Mathematics of Electron Tomography

  • O. Öktem
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This survey starts with a brief description of the scientific relevance of electron tomography in life sciences followed by a survey of image formation models. In the latter, the scattering of electrons against a specimen is modeled by the Schrödinger equation, and the image formation model is completed by adding a description of the transmission electron microscope optics and detector. Electron tomography can then be phrased as an inverse scattering problem and attention is now turned to describing mathematical approaches for solving that reconstruction problem. This part starts out by explaining challenges associated with the aforementioned inverse problem, such as the extremely low signal-to-noise ratio in the data and the severe ill-posedness due to incomplete data, which naturally brings up the issue of choosing a regularization method for reconstruction. Here, the review surveys both methods that have been developed, as well as pointing to new promising approaches. Some of the regularization methods are also tested on simulated and experimental data. As a final note, this is not a traditional mathematical review in the sense that focus here is on the application to electron tomography rather than on describing mathematical techniques that underly proofs of key theorems.


Nuisance Parameter Phase Retrieval Reconstruction Operator Contrast Model Algebraic Reconstruction Technique 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The writing of this review chapter would not be possible without the help of several people. Jan Boman at the Department of Mathematics, Stockholm University and Todd Quinto at the Department of Mathematics, Tufts University provided valuable advice and help, especially regarding material in section “Analytic Methods.” Hans Rullgård at Comsol has provided advice and support regarding the usage of the TEM simulation and the TV-regularization softwares used in section “Examples”. Remco Schoenmakers at FEI generously provided the data used in section “Virions and Bacteriophages in Aqueous Buffer” as well as access to the image in Fig. 1. Milos Vulevic and Bernd Reiger in TU Delft provided valuable insight into the detector modeling in subsection on p. 21, Sergej Masich at the Department of Cell and Molecular Biology, Karolinska Institutet helped out in the alignment of the tilt-series in section “Virions and Bacteriophages in Aqueous Buffer” as well as in running the IMOD reconstructions in section “Examples.” Holger Kohr and Alfred Louis at the Department of Mathematics, Saarlands University contributed to the material on the approximate inverse method and phase contrast tomography. Kohr also provided the approximate inverse reconstructions in section “Examples.” Finally, Günther Uhlmann at the Department of Mathematics, Washington State University provided valuable insight into uniqueness and stability issues discussed in section “Electron–Specimen Interaction.” Work on this chapter is financially supported by the Swedish Foundation for Strategic Research.


  1. 1.
    Aganj, I., Bartesaghi, A., Borgnia, M., Liao, H.Y., Sapiro, G., Subramaniam, S.: Regularization for Inverting the Radon Transform with Wedge Consideration. IMA Preprint Series, vol. 2144. Institute for Mathematics and its Applications, Minneapolis (2006)Google Scholar
  2. 2.
    Alpers, A., Gardner, R.J., König, S., Pennington, R.S., Boothroyd, C.B., Houben, L., Dunin-Borkowski, R.E., Batenburg, K.J.: Geometric reconstruction methods for electron tomography. Ultramicroscopy 128, 42–54 (2013)Google Scholar
  3. 3.
    Amat, F., Castanõ-Diez, D., Lawrence, A., Moussavi, F., Winkler, H., Horowitz, M.: Alignment of cryo-electron tomography datasets. In: Jensen, G.J. (ed.) Cryo-EM, Part B: 3-D Reconstruction. Methods in Enzymology, Chap. 13, vol. 482. pp. 343–367. Academic, San Diego (2010)Google Scholar
  4. 4.
    Ammari, H., Bahouri, H., Dos Santos Ferreira, D., Gallagher, I.: Stability estimates for an inverse scattering problem at high frequencies. J. Math. Anal. Appl. 400, 525–540 (2013)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Andersen, A.H., Kak, A.C.: Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm. Ultrason. Imaging 6(1), 81–94 (1984)Google Scholar
  6. 6.
    Asoubar, D., Zhang, S., Wyrowski, F., Kuhn, M.: Parabasal field decomposition and its application to non-paraxial propagation. Opt. Exp. 20(21), 23502–23517 (2012)Google Scholar
  7. 7.
    Ayache, J., Beaunier, L., Boumendil, J., Ehret, G., Laub, D.: Sample Preparation Handbook for Transmission Electron Microscopy: Methodology. Springer, New York (2010)Google Scholar
  8. 8.
    Ayache, J., Beaunier, L., Boumendil, J., Ehret, G., Laub, D.: Sample Preparation Handbook for Transmission Electron Microscopy: Techniques. Springer, New York (2010)Google Scholar
  9. 9.
    Baker, L.A., Rubinstein, J.L.: Radiation damage in electron cryomicroscopy. In: Jensen, G.J. (ed.) Cryo-EM, Part A: Sample Preparation and Data Collection. Methods in Enzymology, Chap. 15, vol. 481, pp. 371–388. Academic, San Diego (2010)Google Scholar
  10. 10.
    Bakushinsky, A.: Remarks on choosing a regularization parameter using the quasi-optimality and ratio criterion. USSR Comput. Math. Math. Phys. 24(4), 181–182 (1984)Google Scholar
  11. 11.
    Bakushinsky, A.B., Yu, M.: Kokurin. Iterative Methods for Approximate Solution of Inverse Problem. Mathematics and its Applications, vol. 577. Springer, Dordrecht (2004)Google Scholar
  12. 12.
    Batenburg, K.J., Kaiser, U., Kübel, C.: 3D imaging of nanomaterials by discrete tomography. Microsc. Microanal. 12, 1568–1569 (2006). Extended abstract of a paper presented at Microscopy and Microanalysis 2006 in Chicago, IL, USA, 30 July–3 August, 2006Google Scholar
  13. 13.
    Batenburg, K.J., Bals, S., Sijbers, J., Kübel, C., Midgley, P.A., Hernandez, J.C., Kaiser, U., Encina, E.R., Coronado, E.A., Van Tendeloo, G.: 3D imaging of nanomaterials by discrete tomography. Ultramicroscopy 109, 730–740 (2009)Google Scholar
  14. 14.
