The Distorted Born Iterative (DBI) method is used for ultrasound tomography in order to localize and identify malignant breast tissues. This approach begins with the Born approximation to generate an initial prediction of the scattering function. Then, iteratively solves the forward problem for the total field and the inhomogeneous Green’s function, and the inverse problem for the scattering function. The drawback of this method is that the associated inverse scattering problem is ill-posed. We are proposing the Truncated General Singular Value Decomposition (TGSVD) approach as a regularization method for the ill posed inverse problem Xy = b in DBI and comparing it to the well known Truncated Singular Value Decomposition (TSVD). The TGSVD employs generalized SVD (GSVD) of matrix pair (X,L) and is neglecting the smallest, contaminated with noise, generalized singular values, while regularization matrix L (we used the first order derivative operator) is responsible for smoothing the solution. This results in better image quality. We compared the performances of these two methods on simulated phantom and proved that TGSVD produces lower relative error and better reconstructed image.
The distorted Born iterative (DBI) method is a powerful approach for solving the inverse scattering problem for ultrasound tomographic imaging. This method iteratively solves the inverse problem for the scattering function and the forward problem for the inhomogeneous Green’s function and the total field. Because of the ill-posed system from the inverse problem, regularization methods are needed to obtain a smooth solution. The three methods compared are truncated total least squares (TTLS), conjugate gradient for least squares (CGLS), and Tikhonov regularization. This paper uses numerical simulations to compare these three approaches to regularization in terms of both quality of image reconstruction and speed. Noise from both transmitters and receivers is very common in real applications, and is considered in stimulation as well. The solutions are evaluated by residual error of scattering function of region of interest(ROI), convergence of total field solutions in all iteration steps, and accuracy of estimated Green’s functions. By comparing the result of reconstruction quality as well as the computational cost of the three methods under different ultrasound frequency, we prove that TTLS method has the lowest error in solving strongly ill-posed problems. CGLS consumes the shortest computational time but its error is higher than TTLS, but lower than Tikhonov regularization.
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.
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