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EM-TV Methods for Inverse Problems with Poisson Noise

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Level Set and PDE Based Reconstruction Methods in Imaging

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2090))

Abstract

We address the task of reconstructing images corrupted by Poisson noise, which is important in various applications such as fluorescence microscopy (Dey et al., 3D microscopy deconvolution using Richardson-Lucy algorithm with total variation regularization, 2004), positron emission tomography (PET; Vardi et al., J Am Stat Assoc 80:8–20, 1985), or astronomical imaging (Lantéri and Theys, EURASIP J Appl Signal Processing 15:2500–2513, 2005). Here we focus on reconstruction strategies combining the expectation-maximization (EM) algorithm and total variation (TV) based regularization, and present a detailed analysis as well as numerical results. Recently extensions of the well known EM/Richardson-Lucy algorithm received increasing attention for inverse problems with Poisson data (Dey et al., 3D microscopy deconvolution using Richardson-Lucy algorithm with total variation regularization, 2004; Jonsson et al., Total variation regularization in positron emission tomography, 1998; Panin et al., IEEE Trans Nucl Sci 46(6):2202–2210, 1999). However, most of these algorithms for regularizations like TV lead to convergence problems for large regularization parameters, cannot guarantee positivity, and rely on additional approximations (like smoothed TV). The goal of this lecture is to provide accurate, robust and fast EM-TV based methods for computing cartoon reconstructions facilitating post-segmentation and providing a basis for quantification techniques. We illustrate also the performance of the proposed algorithms and confirm the analytical concepts by 2D and 3D synthetic and real-world results in optical nanoscopy and PET.

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References

  1. R. Acar, C.R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems. Inverse Probl. 10(6), 1217–1229 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. H.M. Adorf, R.N. Hook, L.B. Lucy, F.D. Murtagh, Accelerating the Richardson-Lucy restoration algorithm, in Proceedings of the 4th ESO/ST-ECF Data Analysis Workshop, ed. by P.J. Grosboel. (European Southern Observatory, Garching, 1992), pp. 99–103

    Google Scholar 

  3. L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs (Oxford University Press, Oxford, 2000)

    MATH  Google Scholar 

  4. S. Anthoine, J.F. Aujol, Y. Boursier, C. Mélot, On the efficiency of proximal methods in CBCT and PET, in 2011 18th IEEE International Conference on Image Processing (ICIP) (2011). doi: 10.1109/ICIP.2011.6115691

    Google Scholar 

  5. S. Anthoine, J.F. Aujol, Y. Boursier, C. Mélot, Some proximal methods for CBCT and PET, in Proceedings of SPIE (Wavelets and Sparsity XIV), vol. 8138 (2011)

    Google Scholar 

  6. G. Aubert, P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Applied Mathematical Sciences, vol. 147 (Springer, New York, 2002)

    Google Scholar 

  7. J.F. Aujol, Some first-order algorithms for total variation based image restoration. J. Math. Imag. Vis. 34(3), 307–327 (2009)

    Article  MathSciNet  Google Scholar 

  8. M. Bachmayr, M. Burger, Iterative total variation schemes for nonlinear inverse problems. Inverse Probl. 25(10), 105004 (2009)

    MathSciNet  Google Scholar 

  9. D. Baddeley, C. Carl, C. Cremer, 4Pi microscopy deconvolution with a variable point-spread function. Appl. Opt. 45(27), 7056–7064 (2006)

    Article  Google Scholar 

  10. D.L. Bailey, D.W. Townsend, P.E. Valk, M.N. Maisey (eds.), Positron Emission Tomography: Basic Sciences (Springer, New York, 2005)

    Google Scholar 

  11. J.M. Bardsley, An efficient computational method for total variation-penalized Poisson likelihood estimation. Inverse Probl. Imag. 2(2), 167–185 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. J.M. Bardsley, A theoretical framework for the regularization of Poisson likelihood estimation problems. Inverse Probl. Imag. 4(1), 11–17 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. J.M. Bardsley, J. Goldes, Regularization parameter selection methods for ill-posed Poisson maximum likelihood estimation. Inverse Probl. 25(9), 095005 (2009)

