# American Institute of Mathematical Sciences

August  2013, 7(3): 1007-1029. doi: 10.3934/ipi.2013.7.1007

## General convergent expectation maximization (EM)-type algorithms for image reconstruction

 1 Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095, United States 2 Department of Radiological Sciences, University of California, Los Angeles, Los Angeles, CA 90095, United States 3 Department of Computer Sciences, University of California, Los Angeles, Los Angeles, CA 90095, United States 4 Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095-1555, United States

Received  July 2012 Revised  May 2013 Published  September 2013

Obtaining high quality images is very important in many areas of applied sciences, such as medical imaging, optical microscopy, and astronomy. Image reconstruction can be considered as solving the ill-posed and inverse problem $y=Ax+n$, where $x$ is the image to be reconstructed and $n$ is the unknown noise. In this paper, we propose general robust expectation maximization (EM)-type algorithms for image reconstruction. Both Poisson noise and Gaussian noise types are considered. The EM-type algorithms are performed using iteratively EM (or SART for weighted Gaussian noise) and regularization in the image domain. The convergence of these algorithms is proved in several ways: EM with a priori information and alternating minimization methods. To show the efficiency of EM-type algorithms, the application in computerized tomography reconstruction is chosen.
Citation: Ming Yan, Alex A. T. Bui, Jason Cong, Luminita A. Vese. General convergent expectation maximization (EM)-type algorithms for image reconstruction. Inverse Problems & Imaging, 2013, 7 (3) : 1007-1029. doi: 10.3934/ipi.2013.7.1007
##### References:
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Gordon, Component averaging: An efficient iterative parallel algorithm for large and sparse unstructured problems,, Parallel Computing, 27 (2001), 777. doi: 10.1016/S0167-8191(00)00100-9. Google Scholar [11] J. Chen, J. Cong, L. A. Vese, J. D. Villasenor, M. Yan and Y. Zou, A hybrid architecture for compressive sensing 3-D CT reconstruction,, IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 2 (2012), 616. doi: 10.1109/JETCAS.2012.2221530. Google Scholar [12] J. A. Conchello and J. G. McNally, Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy,, in, 2655 (1996), 199. doi: 10.1117/12.237477. Google Scholar [13] A. Dempster, N. Laird and D. Rubin, Maximum likelihood from incomplete data via the EM algorithm,, Journal of the Royal Statistical Society Series B, 39 (1977), 1. Google Scholar [14] N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J. C. Olivo-Marin and J. Zerubia, Richardson-Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution,, Microscopy Research and Technique, 69 (2006), 260. doi: 10.1002/jemt.20294. Google Scholar [15] S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6 (1984), 721. doi: 10.1109/TPAMI.1984.4767596. Google Scholar [16] R. Gordon, R. Bender and G. Herman, Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography,, Journal of Theoretical Biology, 29 (1970), 471. doi: 10.1016/0022-5193(70)90109-8. Google Scholar [17] P. J. Green, On use of the EM algorithm for penalized likelihood estimation,, Journal of the Royal Statistical Society Series B, 52 (1990), 443. doi: 10.2307/2345668. Google Scholar [18] U. Grenander, "Tutorial in Pattern Theory,", Lecture Notes Volume, (1984). Google Scholar [19] Z. T. Harmany, R. F. Marcia and R. M. Willett, Sparse Poisson intensity reconstruction algorithms,, in, (2009), 634. doi: 10.1109/SSP.2009.5278495. Google Scholar [20] G. Herman, "Fundamentals of Computerized Tomography: Image Reconstruction From Projection,", Second edition. Advances in Pattern Recognition. Springer, (2009). doi: 10.1007/978-1-84628-723-7. Google Scholar [21] H. Hurwitz, Entropy reduction in Bayesian analysis of measurements,, Physics Review A, 12 (1975), 698. doi: 10.1103/PhysRevA.12.698. Google Scholar [22] S. Jafarpour, R. Willett, M. Raginsky and R. Calderbank, Performance bounds for expander-based compressed sensing in the presence of Poisson noise,, in, (2009), 513. doi: 