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Handbook of Mathematical Methods in Imaging pp 847–902Cite as

Microlocal Analysis in Tomography

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Abstract

Several limited data problems in tomography will be presented in this chapter, including ones for X-ray tomography, electron microscopy, and radar imaging. First, reconstructions from limited data will be evaluated to observe their strengths and weaknesses. Then, the basic analytic properties of the transforms will be presented. The concept of microlocal analysis will be introduced to make the notion of singularity precise. Finally, the microlocal properties of the tomographic transforms are given and then used to explain the observed strengths and limitations of the reconstructions. This will show that these limitations are intrinsic to these limited data problems themselves.

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References

  1. Ambartsoumian, G., Felea, R., Krishnan, V.P., Nolan, C., Quinto, E.T.: A class of singular Fourier integral operators in synthetic aperture radar imaging. J. Funct. Anal. 264(1), 246–269 (2013). doi:10.1016/j.jfa.2012.10.008, http://dx.doi.org/10.1016/j.jfa.2012.10.008

  2. Anastasio, M.A., Zou, Y., Sidky, E.Y., Pan, X.: Local cone-beam tomography image reconstruction on chords. J. Opt. Soc. Am. A 24, 1569–1579 (2007)

    Article  MathSciNet  Google Scholar 

  3. Bates, R., Lewitt, R.: Image reconstruction from projections: I: general theoretical considerations, II: projection completion methods (theory), III: Projection completion methods (computational examples). Optik 50, I: 19–33, II: 189–204, III: 269–278 (1978)

    Google Scholar 

  4. Bidwell, S.: Limited angle tomography and microlocal analysis. Tech. rep., Tufts University, senior Honors Thesis with Highest Thesis Honors (2012)

    Google Scholar 

  5. Boman, J.: Helgason’s Support Theorem for Radon Transforms – A New Proof and a Generalization. Lecture Notes in Mathematics, vol. 1497, pp. 1–5. Springer, Berlin/New York (1991)

    Google Scholar 

  6. Boman, J., Quinto, E.T.: Support theorems for real analytic Radon transforms. Duke Math. J. 55, 943–948 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cheney, M., Borden, B.: Fundamentals of Radar Imaging, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 79. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2009). doi:10.1137/1.9780898719291, http://dx.doi.org/10.1137/1.9780898719291

  8. Cheney, M., Borden, B.: Problems in synthetic-aperture radar imaging. Inverse Probl. 25(12), 123005, 18 (2009). doi:10.1088/0266-5611/25/12/123005, http://dx.doi.org/10.1088/0266-5611/25/12/123005

  9. Cheney, M., Bordon, B.: Synthetic aperture radar imaging. In: Scherzer, O. (ed.) The Handbook of Mathematical Methods in Imaging, 2nd edn. Springer, New York (2014)

    Google Scholar 

  10. Cormack, A.M.: Representation of a function by its line integrals with some radiological applications. J. Appl. Phys. 34, 2722–2727 (1963)

    Article  MATH  Google Scholar 

  11. Cormack, A.M.: Representation of a function by its line integrals with some radiological applications II. J. Appl. Phys. 35, 2908–2913 (1964)

    Article  MATH  Google Scholar 

  12. Davison, M., Grünbaum, F.: Tomographic reconstruction with arbitrary directions. Commun. Pure Appl. Math. 34, 77–120 (1981)

    Article  MATH  Google Scholar 

  13. de Hoop, M.V.: Microlocal analysis of seismic inverse scattering. In: Inside Out: Inverse Problems and Applications. Mathematical Sciences Research Institute Publications, vol. 47, pp. 219–296. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  14. Duistermaat, J.J., Guillemin, V.W.: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29(1), 39–79 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  15. Faridani, A.: Tomography and sampling theory. In: Olafsson, G., Quinto, E.T. (eds.) The Radon Transform and Applications to Inverse Problems. American Mathematical Society, Providence. AMS Proceedings of Symposia in Applied Mathematics (2006)

    Google Scholar 

  16. Faridani, A., Ritman, E.L.: High-resolution computed tomography from efficient sampling. Inverse Probl. 16, 635–650 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Faridani, A., Ritman, E.L., Smith, K.T.: Local tomography. SIAM J. Appl. Math. 52(2), 459–484 (1992). doi:10.1137/0152026, http://dx.doi.org/10.1137/0152026