    MathSciNet  Google Scholar 

  14. J.M. Bardsley, J. Goldes, Regularization parameter selection and an efficient algorithm for total variation-regularized positron emission tomography. Numer. Algorithms 57(2), 255–271 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. J.M. Bardsley, N. Laobeul, Tikhonov regularized Poisson likelihood estimation: theoretical justification and a computational method. Inverse Probl. Sci. Eng. 16(2), 199–215 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. J.M. Bardsley, N. Laobeul, An analysis of regularization by diffusion for ill-posed Poisson likelihood estimations. Inverse Probl. Sci. Eng. 17(4), 537–550 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. J.M. Bardsley, A. Luttman, Total variation-penalized Poisson likelihood estimation for ill-posed problems. Adv. Comput. Math. 31, 35–59 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Benning, T. Kösters, F. Wübbeling, K. Schäfers, M. Burger, A nonlinear variational method for improved quantification of myocardial blood flow using dynamic H\({_{2}\,}^{15}\) O PET, in IEEE Nuclear Science Symposium Conference Record, 2008, NSS ’08 (2008). doi: 10.1109/NSSMIC.2008.4774274

    Google Scholar 

  19. B. Berkels, M. Burger, M. Droske, O. Nemitz, M. Rumpf, Cartoon extraction based on anisotropic image classification, in Vision, Modeling, and Visualization 2006: Proceedings, ed. by L. Kobbelt, T. Kuhlen, T. Aach, R. Westerman (IOS Press, Aachen, 2006)

    Google Scholar 

  20. M. Bertero, H. Lanteri, L. Zanni, Iterative image reconstruction: a point of view, in Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT), ed. by Y. Censor, M. Jiang, A. Louis. Publications of the Scuola Normale, CRM series, vol. 7 (2008), pp. 37–63

    Google Scholar 

  21. M. Bertero, P. Boccacci, G. Desiderà, G. Vicidomini, Image deblurring with Poisson data: from cells to galaxies. Inverse Probl. 25(12), 123006 (2009)

    Google Scholar 

  22. M. Bertero, P. Boccacci, G. Talenti, R. Zanella, L. Zanni, A discrepancy principle for Poisson data. Inverse Probl. 26(10), 105004 (2010)

    Article  MathSciNet  Google Scholar 

  23. S. Bonettini, V. Ruggiero, An alternating extragradient method for total variation-based image restoration from Poisson data. Inverse Probl. 27(9), 095001 (2011)

    MathSciNet  Google Scholar 

  24. S. Bonettini, R. Zanella, L. Zanni, A scaled gradient projection method for constrained image deblurring. Inverse Probl. 25(1), 015002 (2009)

    MathSciNet  Google Scholar 

  25. S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1–122 (2010)

    Article  MATH  Google Scholar 

  26. K. Bredies, A forward-backward splitting algorithm for the minimization of non-smooth convex functionals in Banach space. Inverse Probl. 25(1), 015005 (2009)

    MathSciNet  Google Scholar 

  27. L.M. Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7, 200–217 (1967)

    Article  Google Scholar 

  28. C. Brune, A. Sawatzky, M. Burger, Bregman-EM-TV methods with application to optical nanoscopy, in Proceedings of the 2nd International Conference on Scale Space and Variational Methods in Computer Vision. Lecture Notes in Computer Science, vol. 5567 (Springer, New York, 2009), pp. 235–246

    Google Scholar 

  29. C. Brune, A. Sawatzky, M. Burger, Primal and dual Bregman methods with application to optical nanoscopy. Int. J. Comput. Vis. 92(2), 211–229 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Burger, S. Osher, Convergence rates of convex variational regularization. Inverse Probl. 20, 1411–1421 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Burger, G. Gilboa, S. Osher, J. Xu, Nonlinear inverse scale space methods. Comm. Math. Sci. 4(1), 179–212 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. M. Burger, K. Frick, S. Osher, O. Scherzer, Inverse total variation flow. Multiscale Model. Simul. 6(2), 366–395 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. A. Chambolle, An algorithm for total variation minimization and applications. J. Math. Imag. Vis. 20, 89–97 (2004)

