EM-TV Methods for Inverse Problems with Poisson Noise

  • Alex Sawatzky
  • Christoph Brune
  • Thomas Kösters
  • Frank Wübbeling
  • Martin Burger
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, 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.

Notes

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.

References

  1. 1.
    R. Acar, C.R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems. Inverse Probl. 10(6), 1217–1229 (1994)MathSciNetCrossRefMATHGoogle Scholar
  2. 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. 99–103Google Scholar
  3. 3.
    L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs (Oxford University Press, Oxford, 2000)MATHGoogle Scholar
  4. 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.2011.6115691Google Scholar
  5. 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. 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. 7.
    J.F. Aujol, Some first-order algorithms for total variation based image restoration. J. Math. Imag. Vis. 34(3), 307–327 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Bachmayr, M. Burger, Iterative total variation schemes for nonlinear inverse problems. Inverse Probl. 25(10), 105004 (2009)MathSciNetGoogle Scholar
  9. 9.
    D. Baddeley, C. Carl, C. Cremer, 4Pi microscopy deconvolution with a variable point-spread function. Appl. Opt. 45(27), 7056–7064 (2006)CrossRefGoogle Scholar
  10. 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. 11.
    J.M. Bardsley, An efficient computational method for total variation-penalized Poisson likelihood estimation. Inverse Probl. Imag. 2(2), 167–185 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    J.M. Bardsley, A theoretical framework for the regularization of Poisson likelihood estimation problems. Inverse Probl. Imag. 4(1), 11–17 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    J.M. Bardsley, J. Goldes, Regularization parameter selection methods for ill-posed Poisson maximum likelihood estimation. Inverse Probl. 25(9), 095005 (2009)MathSciNetGoogle Scholar
  14. 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)MathSciNetCrossRefMATHGoogle Scholar
  15. 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)MathSciNetCrossRefMATHGoogle Scholar
  16. 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)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    J.M. Bardsley, A. Luttman, Total variation-penalized Poisson likelihood estimation for ill-posed problems. Adv. Comput. Math. 31, 35–59 (2009)MathSciNetCrossRefMATHGoogle Scholar
  18. 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.2008.4774274Google Scholar
  19. 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. 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. 37–63Google Scholar
  21. 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. 22.
    M. Bertero, P. Boccacci, G. Talenti, R. Zanella, L. Zanni, A discrepancy principle for Poisson data. Inverse Probl. 26(10), 105004 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    S. Bonettini, V. Ruggiero, An alternating extragradient method for total variation-based image restoration from Poisson data. Inverse Probl. 27(9), 095001 (2011)MathSciNetGoogle Scholar
  24. 24.
    S. Bonettini, R. Zanella, L. Zanni, A scaled gradient projection method for constrained image deblurring. Inverse Probl. 25(1), 015002 (2009)MathSciNetGoogle Scholar
  25. 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)CrossRefMATHGoogle Scholar
  26. 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)MathSciNetGoogle Scholar
  27. 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)CrossRefGoogle Scholar
  28. 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. 235–246Google Scholar
  29. 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)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    M. Burger, S. Osher, Convergence rates of convex variational regularization. Inverse Probl. 20, 1411–1421 (2004)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    M. Burger, G. Gilboa, S. Osher, J. Xu, Nonlinear inverse scale space methods. Comm. Math. Sci. 4(1), 179–212 (2006)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    M. Burger, K. Frick, S. Osher, O. Scherzer, Inverse total variation flow. Multiscale Model. Simul. 6(2), 366–395 (2007)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    A. Chambolle, An algorithm for total variation minimization and applications. J. Math. Imag. Vis. 20, 89–97 (2004)MathSciNetCrossRefGoogle Scholar
  34. 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. 136–152Google Scholar
  35. 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)MathSciNetCrossRefMATHGoogle Scholar
  36. 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. 263–340Google Scholar
  37. 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)MathSciNetCrossRefMATHGoogle Scholar
  38. 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)CrossRefGoogle Scholar
  39. 39.
