# American Institute of Mathematical Sciences

February  2014, 8(1): 223-246. doi: 10.3934/ipi.2014.8.223

## Superiorization of EM algorithm and its application in Single-Photon Emission Computed Tomography(SPECT)

 1 LMAM, School of Mathematical Sciences, Peking University, No.5 Yiheyuan Road Haidian District, Beijing, 100871, China, China

Received  December 2012 Revised  May 2013 Published  March 2014

In this paper, we presented an efficient algorithm to implement the regularization reconstruction of SPECT. Image reconstruction with priori assumptions is usually modeled as a constrained optimization problem. However, there is no efficient algorithm to solve it due to the large scale of the problem. In this paper, we used the superiorization of the expectation maximization (EM) iteration to implement the regularization reconstruction of SPECT. We first investigated the convergent conditions of the EM iteration in the presence of perturbations. Secondly, we designed the superiorized EM algorithm based on the convergent conditions, and then proposed a modified version of it. Furthermore, we gave two methods to generate desired perturbations for two special objective functions. Numerical experiments for SPECT image reconstruction were conducted to validate the performance of the proposed algorithms. The experiments show that the superiorized EM algorithms are more stable and robust for noised projection data and initial image than the classic EM algorithm, and outperform the classic EM algorithm in terms of mean square error and visual quality of the reconstructed images.
Citation: Shousheng Luo, Tie Zhou. Superiorization of EM algorithm and its application in Single-Photon Emission Computed Tomography(SPECT). Inverse Problems and Imaging, 2014, 8 (1) : 223-246. doi: 10.3934/ipi.2014.8.223
##### References:
 [1] M. N. Wernick and J. N. Aarsvold, Emission Tomography: the Fundamentals of PET and SPECT, Elsevier Acdamic Press, California, 2004. [2] G. T. Herman, Image Reconstruction from Projection: the Fundamentals of Computerized Tomography, Academic Press, New York 1980. [3] F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, Society for Industrial and Applied Mathematics, Philadelphia, 2001. doi: 10.1137/1.9780898718324. [4] O. Tretiak and C. Metz, The exponential Radon transform, SIAM Journal on Applied Mathematics, 39(1980), 341-354. doi: 10.1137/0139029. [5] F. Natterer, Inversion of the attenuated Radon transform, Inverse Problems, 17 (2001), 113-119. doi: 10.1088/0266-5611/17/1/309. [6] R. G. Novikov, An inversion formula for the attenuated X-ray transformation, Arkiv för Matematik, 40 (2002), 145-167. doi: 10.1007/BF02384507. [7] F. Noo and J.-M. Wagner, Image reconstruction in 2D SPECT with $180^\circ$ acquisition, Inverse Problems, 17 (2001), 1357-1371. doi: 10.1088/0266-5611/17/5/308. [8] F. Noo, M. Defrise, J. D. Pack and R. Clackdoyle, Image reconstruction from truncated data in single-photon emission computed tomography with uniform attenuation, Inverse Problems, 23 (2007), 645-667. doi: 10.1088/0266-5611/23/2/011. [9] L. A. Shepp and Y. Vardi, Maximum likelihood restoration for emission tomography, IEEE Transactions on Medical Imaging, 1 (1982), 113-122. [10] H. M. Hudson and R. S. Larkin, Accelerated image reconstruction using ordered subsets of projection data, IEEE Transactions on Medical Imaging, 13 (1994), 601-609. doi: 10.1109/42.363108. [11] I. Hsiao and H. Huang, An accelerated ordered subsets reconstruction algorithm using an accelerating power factor for emission tomography, Physics in Medicine and Biology, 55 (2010), 599-614. doi: 10.1088/0031-9155/55/3/003. [12] Y. Vardi, L. A. Shepp and L. Kaufman, A statistical model for positron emission tomography, Journal of the American Statistical Association, 80 (1985), 8-20. doi: 10.1080/01621459.1985.10477119. [13] R. Gordon, R. Bender and G. T. Herman, Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography, Journal of Theoretical Biology, 29 (1970), 471-482. doi: 10.1016/0022-5193(70)90109-8. [14] Y. Censor, T. Elfving and G. T. Herman, Averaging strings of sequential iterations for convex feasibility problems, in Inherently Parallel Algorithms in Feasibility and Optimization and their Applications, (2001), 101-113. doi: 10.1016/S1570-579X(01)80009-4. [15] E. Y. Sidky, C.