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2015 | OriginalPaper | Buchkapitel

Mathematics of Electron Tomography

verfasst von : Ozan Öktem

Erschienen in: Handbook of Mathematical Methods in Imaging

Verlag: Springer New York

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Abstract

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.

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Vorheriges Kapitel Mathematical Methods in PET and SPECT Imaging

Nächstes Kapitel Optical Imaging

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)

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)

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)

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)MATHMathSciNet

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)

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)

Ayache, J., Beaunier, L., Boumendil, J., Ehret, G., Laub, D.: Sample Preparation Handbook for Transmission Electron Microscopy: Methodology. Springer, New York (2010)

Ayache, J., Beaunier, L., Boumendil, J., Ehret, G., Laub, D.: Sample Preparation Handbook for Transmission Electron Microscopy: Techniques. Springer, New York (2010)

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)

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)

11.

Bakushinsky, A.B., Yu, M.: Kokurin. Iterative Methods for Approximate Solution of Inverse Problem. Mathematics and its Applications, vol. 577. Springer, Dordrecht (2004)

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, 2006

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)

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)MATHMathSciNet

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)

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)

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)

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)

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)

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)MATHMathSciNet

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)MATHMathSciNet

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)

23.

Busch, H.: Berechnung der Bahn von Kathodenstrahlen im axialsymmetrischen elektromagnetischen Felde. Ann. Phys. 386, 974–993 (1926)

24.

Busch, H.: Über die Wirkungsweise der Konzentrierungsspule bei der Braunschen Rohre. Electr. Eng. (Archiv fur Elektrotechnik) 18, 583–594 (1927)

25.

Byrne, C.L.: Applied Iterative Methods. A. K. Peters, Wellesley (2008)MATH

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)

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)MATH

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)

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)

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)

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)

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)

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)

34.

Çınlar, E.: Probability and Stochastics. Graduate Texts in Mathematics, vol. 261. Springer, New York (2011)

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)

36.

Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences, vol. 93, 3rd edn. Springer, New York (2013)

37.

Cruz-Uribe, D.V., Fiorenza, A.: Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis. Springer, Basel (2013)

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)MATHMathSciNet

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)

40.

Davidson, M.E.: The ill-conditioned nature of the limited angle tomography problem. SIAM J. Appl. Math. 43(2), 428–448 (1983)MathSciNet

41.

De Rosier, D.J., Klug, A.: Reconstruction of three dimensional structures from electron micrographs. Nature 217, 130–134 (1968)

42.

Defrise, M., Clack, R., Townsend, D.W.: Image reconstruction from truncated, two-dimensional, parallel projections. Inverse Probl. 11(2), 287–313 (1995)MATHMathSciNet

43.

DeRosier, D.J.: The reconstruction of three-dimensional images from electron micrographs. Contemp. Phys. 12(5), 437–452 (1971)

44.

Ebanks, B., Sahoo, P., Sander, W.: Characterization of Information Measures. World Scientific, Singapore (1998)

45.

Egerton, R.F.: Electron Energy-Loss Spectroscopy in the Electron Microscope, 3rd edn. Springer, New York (2011)

46.

Egerton, R.F., Li, P., Malac, M.: Radiation damage in the TEM and SEM. Micron 35, 399–409 (2004)

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)MATHMathSciNet

48.

Elad, M.: Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing. Springer (2010)

49.

Engl, H.W., Landl, G.: Convergence rates for maximum entropy regularization. SIAM J. Numer. Anal. 30(5), 1509–1536 (1993)MATHMathSciNet

50.

Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Mathematics and its Applications, vol. 375. Kluwer Academic Publishers (2000)

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)

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)

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)

54.

Faridani, A., Finch, D., Ritman, E.L., Smith, K.T.: Local tomography II. SIAM J. Appl. Math. 57(4), 1095–1127 (1997)MATHMathSciNet

55.

Faruqi, A.R., McMullan, G.: Electronic detectors for electron microscopy. Q. Rev. Biophys. 44(3), 357–390 (2011)

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)MATHMathSciNet

57.

Fernández, J.J., Li, S., Crowther, R.A.: CTF determination and correction in electron cryotomography. Ultramicroscopy 106(7), 587–596 (2006)

58.

