Total variation superiorized conjugate gradient method for image reconstruction. (23rd January 2018) - Zibetti, Marcelo V W; Lin, Chuan; Herman, Gabor T
Abstract: The conjugate gradient (CG) method is commonly used for the relatively-rapid solution of least squares problems. In image reconstruction, the problem can be ill-posed and also contaminated by noise; due to this, approaches such as regularization should be utilized. Total variation (TV) is a useful regularization penalty, frequently utilized in image reconstruction for generating images with sharp edges. When a non-quadratic norm is selected for regularization, as is the case for TV, then it is no longer possible to use CG. Non-linear CG is an alternative, but it does not share the efficiency that CG shows with least squares and methods such as fast iterative shrinkage-thresholding algorithms (FISTA) are preferred for problems with TV norm. A different approach to including prior information is superiorization. In this paper it is shown that the conjugate gradient method can be superiorized. Five different CG variants are proposed, including preconditioned CG. The CG methods superiorized by the total variation norm are presented and their performance in image reconstruction is demonstrated. It is illustrated that some of the proposed variants of the superiorized CG method can produce reconstructions of superior quality to those produced by FISTA and in less computational time, due to the speed of the original CG for least squares problems. In the Appendix we examine the behavior of one of the superiorized CG methods (we call it S-CG); one of its input parameters is a positive number ε . It is proved that, for any given ε that is greater than the half-squared-residual for the least squares solution, S-CG terminates in a finite number of steps with an output for which the half-squared-residual is less than or equal to ε . Importantly, it is also the case that the output will have a lower value of TV than what would be provided by unsuperiorized CG for the same value ε of the half-squared residual.