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Felea, R.: Composition of Fourier integral operators with fold and blowdown singularities. Commun. Partial Differ. Equ. 30(10–12), 1717–1740 (2005). doi:10.1080/03605300500299968, http://dx.doi.org/10.1080/03605300500299968

  20. Felea, R.: Displacement of artefacts in inverse scattering. Inverse Probl. 23(4), 1519–1531 (2007). doi:10.1088/0266-5611/23/4/009, http://dx.doi.org/10.1088/0266-5611/23/4/009

  21. Felea, R., Greenleaf, A.: An FIO calculus for marine seismic imaging: folds and cross caps. Commun. Partial Differ. Equ. 33(1–3), 45–77 (2008). doi:10.1080/03605300701318716, http://dx.doi.org/10.1080/03605300701318716

  22. Felea, R., Greenleaf, A.: Fourier integral operators with open umbrellas and seismic inversion for cusp caustics. Math. Res. Lett. 17(5), 867–886 (2010). doi:10.4310/MRL.2010.v17.n5.a6, http://dx.doi.org/10.4310/MRL.2010.v17.n5.a6

  23. Felea, R., Quinto, E.T.: The microlocal properties of the local 3-D spect operator. SIAM J. Math. Anal. 43(3), 659–674 (2011)

    Article  MathSciNet  Google Scholar 

  24. Felea, R., Greenleaf, A., Pramanik, M.: An FIO calculus for marine seismic imaging, II: Sobolev estimates. Math. Ann. 352(2), 293–337 (2012). doi:10.1007/s00208-011-0644-5, http://dx.doi.org/10.1007/s00208-011-0644-5

  25. Finch, D.V., Lan, I.R., Uhlmann, G.: Microlocal analysis of the restricted X-ray transform with sources on a curve. In: Uhlmann, G. (ed.) Inside Out, Inverse Problems and Applications. MSRI Publications, vol. 47, pp. 193–218. Cambridge University Press, Cambridge/New York (2003)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. Gelfand, I.M., Graev, M.I.: Integral transformations connected with straight line complexes in a complex affine space. Sov. Math. Dokl. 2, 809–812 (1961)

    Google Scholar 

  28. Gelfand, I.M., Graev, M.I., Vilenkin, N.Y.: Generalized Functions, vol. 5. Academic, New York (1966)

    Google Scholar 

  29. Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. Graduate Texts in Mathematics, vol. 14. Springer, New York (1973)

    Book  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  31. Greenleaf, A., Uhlmann, G.: Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms. Ann. Inst. Fourier (Grenoble) 40(2), 443–466 (1990). http://www.numdam.org/item?id=AIF_1990__40_2_443_0

  32. Greenleaf, A., Uhlmann, G.: Estimates for singular Radon transforms and pseudodifferential operators with singular symbols. J. Funct. Anal. 89(1), 202–232 (1990). doi:10.1016/0022-1236(90)90011-9, http://dx.doi.org/10.1016/0022-1236(90)90011-9

  33. Grigis, A., Sjöstrand, J.: Microlocal Analysis for Differential Operators: An Introduction. London Mathematical Society Lecture Note Series, vol. 196. Cambridge University Press, Cambridge (1994)

    Book  Google Scholar 

  34. Guillemin, V.: Clean intersection theory and Fourier integrals. In: Fourier Integral Operators and Partial Differential Equations. Colloque International, Université de Nice, Nice, 1974. Lecture Notes in Mathematics, vol. 459, pp. 23–35. Springer, Berlin (1975)

    Chapter  Google Scholar 

  35. Guillemin, V.: Some remarks on integral geometry. Technical report, MIT (1975)

    Google Scholar 

  36. Guillemin, V.: On some results of Gelfand in integral geometry. Proc. Symp. Pure Math. 43, 149–155 (1985)

    MathSciNet  Google Scholar 

  37. Guillemin, V.: Cosmology in (2 + 1)-Dimensions, Cyclic Models, and Deformations of M 2, 1. Annals of Mathematics Studies, vol. 121. Princeton University Press, Princeton (1989)