    P. Combettes, V. Wajs, Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4, 1168–1200 (2005)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    P.L. Combettes, J.C. Pesquet, A proximal decomposition method for solving convex variational inverse problems. Inverse Probl. 24(6), 065014 (2008)MathSciNetCrossRefGoogle Scholar
  41. 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)MathSciNetMATHGoogle Scholar
  42. 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)MathSciNetMATHGoogle Scholar
  43. 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. 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)MathSciNetCrossRefMATHGoogle Scholar
  45. 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)MathSciNetCrossRefMATHGoogle Scholar
  46. 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)MathSciNetCrossRefMATHGoogle Scholar
  47. 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. 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)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    H.W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems. Mathematics and Its Applications (Kluwer, Dordrecht, 2000)Google Scholar
  50. 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)MathSciNetCrossRefMATHGoogle Scholar
  51. 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, 2010Google Scholar
  52. 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. 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. 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)MathSciNetCrossRefGoogle Scholar
  55. 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. 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. 299–331Google Scholar
  57. 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)CrossRefMATHGoogle Scholar
  58. 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. 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. 12–18Google Scholar
  60. 60.
    E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80 (Birkhäuser, Basel, 1984)Google Scholar
  61. 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. 62.
    T. Goldstein, S. Osher, The split Bregman method for L 1-regularized problems. SIAM J. Imag. Sci. 2(2), 323–343 (2009)MathSciNetCrossRefMATHGoogle Scholar
  63. 63.
    T. Goldstein, B. O’Donoghue, S. Setzer, Fast alternating direction optimization methods. CAM Report 12–35, UCLA, 2012Google Scholar
  64. 64.
    C.W. Groetsch, Inverse Problems in the Mathematical Sciences (Vieweg, Braunschweig, 1993)CrossRefMATHGoogle Scholar
  65. 65.
    P.C. Hansen, J.G. Nagy, D.P. O’Leary, Deblurring Images: Matrices, Spectra, and Filtering. Fundamentals of Algorithms (SIAM, Philadelphia, 2006)CrossRefGoogle Scholar
  66. 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)MathSciNetCrossRefMATHGoogle Scholar
  67. 67.
    S.W. Hell, Toward fluorescence nanoscopy. Nat. Biotechnol. 21(11), 1347–1355 (2003)CrossRefGoogle Scholar
  68. 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. 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)CrossRefGoogle Scholar
  70. 70.
    S. Hell, E.H.K. Stelzer, Properties of a 4Pi confocal fluorescence microscope. J. Opt. Soc. Am. A 9(12), 2159–2166 (1992)CrossRefGoogle Scholar
  71. 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)CrossRefGoogle Scholar
  72. 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. 73.
    A.O. Hero, J.A. Fessler, Convergence in norm for alternating expectation-maximization (EM) type algorithms. Stat. Sin. 5, 41–54 (1995)MathSciNetMATHGoogle Scholar
  74. 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. 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)CrossRefGoogle Scholar
  76. 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. 77.
    A.N. Iusem, Convergence analysis for a multiplicatively relaxed EM algorithm. Math. Meth. Appl. Sci. 14(8), 573–593 (1991)MathSciNetCrossRefMATHGoogle Scholar
  78. 78.
    E. Jonsson, S.C. Huang, T. Chan, Total variation regularization in positron emission tomography. CAM Report 98–48, UCLA, 1998Google Scholar
  79. 79.
    C.T. Kelley, Iterative Methods for Optimization. Frontiers in Applied Mathematics (SIAM, Philadelphia, 1999)Google Scholar
  80. 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)CrossRefGoogle Scholar
  81. 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.2011.6153840Google Scholar
  82. 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. 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)CrossRefGoogle Scholar
  84. 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)CrossRefMATHGoogle Scholar
  85. 85.
    H. Lantéri, M. Roche, C. Aime, Penalized maximum likelihood image restoration with positivity constraints: multiplicative algorithms. Inverse Probl. 18(5), 1397 (2002)MATHGoogle Scholar
  86. 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)MathSciNetCrossRefGoogle Scholar
  87. 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)MathSciNetCrossRefGoogle Scholar
  88. 88.
    P.L. Lions, B. Mercier, Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964–979 (1979)MathSciNetCrossRefMATHGoogle Scholar
  89. 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)CrossRefGoogle Scholar
