A Post-beamforming 2-D pseudoinverse filter for coarsely sampled ultrasound arrays

Beamforming artifacts due to coarse discretization of imaging apertures represent a significant barrier against the use of array probes in high-frequency applications. Nyquist sampling of array apertures dictates center-to-center spacing of λ/2 for elimination of grating lobes in the array pattern. However, this requirement is hard to achieve using current transducer technologies, even at the lower end of high-frequency ultrasonic imaging (in the range 25-35 MHz). In this paper, we present a new design approach for 2-D regularized pseudoinverse (PIO) filters suitable for restoring imaging contrast in systems employing coarsely sampled arrays. The approach is based on a discretized 2-D imaging model for linear arrays assuming scattering from a Cartesian grid in the imaging field of view (FOV). We show that the discretized imaging operator can be represented with a block Toeplitz matrix with the blocks themselves being Toeplitz. With sufficiently large grid size in the axial and lateral directions, it is possible to replace this Toeplitz-block block Toeplitz (TBBT) operator with its circulant-block block circulant (CBBC) equivalent. This leads to a computationally efficient implementation of the regularized pseudoinverse filtering approach using the 2-D fast Fourier transform (FFT). The derivation of the filtering equation is shown in detail and the regularization procedure is fully described. Using FIELD, we present simulation data to show the 2-D point-spread functions (PSFs) for imaging systems employing linear arrays with fine and coarse sampling of the imaging aperture. PSFs are also computed for a coarsely sampled array with different levels of regularization to demonstrate the tradeoff between contrast and spatial resolution. These results demonstrate the well-behaved nature of the PSF with the variation in a single regularization parameter. Specifically, the 6 dB axial and lateral dimensions of the PSF increase gradually with increasing value of the regularization parameter. On the other hand, the peak grating lobe level decreases gradually with increasing value of the regularization parameter. The results are supported by image reconstructions from a simulated cyst phantom obtained using finely and coarsely sampled apertures with and without the application of the regularized 2-D PIO.