    Article  MathSciNet  Google Scholar 

  34. A. Chambolle, Total variation minimization and a class of binary MRF models, in Energy Minimization Methods in Computer Vision and Pattern Recognition. Lecture Notes in Computer Science, vol. 3757 (Springer, New York, 2005), pp. 136–152

    Google Scholar 

  35. A. Chambolle, T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imag. Vis. 40(1), 120–145 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. A. Chambolle, V. Caselles, D. Cremers, M. Novaga, T. Pock, An introduction to total variation for image analysis, in Theoretical Foundations and Numerical Methods for Sparse Recovery. Radon Series on Computational and Applied Mathematics, vol. 9 (De Gruyter, Berlin, 2010), pp. 263–340

    Google Scholar 

  37. C. Chaux, J.C. Pesquet, N. Pustelnik, Nested iterative algorithms for convex constrained image recovery problems. SIAM J. Imag. Sci. 2(2), 730–762 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. D.Q. Chen, L.Z. Cheng, Deconvolving Poissonian images by a novel hybrid variational model. J. Vis. Comm. Image Represent. 22(7), 643–652 (2011)

    Article  Google Scholar 

  39. P. Combettes, V. Wajs, Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4, 1168–1200 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. P.L. Combettes, J.C. Pesquet, A proximal decomposition method for solving convex variational inverse problems. Inverse Probl. 24(6), 065014 (2008)

    Article  MathSciNet  Google Scholar 

  41. I. Csiszar, Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems. Ann. Stat. 19(4), 2032–2066 (1991)

    MathSciNet  MATH  Google Scholar 

  42. A.P. Dempster, N.M. Laird, D.B. Rubin, Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B 39(1), 1–38 (1977)

    MathSciNet  MATH  Google Scholar 

  43. N. Dey, L. Blanc-Féraud, C. Zimmer, P. Roux, Z. Kam, J.C. Olivio-Marin, J. Zerubia, 3D microscopy deconvolution using Richardson-Lucy algorithm with total variation regularization. Technical Report 5272, Institut National de Recherche en Informatique et en Automatique (2004)

    Google Scholar 

  44. J. Douglas, H.H. Rachford, On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82(2), 421–439 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  45. J. Eckstein, D.P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55(1–3), 293–318 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  46. P.P.B. Eggermont, Maximum entropy regularization for Fredholm integral equations of the first kind. SIAM J. Math. Anal. 24(6), 1557–1576 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  47. I. Ekeland, R. Temam, Convex Analysis and Variational Problems. Studies in Mathematics and Its Applications, vol. 1 (North-Holland, Amsterdam, 1976)

    Google Scholar 

  48. H.C. Elman, G.H. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal. 31(6), 1645–1661 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  49. H.W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems. Mathematics and Its Applications (Kluwer, Dordrecht, 2000)

    Google Scholar 

  50. S. Esedoglu, S.J. Osher, Decomposition of images by the anisotropic Rudin-Osher-Fatemi model. Comm. Pure Appl. Math. 57(12), 1609–1626 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  51. J.E. Esser, Primal dual algorithms for convex models and applications to image restoration, registration and nonlocal inpainting. Ph.D. thesis, University of California, Los Angeles, 2010

    Google Scholar 

  52. L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics (CRC Press, West Palm Beach, 1992)

    Google Scholar 

  53. M.A.T. Figueiredo, J. Bioucas-Dias, Deconvolution of Poissonian images using variable splitting and augmented Lagrangian optimization, in IEEE Workshop on Statistical Signal Processing, Cardiff (2009)

    Google Scholar 

  54. M.A.T. Figueiredo, J.M. Bioucas-Dias, Restoration of Poissonian images using alternating direction optimization. IEEE Trans. Image Process. 19(12), 3133–3145 (2010)

    Article  MathSciNet  Google Scholar 

  55. M. Fortin, R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems. Studies in Mathematics and its Applications, vol. 15 (Elsevier, Amsterdam, 1983)

