February  2013, 7(1): 199-216. doi: 10.3934/ipi.2013.7.199

Constrained SART algorithm for inverse problems in image reconstruction

1. 

Ovidius University of Constanta, Blvd. Mamaia 124, 900527 Constanta, Romania, Romania

Received  July 2011 Revised  September 2012 Published  February 2013

In this paper we integrate the SART (Simultaneous Algebraic Reconstruction Technique) algorithm into a general iterative method, introduced in [8]. This general method offers us the possibility of achieving a new convergence proof of the SART method and prove the convergence of the constrained version of SART. Systematic numerical experiments, comparing SART and Kaczmarz-like algorithms, are made on two phantoms widely used in image reconstruction literature.
Citation: Lacramioara Grecu, Constantin Popa. Constrained SART algorithm for inverse problems in image reconstruction. Inverse Problems and Imaging, 2013, 7 (1) : 199-216. doi: 10.3934/ipi.2013.7.199
References:
[1]

A. H. Andersen and A. C. Kak, Simultaneous algebraic reconstruction techniques (SART): A superior implementation of the ART algorithm, Ultrasonic Imaging, 6 (1984), 81-94.

[2]

Y. Censor and S. A. Zenios, "Parallel Optimization: Theory, Algorithms, and Applications," Numer. Math. and Sci. Comp., Oxford Univ. Press, New York, 1997.

[3]

G. T. Herman, "Image Reconstruction from Projections. The Fundamentals of Computerized Tomography," Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980.

[4]

M. Jiang and G. Wang, Convergence of the simultaneous algebraic reconstruction technique (SART), in "Proc. 35th Asilomar Conf. Signals, Systems, and Computers," Pacific Grove, CA, (2001), 360-364.

[5]

M. Jiang and G. Wang, Convergence studies on iterative algorithms for image reconstruction, IEEE Trans. Medical Imaging, (2003).

[6]

I. Koltracht and P. Lancaster, Constraining strategies for linear iterative processes, IMA Journal of Numerical Analysis, 10 (1990), 555-567. doi: 10.1093/imanum/10.4.555.

[7]

I. Koltracht, P. Lancaster and D. Smith, The structure of some matrices arising in tomography, Linear Algebra and its Applications, 130 (1990), 193-218. doi: 10.1016/0024-3795(90)90212-U.

[8]

A. Nicola, S. Petra, C. Popa and C. Schnörr, On a general extending and constraining procedure for linear iterative methods, Intern. Journal of Computer Mathematics, 89(2) (2012), 231-253.

[9]

X. Pan, E. Y. Sidky and M. Vannier, Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?, Inverse Problems, 25 (2009), 123009, 36 pp. doi: 10.1088/0266-5611/25/12/123009.

[10]

C. Popa, "Projection Algorithms, Classical Results and Developments. Applications to Image Reconstructions," Lambert Academic Publishing - AV Akademikerverlag GmbH & Co.KG,Saarbrücken, Germany, 2012.

[11]

C. Popa, A hybrid Kaczmarz-conjugate gradient algorithm for image reconstruction, Mathematics and Computers in Simulation, 80 (2010), 2272-2285. doi: 10.1016/j.matcom.2010.04.024.

[12]

C. Popa, Constrained Kaczmarz extended algorithm for image reconstruction, Linear Algebra and its Applications, 429 (2008), 2247-2267. doi: 10.1016/j.laa.2008.06.024.

show all references

References:
[1]

A. H. Andersen and A. C. Kak, Simultaneous algebraic reconstruction techniques (SART): A superior implementation of the ART algorithm, Ultrasonic Imaging, 6 (1984), 81-94.

[2]

Y. Censor and S. A. Zenios, "Parallel Optimization: Theory, Algorithms, and Applications," Numer. Math. and Sci. Comp., Oxford Univ. Press, New York, 1997.

[3]

G. T. Herman, "Image Reconstruction from Projections. The Fundamentals of Computerized Tomography," Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980.

[4]

M. Jiang and G. Wang, Convergence of the simultaneous algebraic reconstruction technique (SART), in "Proc. 35th Asilomar Conf. Signals, Systems, and Computers," Pacific Grove, CA, (2001), 360-364.

[5]

M. Jiang and G. Wang, Convergence studies on iterative algorithms for image reconstruction, IEEE Trans. Medical Imaging, (2003).

[6]

I. Koltracht and P. Lancaster, Constraining strategies for linear iterative processes, IMA Journal of Numerical Analysis, 10 (1990), 555-567. doi: 10.1093/imanum/10.4.555.

[7]

I. Koltracht, P. Lancaster and D. Smith, The structure of some matrices arising in tomography, Linear Algebra and its Applications, 130 (1990), 193-218. doi: 10.1016/0024-3795(90)90212-U.

[8]

A. Nicola, S. Petra, C. Popa and C. Schnörr, On a general extending and constraining procedure for linear iterative methods, Intern. Journal of Computer Mathematics, 89(2) (2012), 231-253.

[9]

X. Pan, E. Y. Sidky and M. Vannier, Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?, Inverse Problems, 25 (2009), 123009, 36 pp. doi: 10.1088/0266-5611/25/12/123009.

[10]

C. Popa, "Projection Algorithms, Classical Results and Developments. Applications to Image Reconstructions," Lambert Academic Publishing - AV Akademikerverlag GmbH & Co.KG,Saarbrücken, Germany, 2012.

[11]

C. Popa, A hybrid Kaczmarz-conjugate gradient algorithm for image reconstruction, Mathematics and Computers in Simulation, 80 (2010), 2272-2285. doi: 10.1016/j.matcom.2010.04.024.

[12]

C. Popa, Constrained Kaczmarz extended algorithm for image reconstruction, Linear Algebra and its Applications, 429 (2008), 2247-2267. doi: 10.1016/j.laa.2008.06.024.

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