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

Previous Article
The single-grid multilevel method and its applications
IPI Home
This Issue
Next Article
Fast total variation wavelet inpainting via approximated primal-dual hybrid gradient algorithm

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

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

Ming Yan ^1, , Alex A. T. Bui ^2, , Jason Cong ^3, and Luminita A. Vese ^4,

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

Abstract
Related Papers

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.

Keywords: Richardson-Lucy, computerized tomography., image reconstruction, Expectation maximization (EM), simultaneous algebraic reconstruction technique (SART), total variation (TV).

Mathematics Subject Classification: Primary: 90C30, 94A08; Secondary: 90C26, 92C5.

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:

[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.
[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.
[3]	A. Andersen, Algebraic reconstruction in CT from limited views,, IEEE Transactions on Medical Imaging, 8 (1989), 50. doi: 10.1109/42.20361.
[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.
[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.
[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).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	U. Grenander, "Tutorial in Pattern Theory,", Lecture Notes Volume, (1984).
[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.
[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.
[21]	H. Hurwitz, Entropy reduction in Bayesian analysis of measurements,, Physics Review A, 12 (1975), 698. doi: 10.1103/PhysRevA.12.698.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[29]	W. Karush, "Minima of Functions of Several Variables With Inequalities as Side Constraints,'', Master's thesis, (1939).
[30]	H. Kuhn and A. Tucker, Nonlinear programming,, in, (1950).
[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.
[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.
[33]	L. B. Lucy, An iterative technique for the rectification of observed distributions,, Astronomical Journal, 79 (1974), 745. doi: 10.1086/111605.
[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.
[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.
[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.
[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.
[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.
[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.
[40]	H. Shah, A common framework for curve evolution, segmentation and anisotropic diffusion,, in, (1996), 136. doi: 10.1109/CVPR.1996.517065.
[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.
[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.
[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.
[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.
[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.
[46]	A. N. Tychonoff and V. Y. Arsenin, "Solution of Ill-posed Problems,", Winston & Sons, (1977).
[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.
[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.
[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.
[50]	M. Yan, Convergence analysis of SART: Optimization and statistics,, International Journal of Computer Mathematics, 90 (2013), 30. doi: 10.1080/00207160.2012.709933.
[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.
[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.
[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.
[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.
[55]	, Compressive Sensing Resources,, , ().

show all references

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.
[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.
[3]	A. Andersen, Algebraic reconstruction in CT from limited views,, IEEE Transactions on Medical Imaging, 8 (1989), 50. doi: 10.1109/42.20361.
[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.
[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.
[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).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	U. Grenander, "Tutorial in Pattern Theory,", Lecture Notes Volume, (1984).
[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.
[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.
[21]	H. Hurwitz, Entropy reduction in Bayesian analysis of measurements,, Physics Review A, 12 (1975), 698. doi: 10.1103/PhysRevA.12.698.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[29]	W. Karush, "Minima of Functions of Several Variables With Inequalities as Side Constraints,'', Master's thesis, (1939).
[30]	H. Kuhn and A. Tucker, Nonlinear programming,, in, (1950).
[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.
[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.
[33]	L. B. Lucy, An iterative technique for the rectification of observed distributions,, Astronomical Journal, 79 (1974), 745. doi: 10.1086/111605.
[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.
[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.
[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.
[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.
[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.
[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.
[40]	H. Shah, A common framework for curve evolution, segmentation and anisotropic diffusion,, in, (1996), 136. doi: 10.1109/CVPR.1996.517065.
[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.
[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.
[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.
[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.
[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.
[46]	A. N. Tychonoff and V. Y. Arsenin, "Solution of Ill-posed Problems,", Winston & Sons, (1977).
[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.
[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.
[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.
[50]	M. Yan, Convergence analysis of SART: Optimization and statistics,, International Journal of Computer Mathematics, 90 (2013), 30. doi: 10.1080/00207160.2012.709933.
[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.
[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.
[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.
[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.
[55]	, Compressive Sensing Resources,, , ().