    Google Scholar 

  56. D. Gabay, Applications of the method of multipliers to variational inequalities, in Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems. Studies in Mathematics and its Applications, vol. 15 (Elsevier, Amsterdam, 1983),pp. 299–331

    Google Scholar 

  57. D. Gabay, B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Comput. Math. Appl. 2(1), 17–40 (1976)

    Article  MATH  Google Scholar 

  58. S. Geman, D. Geman, Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. J. Appl. Stat. 20(5), 25–62 (1993)

    Google Scholar 

  59. S. Geman, D.E. McClure, Bayesian image analysis: an application to single photon emission tomography, in Proceedings of Statistical Computation Section (American Statistical Association, Alexandria, 1985), pp. 12–18

    Google Scholar 

  60. E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80 (Birkhäuser, Basel, 1984)

    Google Scholar 

  61. R. Glowinski, P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. Studies in Applied Mathematics, vol. 9 (SIAM, Philadelphia, 1989)

    Google Scholar 

  62. T. Goldstein, S. Osher, The split Bregman method for L 1-regularized problems. SIAM J. Imag. Sci. 2(2), 323–343 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  63. T. Goldstein, B. O’Donoghue, S. Setzer, Fast alternating direction optimization methods. CAM Report 12–35, UCLA, 2012

    Google Scholar 

  64. C.W. Groetsch, Inverse Problems in the Mathematical Sciences (Vieweg, Braunschweig, 1993)

    Book  MATH  Google Scholar 

  65. P.C. Hansen, J.G. Nagy, D.P. O’Leary, Deblurring Images: Matrices, Spectra, and Filtering. Fundamentals of Algorithms (SIAM, Philadelphia, 2006)

    Book  Google Scholar 

  66. B.S. He, H. Yang, S.L. Wang, Alternating direction method with self-adaptive penalty parameters for monotone variational inequalities. J. Optim. Theor. Appl. 106(2), 337–356 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  67. S.W. Hell, Toward fluorescence nanoscopy. Nat. Biotechnol. 21(11), 1347–1355 (2003)

    Article  Google Scholar 

  68. S. Hell, A. Schönle, Nanoscale resolution in far-field fluorescence microscopy, in Science of Microscopy, ed. by P.W. Hawkes, J.C.H. Spence (Springer, New York, 2006)

    Google Scholar 

  69. S. Hell, E.H.K. Stelzer, Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation. Opt. Comm. 93(5–6), 277–282 (1992)

    Article  Google Scholar 

  70. S. Hell, E.H.K. Stelzer, Properties of a 4Pi confocal fluorescence microscope. J. Opt. Soc. Am. A 9(12), 2159–2166 (1992)

    Article  Google Scholar 

  71. S. Hell, J. Wichmann, Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy. Opt. Lett. 19(11), 780–782 (1994)

    Article  Google Scholar 

  72. F.M. Henderson, A.J. Lewis, Principles and Applications of Imaging Radar: Manual of Remote Sensing, vol. 2 (Wiley, London, 1998)

    Google Scholar 

  73. A.O. Hero, J.A. Fessler, Convergence in norm for alternating expectation-maximization (EM) type algorithms. Stat. Sin. 5, 41–54 (1995)

    MathSciNet  MATH  Google Scholar 

  74. J.B. Hiriart-Urruty, C. Lemaréchal, Convex Analysis and Minimization Algorithms I. Grundlehren der mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 305 (Springer, New York, 1993)

    Google Scholar 

  75. T.J. Holmes, Y.H. Liu, Acceleration of maximum-likelihood image restoration for fluorescence microscopy and other noncoherent imagery. J. Opt. Soc. Am. A 8(6), 893–907 (1991)

    Article  Google Scholar 

  76. K. Ito, K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications. Advances in Design and Control, vol. 15 (SIAM, Philadelphia, 2008)

    Google Scholar 

  77. A.N. Iusem, Convergence analysis for a multiplicatively relaxed EM algorithm. Math. Meth. Appl. Sci. 14(8), 573–593 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  78. E. Jonsson, S.C. Huang, T. Chan, Total variation regularization in positron emission tomography. CAM Report 98–48, UCLA, 1998