10.1109/ACSSC.2009.5469879. Google Scholar [23] X. Jia, Y. Lou, R. Li, W. Y. Song and S. B. Jiang, GPU-based fast cone beam CT reconstruction from undersampled and noisy projection data via total variation,, Medical Physics, 37 (2010), 1757. doi: 10.1118/1.3371691. Google Scholar [24] M. Jiang and G. Wang, Convergence of the simultaneous algebraic reconstruction technique (SART),, IEEE Transaction on Image Processing, 12 (2003), 957. doi: 10.1109/TIP.2003.815295. Google Scholar [25] M. Jiang and G. Wang, Convergence studies on iterative algorithms for image reconstruction,, IEEE Transactions on Medical Imaging, 22 (2003), 569. doi: 10.1109/TMI.2003.812253. Google Scholar [26] S. Joshi and M. I. Miller, Maximum a posteriori estimation with Good's roughness for three-dimensional optical sectioning microscopy,, Journal of the Optical Society of America A, 10 (1993), 1078. doi: 10.1364/JOSAA.10.001078. Google Scholar [27] M. Jung, E. Resmerita and L. A. Vese, Dual norm based iterative methods for image restoration,, Journal of Mathematical Imaging and Vision, 44 (2012), 128. doi: 10.1007/s10851-011-0318-7. Google Scholar [28] A. Kak and M. Slaney, "Principles of Computerized Tomographic Imaging,", Reprint of the 1988 original. Classics in Applied Mathematics, (1988). doi: 10.1137/1.9780898719277. Google Scholar [29] W. Karush, "Minima of Functions of Several Variables With Inequalities as Side Constraints,'', Master's thesis, (1939). Google Scholar [30] H. Kuhn and A. Tucker, Nonlinear programming,, in, (1950). Google Scholar [31] T. Le, R. Chartrand and T. J. Asaki, A variational approach to reconstructing images corrupted by Poisson noise,, Journal of Mathematical Imaging and Vision, 27 (2007), 257. doi: 10.1007/s10851-007-0652-y. Google Scholar [32] E. Levitan and G. T. Herman, A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography,, IEEE Transactions on Medial Imaging, 6 (1987), 185. doi: 10.1109/TMI.1987.4307826. Google Scholar [33] L. B. Lucy, An iterative technique for the rectification of observed distributions,, Astronomical Journal, 79 (1974), 745. doi: 10.1086/111605. Google Scholar [34] J. Markham and J. A. Conchello, Fast maximum-likelihood image-restoration algorithms for three-dimensional fluorescence microscopy,, Journal of the Optical Society America A, 18 (2001), 1062. doi: 10.1364/JOSAA.18.001062. Google Scholar [35] F. Natterer and F. Wubbeling, "Mathematical Methods in Image Reconstruction,", SIAM Monographs on Mathematical Modeling and Computation. Society for Industrial and Applied Mathematics, (2001). doi: 10.1137/1.9780898718324. Google Scholar [36] Y. Pan, R. Whitaker, A. Cheryauka and D. Ferguson, Feasibility of GPU-assisted iterative image reconstruction for mobile C-arm CT,, in, 7258 (2009). doi: 10.1117/12.812162. Google Scholar [37] W. H. Richardson, Bayesian-based iterative method of image restoration,, Journal of the Optical Society America, 62 (1972), 55. doi: 10.1364/JOSA.62.000055. Google Scholar [38] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. Google Scholar [39] S. Setzer, G. Steidl and T. Teuber, Deblurring Poissonian images by split Bregman techniques,, Journal of Visual Communication and Image Representation, 21 (2010), 193. doi: 10.1016/j.jvcir.2009.10.006. Google Scholar [40] H. Shah, A common framework for curve evolution, segmentation and anisotropic diffusion,, in, (1996), 136. doi: 10.1109/CVPR.1996.517065. Google Scholar [41] L. Shepp and B. Logan, The Fourier reconstruction of a head section,, IEEE Transaction on Nuclear Science, 21 (1974), 21. doi: 10.1109/TNS.1974.6499235. Google Scholar [42] L. Shepp and Y. Vardi, Maximum likelihood reconstruction for emission tomography,, IEEE Transaction on Medical Imaging, 1 (1982), 113. doi: 10.1109/tmi.1982.4307558. Google Scholar [43] R. Siddon, Fast calculation of the exact radiological path for a three-dimensional CT array,, Medical Physics, 12 (1985), 252. doi: 10.1118/1.595715. Google Scholar [44] E. Y. Sidky, R. Chartrand and X. Pan, Image reconstruction from few views by non-convex optimization,, in, 5 (2007), 3526. doi: 10.1109/NSSMIC.2007.4436889. Google Scholar [45] E. Y Sidky, J. H. Jorgensen 