Foley, J.T., Butts, R.R.: Uniqueness of phase retrieval from intensity measurements. J. Opt. Soc. Am. A 71(8), 1008–1014 (1981)

59.

Foucart, S., Rahut, H.: A Mathematical Introduction to Compressive Sensing. Applied and Numerical Harmonic Analysis. Springer, New York (2013)MATH

60.

Frank, J.: Three-Dimensional Electron Microscopy of Macromolecular Assemblies, 2nd edn. Oxford University Press, Oxford (2006)

61.

Frank, J.: Single-particle reconstruction of biological macromoleculses in electron micrscopy – 30 years. Q. Rev. Biophys. 42, 139–158 (2009)

62.

Frikel, J., Quinto, T.: Characterization and reduction of artifacts in limited angle tomography. Inverse Probl. 29(12), 125007 (2013)MathSciNet

63.

Fultz, B., Howe, J.: Transmission Electron Microscopy and Diffractometry of Materials. Graduate Texts in Physics, 4th edn. Springer, Berlin (2013)

64.

Gardueno, E., Herman, G.T.: Optimization of basis functions for both reconstruction and visualization. Discrete Appl. Math. 139, 95–111 (2004)MathSciNet

65.

Gilbert, P.: Iterative methods for the three-dimensional reconstruction of an object from projections. J. Theor. Biol. 36(1), 105–117 (1972)

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)MATHMathSciNet

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)

68.

Glaeser, R.M., Downing, K.H., DeRozier, D., Chu, W., Frank, J.: Electron Crystallography of Biological Macromolecules. Oxford University Press, Oxford (2006)

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)

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)

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)

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)

73.

Greenleaf, A., Uhlmann, G.: Non-local inversion formulas for the X-ray transform. Duke Math. J. 58, 205–240 (1989)MATHMathSciNet

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)MATHMathSciNet

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)

76.

Guillemin, V., Sternberg, S.: Geometric Asymptotics. Mathematical Surveys and Monographs, vol. 14. American Mathematical Society, Providence (1977). Revised edition (June 1990) edition

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)

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)

79.

Hawkes, P.W., Kasper, E.: Principles of Electron Optics. Wave Optics, vol. 3. Academic, San Diego (1995)

80.

Hawkes, P.W., Kasper, E.: Principles of Electron Optics. Applied Geometrical Optics, vol. 2. Academic, San Diego, (1996)

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)

82.

Hermann, U., Noll, D.: Adaptive image reconstruction using information measures. SIAM J. Control Optim. 38(4), 1223–1240 (2000)MATHMathSciNet

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)MATHMathSciNet

84.

Hohage, T., Werner, F.: Convergence rates for inverse problems with impulsive noise. SIAM J. Numer. Anal. 52(3), 1203–1221 (2014)MathSciNet

85.

Hoppe, W., Langer, R., Knesch, G., Poppe, C.: Protein-kristallstrukturanalyse mit elektronenstrahlen. Naturwissenschaften 55, 333–336 (1968)

86.

Hörmander L.: Fourier integral operators I. Acta Math. 127(1–2), 79–183 (1971)MATHMathSciNet

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)

88.

Ivanyshyn, O., Kress, R.: Inverse scattering for surface impedance from phase-less far field data. J. Comput. Phys. 230, 3443–3452 (2011)MATHMathSciNet

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)MATHMathSciNet

90.

Jonas, P., Louis, A.K.: Phase contrast tomography using holographic measurements. Inverse Probl. 20(1), 75–102 (2004)MATHMathSciNet

91.

Kaipio, J., Somersalo, E.: Statistical and Computational Inverse Problems. Applied Mathematical Sciences, vol. 160. Springer, New York (2005)

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)

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)

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)MATHMathSciNet

95.

Klibanov, M.V.: Phaseless inverse scattering problems in 3-d. Technical Report March (2013). arXiv:1303.0923v1, arXiv

96.

Klibanov, M.V.: Uniqueness of two phaseless inverse acoustics problems in 3-d. Technical Report March (2013). arXiv:1303.7384v1, arXiv

97.