-M. Kao and X. Pan, Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT, X-Ray Science and Technology, 14 (2006), 119-139. [16] S. J. LaRoque, E. Y. Sidky and X. Pan, Accurate image reconstruction from few-view and limited-angle data in diffraction tomography, Journal of the Optical Society of America, 25 (2008), 1772-1782. doi: 10.1364/JOSAA.25.001772. [17] X. Pan, E. Y. Sidky and M. Vannier, Why do commercial CT ccanners still employ traditional, filtered backprojection for image reconstruction? Inverse Problems, 25 (2009), 123009. doi: 10.1088/0266-5611/25/12/123009. [18] M. Defrise, C. Vanhove and X. Liu, An algorithm for total variation regularization in high-dimensional linear problems, Inverse Problems, 27 (2011), 065002. doi: 10.1088/0266-5611/27/6/065002. [19] E. Y. Sidky, J. H. Jøgensen and X. Pan, Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle-Pock algorithm, Physics in Medicine and Biology, 57 (2012), 3065-3091. doi: 10.1088/0031-9155/57/10/3065. [20] B. Dong, J. Li and Z. Shen, X-ray CT image reconstruction via wavelet frame based regularization and Radon domain inpainting, Journal of Scientific Computing, 54 (2013), 333-349. doi: 10.1007/s10915-012-9579-6. [21] D. Butnariu, R. Davidi, G. T. Herman and I. G. Kazantsev, Stable convergence behavior under summable perturbations of a class of projection methods for convex feasibility and optimization problem, IEEE Journal of Selected Topics Signal Processing, 1 (2007), 540-547. doi: 10.1109/JSTSP.2007.910263. [22] G. T. Herman and R. Davidi, Image reconstruction from a small number of projections, Inverse Problems, 24 (2008), 045011. doi: 10.1088/0266-5611/24/4/045011. [23] E. Garduo, G. T. Herman and R. Davidi, Reconstruction from a few projections by $l^1$ minimization of the Haar transform, Inverse Problems, 24 (2011), 0550061. doi: 10.1088/0266-5611/24/4/045011. [24] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problem with apllications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145. doi: 10.1007/s10851-010-0251-1. [25] Y. Censor, R. Davidi and G. T. Herman, Perturbation resilience and superiorization of iteration algorithm, Inverse Problems, 25 (2010), 065008. doi: 10.1088/0266-5611/26/6/065008. [26] R. Davidi, G. T. Herman and Y. Censor, Perturbation-resilient block-iterative projection methods with application to image reconstruction from projection, International Transactions in Operational Research, 16 (2009), 505-524. doi: 10.1111/j.1475-3995.2009.00695.x. [27] T. Nikazad, R. Davidi and G. T. Herman, Accelerated perturbation-resilient block-iterative projection methods with application to image reconstruction, Inverse Problems, 28 (2012), 035005. doi: 10.1088/0266-5611/28/3/035005. [28] W. Jin, Y. Censor and M. Jiang, A heuristic superiorization-like approach to bioluminescence tomography, World Congress on Medical Physics and Biomedical Engineering in Proceedings of the International Federation for Medical and Biological Engineering (IFMBE), 39 (2013), 1026-1029 . [29] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physca D., 60 (1992), 259-268. [30] E. Candes, J. Romberg and T. Tao, The dantzig selector: statistical estimation when $p$ is much larger than $n$, the Annals of Statistics, 35 (2007), 2313-2351. doi: 10.1214/009053606000001523. [31] D. L. Snyder, T. J. Schulz and J. A. O. Sullivan, Deblurring subject to nonnegativity constraints, IEEE Transactions on Signal Processing, 40 (1992), 1143-1150. [32] E. Candes, J. Romberg and T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Transactions on Information Theory, 52 (2006), 489-509. doi: 10.1109/TIT.2005.862083. [33] H. Yu and G. Wang, Compressed sensing based interior tomography, Physics in Medicine and Biology, 54 (2009), 2791-2805. [34] J. A. fessler, Matlab code for emission tomograph,, Available from: , ().