    Benning, M., Brune, C., Burger, M., Müller, J.: Higher-order tv methods—enhancement via Bregman iteration. J. Sci. Comput. 54(2–3), 269–310 (2013)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Benvenuto, F., La Camera, A., Theys, C., Ferrari, A., Lantéri, H., Bertero, M.: The study of an iterative method for the reconstruction of images corrupted by Poisson and Gaussian noise. Inverse Prob. 24, 035016 (20 pp.) (2008)Google Scholar
  16. 16.
    Betero, M., Lantéri, H., Zanni, L.: Iterative image reconstruction: a point of view. In: Censor, Y., Jiang, M., Louis, A.K. (eds.) Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT). Publications of the Scuola Normale Superiore: CRM Series, vol. 7, pp. 37–63. Springer, Pisa (2008)Google Scholar
  17. 17.
    Binev, P., Dahmen, W., DeVore, R., Lamby, P., Savu, D., Sharpley, R.: Compressed sensing and electron microscopy. In: Vogt, T., Dahmen, W., Binev, P. (eds.) Modeling Nanoscale Imaging in Electron Microscopy, Nanostructure Science and Technology, pp. 73–126. Springer, New York (2012)Google Scholar
  18. 18.
    Blomgren, P., Chan, T., Mulet, P., Wong, C.K.: Total variation image restoration: numerical methods and extensions. In Proceedings of the 1997 IEEE International Conference on Image Processing, Santa Barbara, vol. 3, pp. 384–387 (1997)Google Scholar
  19. 19.
    Böhm, J., Frangakis, A.S., Hegerl, R., Nickell, S., Typke, D., Baumeister, W.: Toward detecting and identifying macromolecules in a cellular context: template matching applied to electron tomograms. Proc. Natl. Acad. Sci. 97(26), 14245–14250 (2000)Google Scholar
  20. 20.
    Boman, J., Quinto, E.T.: Support theorems for real analytic Radon transforms on line complexes in r 3. Trans. Am. Math. Soc. 335, 877–890 (1993)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Bourgain, J., Dilworth, S., Ford, K., Konyagin, S., Kutzarova, D.: Explicit constructions of RIP matrices and related problems. Duke Math. J. 159(1), 145–185 (2011)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Burvall, A., Lundström, U., Takman, P.A.C., Larsson, D.H., Hertz, H.M.: Phase retrieval in x-ray phase-contrast imaging suitable for tomography. Opt. Exp. 19(11), 10359–10376 (2011)Google Scholar
  23. 23.
    Busch, H.: Berechnung der Bahn von Kathodenstrahlen im axialsymmetrischen elektromagnetischen Felde. Ann. Phys. 386, 974–993 (1926)Google Scholar
  24. 24.
    Busch, H.: Über die Wirkungsweise der Konzentrierungsspule bei der Braunschen Rohre. Electr. Eng. (Archiv fur Elektrotechnik) 18, 583–594 (1927)Google Scholar
  25. 25.
    Byrne, C.L.: Applied Iterative Methods. A. K. Peters, Wellesley (2008)zbMATHGoogle Scholar
  26. 26.
    Cai, Y., Zhao, Y., Tang, Y.: Exponential convergence of a randomized Kaczmarz algorithm with relaxation. In: Gaol, F.L., Nguyen, Q.V. (eds.) Proceedings of the 2011–2nd International Congress on Computer Applications and Computational Science. Volume 2. Advances in Intelligent and Soft Computing, vol. 145, pp. 467–473 (2012)Google Scholar
  27. 27.
    Candès, E.J., Romberg, J., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006)zbMATHGoogle Scholar
  28. 28.
    Carazo, J.-M., Carrascosa, J.L.: Restoration of direct Fourier three-dimensional reconstructions of crystalline specimens by the method of convex projections. J. Microsc. 145(2), 159–177 (1987)Google Scholar
  29. 29.
    Carazo, J.-M., Sorzano, C.O.S., Reitzel, E., Schröder, R., Marabini, R.: Discrete tomography in electron microscopy. In: Herman, G.T., Kuba, A. (eds.) Discrete Tomography. Foundations, Algorithms and Applications, Chap. 18, pp. 405–416. Birkhäuser, Boston (1999)Google Scholar
  30. 30.
    Carazo, J.-M., Herman, G.T., Sorzano, C.O.S., Marabini, R.: Algorithms for three-dimensional reconstruction from the imperfect projection data provided by electron microscopy. In: Frank, J. (ed.) Electron Tomography: Methods for Three-Dimensional Visualization of Structures in the Cell, Chap. 7, 2nd edn., pp. 217–243. Springer, Boston (2006)Google Scholar
  31. 31.
    Chadan, K.: Inverse problems in potential scattering. In: Chadan, K., Colton, D., Päivärinta, L., Rundell, W. (eds.) An Introduction to Inverse Scattering and Inverse Spectral Problems. SIAM Monographs on Mathematical Modeling and Computation, Chap. 4, vol. 2. SIAM, Philadelphia (1997)Google Scholar
  32. 32.
    Chakrabarti, A., Zickler, T.: Depth and deblurring from a spectrally-varying depth-of-field. In: Proceedings of the European Conference on Computer Vision 2012. Springer, New York (2012)Google Scholar
  33. 33.
    Chan, T.F., Esedoglu, S., Park, F., Yip, A.: Recent developments in total variation image restoration. In: Paragios, N., Chen, Y., Faugeras, O. (eds.) Handbook of Mathematical Models in Computer Vision. Springer, New York (2005)Google Scholar
  34. 34.
    Çınlar, E.: Probability and Stochastics. Graduate Texts in Mathematics, vol. 261. Springer, New York (2011)Google Scholar
  35. 35.
    Colton, D., Kress, R.: Inverse scattering. In: Scherzer, O. (ed.) Handbook of Mathematical Methods in Imaging, Chap. 13, pp. 551–598. Springer, Berlin (2011)Google Scholar
  36. 36.
    Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences, vol. 93, 3rd edn. Springer, New York (2013)Google Scholar
  37. 37.
    Cruz-Uribe, D.V., Fiorenza, A.: Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis. Springer, Basel (2013)Google Scholar
  38. 38.
    Csiszar, I.: Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems. Ann. Stat. 19(4), 2032–2066 (1991)zbMATHMathSciNetGoogle Scholar
  39. 39.
    Davenport, M., Duarte, M., Eldar, Y., Kutyniok, G.: Introduction to compressed sensing. In: Eldar, Y.C., Kutyniok, G. (eds.) Compressed Sensing: Theory and Applications, Chap. 1, pp. 1–65. Cambridge University Press, Cambridge (2011)Google Scholar
  40. 40.