Klug, A., DeRosier, D.J.: Reconstruction of three dimensional structures from electron micrographs. Nature 217, 130–134 (1968)

98.

Knoll, M., Ruska, E.: Das Elektronenmikroskop. Z. Phys. A: Hadrons Nucl. 78, 318–339 (1932)

99.

Koch, C.T., Lubk, A.: Off-axis and inline electron holography: a quantitative comparison. Ultramicroscopy 110(5), 460–471 (2010)

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)

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)

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)

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)

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)MATHMathSciNet

105.

Lassas, M., Saksman, E., Siltanen, S.: Discretization-invariant bayesian inversion and besov space priors. Inverse Probl. Imaging 3(1), 87–122 (2009)MATHMathSciNet

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)

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)

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)

109.

Lewitt, R.M.: Alternatives to voxels for image representation in iterative reconstruction algorithms. Phys. Med. Biol. 37(3), 705–716 (1992)

110.

Li, J., Shen, Z., Yin, R., Zhang, X.: A reweighted ℓ 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)

111.

Lin, P.D.: New Computation Methods for Geometrical Optics. Springer Series in Optical Sciences, vol. 178. Springer (2014)

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)

113.

Liu, H., Ralston, J., Runborg, O., Tanushev, N.M.: Gaussian beam methods for the helmholtz equation (2013). arXiv [math.NA] 1304.1291v1, arXiv

114.

Louis, A.K.: Incomplete data problems in x-ray computerized tomography. Numer. Math. 48(3), 251–262 (1986)MATHMathSciNet

115.

Louis, A.K.: A unified approach to regularization methods for linear ill-posed problems. Inverse Probl. 15(2), 489–498 (1999)MATHMathSciNet

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)

117.

Louis, A.K.: Feature reconstruction in inverse problems. Inverse Probl. 27(6), 065010 (2011)MathSciNet

118.

Louis, A.K., Maass, P.: A mollifier method for linear operator equations of the first kind. Inverse Probl. 6(3), 427–440 (1990)MATHMathSciNet

119.

Macaulay, V.A., Buck, B.: Linear inversion by the method of maximum entropy. Inverse Probl. 5, 859–874 (1989)MATHMathSciNet

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)

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)

122.

Markoe, A.: Analytic Tomography. Encyclopedia of Mathematics and its Applications, vol. 106. Cambridge University Press, Cambridge (2006)

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)

124.

Mastronarde, D.N.: Dual-axis tomography: an approach with alignment methods that preserve resolution. J. Struct. Biol. 120, 343–352 (1997)

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)

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)

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)

128.

Muga, J.G., Palao, J.P., Navarro, B., Egusquiza, I.L.: Complex absorbing potentials. Phys. Rep. 395, 357–426 (2004)MathSciNet

129.

Müller, H.: A coherence function approach to image simulation. Ph.D. thesis, Fachbereich Physik, Technischen Universität Darmstadt, Darmstadt, Germany (2000)

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)

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)

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)

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)

134.

Natterer, F.: The Mathematics of Computerized Tomography. Classics in Applied Mathematics, vol. 32. SIAM, Philadelphia (2001)

135.

Natterer, F.: An error bound for the Born approximation. Inverse Probl. 20, 447–452 (2004)MATHMathSciNet

136.

Natterer, F., Wübbeling, F.: Mathematical Methods in Image Reconstruction. SIAM Monographs on Mathematical Modeling and Computation, vol. 5. SIAM, Philadelphia (2001)

137.

Noll, D.: Consistency of a nonlinear deconvolution method with applications in image restoration. Adv. Math. Sci. Appl. 7(2), 789–808 (1997)MATHMathSciNet

138.

Öktem, O., Bajaj, C., Ravikumar, P.: Summary of results regarding shape based regularization for de-noising and 2D tomography. Preliminary Progress Report (2013)

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)

140.

Palamodov, V.P.: Stability in diffraction tomography and a nonlinear “basic theorem”. J. d’Analyse Mathématique 91(1), 247–268 (2003)MATHMathSciNet

141.

Palamodov, V.P.: Reconstructive Integral Geometry. Monographs in Mathematics, vol. 98. Birkhäuser, Basel (2004)

142.

