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 lambda/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 regu- arization 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.