    Davidson, M.E.: The ill-conditioned nature of the limited angle tomography problem. SIAM J. Appl. Math. 43(2), 428–448 (1983)MathSciNetGoogle Scholar
  41. 41.
    De Rosier, D.J., Klug, A.: Reconstruction of three dimensional structures from electron micrographs. Nature 217, 130–134 (1968)Google Scholar
  42. 42.
    Defrise, M., Clack, R., Townsend, D.W.: Image reconstruction from truncated, two-dimensional, parallel projections. Inverse Probl. 11(2), 287–313 (1995)zbMATHMathSciNetGoogle Scholar
  43. 43.
    DeRosier, D.J.: The reconstruction of three-dimensional images from electron micrographs. Contemp. Phys. 12(5), 437–452 (1971)Google Scholar
  44. 44.
    Ebanks, B., Sahoo, P., Sander, W.: Characterization of Information Measures. World Scientific, Singapore (1998)Google Scholar
  45. 45.
    Egerton, R.F.: Electron Energy-Loss Spectroscopy in the Electron Microscope, 3rd edn. Springer, New York (2011)Google Scholar
  46. 46.
    Egerton, R.F., Li, P., Malac, M.: Radiation damage in the TEM and SEM. Micron 35, 399–409 (2004)Google Scholar
  47. 47.
    Eggermont, P.P.B.: Maximum entropy regularization for Fredholm integral equations of the first kind. SIAM J. Math. Anal. 24(6), 1557–1576 (1993)zbMATHMathSciNetGoogle Scholar
  48. 48.
    Elad, M.: Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing. Springer (2010)Google Scholar
  49. 49.
    Engl, H.W., Landl, G.: Convergence rates for maximum entropy regularization. SIAM J. Numer. Anal. 30(5), 1509–1536 (1993)zbMATHMathSciNetGoogle Scholar
  50. 50.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Mathematics and its Applications, vol. 375. Kluwer Academic Publishers (2000)Google Scholar
  51. 51.
    Fanelli, D., Öktem, O.: Electron tomography: a short overview with an emphasis on the absorption potential model for the forward problem. Inverse Probl. 24(1), 013001 (51 pp.) (2008)Google Scholar
  52. 52.
    Faridani, A.: Introduction to the mathematics of computed tomography. In: Uhlmann, G. (ed.) Inside Out: Inverse Problems and Applications. MSRI Publications, vol. 47, pp. 1–46. Cambridge University Press, Cambridge (2003)Google Scholar
  53. 53.
    Faridani, A.: Fan-beam tomography and sampling theory. In: Ólafsson, G., Quinto, E.T. (eds.) The Radon Transform, Inverse Problems, and Tomography. Proceedings of Symposia in Applied Mathematics, vol. 63, pp. 43–66. American Mathematical Society, Providence (2006)Google Scholar
  54. 54.
    Faridani, A., Finch, D., Ritman, E.L., Smith, K.T.: Local tomography II. SIAM J. Appl. Math. 57(4), 1095–1127 (1997)zbMATHMathSciNetGoogle Scholar
  55. 55.
    Faruqi, A.R., McMullan, G.: Electronic detectors for electron microscopy. Q. Rev. Biophys. 44(3), 357–390 (2011)Google Scholar
  56. 56.
    Felea, R., Quinto, E.T.: The microlocal properties of the local 3-D SPECT operator. SIAM J. Math. Anal. 43(3), 1145–1157 (2011)zbMATHMathSciNetGoogle Scholar
  57. 57.
    Fernández, J.J., Li, S., Crowther, R.A.: CTF determination and correction in electron cryotomography. Ultramicroscopy 106(7), 587–596 (2006)Google Scholar
  58. 58.
    Foley, J.T., Butts, R.R.: Uniqueness of phase retrieval from intensity measurements. J. Opt. Soc. Am. A 71(8), 1008–1014 (1981)Google Scholar
  59. 59.
    Foucart, S., Rahut, H.: A Mathematical Introduction to Compressive Sensing. Applied and Numerical Harmonic Analysis. Springer, New York (2013)zbMATHGoogle Scholar
  60. 60.
    Frank, J.: Three-Dimensional Electron Microscopy of Macromolecular Assemblies, 2nd edn. Oxford University Press, Oxford (2006)Google Scholar
  61. 61.
    Frank, J.: Single-particle reconstruction of biological macromoleculses in electron micrscopy – 30 years. Q. Rev. Biophys. 42, 139–158 (2009)Google Scholar
  62. 62.
    Frikel, J., Quinto, T.: Characterization and reduction of artifacts in limited angle tomography. Inverse Probl. 29(12), 125007 (2013)MathSciNetGoogle Scholar
  63. 63.
    Fultz, B., Howe, J.: Transmission Electron Microscopy and Diffractometry of Materials. Graduate Texts in Physics, 4th edn. Springer, Berlin (2013)Google Scholar
  64. 64.
    Gardueno, E., Herman, G.T.: Optimization of basis functions for both reconstruction and visualization. Discrete Appl. Math. 139, 95–111 (2004)MathSciNetGoogle Scholar
  65. 65.
    Gilbert, P.: Iterative methods for the three-dimensional reconstruction of an object from projections. J. Theor. Biol. 36(1), 105–117 (1972)Google Scholar
  66. 66.
    Gilbert, R., Hackl, K., Xu, Y.: Inverse problem for wave propagation in a perturbed layered half-space. Math. Comput. Model. 45(1–2), 21–33 (2007)zbMATHMathSciNetGoogle Scholar
  67. 67.
    Gil-Rodrigo, E., Portilla, J., Miraut, D., Suarez-Mesa, R.: Efficient joint poisson-gauss restoration using multi-frame l2-relaxed-l0 analysis-based sparsity. In: 18th IEEE International Conference on Image Processing (ICIP), 2011, pp. 1385–1388 (2011)Google Scholar
  68. 68.
    Glaeser, R.M., Downing, K.H., DeRozier, D., Chu, W., Frank, J.: Electron Crystallography of Biological Macromolecules. Oxford University Press, Oxford (2006)Google Scholar
  69. 69.