show all references

##### References:
 [1] M. N. Wernick and J. N. Aarsvold, Emission Tomography: the Fundamentals of PET and SPECT, Elsevier Acdamic Press, California, 2004. [2] G. T. Herman, Image Reconstruction from Projection: the Fundamentals of Computerized Tomography, Academic Press, New York 1980. [3] F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, Society for Industrial and Applied Mathematics, Philadelphia, 2001. doi: 10.1137/1.9780898718324. [4] O. Tretiak and C. Metz, The exponential Radon transform, SIAM Journal on Applied Mathematics, 39(1980), 341-354. doi: 10.1137/0139029. [5] F. Natterer, Inversion of the attenuated Radon transform, Inverse Problems, 17 (2001), 113-119. doi: 10.1088/0266-5611/17/1/309. [6] R. G. Novikov, An inversion formula for the attenuated X-ray transformation, Arkiv för Matematik, 40 (2002), 145-167. doi: 10.1007/BF02384507. [7] F. Noo and J.-M. Wagner, Image reconstruction in 2D SPECT with $180^\circ$ acquisition, Inverse Problems, 17 (2001), 1357-1371. doi: 10.1088/0266-5611/17/5/308. [8] F. Noo, M. Defrise, J. D. Pack and R. Clackdoyle, Image reconstruction from truncated data in single-photon emission computed tomography with uniform attenuation, Inverse Problems, 23 (2007), 645-667. doi: 10.1088/0266-5611/23/2/011. [9] L. A. Shepp and Y. Vardi, Maximum likelihood restoration for emission tomography, IEEE Transactions on Medical Imaging, 1 (1982), 113-122. [10] H. M. Hudson and R. S. Larkin, Accelerated image reconstruction using ordered subsets of projection data, IEEE Transactions on Medical Imaging, 13 (1994), 601-609. doi: 10.1109/42.363108. [11] I. Hsiao and H. Huang, An accelerated ordered subsets reconstruction algorithm using an accelerating power factor for emission tomography, Physics in Medicine and Biology, 55 (2010), 599-614. doi: 10.1088/0031-9155/55/3/003. [12] Y. Vardi, L. A. Shepp and L. Kaufman, A statistical model for positron emission tomography, Journal of the American Statistical Association, 80 (1985), 8-20. doi: 10.1080/01621459.1985.10477119. [13] R. Gordon, R. Bender and G. T. Herman, Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography, Journal of Theoretical Biology, 29 (1970), 471-482. doi: 10.1016/0022-5193(70)90109-8. [14] Y. Censor, T. Elfving and G. T. Herman, Averaging strings of sequential iterations for convex feasibility problems, in Inherently Parallel Algorithms in Feasibility and Optimization and their Applications, (2001), 101-113. doi: 10.1016/S1570-579X(01)80009-4. [15] E. Y. Sidky, C.-M. Kao and X. Pan, Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT, X-Ray Science and Technology, 14 (2006), 119-139. [16] S. J. LaRoque, E. Y. Sidky and X. Pan, Accurate image reconstruction from few-view and limited-angle data in diffraction tomography, Journal of the Optical Society of America, 25 (2008), 1772-1782. doi: 10.1364/JOSAA.25.001772. [17] X. Pan, E. Y. Sidky and M. Vannier, Why do commercial CT ccanners still employ traditional, filtered backprojection for image reconstruction? Inverse Problems, 25 (2009), 123009. doi: 10.1088/0266-5611/25/12/123009. [18] M. Defrise, C. Vanhove and X. Liu, An algorithm for total variation regularization in high-dimensional linear problems, Inverse Problems, 27 (2011), 065002. doi: 10.1088/0266-5611/27/6/065002. [19] E. Y. Sidky, J. H. Jøgensen and X. Pan, Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle-Pock algorithm, Physics in Medicine and Biology, 57 (2012), 3065-3091. doi: 10.1088/0031-9155/57/10/3065. [20] B. Dong, J. Li and Z. Shen, X-ray CT image reconstruction via wavelet frame based regularization and Radon domain inpainting, Journal of Scientific Computing, 54 (2013), 333-349. doi: 10.1007/s10915-012-9579-6. [21] D. Butnariu, R. Davidi, G. T. Herman and I. G. Kazantsev, Stable convergence behavior under summable perturbations of a class of projection methods for convex feasibility and optimization problem, IEEE Journal of Selected Topics Signal Processing, 1 (2007), 540-547. doi: 10.1109/JSTSP.2007.910263. [22] G. T. Herman and R. Davidi, Image reconstruction from a small number of projections, Inverse Problems, 24 (2008), 045011. doi: 10.1088/0266-5611/24/4/045011. [23] E. Garduo, G. T. Herman and R. Davidi, Reconstruction from a few projections by $l^1$ minimization of the Haar transform, Inverse Problems, 24 (2011), 0550061. doi: 10.1088/0266-5611/24/4/045011. [24] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problem with apllications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145. doi: 10.1007/s10851-010-0251-1. [25] Y. Censor, R. Davidi and G. T. Herman, Perturbation resilience and superiorization of iteration algorithm, Inverse Problems, 25 (2010), 065008. doi: 10.1088/0266-5611/26/6/065008. [26] R. Davidi, G. T. Herman and Y. Censor, Perturbation-resilient block-iterative projection methods with application to image reconstruction from projection, International Transactions in Operational Research, 16 (2009), 505-524. doi: 10.1111/j.1475-3995.2009.00695.x. [27] T. Nikazad, R. Davidi and G. T. Herman, Accelerated perturbation-resilient block-iterative projection methods with application to image reconstruction, Inverse Problems, 28 (2012), 035005. doi: 10.1088/0266-5611/28/3/035005. [28] W. Jin, Y. Censor and M. Jiang, A heuristic superiorization-like approach to bioluminescence tomography, World Congress on Medical Physics and Biomedical Engineering in Proceedings of the International Federation for Medical and Biological Engineering (IFMBE), 39 (2013), 1026-1029 . [29] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physca D., 60 (1992), 259-268. [30] E. Candes, J. Romberg and T. Tao, The dantzig selector: statistical estimation when $p$ is much larger than $n$, the Annals of Statistics, 35 (2007), 2313-2351. doi: 10.1214/009053606000001523. [31] D. L. Snyder, T. J. Schulz and J. A. O. Sullivan, Deblurring subject to nonnegativity constraints, IEEE Transactions on Signal Processing, 40 (1992), 1143-1150. [32] E. Candes, J. Romberg and T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Transactions on Information Theory, 52 (2006), 489-509. doi: 10.1109/TIT.2005.862083. [33] H. Yu and G. Wang, Compressed sensing based interior tomography, Physics in Medicine and Biology, 54 (2009), 2791-2805. [34] J. A. fessler, Matlab code for emission tomograph,, Available from: , ().
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