Determination of time-of-flight surfaces using the method of moments [US imaging]

To produce an image from backscattered signals requires a knowledge of the time of flight from each source to each sensor in a transducer array. The authors reduced the determination of the rectangular coordinates of a new point on any time-of-flight surface, which requires finding the square root of the sum of squares, to the time of a single addition. To do this, the time-of-flight surface was represented by a two-dimensional, positive-integer-degree polynomial and then that polynomial was implemented in its forward-difference form. Two solutions for such a polynomial were found using the method of moments. A minimum-mean-square-error constraint yields polynomial coefficients from a numeric evaluation of the moments of arbitrary surfaces over rectangular regions, and an analytical solution for moments of time-of-flight surfaces over sector-shaped regions. For a 7x6-cm region at a minimum range of 4 cm centered in a 74 degrees sector, the maximum error for a second-degree polynomial was 0.30% of the average time of flight over the region for the numeric solution and 0.32% for the analytic solution, assuming a 1500-m/s background velocity.