    Gopinath, A., Xu, G., Ress, D., Öktem, O., Subramaniam, S., Bajaj, C.: Shape-based regularization of electron tomographic reconstruction. IEEE Trans. Med. Imaging 31(12), 2241–2252 (2012)Google Scholar
  70. 70.
    Gordon, R., Bender, R., Herman, G.T.: Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography. J. Theor. Biol. 29(3), 471–481 (1970)Google Scholar
  71. 71.
    Goris, B., Van den Broek, W., Batenburg, K.J., Mezerji, H.H., Bals, S.: Electron tomography based on a total variation minimization reconstruction technique. Ultramicroscopy 113, 120–130 (2012)Google Scholar
  72. 72.
    Goris, B., Roelandts, T., Batenburg, K.J., Mezerji, H.H., Bals, S.: Advanced reconstruction algorithms for electron tomography: from comparison to combination. Ultramicroscopy 127, 40–47 (2013)Google Scholar
  73. 73.
    Greenleaf, A., Uhlmann, G.: Non-local inversion formulas for the X-ray transform. Duke Math. J. 58, 205–240 (1989)zbMATHMathSciNetGoogle Scholar
  74. 74.
    Greenleaf, A., Uhlmann, G.: Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms. Annales de l’Institut Fourier 40, 443–466 (1990)zbMATHMathSciNetGoogle Scholar
  75. 75.
    Greenleaf, A., Uhlmann, G.: Microlocal techniques in integral geometry. In: Grinberg, E., Quinto, E.T. (eds.) Integral Geometry and Tomography. Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference. Contemporary Mathematics, vol. 113, pp. 121–136. American Mathematical Society, Providence (1990)Google Scholar
  76. 76.
    Guillemin, V., Sternberg, S.: Geometric Asymptotics. Mathematical Surveys and Monographs, vol. 14. American Mathematical Society, Providence (1977). Revised edition (June 1990) editionGoogle Scholar
  77. 77.
    Hansen, P.-C.: Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. SIAM Monographs on Mathematical Modeling and Computation, vol. 4. SIAM, Philadelphia (1997)Google Scholar
  78. 78.
    Hawkes, P.W.: The electron microscope as a structure projector. In: Frank, J. (ed.) Electron Tomography. Methods for Three-Dimensional Visualization of Structures in the Cell, Chap. 3, 2nd edn., pp. 83–111. Springer, New York (2006)Google Scholar
  79. 79.
    Hawkes, P.W., Kasper, E.: Principles of Electron Optics. Wave Optics, vol. 3. Academic, San Diego (1995)Google Scholar
  80. 80.
    Hawkes, P.W., Kasper, E.: Principles of Electron Optics. Applied Geometrical Optics, vol. 2. Academic, San Diego, (1996)Google Scholar
  81. 81.
    Herman, G.T.: Fundamentals of Computerized Tomography: Image Reconstruction from Projections. Advances in Computer Vision and Pattern Recognition, 2nd edn. Springer, New York (2010)Google Scholar
  82. 82.
    Hermann, U., Noll, D.: Adaptive image reconstruction using information measures. SIAM J. Control Optim. 38(4), 1223–1240 (2000)zbMATHMathSciNetGoogle Scholar
  83. 83.
    Hohage, T., Werner, F.: Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data. Numerische Mathematik, 123(4), 745–779 (2013)zbMATHMathSciNetGoogle Scholar
  84. 84.
    Hohage, T., Werner, F.: Convergence rates for inverse problems with impulsive noise. SIAM J. Numer. Anal. 52(3), 1203–1221 (2014)MathSciNetGoogle Scholar
  85. 85.
    Hoppe, W., Langer, R., Knesch, G., Poppe, C.: Protein-kristallstrukturanalyse mit elektronenstrahlen. Naturwissenschaften 55, 333–336 (1968)Google Scholar
  86. 86.
    Hörmander L.: Fourier integral operators I. Acta Math. 127(1–2), 79–183 (1971)zbMATHMathSciNetGoogle Scholar
  87. 87.
    Isakov, V.: Increased stability in the continuation for the Helmholtz equation with variable coefficient. In: Ancona, F., Lasiecka, I., Littman, W., Triggiani, R. (eds.) Control Methods in PDE-Dynamical Systems. Contemporary Mathematics, vol. 426, pp. 243–254. American Mathematical Society, Providence (2007)Google Scholar
  88. 88.
    Ivanyshyn, O., Kress, R.: Inverse scattering for surface impedance from phase-less far field data. J. Comput. Phys. 230, 3443–3452 (2011)zbMATHMathSciNetGoogle Scholar
  89. 89.
    Jin, S., Yang, X.: Computation of the semiclassical limit of the Schrödinger equation with phase shift by a level set method. J. Sci. Comput. 35(2), 144–169 (2008)zbMATHMathSciNetGoogle Scholar
  90. 90.
    Jonas, P., Louis, A.K.: Phase contrast tomography using holographic measurements. Inverse Probl. 20(1), 75–102 (2004)zbMATHMathSciNetGoogle Scholar
  91. 91.
    Kaipio, J., Somersalo, E.: Statistical and Computational Inverse Problems. Applied Mathematical Sciences, vol. 160. Springer, New York (2005)Google Scholar
  92. 92.
    Kaltenbacher, B., Neubauer, A., Scherzer, O.: Iterative Regularization Methods for Nonlinear Ill-Posed Problems. Radon Series on Computational and Applied Mathematics, vol. 6. Walter de Gruyter, Berlin (2008)Google Scholar
  93. 93.
    Kisielowski, C., Freitag, B., Bischoff, M., van Lin, H., Lazar, S., Knippels, G., Tiemeijer, P., van der Stam, M., von Harrach, S., Stekelenburg, M., Haider, M., Müller, H., Hartel, P., Kabius, B., Miller, D., Petrov, I., Olson, E., Donchev, T., Kenik, E.A., Lupini, A., Bentley, J., Pennycook, S., Minor, A.M., Schmid, A.K., Duden, T., Radmilovic, V., Ramasse, Q., Erni, R., Watanabe, M., Stach, E., Denes, P., Dahmen, U.: Detection of single atoms and buried defects in three dimensions by aberration-corrected electron microscopy with 0.5 Å information limit. Microsc. Microanal. 14, 454–462 (2008)Google Scholar
  94. 94.
    Klann, E.: A Mumford–Shah-like method for limited data tomography with an application to electron tomography. SIAM J. Imaging Sci. 4(4), 1029–1048 (2011)zbMATHMathSciNetGoogle Scholar
  95. 95.
