Solving Least Square Problem in Tomography

Abstract

The efficacy of the tomographic process depends upon the image reconstruction. Utmost mathematical problems encounter in tomography are systems of large linear equations. Krylov solvers for linear systems have sophisticated and straightforward formulae for the residual norm. Two Krylov solvers CGLS and LSQR are the variations of steep descent. The steep descent is one of the fundamental iterative technique used exclusively for the solution of large sparse square matrices. However, CGLS and LSQR the variations of steep descent also solve least square problems. This work involves the comparison of CGLS and LSQR. CGLS and LSQR are mathematically equivalent, but LSQR is robust and difficult to apply. Large sparse linear least square problem solved by LSQR, that is Krylov space solver in-fact based on Lanczos's bidiagonalization. This work applies said variations (CGLS and LSQR) of the steep descent on a tomographic test problem and compare the two algorithms on the basis of accuracy using MATLAB.