Asaki, Thomas J.; Campbell, Patrick R.; Chartrand, Rick; Powell, Collin E.; Vixie, Kevin R.; Wohlberg, Brendt E. Abel inversion using total variation regularization: applications. (English) Zbl 1200.65115 Inverse Probl. Sci. Eng. 14, No. 8, 873-885 (2006). Summary: 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. Cited in 10 Documents MSC: 65R32 Numerical methods for inverse problems for integral equations 45Q05 Inverse problems for integral equations 92C55 Biomedical imaging and signal processing 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 15A29 Inverse problems in linear algebra 49M37 Numerical methods based on nonlinear programming 65F22 Ill-posedness and regularization problems in numerical linear algebra 78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory Keywords:Abel transform; inverse problems; total variation; tomography; regularization PDFBibTeX XMLCite \textit{T. J. Asaki} et al., Inverse Probl. Sci. Eng. 14, No. 8, 873--885 (2006; Zbl 1200.65115) Full Text: DOI References: [1] DOI: 10.1137/1.9780898719277 · doi:10.1137/1.9780898719277 [2] DOI: 10.1137/1.9780898717570 · Zbl 1008.65103 · doi:10.1137/1.9780898717570 [3] DOI: 10.1088/0266-5611/21/6/006 · Zbl 1094.65129 · doi:10.1088/0266-5611/21/6/006 [4] DOI: 10.1137/S0036142997327075 · Zbl 0923.65037 · doi:10.1137/S0036142997327075 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.