    Klibanov, M.V.: Phaseless inverse scattering problems in 3-d. Technical Report March (2013). arXiv:1303.0923v1, arXivGoogle Scholar
  96. 96.
    Klibanov, M.V.: Uniqueness of two phaseless inverse acoustics problems in 3-d. Technical Report March (2013). arXiv:1303.7384v1, arXivGoogle Scholar
  97. 97.
    Klug, A., DeRosier, D.J.: Reconstruction of three dimensional structures from electron micrographs. Nature 217, 130–134 (1968)Google Scholar
  98. 98.
    Knoll, M., Ruska, E.: Das Elektronenmikroskop. Z. Phys. A: Hadrons Nucl. 78, 318–339 (1932)Google Scholar
  99. 99.
    Koch, C.T., Lubk, A.: Off-axis and inline electron holography: a quantitative comparison. Ultramicroscopy 110(5), 460–471 (2010)Google Scholar
  100. 100.
    Kohl, H., Rose, H.: Theory of image formation by inelastically scattered electrons in the electron microscope. Adv. Electr. Electron Phys. 65, 173–227 (1985)Google Scholar
  101. 101.
    Kohr, H.: Fast and high-quality reconstructions in electron tomography. In: Simos, T.E., Psihoyios, G., Tsitouras, Ch. (eds.) ICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics 2010, 19–25 September 2010, Rhodes (Greece). AIP Conference Proceedings, vol. 1281, pp. 1979–1981 (2010)Google Scholar
  102. 102.
    Kohr, H., Louis, A.K.: Fast and high-quality reconstruction in electron tomography based on an enhanced linear forward model. Inverse Probl. 27 045008 (20 pp.) (2011)Google Scholar
  103. 103.
    Koster, A.J., Bárcena, M.: Cryotomography: Low-dose automated tomography of frozen-hydrated specimens. In: Frank, J. (ed.) Electron Tomography. Methods for Three-Dimensional Visualization of Structures in the Cell, Chap. 4, 2nd edn., pp. 113–161. Springer, New York (2006)Google Scholar
  104. 104.
    Krupchyk, K., Lassas, M., Uhlmann, G.: Inverse problems with partial data for a magnetic Schrödinger operator in an infinite slab and on a bounded domain. Commun. Math. Phys. 312, 87–126 (2012)zbMATHMathSciNetGoogle Scholar
  105. 105.
    Lassas, M., Saksman, E., Siltanen, S.: Discretization-invariant bayesian inversion and besov space priors. Inverse Probl. Imaging 3(1), 87–122 (2009)zbMATHMathSciNetGoogle Scholar
  106. 106.
    Lawrence, A., Bouwer, J.C., Perkins, G.A., Ellisman, M.H.: Transform-based backprojection for volume reconstruction of large format electron microscope tilt series. J. Struct. Biol. 154, 144–167 (2006)Google Scholar
  107. 107.
    Lee, H., Rose, Z., Hambach, H., Wachsmuth, P., Kaiser, U.: The influence of inelastic scattering on EFTEM images – exemplified at 20 kV for graphene and silicon. Ultramicroscopy 134, 102–112 (2013)Google Scholar
  108. 108.
    Leimana, P.G., Kanamaru, S., Mesyanzhinov, V.V., Arisaka, F., Rossmann, M.G.: Structure and morphogenesis of bacteriophage T4. Cell. Mol Life Sci. 60, 2356–2370 (2003)Google Scholar
  109. 109.
    Lewitt, R.M.: Alternatives to voxels for image representation in iterative reconstruction algorithms. Phys. Med. Biol. 37(3), 705–716 (1992)Google Scholar
  110. 110.
    Li, J., Shen, Z., Yin, R., Zhang, X.: A reweighted 2-method for image restoration with poisson and mixed Poisson–Gaussian noise. UCLA Computational and Applied Mathematics Reports 12–84. Department of Mathematics University of California, Los Angeles (2012)Google Scholar
  111. 111.
    Lin, P.D.: New Computation Methods for Geometrical Optics. Springer Series in Optical Sciences, vol. 178. Springer (2014)Google Scholar
  112. 112.
    Lin, T., Chen, Z., Usha, R., Stauffacher, C.V., Dai, J.-B., Schmidt, T., Johnson, J.E.: The refined crystal structure of cowpea mosaic virus at 2.8 A resolution. Virology 265, 20–34 (1999)Google Scholar
  113. 113.
    Liu, H., Ralston, J., Runborg, O., Tanushev, N.M.: Gaussian beam methods for the helmholtz equation (2013). arXiv [math.NA] 1304.1291v1, arXivGoogle Scholar
  114. 114.
    Louis, A.K.: Incomplete data problems in x-ray computerized tomography. Numer. Math. 48(3), 251–262 (1986)zbMATHMathSciNetGoogle Scholar
  115. 115.
    Louis, A.K.: A unified approach to regularization methods for linear ill-posed problems. Inverse Probl. 15(2), 489–498 (1999)zbMATHMathSciNetGoogle Scholar
  116. 116.
    Louis, A.K.: Development of algorithms in computerized tomography. In: Ólafsson, G., Quinto, E.T. (eds.) The Radon Transform, Inverse Problems, and Tomography. Proceedings of Symposia in Applied Mathematics, vol. 63, pp. 25–42. American Mathematical Society, Providence (2006)Google Scholar
  117. 117.
    Louis, A.K.: Feature reconstruction in inverse problems. Inverse Probl. 27(6), 065010 (2011)MathSciNetGoogle Scholar
  118. 118.
    Louis, A.K., Maass, P.: A mollifier method for linear operator equations of the first kind. Inverse Probl. 6(3), 427–440 (1990)zbMATHMathSciNetGoogle Scholar
  119. 119.
    Macaulay, V.A., Buck, B.: Linear inversion by the method of maximum entropy. Inverse Probl. 5, 859–874 (1989)zbMATHMathSciNetGoogle Scholar
  120. 120.
    Marabini, R., Herman, G.T., Carazo, J.-M.: 3D reconstruction in electron microscopy using ART with smooth spherically symmetric volume elements (blobs). Ultramicroscopy 72, 53–65 (1998)Google Scholar
  121. 121.
