperture. In this study, to gain insight into the ideal shape of the PSF, the accuracies of the two-dimensional (2D) displacement vector measurements were compared for typical PSFs with distinct lateral envelope shapes, particularly, in terms of full width at half maximum (FWHM) and the length of the feet, i.e., the Gaussian function, Hanning window and parabolic function. It was confirmed that a PSF having a wide FWHM and short feet was ideal. Such a PSF yielded an echo with a high signal-to-noise ratio (SNR), a large bandwidth and a large maximum spectrum of the center frequency. Moreover, for the three PSFs used, by calculating the PSFs using a typical transducer model and the apodization functions obtained by the respective LO and NLO methods and FA, we compare the approximation accuracies of the realized PSFs. NLO was effective for realizing such an ideal PSF. In addition, NLO allowed the significant decrease in the number of channels or the confinement of the effective aperture. Thus, in the comparisons of the three distinct PSFs, we obtain an appropriate apodization function. This study will assist the realization of the best lateral modulation.