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Abel inversion using total variation regularization: applications. (English) Zbl 1200.65115

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.

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
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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
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