    Marburg, S.: Discretization requirements: how many elements per wavelength are necessary? In: Marburg, S., Nolte, B. (eds.) Computational Acoustics of Noise Propagation in Fluids – Finite and Boundary Element Methods, Chap. 11, pp. 309–332. Springer, Berlin (2008)Google Scholar
  122. 122.
    Markoe, A.: Analytic Tomography. Encyclopedia of Mathematics and its Applications, vol. 106. Cambridge University Press, Cambridge (2006)Google Scholar
  123. 123.
    Marone, F., Münch, B., Stampanoni, M.: Fast reconstruction algorithm dealing with tomography artifacs. In: Stock, S.R. (ed.) Developments in X-Ray Tomography VII, San Diego, CA, 1 August 2010. Proceedings of SPIE, vol. 7804, pp. 780410-1–780410-11 (2010)Google Scholar
  124. 124.
    Mastronarde, D.N.: Dual-axis tomography: an approach with alignment methods that preserve resolution. J. Struct. Biol. 120, 343–352 (1997)Google Scholar
  125. 125.
    Matsushima, K., Schimmel, H., Buehling, S., Wyrowski, F.: Propagation of electromagnetic fields between nonparallel planes. In: Wyrowski, F. (ed.) Wave-Optical Systems Engineering II. Proceedings of SPIE, vol. 5182, pp. 55–62. SPIE, Bellingham (2003)Google Scholar
  126. 126.
    Midgley, P.A., Weyland, M.: 3D electron microscopy in the physical sciences. The development of Z-contrast and EFTEM tomography. Ultramicroscopy 96, 413–431 (2003)Google Scholar
  127. 127.
    Midgley, P.A., Weyland, M., Stegmann, H.: Applications of electron tomography. In: Banhart, J. (ed.) Advanced Tomographic Methods in Materials Research and Engineering. Monographs on the Physics and Chemistry of Materials, Chap. 12, vol. 66, pp. 335–372. Oxford University Press, Oxford (2008)Google Scholar
  128. 128.
    Muga, J.G., Palao, J.P., Navarro, B., Egusquiza, I.L.: Complex absorbing potentials. Phys. Rep. 395, 357–426 (2004)MathSciNetGoogle Scholar
  129. 129.
    Müller, H.: A coherence function approach to image simulation. Ph.D. thesis, Fachbereich Physik, Technischen Universität Darmstadt, Darmstadt, Germany (2000)Google Scholar
  130. 130.
    Myers, G.R., Gureyev, T.E., Paganin, D.M.: Stability of phase-contrast tomography. J. Opt. Soc. Am. A: Opt. Image Sci. Vis. 24(9), 2516–2526 (2007)Google Scholar
  131. 131.
    Nagayasu, S., Uhlmann, G., Wang, J.-N.: Increasing stability in an inverse problem for the acoustic equation. Inverse Probl. 29(2), 025012 (11 pp.) (2013)Google Scholar
  132. 132.
    Namba, K., Pattanayek, R., Stubbs, G.: Visualization of protein-nucleic acid interactions in a virus. Refined structure of intact tobacco mosaic virus at 2.9 Å resolution by X-ray fiber diffraction. J. Mol. Biol. 208(2), 307–325 (1989)Google Scholar
  133. 133.
    Narasimha, R., Aganj, I., Bennett, A., Borgnia, M.J., Zabransky, D., Sapiro, G., McLaughlin, S.W., Milne, J.L.S., Subramaniam, S.: Evaluation of denoising algorithms for biological electron tomography. J. Struct. Biol. 164(1), 7–17 (2008)Google Scholar
  134. 134.
    Natterer, F.: The Mathematics of Computerized Tomography. Classics in Applied Mathematics, vol. 32. SIAM, Philadelphia (2001)Google Scholar
  135. 135.
    Natterer, F.: An error bound for the Born approximation. Inverse Probl. 20, 447–452 (2004)zbMATHMathSciNetGoogle Scholar
  136. 136.
    Natterer, F., Wübbeling, F.: Mathematical Methods in Image Reconstruction. SIAM Monographs on Mathematical Modeling and Computation, vol. 5. SIAM, Philadelphia (2001)Google Scholar
  137. 137.
    Noll, D.: Consistency of a nonlinear deconvolution method with applications in image restoration. Adv. Math. Sci. Appl. 7(2), 789–808 (1997)zbMATHMathSciNetGoogle Scholar
  138. 138.
    Öktem, O., Bajaj, C., Ravikumar, P.: Summary of results regarding shape based regularization for de-noising and 2D tomography. Preliminary Progress Report (2013)Google Scholar
  139. 139.
    Paganin, D., Mayo, S.C., Gureyev, T.E., Miller, P.R., Wilkins, S.W.: Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object. J. Microsc. 206(Part 1), 33–40 (2002)Google Scholar
  140. 140.
    Palamodov, V.P.: Stability in diffraction tomography and a nonlinear “basic theorem”. J. d’Analyse Mathématique 91(1), 247–268 (2003)zbMATHMathSciNetGoogle Scholar
  141. 141.
    Palamodov, V.P.: Reconstructive Integral Geometry. Monographs in Mathematics, vol. 98. Birkhäuser, Basel (2004)Google Scholar
  142. 142.
    Palamodov, V.P.: Inverse scattering as nonlinear tomography. Wave Motion 47, 635–640 (2010)zbMATHMathSciNetGoogle Scholar
  143. 143.
    Penczek, P.A.: Fundamentals of three-dimensional reconstruction from projections. In: Jensen, G.J. (ed.) Cryo-EM, Part B: 3-D Reconstruction. Methods in Enzymology, Chap. 1, vol. 482, pp. 1–33. Academic, San Diego (2010)Google Scholar
  144. 144.
    Penczek, P.A.: Resolution measures in molecular electron microscopy. In: Jensen, G.J. (ed.) Cryo-EM, Part B: 3-D Reconstruction. Methods in Enzymology, Chap. 3, vol. 482, pp. 73–100. Academic, San Diego (2010)Google Scholar
  145. 145.
    Penczek, P.A., Frank, J.: Resolution in electron tomography. In: Frank, J. (ed.) Electron Tomography. Methods for Three-Dimensional Visualization of Structures in the Cell, Chap. 10, 2nd edn., pp. 307–330. Springer, New York (2006)Google Scholar
  146. 146.
