×

Tomographic reconstruction from a small number of projections by an efficient sum-product reasoning method. (English) Zbl 1455.92081

Summary: Tomographic reconstruction from a small number of projections is still a challenging problem. In the paper, we formulate this problem as a statistical graphical model by the smooth assumption that the image has a structure where neighbor pixels have a larger probability to take a closer value. This Markov random filed framework allows easily integrating other prior information. Reasoning in the model can be solved using belief propagation algorithm. However, one projection line involves multiple pixels. This leads to high order cliques and exponential computation in the message passing procedure. A variable-change strategy is used to largely reduce the computation and forms an efficient sum-product reasoning algorithm. Numerical simulation examples show that the proposed method greatly surpasses traditional methods, such as FBP, EM and ART. Our method is suitable not only for the case of a very small amount of projection, but also for the multi-pixel-value case.

MSC:

92C55 Biomedical imaging and signal processing
62-08 Computational methods for problems pertaining to statistics
Full Text: DOI

References:

[1] Bouman C, Sauer K (1996) A unified approach to statistical tomography using coordinate descent optimization [J]. Image Process IEEE Trans 5(3):480-492 · doi:10.1109/83.491321
[2] Fessler J (2006) Tutorial I. Iterative methods for image reconstruction. In: IEEE international symposium on biomedical imaging, Arlington, Virginia
[3] Ganan S, McClure D (1985) Bayesian image analysis: an application to single photon emission tomography. In: Proceedings of American Statistical Association, pp 12-18
[4] Gordon R, Bender R, Herman GT (1970) Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography. J Theor Biol 29(3):471-481 · doi:10.1016/0022-5193(70)90109-8
[5] Gouillart E, Krzakala F, Mezard M et al (2013) Belief-propagation reconstruction for discrete tomography. Inverse Probl 29(3):035003 · Zbl 1318.92030 · doi:10.1088/0266-5611/29/3/035003
[6] Jeng FC, Woods JW (1991) Compound Gauss-Markov random fields for image estimation. Signal Process IEEE Trans 39(3):683-697 · doi:10.1109/78.80887
[7] Kak AC, Slaney M (2001) Principles of computerized tomographic imaging. Society for Industrial and Applied Mathematics, Philadelphia · Zbl 0721.92011
[8] Kschischang FR, Frey BJ, Loeliger HA (2001) Factor graphs and the sum-product algorithm. Inf Theory IEEE Trans 47(2):498-519 · Zbl 0998.68234 · doi:10.1109/18.910572
[9] López A, Martín JM, Molina R et al (2006) Transmission tomography reconstruction using compound gauss-markov random fields and ordered subsets. In: Proceedings of image analysis and recognition 2006. Springer, Berlin, Heidelberg, pp 559-569
[10] Potetz B (2007) Efficient belief propagation for vision using linear constraint nodes. In: IEEE conference on computer vision and pattern recognition. CVPR’07. IEEE, pp 1-8
[11] Roux S, Leclerc H, Hild F (2014) Efficient binary tomographic reconstruction. J Math Imaging Vis 49(2):35-351 · doi:10.1007/s10851-013-0465-0
[12] Shi J, Zhang B, Liu F et al (2013) Efficient L1 regularization-based reconstruction for fluorescent molecular tomography using restarted nonlinear conjugate gradient. Opt Lett 38(18):3696-3699 · doi:10.1364/OL.38.003696
[13] Singh S, Kalra MK, Hsieh J et al (2010) Abdominal CT: comparison of adaptive statistical iterative and filtered back projection reconstruction techniques 1. Radiology 257(2):373-383 · doi:10.1148/radiol.10092212
[14] Tang J, Nett BE, Chen GH (2009) Performance comparison between total variation (TV)-based compressed sensing and statistical iterative reconstruction algorithms. Phys Med Biol 54(19):5781 · doi:10.1088/0031-9155/54/19/008
[15] Van Sloun R, Pandharipande A, Mischi M et al (2015) Compressed sensing for ultrasound computed tomography. IEEE Trans Bio-med Eng 62(6):1660 · doi:10.1109/TBME.2015.2422135
[16] Vest CM (1979) Holographic interferometry. Wiley, New York
[17] Yanover C, Weiss Y (2004) Finding the AI most probable configurations using loopy belief propagation. Adv Neural Inf Process Syst 16:289
[18] Yan M, Vese L A (2011) Expectation maximization and total variation-based model for computed tomography reconstruction from undersampled data. In: SPIE medical imaging. International Society for Optics and Photonics, p 79612X-79612X-8
[19] Zeng GL (2015) The ML-EM algorithm is not Ooptimal for Poisson noise [J]. Nucl Sci IEEE Tran 62(5):2096-2101 · doi:10.1109/TNS.2015.2475128
[20] Zeng W, Zhong X, Li J (2013) Eliminating sign ambiguity for phase extraction from a single interferogram. Opt Eng 52(12):124102 · doi:10.1117/1.OE.52.12.124102
[21] Zhao R, Li X, Sun P (2015) An improved windowed Fourier transform filter algorithm. Opt Laser Technol 74:103-107 · doi:10.1016/j.optlastec.2015.06.005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.