    Google Scholar 

  79. C.T. Kelley, Iterative Methods for Optimization. Frontiers in Applied Mathematics (SIAM, Philadelphia, 1999)

    Google Scholar 

  80. T.A. Klar, S. Jakobs, M. Dyba, A. Egner, S.W. Hell, Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission. Proc. Natl. Acad. Sci. USA 97(15),8206–8210 (2000)

    Article  Google Scholar 

  81. T. Kösters, K. Schäfers, F. Wübbeling, EMrecon: An expectation maximization based image reconstruction framework for emission tomography data, in 2011 IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC) (2011), pp. 4365–4368. doi: 10.1109/NSSMIC.2011.6153840

    Google Scholar 

  82. K. Lange, R. Carson, EM reconstruction algorithms for emission and transmission tomography. J. Comput. Assist. Tomogr. 8(2), 306–316 (1984)

    Google Scholar 

  83. H. Lantéri, C. Theys, Restoration of astrophysical images - the case of Poisson data with additive Gaussian noise. EURASIP J. Appl. Signal Process. 15, 2500–2513 (2005)

    Article  Google Scholar 

  84. H. Lantéri, M. Roche, O. Cuevas, C. Aime, A general method to devise maximum-likelihood signal restoration multiplicative algorithms with non-negativity constraints. Signal Process. 81(5), 945–974 (2001)

    Article  MATH  Google Scholar 

  85. H. Lantéri, M. Roche, C. Aime, Penalized maximum likelihood image restoration with positivity constraints: multiplicative algorithms. Inverse Probl. 18(5), 1397 (2002)

    MATH  Google Scholar 

  86. T. Le, R. Chartrand, T.J. Asaki, A variational approach to reconstructing images corrupted by Poisson noise. J. Math. Imag. Vis. 27(3), 257–263 (2007)

    Article  MathSciNet  Google Scholar 

  87. H. Liao, F. Li, M.K. Ng, Selection of regularization parameter in total variation image restoration. J. Opt. Soc. Am. A 26(11), 2311–2320 (2009)

    Article  MathSciNet  Google Scholar 

  88. P.L. Lions, B. Mercier, Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964–979 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  89. X. Liu, C. Comtat, C. Michel, P. Kinahan, M. Defrise, D. Townsend, Comparison of 3-D reconstruction with 3D-OSEM and with FORE + OSEM for PET. IEEE Trans. Med. Imag. 20(8), 804–814 (2001)

    Article  Google Scholar 

  90. L.B. Lucy, An iterative technique for the rectification of observed distributions. Astron. J. 79, 745–754 (1974)

    Article  Google Scholar 

  91. A. Luttman, A theoretical analysis of L 1 regularized Poisson likelihood estimation. Inverse Prob. Sci. Eng. 18(2), 251–264 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  92. M.J. Martínez, Y. Bercier, M. Schwaiger, S.I. Ziegler, PET/CT Biograph TM Sensation 16 - Performance improvement using faster electronics. Nuklearmedizin 45(3), 126–133 (2006)

    Google Scholar 

  93. R.E. Megginson, An Introduction to Banach Space Theory. Graduate Texts in Mathematics, vol. 183 (Springer, New York, 1998)

    Google Scholar 

  94. Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures, University Lecture Series, vol. 22 (American Mathematical Society, Boston, 2001)

    Google Scholar 

  95. H.N. Mülthei, Iterative continuous maximum-likelihood reconstruction method. Math. Meth. Appl. Sci. 15(4), 275–286 (1992)

    Article  MATH  Google Scholar 

  96. H.N. Mülthei, B. Schorr, On an iterative method for a class of integral equations of the first kind. Math. Meth. Appl. Sci. 9(1), 137–168 (1987)

    Article  MATH  Google Scholar 

  97. H.N. Mülthei, B. Schorr, On properties of the iterative maximum likelihood reconstruction method. Math. Meth. Appl. Sci. 11(3), 331–342 (1989)