    Peng, L.-M., Dudarev, S.L., Whelan, M.J.: High-Energy Electron Diffraction and Microscopy. Monographs on the Physics and Chemistry of Materials, vol. 61, Oxford University Press, Oxford (2004)Google Scholar
  147. 147.
    Pereyra, V.: Ray tracing methods for inverse problems. Inverse Probl. 16(6), R1–R35 (2000)zbMATHMathSciNetGoogle Scholar
  148. 148.
    Plitzko, J.M., Baumeister, W.: Cryoelectron tomography (CET). In: Hawkes, P.W., Spence, J.C.H. (eds.) Science of Microscopy, Chap. 7, pp. 535–604. Springer (2008)Google Scholar
  149. 149.
    Pohjola, V.: An Inverse Boundary Value Problem for the Magnetic Schrödinger Operator on a Half Space. Licentiate thesis. Department of Mathematics and Statistics, University of Helsinki (2012)Google Scholar
  150. 150.
    Quinto, E.T.: Singularities of the X-ray transform and limited data tomography in r 2 and r 3. SIAM J. Math. Anal. 24, 1215–1225 (1993)zbMATHMathSciNetGoogle Scholar
  151. 151.
    Quinto, E.T., Öktem, O.: Local tomography in electron microscopy. SIAM J. Appl. Math. 68(5), 1282–1303 (2008)zbMATHMathSciNetGoogle Scholar
  152. 152.
    Quinto, E.T., Rullgård, H.: Electron microscope tomography over curves. Oberwolfach Reports 18/2010, Mathematisches Forschungsinstitut Oberwolfach (2010)Google Scholar
  153. 153.
    Quinto, E.T., Rullgård, H.: Local sobolev estimates of a function by means of its Radon transform. Inverse Probl. Imaging 4(4), 721–734 (2010)zbMATHMathSciNetGoogle Scholar
  154. 154.
    Quinto, E.T., Rullgård, H.: Local singularity reconstruction from integrals over curves in r 3. Inverse Probl. Imaging 7(2), 585–609 (2013)zbMATHMathSciNetGoogle Scholar
  155. 155.
    Quinto, E.T., Skoglund, U., Öktem, O.: Electron lambda-tomography. Proc. Natl. Acad. Sci. 106(51), 21842–21847 (2009)Google Scholar
  156. 156.
    Radermacher, M.: Weighted back-projection methods. In: Frank, J. (ed.) Electron Tomography. Methods for Three-Dimensional Visualization of Structures in the Cell, Chap. 8, 2nd edn., pp. 245–273. Springer, New York (2006)Google Scholar
  157. 157.
    Ram, S., Ward, S.E., Ober, R.J.: A stochastic analysis of distance estimation approaches in single molecule microscopy: quantifying the resolution limits of photon-limited imaging systems. Multidimension. Syst. Signal Process. 24, 503–542 (2013)zbMATHMathSciNetGoogle Scholar
  158. 158.
    Reich, E.S.: Imaging hits noise barrier: physical limits mean that electron microscopy may be nearing highest possible resolution. Nature 499(7457), 135–136 (2013)Google Scholar
  159. 159.
    Resmerita, E.: Regularization of ill-posed problems in Banach spaces: convergence rates. Inverse Probl. 21, 1303–1314 (2005)zbMATHMathSciNetGoogle Scholar
  160. 160.
    Rose, H.: Information transfer in transmission electron microscopy. Ultramicroscopy 15(3), 173–192 (1984)Google Scholar
  161. 161.
    Rose, H.: Advances in electron optics. In: Ernst, F., Rühle, M. (eds.) High-Resolution Imaging and Spectrometry of Materials. Springer Series in Materials Science, Chap. 5, pp. 189–270. Springer, Berlin (2003)Google Scholar
  162. 162.
    Rose, H.: Geometrical Charged-Particle Optics. Springer Series in Optical Sciences, vol. 142, 2nd edn. Springer, New York (2012)Google Scholar
  163. 163.
    Rubinstein, R., Bruckstein, A.M., Elad, M.: Dictionaries for sparse representation modeling. Proc. IEEE 98(6), 1045–1057 (2010)Google Scholar
  164. 164.
    Rullgård, H.: A new principle for choosing regularization parameter in certain inverse problems. Department of Mathematics, Stockholm University (2008). arXiv report in math.NA arXiv:0803.3713v2Google Scholar
  165. 165.
    Rullgård, H., Öktem, O., Skoglund, U.: A componentwise iterated relative entropy regularization method with updated prior and regularization parameter. Inverse Prob. 23, 2121–2139 (2007)zbMATHGoogle Scholar
  166. 166.
    Rullgård, H., Öfverstedt, L.-G., Masich, S., Daneholt, B., Öktem, O.: Simulation of transmission electron microscope images of biological specimens. J. Microsc. 243(3), 234–256 (2011)Google Scholar
  167. 167.
    Runborg, O.: Mathematical models and numerical methods for high frequency waves. Commun. Comput. Phys. 2(5), 827–880 (2007)zbMATHMathSciNetGoogle Scholar
  168. 168.
    Sato, M.: Hyperfunctions and partial differential equations. In: Proceedings of the 2nd Conference on Functional Analysis and Related Topics, Tokyo, pp. 91–94. Tokyo University, Tokyo University Press, Tokyo (1969)Google Scholar
  169. 169.
    Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational Methods in Imaging. Applied Mathematical Sciences, vol. 167. Springer, New York (2009)Google Scholar
  170. 170.
    Schuster, T.: The Method of Approximate Inverse: Theory and Applications. Lecture Notes in Mathematics, vol. 1906. Springer, Heidelberg (2007)Google Scholar
  171. 171.
    Schuster, T., Kaltenbacher, B., Hofmann, B., Kazimierski, K.S.: Regularization Methods in Banach Spaces. Radon Series on Computational and Applied Mathematics, vol. 10. Walter de Gruyter, Berlin (2012)Google Scholar
  172. 172.
    Skoglund, U., Öfverstedt, L.-G., Burnett, R.M., Bricogne, G.: Maximum-entropy three-dimensional reconstruction with deconvolution of the contrast transfer function: a test application with Adenovirus. J. Struct. Biol. 117, 173–188 (1996)Google Scholar
  173. 173.