    Google Scholar 

  38. Guillemin, V., Sternberg, S.: Geometric Asymptotics. Mathematical Surveys, vol. 14. American Mathematical Society, Providence (1977)

    Book  Google Scholar 

  39. Guillemin, V., Uhlmann, G.: Oscillatory integrals with singular symbols. Duke Math. J. 48(1), 251–267 (1981). http://projecteuclid.org/getRecord?id=euclid.dmj/1077314493

  40. Hahn, M.G., Quinto, E.T.: Distances between measures from 1-dimensional projections as implied by continuity of the inverse Radon transform. Zeitschrift Wahrscheinlichkeit 70, 361–380 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  41. Helgason, S.: The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassman manifolds. Acta Math. 113, 153–180 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  42. Helgason, S.: Support of Radon transforms. Adv. Math. 38, 91–100 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  43. Helgason, S.: Integral Geometry and Radon Transforms. Springer, New York (2011). doi:10.1007/978-1-4419-6055-9

    Book  MATH  Google Scholar 

  44. Herman, G.: Tomography. In: Scherzer, O. (ed.) The Handbook of Mathematical Methods in Imaging, 2nd edn. Springer, New York (2014)

    Google Scholar 

  45. Hertle, A.: Continuity of the Radon transform and its inverse on Euclidean space. Mathematische Zeitschrift 184, 165–192 (1983)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  47. Hörmander, L.: The Analysis of Linear Partial Differential Operators. I. Classics in Mathematics. Springer, Berlin (2003). Distribution theory and Fourier analysis, Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)]

    Google Scholar 

  48. Horne, A., Yates, G.: Bistatic synthetic aperture radar. In: 2002 International Radar Conference, pp. 6–10(4) (2002). doi:10.1049/cp:20020238, http://digital-library.theiet.org/content/conferences/10.1049/cp_20020238

  49. Katsevich, A.I.: Local tomography for the limited-angle problem. J. Math. Anal. Appl. 213, 160–182 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  50. Katsevich, A.I.: An improved exact filtered backprojection algorithm for spiral computed tomography. Adv. Appl. Math. 32, 681–697 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  51. Katsevich, A.: Improved cone beam local tomography. Inverse Probl. 22, 627–643 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  52. Katsevich, A., Ramm, A.: The Radon Transform and Local Tomography. CRC Press, Boca Raton (1996)

    MATH  Google Scholar 

  53. Krishnan, V.P.: A support theorem for the geodesic ray transform on functions. J. Fourier Anal. Appl. 15(4), 515–520 (2009). doi:10.1007/s00041-009-9061-5

    Article  MATH  MathSciNet  Google Scholar 

  54. Krishnan, V.P., Quinto, E.T.: Microlocal aspects of common offset synthetic aperture radar imaging. Inverse Probl. Imaging 5(3), 659–674 (2011). doi:10.3934/ipi.2011.5.659, http://dx.doi.org/10.3934/ipi.2011.5.659

  55. Krishnan, V.P., Levinson, H., Quinto, E.T.: Microlocal analysis of elliptical Radon transforms with foci on a line. In: Sabadini, I., Struppa, D.C. (eds.) The Mathematical Legacy of Leon Ehrenpreis. Springer Proceedings in Mathematics, vol. 16, pp. 163–182. Springer, Berlin/New York (2012)

    Chapter  Google Scholar 

  56. Kuchment, P., Lancaster, K., Mogilevskaya, L.: On local tomography. Inverse Probl. 11, 571–589 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  57. Kurusa, Á.: Support theorems for totally geodesic Radon transforms on constant curvature spaces. Proc. Am. Math. Soc. 122(2), 429–435 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  58. Lissianoi, S.: On stability estimates in the exterior problem for the Radon transform. In: Quinto, E., Ehrenpreis, L., Faridani, A., Gonzalez, F., Grinberg, E. (eds.) Tomography, Impedance Imaging, and Integral Geometry, South Hadley, 1993. Lectures in Applied Mathematics, vol. 30, pp. 143–147. American Mathematical Society, Providence (1994)

