Solving the ultrasound inverse scattering problem of inhomogeneous media using different approaches of total least squares algorithms

Abstract

The distorted Born iterative method (DBI) is used to solve the inverse scattering problem in the ultrasound tomography with the objective of determining a scattering function that is related to the acoustical properties of the region of interest (ROI) from the disturbed waves measured by transducers outside the ROI. Since the method is iterative, we use Born approximation for the first estimate of the scattering function. The main problem with the DBI is that the linear system of the inverse scattering equations is ill-posed. To deal with that, we use two different algorithms and compare the relative errors and execution times. The first one is Truncated Total Least Squares (TTLS). The second one is Regularized Total Least Squares method (RTLS-Newton) where the parameters for regularization were found by solving a nonlinear system with Newton method. We simulated the data for the DBI method in a way that leads to the overdetermined system. The advantage of RTLS-Newton is that the computation of singular value decomposition for a matrix is avoided, so it is faster than TTLS, but it still solves the similar minimization problem. For the exact scattering function we used Modified Shepp-Logan phantom. For finding the Born approximation, RTLS-Newton is 10 times faster than TTLS. In addition, the relative error in L2-norm is smaller using RTLS-Newton than TTLS after 10 iterations of the DBI method and it takes less time.