    Article  MATH  Google Scholar 

  98. A. Myronenko, Free DCTN and IDCTN Matlab code (2011). https://sites.google.com/site/myronenko/software

  99. F. Natterer, F. Wübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, 2001)

    Book  MATH  Google Scholar 

  100. S. Osher, M. Burger, D. Goldfarb, J. Xu, W. Yin, An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul. 4(2), 460–489 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  101. V.Y. Panin, G.L. Zeng, G.T. Gullberg, Total variation regulated EM algorithm [SPECT reconstruction]. IEEE Trans. Nucl. Sci. 46(6), 2202–2210 (1999)

    Article  Google Scholar 

  102. G.B. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert spaces. J. Math. Anal. Appl. 72, 383–390 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  103. D. Potts, G. Steidl, Optimal trigonometric preconditioners for nonsymmetric Toeplitz systems. Linear Algebra Appl. 281(1–3), 265–292 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  104. E. Resmerita, R.S. Anderssen, Joint additive Kullback-Leibler residual minimization and regularization for linear inverse problems. Math. Meth. Appl. Sci. 30, 1527–1544 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  105. E. Resmerita, H.W. Engl, A.N. Iusem, The expectation-maximization algorithm for ill-posed integral equations: a convergence analysis. Inverse Probl. 23(6), 2575–2588 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  106. W.H. Richardson, Bayesian-based iterative method of image restoration. J. Opt. Soc. Am. 62(1), 55–59 (1972)

    Article  Google Scholar 

  107. L.I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms. Phys. D 60, 259–268 (1992)

    Article  MATH  Google Scholar 

  108. A. Sawatzky, (Nonlocal) total variation in medical imaging. Ph.D. thesis, University of Münster, 2011. CAM Report 11–47, UCLA

    Google Scholar 

  109. A. Sawatzky, C., Brune, F. Wübbeling, T. Kösters, K. Schäfers, M. Burger, Accurate EM-TV algorithm in PET with low SNR, in IEEE Nuclear Science Symposium Conference Record, 2008, NSS ’08. doi: 10.1109/NSSMIC.2008.4774392

    Google Scholar 

  110. K.P. Schäfers, T.J. Spinks, P.G. Camici, P.M. Bloomfield, C.G. Rhodes, M.P. Law, C.S.R. Baker, O. Rimoldi, Absolute quantification of myocardial blood flow with H\({_{2}\,}^{15}\) O and 3-dimensional PET: an experimental validation. J. Nucl. Med. 43(8), 1031–1040 (2002)

    Google Scholar 

  111. S. Setzer, Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage, in Proceedings of the 2nd International Conference on Scale Space and Variational Methods in Computer Vision. Lecture Notes in Computer Science, vol. 5567 (Springer, New York, 2009), pp. 464–476

    Google Scholar 

  112. S. Setzer, Splitting methods in image processing. Ph.D. thesis, University of Mannheim, 2009. http://ub-madoc.bib.uni-mannheim.de/2924/

  113. S. Setzer, G. Steidl, T. Teuber, Deblurring Poissonian images by split Bregman techniques. J. Vis. Comm. Image Represent. 21(3), 193–199 (2010)

    Article  MathSciNet  Google Scholar 

  114. L.A. Shepp, Y. Vardi, Maximum likelihood reconstruction for emission tomography. IEEE Trans. Med. Imag. 1(2), 113–122 (1982)

    Article  Google Scholar 

  115. J.J. Sieber, K.I. Willig, C. Kutzner, C. Gerding-Reimers, B. Harke, G. Donnert, B. Rammner, C. Eggeling, S.W. Hell, H. Grubmüller, T. Lang, Anatomy and dynamics of a supramolecular membrane protein cluster. Science 317, 1072–1076 (2007)

    Article  Google Scholar 

  116. T.J. Spinks, T. Jones, P.M. Bloomfield, D.L. Bailey, D. Hogg, W.F. Jones, K. Vaigneur, J. Reed, J. Young, D. Newport, C. Moyers, M.E. Casey, R. Nutt, Physical characteristics of the ECAT EXACT3D positron tomograph. Phys. Med. Biol. 45(9), 2601–2618 (2000)

