A new step size rule for the superiorization method and its application in computerized tomography

• Published in: Numerical algorithms Vol. 90; no. 3; pp. 1253 - 1277
• Main Authors: Nikazad, T., Abbasi, M., Afzalipour, L., Elfving, T.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2022; Springer Nature B.V
• Subjects: Algebra; Algorithms; Computed tomography; Computer Science; Conjugate gradient method; Convergence; Iterative methods; Least squares method; Numeric Computing; Numerical Analysis; Original Paper; Regularization; Theory of Computation; Tomography; Strictly quasi-nonexpansive operator; Convex feasibility problem; Computed tomography; 65F10; Convex optimization; Sequential block method; Perturbation resilient iterative method; 47J25; Superiorization; 47H14
• ISSN: 1017-1398; 1572-9265; 1572-9265
• DOI: 10.1007/s11075-021-01229-z

In this paper, we consider a regularized least squares problem subject to convex constraints. Our algorithm is based on the superiorization technique, equipped with a new step size rule which uses subgradient projections. The superiorization method is a two-step method where one step reduces the value of the penalty term and the other step reduces the residual of the underlying linear system (using an algorithmic operator T ). For the new step size rule, we present a convergence analysis for the case when T belongs to a large subclass of strictly quasi-nonexpansive operators. To examine our algorithm numerically, we consider box constraints and use the total variation (TV) functional as a regularization term. The specific test cases are chosen from computed tomography using both noisy and noiseless data. We compare our algorithm with previously used parameters in superiorization. The T operator is based on sequential block iteration (for which our convergence analysis is valid), but we also use the conjugate gradient method (without theoretical support). Finally, we compare with the well-known “fast iterative shrinkage-thresholding algorithm” (FISTA). The numerical results demonstrate that our new step size rule improves previous step size rules for the superiorization methodology and is competitive with, and in several instances behaves better than, the other methods.