    Smith, K.T.: Inversion of the X-ray transform. In: McLaughlin, D.W. (ed.) Inverse Problems. SIAM-AMS Proceedings, vol. 14, pp. 41–52. American Mathematical Society, Providence (1984)Google Scholar
  174. 174.
    Snyder, D.L., Schutz, T.J., O’Sullivan, J.A.: Deblurring subject to nonnegative constraint. IEEE Trans. Signal Process. 40, 1143–1150 (1992)zbMATHGoogle Scholar
  175. 175.
    Sorzano, C.O.S., Otero, A., Olmos, E.M., Carazo, J.-M.: Error analysis in the determination of the electron microscopical contrast transfer function parameters from experimental power spectra. BMC Struct. Biol. 9(18) (2009)Google Scholar
  176. 176.
    Spence, J.C.H.: High-Resolution Electron Microscopy. Monographs on the Physics and Chemistry of Materials, vol. 60, 3rd edn. Oxford University Press, New York (2003)Google Scholar
  177. 177.
    Turonǒvá, B.: Simultaneous Algebraic Reconstruction Technique for Electron Tomography Using OpenCL. MSc thesis. Saarland University, Faculty of Natural Sciences and Technology I, Department of Computer Science (2011)Google Scholar
  178. 178.
    Vainberg, E.I., Kazak, I.A., Kurozaev, V.P.: Reconstruction of the internal three-dimensional structure of objects based on real-time integral projections. Sov. J. Nondestruct. Test. 17, 415–423 (1981)Google Scholar
  179. 179.
    Vainshtein, B.K., Barynin, V.V., Gurskaya, G.V.: The hexagonal crystalline structure of catalase and its molecular structure. Dokl. Akad. Nauk SSSR 182, 569–572 (1968). See also Sov. Phys. Dokl. 13, 838–841 (1969)Google Scholar
  180. 180.
    Van Aert, S., den Dekker, A.J., Van Dyck, D., van den Bos, A.: The notion of resolution. In: Hawkes, P.W., Spence, J.C.H. (eds.) Science of Microscopy, Chap. 20, pp. 1228–1265. Springer, New York (2007)Google Scholar
  181. 181.
    Van den Broek, W., Koch, C.T.: General framework for quantitative three-dimensional reconstruction from arbitrary detection geometries in TEM. Phys. Rev. B: Condens. Matter Mater. Phys. 87(18), 184108 (11 pp.) (2013)Google Scholar
  182. 182.
    Vanska, S., Lassas, M., Siltanen, S.: Statistical X-ray tomography using empirical Besov priors. Int. J. Tomography Stat. 30, 3–32 (2009)MathSciNetGoogle Scholar
  183. 183.
    Verbeeck, J., Schattschneider, P., Rosenauer, A.: Image simulation of high resolution energy filtered TEM images. Ultramicroscopy 109, 350–360 (2009)Google Scholar
  184. 184.
    Voortman, L.M., Stallinga, S., Schoenmakers, R.H., van Vliet, L.J., Rieger, B.: A fast algorithm for computing and correcting the CTF for tilted, thick specimens in TEM. Ultramicroscopy 111(8), 1029–1036 (2011)Google Scholar
  185. 185.
    Voortman, L.M., Franken, E.M., van Vliet, L.J., Rieger, B.: Fast, spatially varying CTF correction in TEM. Ultramicroscopy 118, 26–34 (2012)Google Scholar
  186. 186.
    Vulović, M., Franken, E.M., Ravelli, R.B., van Vliet, L.J., Rieger, B.: Precise and unbiased estimation of astigmatism and defocus in transmission electron microscopy. Ultramicroscopy 116, 115–134 (2012)Google Scholar
  187. 187.
    Vulović, M., Ravelli, R.B., van Vliet, L.J., Koster, A.J., Lazić, I., Lücken, U., Rullgård, H., Öktem, O., Rieger, B.B.: Image formation modeling in cryo-electron microscopy. J. Struct. Biol. 183(1), 19–32 (2013)Google Scholar
  188. 188.
    Vulović, M., Rieger, B., van Vliet, L.J., Koster, A.J., Ravelli, R.B.: A toolkit for the characterization of CCD cameras for transmission electron microscopy. Acta Crystallogr. Sect. D: Biol. Crystallogr. 66, 97–109 (2010)Google Scholar
  189. 189.
    Vulović, M., Voortman, L.M., van Vliet, L.J., Rieger, B.: When to use the projection assumption and the weak-phase object approximation in phase contrast cryo-EM. Ultramicroscopy 136, 61–66 (2014)Google Scholar
  190. 190.
    Wang, A., Turner, S., Van Aert, S., Van Dyck, D.: An alternative approach to determine attainable resolution directly from HREM images. Ultramicroscopy 133, 50–61 (2013)Google Scholar
  191. 191.
    Xiong, Q., Morphew, M.K., Schwartz, C.L., Hoenger, A.H., Mastronarde, D.N.: CTF determination and correction for low dose tomographic tilt series. J. Struct. Biol. 168, 378–387 (2009)Google Scholar
  192. 192.
    Xu, W., Xu, F., Jones, M., Keszthelyi, B., Sedat, J., Agard, D., Mueller, K.: High-performance iterative electron tomography reconstruction with long-object compensation using graphics processing units (gpus). J. Struct. Biol. 171(2), 142–153 (2010)Google Scholar
  193. 193.
    Xu, G., Li, M., Gopinath, A., Bajaj, C.: Inversion of electron tomography images using l2-gradient flows – computational methods. J. Comput. Math. 29(5), 501–525 (2011)zbMATHMathSciNetGoogle Scholar
  194. 194.
    Younes, L.: Shapes and Diffeomorphisms. Applied Mathematical Sciences, vol. 171. Springer, New York (2010)Google Scholar
  195. 195.
    Yserentant, H.: Regularity and Approximability of Electronic Wave Functions. Lecture Notes in Mathematics, vol. 2000. Springer, Berlin (2010)Google Scholar
  196. 196.
    Zeidler, E.: Nonlinear Functional Analysis and Its Applications. Part 3: Variational Methods and Optimization. Springer, New York (1985)Google Scholar
  197. 197.
    Zuo, J.M.: Electron detection characteristics of a slow-scan CCD camera, imaging plates and film, and electron image restoration. Microsc. Res. Tech. 49, 245–268 (2000)Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsKTH Royal Institute of TechnologyStockholmSweden

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