# 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 & 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. Google Scholar [2] G. T. Herman, Image Reconstruction from Projection: the Fundamentals of Computerized Tomography, Academic Press, New York 1980.  Google Scholar [3] F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, Society for Industrial and Applied Mathematics, Philadelphia, 2001. doi: 10.1137/1.9780898718324.  Google Scholar [4] O. Tretiak and C. Metz, The exponential Radon transform, SIAM Journal on Applied Mathematics, 39(1980), 341-354. doi: 10.1137/0139029.  Google Scholar [5] F. Natterer, Inversion of the attenuated Radon transform, Inverse Problems, 17 (2001), 113-119. doi: 10.1088/0266-5611/17/1/309.  Google Scholar [6] R. G. Novikov, An inversion formula for the attenuated X-ray transformation, Arkiv för Matematik, 40 (2002), 145-167. doi: 10.1007/BF02384507.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] L. A. Shepp and Y. Vardi, Maximum likelihood restoration for emission tomography, IEEE Transactions on Medical Imaging, 1 (1982), 113-122. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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 . Google Scholar [29] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physca D., 60 (1992), 259-268. Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [33] H. Yu and G. Wang, Compressed sensing based interior tomography, Physics in Medicine and Biology, 54 (2009), 2791-2805. Google Scholar [34] J. A. fessler, Matlab code for emission tomograph,, Available from: , ().   Google Scholar

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. Google Scholar [2] G. T. Herman, Image Reconstruction from Projection: the Fundamentals of Computerized Tomography, Academic Press, New York 1980.  Google Scholar [3] F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, Society for Industrial and Applied Mathematics, Philadelphia, 2001. doi: 10.1137/1.9780898718324.  Google Scholar [4] O. Tretiak and C. Metz, The exponential Radon transform, SIAM Journal on Applied Mathematics, 39(1980), 341-354. doi: 10.1137/0139029.  Google Scholar [5] F. Natterer, Inversion of the attenuated Radon transform, Inverse Problems, 17 (2001), 113-119. doi: 10.1088/0266-5611/17/1/309.  Google Scholar [6] R. G. Novikov, An inversion formula for the attenuated X-ray transformation, Arkiv för Matematik, 40 (2002), 145-167. doi: 10.1007/BF02384507.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] L. A. Shepp and Y. Vardi, Maximum likelihood restoration for emission tomography, IEEE Transactions on Medical Imaging, 1 (1982), 113-122. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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 . Google Scholar [29] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physca D., 60 (1992), 259-268. Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [33] H. Yu and G. Wang, Compressed sensing based interior tomography, Physics in Medicine and Biology, 54 (2009), 2791-2805. Google Scholar [34] J. A. fessler, Matlab code for emission tomograph,, Available from: , ().   Google Scholar
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