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). doi: 10.1088/0031-9155/57/10/3065. Google Scholar [46] A. N. Tychonoff and V. Y. Arsenin, "Solution of Ill-posed Problems,", Winston & Sons, (1977). Google Scholar [47] J. Wang and Y. Zheng, On the convergence of generalized simultaneous iterative reconstruction algorithms,, IEEE Transaction on Image Processing, 16 (2007), 1. doi: 10.1109/TIP.2006.887725. Google Scholar [48] R. M. Willett, Z. T. Harmany and R. F. Marcia, Poisson image reconstruction with total variation regularization,, Proceedings of 17th IEEE International Conference on Image Processing, (2010), 4177. doi: 10.1109/ICIP.2010.5649600. Google Scholar [49] M. Yan and L. A. Vese, Expectation maximization and total variation based model for computed tomography reconstruction from undersampled data,, in, 7961 (2011). doi: 10.1117/12.878238. Google Scholar [50] M. Yan, Convergence analysis of SART: Optimization and statistics,, International Journal of Computer Mathematics, 90 (2013), 30. doi: 10.1080/00207160.2012.709933. Google Scholar [51] M. Yan, J. Chen, L. A. Vese, J. D. Villasenor, A. A. T. Bui and J. Cong, EM+TV based reconstruction for cone-beam CT with reduced radiation,, in, 6938 (2011), 1. doi: 10.1007/978-3-642-24028-7_1. Google Scholar [52] H. Yu and G. Wang, SART-type image reconstruction from a limited number of projections with the sparsity constraint,, Journal of Biomedical Imaging, 2010 (2010), 1. doi: 10.1155/2010/934847. Google Scholar [53] H. Zhao and A. J. Reader, Fast ray-tracing technique to calculate line integral paths in voxel arrays,, in IEEE Nuclear Science Symposium Conference Record, 4 (2003), 2808. doi: 10.1109/NSSMIC.2003.1352469. Google Scholar [54] D. Zhu, M. Razaz and R. Lee, Adaptive penalty likelihood for reconstruction of multi-dimensional confocal microscopy images,, Computerized Medical Imaging and Graphics, 29 (2005), 319. doi: 10.1016/j.compmedimag.2004.12.004. Google Scholar [55] , Compressive Sensing Resources,, , (). Google Scholar

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##### References:
 [1] R. Acar and C. R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems,, Inverse Problems, 10 (1994), 1217. doi: 10.1088/0266-5611/10/6/003. Google Scholar [2] R. Alicandro, A. braides and J. Shah, Free-discontinuity problems via functionals involving the $L^1$-norm of the gradient and their approximation,, Interfaces and Free Boundaries, 1 (1999), 17. doi: 10.4171/IFB/2. Google Scholar [3] A. Andersen, Algebraic reconstruction in CT from limited views,, IEEE Transactions on Medical Imaging, 8 (1989), 50. doi: 10.1109/42.20361. Google Scholar [4] A. Andersen and A. Kak, Simultaneous algebraic reconstruction technique (SART): A superior implementation of the ART algorithm,, Ultrasonic Imaging, 6 (1984), 81. doi: 10.1177/016173468400600107. Google Scholar [5] C. Atkinson and J. Soria, An efficient simultaneous reconstruction technique for tomographic particle image velocimetry,, Experiments in Fluids, 47 (2009), 553. doi: 10.1007/s00348-009-0728-0. Google Scholar [6] C. Brune, M. Burger, A. Sawatzky, T. Kosters and F. Wubbeling, Forward-Backward EM-tV methods for inverse problems with poisson noise,, Preprint, (2009). Google Scholar [7] C. Brune, A. Sawatzky and M. Burger, Bregman-EM-TV methods with application to optical nanoscopy,, Lecture Notes in Computer Science, 5567 (2009), 235. doi: 10.1007/978-3-642-02256-2_20. Google Scholar [8] C. Brune, A. Sawatzky and M. Burger, Primal and dual Bregman methods with application to optical nanoscopy,, International Journal of Computer Vision, 92 (2011), 211. doi: 10.1007/s11263-010-0339-5. Google Scholar [9] Y. Censor and T. Elfving, Block-iterative algorithms with diagonally scaled oblique projections for the linear feasibility problem,, SIAM Journal on Matrix Analysis and Applications, 24 (2002), 40. doi: 10.1137/S089547980138705X. Google Scholar [10] Y. Censor, D. Gordon and R. Gordon, Component averaging: An efficient iterative parallel algorithm for large and sparse unstructured problems,, Parallel Computing, 27 (2001), 777. doi: 10.1016/S0167-8191(00)00100-9. Google Scholar [11] J. Chen, J. Cong, L. A. Vese, J. D. Villasenor, M. Yan