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February  2015, 9(1): 55-77. doi: 10.3934/ipi.2015.9.55

## High-order total variation regularization approach for axially symmetric object tomography from a single radiograph

 1 Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong, China 2 Department of Mathematical Sciences, Xi'an Jiaotong-Liverpool University, No. 111 Ren'ai Road, Suzhou Industrial Park, Jiangsu Province, China 3 Institute of Applied Physics and Computational Mathematics, Beijing, China 4 Centre de Mathematiques et de Leurs Applications, CNRS, ENS de Cachan, PRES UniverSud, 61 av. du President Wilson, 94235 Cachan Cedex, France 5 Department of Mathematics, University of Bergen, P. O. Box 7800, N-5020, Bergen, Norway

Received  June 2012 Revised  March 2014 Published  January 2015

In this paper, we consider tomographic reconstruction for axially symmetric objects from a single radiograph formed by fan-beam X-rays. All contemporary methods are based on the assumption that the density is piecewise constant or linear. From a practical viewpoint, this is quite a restrictive approximation. The method we propose is based on high-order total variation regularization. Its main advantage is to reduce the staircase effect while keeping sharp edges and enable the recovery of smoothly varying regions. The optimization problem is solved using the augmented Lagrangian method which has been recently applied in image processing. Furthermore, we use a one-dimensional (1D) technique for fan-beam X-rays to approximate 2D tomographic reconstruction for cone-beam X-rays. For the 2D problem, we treat the cone beam as fan beam located at parallel planes perpendicular to the symmetric axis. Then the density of the whole object is recovered layer by layer. Numerical results in 1D show that the proposed method has improved the preservation of edge location and the accuracy of the density level when compared with several other contemporary methods. The 2D numerical tests show that cylindrical symmetric objects can be recovered rather accurately by our high-order regularization model.
Citation: Raymond H. Chan, Haixia Liang, Suhua Wei, Mila Nikolova, Xue-Cheng Tai. High-order total variation regularization approach for axially symmetric object tomography from a single radiograph. Inverse Problems & Imaging, 2015, 9 (1) : 55-77. doi: 10.3934/ipi.2015.9.55
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##### References:
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