    Google Scholar 

  59. Louis, A.K.: Analytische Methoden in der Computer Tomographie. Universität Münster, habilitationsschrift (1981)

    Google Scholar 

  60. Louis, A.K.: Ghosts in tomography, the null space of the Radon transform. Math. Methods Appl. Sci. 3, 1–10 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  61. Louis, A.K.: Incomplete data problems in X-ray computerized tomography I. Singular value decomposition of the limited angle transform. Numerische Mathematik 48, 251–262 (1986)

    MATH  MathSciNet  Google Scholar 

  62. Louis, A.K., Maaß, P.: Contour reconstruction in 3-D X-Ray CT. IEEE Trans. Med. Imaging 12(4), 764–769 (1993)

    Article  Google Scholar 

  63. Louis, A.K., Rieder, A.: Incomplete data problems in X-ray computerized tomography II. Truncated projections and region-of-interest tomography. Numerische Mathematik 56, 371–383 (1986)

    MathSciNet  Google Scholar 

  64. Melrose, R.B., Uhlmann, G.A.: Lagrangian intersection and the Cauchy problem. Commun. Pure Appl. Math. 32(4), 483–519 (1979). doi:10.1002/cpa.3160320403, http://dx.doi.org/10.1002/cpa.3160320403

  65. Natterer, F.: Efficient implementation of ‘optimal’ algorithms in computerized tomography. Math. Methods Appl. Sci. 2, 545–555 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  66. Natterer, F.: The Mathematics of Computerized Tomography. Classics in Applied Mathematics, vol. 32. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001). Reprint of the 1986 original

    Google Scholar 

  67. Natterer, F., Wübbeling, F.: Mathematical Methods in Image Reconstruction. Monographs on Mathematical Modeling and Computation. Society for Industrial and Applied Mathematics, New York (2001)

    Book  MATH  Google Scholar 

  68. Niinimaki, K., Siltanen, S., Kolehmainen, V.: Bayesian multiresolution method for local tomography in dental X-ray imaging. Phys. Med. Biol. 52, 6663–6678 (2007)

    Article  Google Scholar 

  69. Nolan, C.J.: Scattering in the presence of fold caustics. SIAM J. Appl. Math. 61(2), 659–672 (2000). doi:10.1137/S0036139999356107, http://dx.doi.org/10.1137/S0036139999356107

  70. Nolan, C.J., Cheney, M.: Microlocal analysis of synthetic aperture radar imaging. J. Fourier Anal. Appl. 10(2), 133–148 (2004). doi:10.1007/s00041-004-8008-0, http://dx.doi.org/10.1007/s00041-004-8008-0

  71. Öktem, O.: The mathematics of electron microscopy. In: Scherzer, O. (ed.) The Handbook of Mathematical Methods in Imaging, 2nd edn. Springer, New York (2014)

    Google Scholar 

  72. Park, J.M., Franken, E.A., Jr., Garg, M., Fajardo, L.L., Niklason, L.T.: Breast tomosynthesis: present considerations and future applications. RadioGraphics 27, S231–S240 (2007)

    Article  Google Scholar 

  73. Perry, R.M.: On reconstructing a function on the exterior of a disc from its Radon transform. J. Math. Anal. Appl. 59, 324–341 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  74. Petersen, B.: Introduction to the Fourier Transform and Pseudo-differential Operators. Pittman, Boston (1983)

    MATH  Google Scholar 

  75. Quinto, E.T.: The dependence of the generalized Radon transform on defining measures. Trans. Am. Math. Soc. 257, 331–346 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  76. Quinto, E.T.: Singular value decompositions and inversion methods for the exterior Radon transform and a spherical transform. J. Math. Anal. Appl. 95, 437–448 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  77. Quinto, E.T.: Tomographic reconstructions from incomplete data–numerical inversion of the exterior Radon transform. Inverse Probl. 4, 867–876 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  78. Quinto, E.T.: Support theorems for the spherical Radon transform on manifolds. Int. Math. Res. Not. 2006, 1–17 (2006). Article ID = 67205

    Google Scholar 

  79. Quinto, E.T.: Local algorithms in exterior tomography. J. Comput. Appl. Math. 199, 141–148 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  82. Quinto, E.T., Bakhos, T., Chung, S.: A local algorithm for Slant Hole SPECT. In: Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT), Centro De Georgi, Pisa. cRM Series, vol. 7, pp. 321–348 (2008)

