Abel inversion using total variation regularization: applications

• Published in: Inverse problems in science and engineering Vol. 14; no. 8; pp. 873 - 885
• Main Authors: Asaki, Thomas J., Campbell, Patrick R., Chartrand, Rick, Powell, Collin E., Vixie, Kevin R., Wohlberg, Brendt E.
• Format: Journal Article
• Language: English
• Published: Taylor & Francis 01.12.2006
• Subjects: 2000 Mathematics Subject Classifications: 45E10; Abel transform; Inverse problems; Regularization; Tomography; Total variation
• ISSN: 1741-5977; 1741-5985
• DOI: 10.1080/17415970600882549

We apply total-variation (TV) regularization methods to Abel inversion tomography. Inversions are performed using the fixed-point iteration method and the regularization parameter is chosen such that the resulting data fidelity approximates the known or estimated statistical character of the noisy data. Five one-dimensional (1D) examples illustrate the favorable characteristics of TV-regularized solutions: noise suppression and density discontinuity preservation. Experimental and simulated examples from X-ray radiography also illustrate limitations due to a linear projection approximation. TV-regularized inversions are shown to be superior to squared gradient (Tikhonov) regularized inversions for objects with density discontinuities. We also introduce an adaptive TV method that utilizes a modified discrete gradient operator acting only apart from data-determined density discontinuities. This method provides improved density level preservation relative to the basic TV method.