[1]	Yunhai Xiao, Junfeng Yang, Xiaoming Yuan. Alternating algorithms for total variation image reconstruction from random projections. Inverse Problems & Imaging, 2012, 6 (3) : 547-563. doi: 10.3934/ipi.2012.6.547
[2]	Lacramioara Grecu, Constantin Popa. Constrained SART algorithm for inverse problems in image reconstruction. Inverse Problems & Imaging, 2013, 7 (1) : 199-216. doi: 10.3934/ipi.2013.7.199
[3]	Tim Kreutzmann, Andreas Rieder. Geometric reconstruction in bioluminescence tomography. Inverse Problems & Imaging, 2014, 8 (1) : 173-197. doi: 10.3934/ipi.2014.8.173
[4]	Jianjun Zhang, Yunyi Hu, James G. Nagy. A scaled gradient method for digital tomographic image reconstruction. Inverse Problems & Imaging, 2018, 12 (1) : 239-259. doi: 10.3934/ipi.2018010
[5]	Adriana González, Laurent Jacques, Christophe De Vleeschouwer, Philippe Antoine. Compressive optical deflectometric tomography: A constrained total-variation minimization approach. Inverse Problems & Imaging, 2014, 8 (2) : 421-457. doi: 10.3934/ipi.2014.8.421
[6]	Juan C. Moreno, V. B. Surya Prasath, João C. Neves. Color image processing by vectorial total variation with gradient channels coupling. Inverse Problems & Imaging, 2016, 10 (2) : 461-497. doi: 10.3934/ipi.2016008
[7]	Zhengmeng Jin, Chen Zhou, Michael K. Ng. A coupled total variation model with curvature driven for image colorization. Inverse Problems & Imaging, 2016, 10 (4) : 1037-1055. doi: 10.3934/ipi.2016031
[8]	Leonid Kunyansky. Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries. Inverse Problems & Imaging, 2012, 6 (1) : 111-131. doi: 10.3934/ipi.2012.6.111
[9]	Larisa Beilina, Michel Cristofol, Kati Niinimäki. Optimization approach for the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions from limited observations. Inverse Problems & Imaging, 2015, 9 (1) : 1-25. doi: 10.3934/ipi.2015.9.1
[10]	Chengxiang Wang, Li Zeng, Yumeng Guo, Lingli Zhang. Wavelet tight frame and prior image-based image reconstruction from limited-angle projection data. Inverse Problems & Imaging, 2017, 11 (6) : 917-948. doi: 10.3934/ipi.2017043
[11]	Raymond H. Chan, Haixia Liang, Suhua Wei, Mila Nikolova, Xue-Cheng Tai. High-order total variation regularization approach for axially symmetric object tomography from a single radiograph. Inverse Problems & Imaging, 2015, 9 (1) : 55-77. doi: 10.3934/ipi.2015.9.55
[12]	Baoli Shi, Zhi-Feng Pang, Jing Xu. Image segmentation based on the hybrid total variation model and the K-means clustering strategy. Inverse Problems & Imaging, 2016, 10 (3) : 807-828. doi: 10.3934/ipi.2016022
[13]	Xavier Bresson, Tony F. Chan. Fast dual minimization of the vectorial total variation norm and applications to color image processing. Inverse Problems & Imaging, 2008, 2 (4) : 455-484. doi: 10.3934/ipi.2008.2.455
[14]	Henrik Garde, Kim Knudsen. 3D reconstruction for partial data electrical impedance tomography using a sparsity prior. Conference Publications, 2015, 2015 (special) : 495-504. doi: 10.3934/proc.2015.0495
[15]	Jianhong (Jackie) Shen, Sung Ha Kang. Quantum TV and applications in image processing. Inverse Problems & Imaging, 2007, 1 (3) : 557-575. doi: 10.3934/ipi.2007.1.557
[16]	Yonggui Zhu, Yuying Shi, Bin Zhang, Xinyan Yu. Weighted-average alternating minimization method for magnetic resonance image reconstruction based on compressive sensing. Inverse Problems & Imaging, 2014, 8 (3) : 925-937. doi: 10.3934/ipi.2014.8.925
[17]	Yunmei Chen, Xiaojing Ye, Feng Huang. A novel method and fast algorithm for MR image reconstruction with significantly under-sampled data. Inverse Problems & Imaging, 2010, 4 (2) : 223-240. doi: 10.3934/ipi.2010.4.223
[18]	Ke Chen, Yiqiu Dong, Michael Hintermüller. A nonlinear multigrid solver with line Gauss-Seidel-semismooth-Newton smoother for the Fenchel pre-dual in total variation based image restoration. Inverse Problems & Imaging, 2011, 5 (2) : 323-339. doi: 10.3934/ipi.2011.5.323
[19]	Jiaqing Yang, Bo Zhang, Ruming Zhang. Reconstruction of penetrable grating profiles. Inverse Problems & Imaging, 2013, 7 (4) : 1393-1407. doi: 10.3934/ipi.2013.7.1393
[20]	Xiaoqun Zhang, Tony F. Chan. Wavelet inpainting by nonlocal total variation. Inverse Problems & Imaging, 2010, 4 (1) : 191-210. doi: 10.3934/ipi.2010.4.191

2016 Impact Factor: 1.094

American Institute of Mathematical Sciences