Convergence and Speed of the Sinc Basis Moment Method

Abstract

A method for applying the fast Fourier t ransform to the convolutions arising in the internal field equations of the sinc basis moment method diffraction tomography algorithm is described. Exactly e quivalent results are obtained, whil redu ing the order of those computations from n f to n s log n for an nxn reconstruction. As originally derived, the sinc ba is moment method has an order of computation of ns for an nxn reconstruction. To reduce this order, spatial convolutions in the internal field equations were performed by a somewhat unusual application of the FFT. Specifically, if f $1 and f 41lnc represent, respectively, the total and incident ultrasonic fields evaluated at pixel 1 due to a transmitter at 1 ocati on $ , is the object function evaluated at pixel j, ',id Clj are coefficients resulting from the s inc basis expansions and integrations (Johnson and Tracy, 1983),