CONVERGENCE ANALYSIS OF THE LOPING OS-EM ITERATIVE VERSION OF THE CIRCULAR RADON TRANSFORM

doi:10.1016/S0252-9602(14)60131-4

Abstract

Abstract:

The loping OS-EM iteration is a numerically efficient regularization method for solving ill-posed problems. In this article we investigate the loping OS-EM iterative method in connection with the circular Radon transform. We show that the proposed method converges weakly for the noisy data. Numerical tests are presented for a linear problem related to
photoacoustic tomography.

Key words: ill-posed equations, regularization, loping OS-EM iteration, circular Radon transform

CLC Number:

44A15

GUO Juan, WANG Jin-Ping. CONVERGENCE ANALYSIS OF THE LOPING OS-EM ITERATIVE VERSION OF THE CIRCULAR RADON TRANSFORM[J].Acta mathematica scientia,Series B, 2014, 34(6): 1875-1884.

Trendmd

References

[1] Haltmeier M, Leitao A, Resmerita E. On regularization methods of EM-Kaczmarz type. Inverse Problems, 2009, 25: 075008

[2] Resmerita E, Engl H W, Iusem A N. The expectation-maximization algorithm for ill-posed integral equa-tions: a convergence analysis. Inverse Problems, 2007, 23: 2575–2588

[3] Haltmeier M. Convergence analysis of a block iterative version of the loping Landweber-Kaczmarz iteration. Nonlinear Analysis-Theory, Methods and Applications, 2009, 71: 2912–2919

[4] Borwein J M, Lewis A S. Convergence of the best entropy estimates. SIAM J Optim, 2002, 1: 191–205

[5] Resmerita E, Anderssen R S. Joint additive Kullback-Leibler residual minimization and regularization for linear inverse problems. Math Methods Appl Sci, 2007, 30: 1527–1544

[6] Iusem A N. A short convergence proof of the EM algorithm for a specific Poisson model. Rev Bras Probab Estat, 1992, 6: 57–67

[7] Engl H W, Hanke M, Neubauer A. Regularization of Inverse Problems. Dordrecht: Kluwer Academic Publishers Group, 1996

[8] Bakushinsky A B, Kokurin M Y. Iterative Methods for Approximate Solution of Inverse Problems. Dor-drecht: Springer, 2004

[9] Ambartsoumian G, Gouia-Zarrad, Lewis M. Inversion of the circular Radon transform on annulus. Inverse Problems, 2010, 26: 105015

[10] Ambartsoumian G, Kuchment P. On the injectivity of the circular Radon transform. Inverse Problems, 2005, 21: 473–485

[11] Ambatrsoumian G, Kuchment P. A range description for the planar circular Radon transform. SIAM J Math Anal, 2006, 38: 2681–2692

[12] Wang J P. Inversion and property characterization on generalized transform of Radon type. Acta Math Sci, 2011, 31A(3): 636–643

[13] Shi T T, Wang J P. Reconstruction of the attenuated Radon transform in -scheme short-scan SPECT. Acta Math Sci, 2013, 33B(6): 1615–1626

[14] Hudson H M, Larkin R S. Accelerated image reconstruction using ordered subsets projection data. IEEE Trans Med Imaging, 1994, 13: 601–609

[15] Wang J P, Du J Y. A note on singular value decomposition for Radon transform in Rn. Acta Math Sci, 2002, 22B(3): 311–318

[16] Byrne C. Block-iterative algorithms. Intl Trans Oper Res, 2009, 16: 427–463

[17] Wallis J W, Miller T R. Rapidly converging iterative reconstruction algorithms in Single-Photon Emission Computed Tomography. J Nucl Med, 1993, 34(10): 1793–1800

[18] Yin Y, Liu L. Study of OS-EM with different subsets in fan beam reconstruction. CT Theory Appl, 2003, 12(3): 1–8