    Article  Google Scholar 

  117. G. Steidl, T. Teuber, Anisotropic smoothing using double orientations, in Proceedings of the 2nd International Conference on Scale Space and Variational Methods in Computer Vision. Lecture Notes in Computer Science, vol. 5567 (Springer, New York, 2009), pp. 477–489

    Google Scholar 

  118. D.M. Strong, J.F. Aujol, T.F. Chan, Scale recognition, regularization parameter selection, and Meyer’s G norm in total variation regularization. Multiscale Model. Simul. 5(1), 273–303 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  119. P. Tseng, Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM J. Contr. Optim. 29(1), 119–138 (1991)

    Article  MATH  Google Scholar 

  120. Y. Vardi, L.A. Shepp, L. Kaufman, A statistical model for positron emission tomography. J. Am. Stat. Assoc. 80, 8–20 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  121. L.A. Vese, S.J. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput. 19, 553–572 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  122. C.R. Vogel, Computational Methods for Inverse Problems. Frontiers in Applied Mathematics (SIAM, Philadelphia, 2002)

    Google Scholar 

  123. C.R. Vogel, M.E. Oman, Fast, robust total variation-based reconstruction of noisy, blurred images. IEEE Trans. Image Process. 7(6), 813–824 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  124. Y. Wang, J. Yang, W. Yin, Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imag. Sci. 1(3), 248–272 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  125. M.N. Wernick, J.N. Aarsvold (eds.), Emission Tomography: The Fundamentals of PET and SPECT (Elsevier, Amsterdam, 2004)

    Google Scholar 

  126. K.I. Willig, B. Harke, R. Medda, S.W. Hell, STED microscopy with continuous wave beams. Nat. Meth. 4(11), 915–918 (2007)

    Article  Google Scholar 

  127. M. Yan, L.A. Vese, Expectation maximization and total variation based model for computed tomography reconstruction from undersampled data, in Proceedings of SPIE 7961, Medical Imaging 2011: Physics of Medical Imaging, 79612X, 16 March, 2011. doi:10.1117/12.878238 [From Conference Volume 7961 Medical Imaging 2011: Physics of Medical Imaging Norbert J. Pelc, Ehsan Samei, Robert M. Nishikawa, Lake Buena Vista, Florida, 12 February 2011]

    Google Scholar 

  128. R. Zanella, P. Boccacci, L. Zanni, M. Bertero, Efficient gradient projection methods for edge-preserving removal of Poisson noise. Inverse Probl. 25(4), 045010 (2009)

    MathSciNet  Google Scholar 

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Acknowledgements

This work has been supported by the German Ministry of Education and Research (BMBF) through the project INVERS: Deconvolution problems with sparsity constraints in nanoscopy and mass spectrometry. This research was performed when the second author was with the Mathematical Imaging Group at University of Münster. C. Brune acknowledges further support by the Deutsche Telekom Foundation. The work has been further supported by the German Science Foundation DFG through the SFB 656 Molecular Cardiovascular Imaging and the project Regularization with singular energies. The authors thank Florian Büther and Klaus Schäfers (both European Institute for Molecular Imaging, University of Münster) for providing real data in PET and useful discussions. The authors thank Katrin Willig and Andreas Schönle (both Max Planck Institute for Biophysical Chemistry, Göttingen) for providing real data in optical nanoscopy.

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  1. Institute for Computational and Applied Mathematics, University of Münster, Orléans-Ring 10, 48149, Münster, Germany

    Alex Sawatzky, Frank Wübbeling & Martin Burger

  2. Department of Mathematics, University of California, Los Angeles, CA, 90095-1555, USA

    Christoph Brune

  3. European Institute for Molecular Imaging, University of Münster, Mendelstr. 11, 48149, Münster, Germany

    Thomas Kösters

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Correspondence to Alex Sawatzky .

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Sawatzky, A., Brune, C., Kösters, T., Wübbeling, F., Burger, M. (2013). EM-TV Methods for Inverse Problems with Poisson Noise. In: Level Set and PDE Based Reconstruction Methods in Imaging. Lecture Notes in Mathematics(), vol 2090. Springer, Cham. https://doi.org/10.1007/978-3-319-01712-9_2

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