Palamodov, V.P.: Inverse scattering as nonlinear tomography. Wave Motion 47, 635–640 (2010)MATHMathSciNet

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)

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)

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)

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)

147.

Pereyra, V.: Ray tracing methods for inverse problems. Inverse Probl. 16(6), R1–R35 (2000)MATHMathSciNet

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)

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)

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)MATHMathSciNet

151.

Quinto, E.T., Öktem, O.: Local tomography in electron microscopy. SIAM J. Appl. Math. 68(5), 1282–1303 (2008)MATHMathSciNet

152.

Quinto, E.T., Rullgård, H.: Electron microscope tomography over curves. Oberwolfach Reports 18/2010, Mathematisches Forschungsinstitut Oberwolfach (2010)

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)MATHMathSciNet

154.

Quinto, E.T., Rullgård, H.: Local singularity reconstruction from integrals over curves in r ³. Inverse Probl. Imaging 7(2), 585–609 (2013)MATHMathSciNet

155.

Quinto, E.T., Skoglund, U., Öktem, O.: Electron lambda-tomography. Proc. Natl. Acad. Sci. 106(51), 21842–21847 (2009)

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)

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)MATHMathSciNet

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)

159.

Resmerita, E.: Regularization of ill-posed problems in Banach spaces: convergence rates. Inverse Probl. 21, 1303–1314 (2005)MATHMathSciNet

160.

Rose, H.: Information transfer in transmission electron microscopy. Ultramicroscopy 15(3), 173–192 (1984)

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)

162.

Rose, H.: Geometrical Charged-Particle Optics. Springer Series in Optical Sciences, vol. 142, 2nd edn. Springer, New York (2012)

163.

Rubinstein, R., Bruckstein, A.M., Elad, M.: Dictionaries for sparse representation modeling. Proc. IEEE 98(6), 1045–1057 (2010)

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:0803.3713v2

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)MATH

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)

167.

Runborg, O.: Mathematical models and numerical methods for high frequency waves. Commun. Comput. Phys. 2(5), 827–880 (2007)MATHMathSciNet

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)

169.

Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational Methods in Imaging. Applied Mathematical Sciences, vol. 167. Springer, New York (2009)

170.

Schuster, T.: The Method of Approximate Inverse: Theory and Applications. Lecture Notes in Mathematics, vol. 1906. Springer, Heidelberg (2007)

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)

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)

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)

174.

Snyder, D.L., Schutz, T.J., O’Sullivan, J.A.: Deblurring subject to nonnegative constraint. IEEE Trans. Signal Process. 40, 1143–1150 (1992)MATH

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)

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)

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)

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)

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)

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)

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)

182.

Vanska, S., Lassas, M., Siltanen, S.: Statistical X-ray tomography using empirical Besov priors. Int. J. Tomography Stat. 30, 3–32 (2009)MathSciNet

183.

Verbeeck, J., Schattschneider, P., Rosenauer, A.: Image simulation of high resolution energy filtered TEM images. Ultramicroscopy 109, 350–360 (2009)

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)

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)

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)

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)

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)

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)

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)

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)

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)

193.

Xu, G., Li, M., Gopinath, A., Bajaj, C.: Inversion of electron tomography images using l ²-gradient flows – computational methods. J. Comput. Math. 29(5), 501–525 (2011)MATHMathSciNet

194.

Younes, L.: Shapes and Diffeomorphisms. Applied Mathematical Sciences, vol. 171. Springer, New York (2010)

195.

Yserentant, H.: Regularity and Approximability of Electronic Wave Functions. Lecture Notes in Mathematics, vol. 2000. Springer, Berlin (2010)

196.

Zeidler, E.: Nonlinear Functional Analysis and Its Applications. Part 3: Variational Methods and Optimization. Springer, New York (1985)

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)

Titel: Mathematics of Electron Tomography
verfasst von: Ozan Öktem
Verlag: Springer New York
Buch: Handbook of Mathematical Methods in Imaging
Print ISBN: 978-1-4939-0789-2

Electronic ISBN: 978-1-4939-0790-8

Copyright-Jahr: 2015
DOI: https://doi.org/10.1007/978-1-4939-0790-8_43

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