A modified gradient method for two- dimensional problems in tomography

AbstractA method for reconstructing the complex index of refraction of a bounded two-dimensional inhomogeneous object of known geometric configuration from measured scattered field data is presented. This work is an extension of recent results on the direct scattering problem wherein the governing domain integral equation was solved iteratively by a successive over-relaxation technique. The relaxation parameter was chosen to minimize the residual error at each step. Convergence of this process was established for indices of refraction much larger than required for convergence of the Born approximation. For the inverse problem the same technique is applied except in this case both the index of refraction and the field are unknown. Iterative solutions for both unknowns are postulated with two relaxation parameters at each step. They are determined by simultaneously minimizing the residual errors in satisfying the domain integral equation and matching the measured data. This procedure retains the nonlinear relation between the two unknowns. Numerical results are presented for a number of representative two-dimensional objects. The algorithm is shown to be effective in cases where the iterative solution of the direct problem is rapidly convergent