and Y. Zou, A hybrid architecture for compressive sensing 3-D CT reconstruction,, IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 2 (2012), 616. doi: 10.1109/JETCAS.2012.2221530. Google Scholar [12] J. A. Conchello and J. G. McNally, Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy,, in, 2655 (1996), 199. doi: 10.1117/12.237477. Google Scholar [13] A. Dempster, N. Laird and D. Rubin, Maximum likelihood from incomplete data via the EM algorithm,, Journal of the Royal Statistical Society Series B, 39 (1977), 1. Google Scholar [14] N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J. C. Olivo-Marin and J. Zerubia, Richardson-Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution,, Microscopy Research and Technique, 69 (2006), 260. doi: 10.1002/jemt.20294. Google Scholar [15] S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6 (1984), 721. doi: 10.1109/TPAMI.1984.4767596. Google Scholar [16] R. Gordon, R. Bender and G. Herman, Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography,, Journal of Theoretical Biology, 29 (1970), 471. doi: 10.1016/0022-5193(70)90109-8. Google Scholar [17] P. J. Green, On use of the EM algorithm for penalized likelihood estimation,, Journal of the Royal Statistical Society Series B, 52 (1990), 443. doi: 10.2307/2345668. Google Scholar [18] U. Grenander, "Tutorial in Pattern Theory,", Lecture Notes Volume, (1984). Google Scholar [19] Z. T. Harmany, R. F. Marcia and R. M. Willett, Sparse Poisson intensity reconstruction algorithms,, in, (2009), 634. doi: 10.1109/SSP.2009.5278495. Google Scholar [20] G. Herman, "Fundamentals of Computerized Tomography: Image Reconstruction From Projection,", Second edition. Advances in Pattern Recognition. Springer, (2009). doi: 10.1007/978-1-84628-723-7. Google Scholar [21] H. Hurwitz, Entropy reduction in Bayesian analysis of measurements,, Physics Review A, 12 (1975), 698. doi: 10.1103/PhysRevA.12.698. Google Scholar [22] S. Jafarpour, R. Willett, M. Raginsky and R. Calderbank, Performance bounds for expander-based compressed sensing in the presence of Poisson noise,, in, (2009), 513. doi: 10.1109/ACSSC.2009.5469879. Google Scholar [23] X. Jia, Y. Lou, R. Li, W. Y. Song and S. B. Jiang, GPU-based fast cone beam CT reconstruction from undersampled and noisy projection data via total variation,, Medical Physics, 37 (2010), 1757. doi: 10.1118/1.3371691. Google Scholar [24] M. Jiang and G. Wang, Convergence of the simultaneous algebraic reconstruction technique (SART),, IEEE Transaction on Image Processing, 12 (2003), 957. doi: 10.1109/TIP.2003.815295. Google Scholar [25] M. Jiang and G. Wang, Convergence studies on iterative algorithms for image reconstruction,, IEEE Transactions on Medical Imaging, 22 (2003), 569. doi: 10.1109/TMI.2003.812253. Google Scholar [26] S. Joshi and M. I. Miller, Maximum a posteriori estimation with Good's roughness for three-dimensional optical sectioning microscopy,, Journal of the Optical Society of America A, 10 (1993), 1078. doi: 10.1364/JOSAA.10.001078. Google Scholar [27] M. Jung, E. Resmerita and L. A. Vese, Dual norm based iterative methods for image restoration,, Journal of Mathematical Imaging and Vision, 44 (2012), 128. doi: 10.1007/s10851-011-0318-7. Google Scholar [28] A. Kak and M. Slaney, "Principles of Computerized Tomographic Imaging,", Reprint of the 1988 original. Classics in Applied Mathematics, (1988). doi: 10.1137/1.9780898719277. Google Scholar [29] W. Karush, "Minima of Functions of Several Variables With Inequalities as Side Constraints,'', Master's thesis, (1939). Google Scholar [30] H. Kuhn and A. Tucker, Nonlinear programming,, in, (1950). Google Scholar [31] T. Le, R. Chartrand and T. J. Asaki, A variational approach to reconstructing images corrupted by Poisson noise,, Journal of Mathematical Imaging and Vision, 27 (2007), 257. doi: 10.1007/s10851-007-0652-y. Google Scholar [32] E. Levitan and G. T. Herman, A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography,, IEEE Transactions on Medial Imaging, 6 (1987), 185. doi: 10.1109/TMI.1987.4307826. Google Scholar [33] L. B. Lucy, An iterative technique for the rectification of observed distributions,, Astronomical