High-order total variation regularizationapproach  for axially symmetricobject tomography from a single radiograph

Raymond H. Chan; Haixia Liang; Suhua Wei; Mila Nikolova; Xue-Cheng Tai

doi:10.3934/ipi.2015.9.55

Article Contents

2015, Volume 9, Issue 1: 55-77. Doi: 10.3934/ipi.2015.9.55

This issue Previous Article A scalable algorithm for MAP estimators in Bayesian inverse problems with Besov priors Next Article Deformable multi-modal image registration by maximizing Rényi's statistical dependence measure

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

Received: May 31, 2012

Revised: February 28, 2014

Published: January 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 65R32, 65K10; Secondary: 49M25, 93B40.

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