Employing methods with generalized singular value decomposition for regularization in ultrasound tomography

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.