  90. 90.
    L.B. Lucy, An iterative technique for the rectification of observed distributions. Astron. J. 79, 745–754 (1974)CrossRefGoogle Scholar
  91. 91.
    A. Luttman, A theoretical analysis of L 1 regularized Poisson likelihood estimation. Inverse Prob. Sci. Eng. 18(2), 251–264 (2010)MathSciNetCrossRefMATHGoogle Scholar
  92. 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. 93.
    R.E. Megginson, An Introduction to Banach Space Theory. Graduate Texts in Mathematics, vol. 183 (Springer, New York, 1998)Google Scholar
  94. 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. 95.
    H.N. Mülthei, Iterative continuous maximum-likelihood reconstruction method. Math. Meth. Appl. Sci. 15(4), 275–286 (1992)CrossRefMATHGoogle Scholar
  96. 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)CrossRefMATHGoogle Scholar
  97. 97.
    H.N. Mülthei, B. Schorr, On properties of the iterative maximum likelihood reconstruction method. Math. Meth. Appl. Sci. 11(3), 331–342 (1989)CrossRefMATHGoogle Scholar
  98. 98.
    A. Myronenko, Free DCTN and IDCTN Matlab code (2011). https://sites.google.com/site/myronenko/software
  99. 99.
    F. Natterer, F. Wübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, 2001)CrossRefMATHGoogle Scholar
  100. 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)MathSciNetCrossRefMATHGoogle Scholar
  101. 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)CrossRefGoogle Scholar
  102. 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)MathSciNetCrossRefMATHGoogle Scholar
  103. 103.
    D. Potts, G. Steidl, Optimal trigonometric preconditioners for nonsymmetric Toeplitz systems. Linear Algebra Appl. 281(1–3), 265–292 (1998)MathSciNetCrossRefMATHGoogle Scholar
  104. 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)MathSciNetCrossRefMATHGoogle Scholar
  105. 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)MathSciNetCrossRefMATHGoogle Scholar
  106. 106.
    W.H. Richardson, Bayesian-based iterative method of image restoration. J. Opt. Soc. Am. 62(1), 55–59 (1972)CrossRefGoogle Scholar
  107. 107.
    L.I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms. Phys. D 60, 259–268 (1992)CrossRefMATHGoogle Scholar
  108. 108.
    A. Sawatzky, (Nonlocal) total variation in medical imaging. Ph.D. thesis, University of Münster, 2011. CAM Report 11–47, UCLAGoogle Scholar
  109. 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.2008.4774392Google Scholar
  110. 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. 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. 464–476Google Scholar
  112. 112.
    S. Setzer, Splitting methods in image processing. Ph.D. thesis, University of Mannheim, 2009. http://ub-madoc.bib.uni-mannheim.de/2924/
  113. 113.
    S. Setzer, G. Steidl, T. Teuber, Deblurring Poissonian images by split Bregman techniques. J. Vis. Comm. Image Represent. 21(3), 193–199 (2010)MathSciNetCrossRefGoogle Scholar
  114. 114.
    L.A. Shepp, Y. Vardi, Maximum likelihood reconstruction for emission tomography. IEEE Trans. Med. Imag. 1(2), 113–122 (1982)CrossRefGoogle Scholar
  115. 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)CrossRefGoogle Scholar
  116. 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)CrossRefGoogle Scholar
  117. 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. 477–489Google Scholar
  118. 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)MathSciNetCrossRefMATHGoogle Scholar
  119. 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)CrossRefMATHGoogle Scholar
  120. 120.
    Y. Vardi, L.A. Shepp, L. Kaufman, A statistical model for positron emission tomography. J. Am. Stat. Assoc. 80, 8–20 (1985)MathSciNetCrossRefMATHGoogle Scholar
  121. 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)MathSciNetCrossRefMATHGoogle Scholar
  122. 122.
    C.R. Vogel, Computational Methods for Inverse Problems. Frontiers in Applied Mathematics (SIAM, Philadelphia, 2002)Google Scholar
  123. 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)MathSciNetCrossRefMATHGoogle Scholar
  124. 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)MathSciNetCrossRefMATHGoogle Scholar
  125. 125.
    M.N. Wernick, J.N. Aarsvold (eds.), Emission Tomography: The Fundamentals of PET and SPECT (Elsevier, Amsterdam, 2004)Google Scholar
  126. 126.
    K.I. Willig, B. Harke, R. Medda, S.W. Hell, STED microscopy with continuous wave beams. Nat. Meth. 4(11), 915–918 (2007)CrossRefGoogle Scholar
  127. 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. 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)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Alex Sawatzky
    • 1
  • Christoph Brune
    • 2
  • Thomas Kösters
    • 3
  • Frank Wübbeling
    • 1
  • Martin Burger
    • 1
  1. 1.Institute for Computational and Applied MathematicsUniversity of MünsterMünsterGermany
  2. 2.Department of MathematicsUniversity of CaliforniaLos AngelesUSA
  3. 3.European Institute for Molecular ImagingUniversity of MünsterMünsterGermany

Personalised recommendations