Journal, 79 (1974), 745. doi: 10.1086/111605. Google Scholar [34] J. Markham and J. A. Conchello, Fast maximum-likelihood image-restoration algorithms for three-dimensional fluorescence microscopy,, Journal of the Optical Society America A, 18 (2001), 1062. doi: 10.1364/JOSAA.18.001062. Google Scholar [35] F. Natterer and F. Wubbeling, "Mathematical Methods in Image Reconstruction,", SIAM Monographs on Mathematical Modeling and Computation. Society for Industrial and Applied Mathematics, (2001). doi: 10.1137/1.9780898718324. Google Scholar [36] Y. Pan, R. Whitaker, A. Cheryauka and D. Ferguson, Feasibility of GPU-assisted iterative image reconstruction for mobile C-arm CT,, in, 7258 (2009). doi: 10.1117/12.812162. Google Scholar [37] W. H. Richardson, Bayesian-based iterative method of image restoration,, Journal of the Optical Society America, 62 (1972), 55. doi: 10.1364/JOSA.62.000055. Google Scholar [38] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. Google Scholar [39] S. Setzer, G. Steidl and T. Teuber, Deblurring Poissonian images by split Bregman techniques,, Journal of Visual Communication and Image Representation, 21 (2010), 193. doi: 10.1016/j.jvcir.2009.10.006. Google Scholar [40] H. Shah, A common framework for curve evolution, segmentation and anisotropic diffusion,, in, (1996), 136. doi: 10.1109/CVPR.1996.517065. Google Scholar [41] L. Shepp and B. Logan, The Fourier reconstruction of a head section,, IEEE Transaction on Nuclear Science, 21 (1974), 21. doi: 10.1109/TNS.1974.6499235. Google Scholar [42] L. Shepp and Y. Vardi, Maximum likelihood reconstruction for emission tomography,, IEEE Transaction on Medical Imaging, 1 (1982), 113. doi: 10.1109/tmi.1982.4307558. Google Scholar [43] R. Siddon, Fast calculation of the exact radiological path for a three-dimensional CT array,, Medical Physics, 12 (1985), 252. doi: 10.1118/1.595715. Google Scholar [44] E. Y. Sidky, R. Chartrand and X. Pan, Image reconstruction from few views by non-convex optimization,, in, 5 (2007), 3526. doi: 10.1109/NSSMIC.2007.4436889. Google Scholar [45] E. Y Sidky, J. H. Jorgensen 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). doi: 10.1088/0031-9155/57/10/3065. Google Scholar [46] A. N. Tychonoff and V. Y. Arsenin, "Solution of Ill-posed Problems,", Winston & Sons, (1977). Google Scholar [47] J. Wang and Y. Zheng, On the convergence of generalized simultaneous iterative reconstruction algorithms,, IEEE Transaction on Image Processing, 16 (2007), 1. doi: 10.1109/TIP.2006.887725. Google Scholar [48] R. M. Willett, Z. T. Harmany and R. F. Marcia, Poisson image reconstruction with total variation regularization,, Proceedings of 17th IEEE International Conference on Image Processing, (2010), 4177. doi: 10.1109/ICIP.2010.5649600. Google Scholar [49] M. Yan and L. A. Vese, Expectation maximization and total variation based model for computed tomography reconstruction from undersampled data,, in, 7961 (2011). doi: 10.1117/12.878238. Google Scholar [50] M. Yan, Convergence analysis of SART: Optimization and statistics,, International Journal of Computer Mathematics, 90 (2013), 30. doi: 10.1080/00207160.2012.709933. Google Scholar [51] M. Yan, J. Chen, L. A. Vese, J. D. Villasenor, A. A. T. Bui and J. Cong, EM+TV based reconstruction for cone-beam CT with reduced radiation,, in, 6938 (2011), 1. doi: 10.1007/978-3-642-24028-7_1. Google Scholar [52] H. Yu and G. Wang, SART-type image reconstruction from a limited number of projections with the sparsity constraint,, Journal of Biomedical Imaging, 2010 (2010), 1. doi: 10.1155/2010/934847. Google Scholar [53] H. Zhao and A. J. Reader, Fast ray-tracing technique to calculate line integral paths in voxel arrays,, in IEEE Nuclear Science Symposium Conference Record, 4 (2003), 2808. doi: 10.1109/NSSMIC.2003.1352469. Google Scholar [54] D. Zhu, M. Razaz and R. Lee, Adaptive penalty likelihood for reconstruction of multi-dimensional confocal microscopy images,, Computerized Medical Imaging and Graphics, 29 (2005), 319. doi: 10.1016/j.compmedimag.2004.12.004. Google Scholar [55] , Compressive Sensing Resources,, , (). Google Scholar
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