    Google Scholar 

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

    Article  Google Scholar 

  84. Radon, J.: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber. Verh. Sach. Akad. 69, 262–277 (1917)

    Google Scholar 

  85. Ramachandran, G., Lakshminarayanan, A.: Three dimensional reconstruction from radiographs and electron micrographs: applications of convolutions instead of Fourier transforms. Proc. Natl. Acad. Sci. USA 68, 262–277 (1971)

    Article  MathSciNet  Google Scholar 

  86. Rudin, W.: Functional Analysis. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York (1973)

    MATH  Google Scholar 

  87. Shepp, L.A., Kruskal, J.B.: Computerized tomography: the new medical X-ray technology. Am. Math. Mon. 85, 420–439 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  88. Shepp, L.A., Srivastava, S.: Computed tomography of PKM and AKM exit cones. AT & T Tech. J. 65, 78–88 (1986)

    Article  Google Scholar 

  89. Shubin, M.A.: Pseudodifferential Operators and Spectral Theory, 2nd edn. Springer, Berlin (2001) (Translated from the 1978 Russian original by Stig I. Andersson)

    Book  MATH  Google Scholar 

  90. Smith, K.T., Solmon, D.C., Wagner, S.L.: Practical and mathematical aspects of the problem of reconstructing objects from radiographs. Bull. Am. Math. Soc. 83, 1227–1270 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  91. Stefanov, P., Uhlmann, G.: Is a curved flight path in SAR better than a straight one? SIAM J. Appl. Math. 73(4), 1596–1612 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  92. Strichartz, R.S.: Radon inversion–variations on a theme. Am. Math. Mon. 89, 377–384 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  93. Taylor, M.E.: Pseudodifferential Operators. Princeton Mathematical Series, vol. 34. Princeton University Press, Princeton (1981)

    Google Scholar 

  94. Trèves, F.: Introduction to Pseudodifferential and Fourier Integral Operators. The University Series in Mathematics, vol. 1. Pseudodifferential operators. Plenum Press, New York (1980).

    Book  MATH  Google Scholar 

  95. Trèves, F.: Introduction to Pseudodifferential and Fourier Integral Operators. The University Series in Mathematics, vol. 2. Fourier integral operators. Plenum Press, New York (1980).

    Book  MATH  Google Scholar 

  96. Vainberg, E., 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 

  97. Vaughan, C.L.: Imagining The Elephant: A Biography of Allan Macleod Cormack. Imperial College Press, London (2008)

    Book  Google Scholar 

  98. Weinstein, A.: On Maslov’s quantization condition. In: Fourier Integral Operators and Partial Differential Equations. Colloque International, Université de Nice, Nice, 1974. Lecture Notes in Mathematics, vol. 459, pp. 341–372. Springer, Berlin (1975)

    Google Scholar 

  99. Yarman, C.E., Yazıcı, B.: Synthetic aperture hitchhiker imaging. IEEE Trans. Image Process. 17(11), 2156–2173 (2008). doi:10.1109/TIP.2008.2002839, http://dx.doi.org/10.1109/TIP.2008.2002839

  100. Yarman, C.E., Yazıcı, B., Cheney, M.: Bistatic synthetic aperture radar imaging for arbitrary flight trajectories. IEEE Trans. Image Process. 17(1), 84–93 (2008)

    Article  MathSciNet  Google Scholar 

  101. Yarman, C.E., Wang, L., Yazıcı, B.: Doppler synthetic aperture hitchhiker imaging. Inverse Probl. 26(6), 065006, 26 (2010). doi:10.1088/0266-5611/26/6/065006, http://dx.doi.org/10.1088/0266-5611/26/6/065006

  102. Yazici, B., Cheney, M., Yarman, C.: Synthetic-aperture inversion in the presence of noise and clutter. Inverse Probl. 22, 1705–1729 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  103. Ye, Y., Yu, H., Wang, G.: Cone beam pseudo-lambda tomography. Inverse Probl. 23, 203–215 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  104. Ye, Y., Yu, H., Wang, G.: Exact Interior Reconstruction from Truncated Limited-Angle Projection Data. Int. J. Biomed. Imaging 2008, Article ID 427989, 6p (2008). doi:10.1155/2008/427989

    Google Scholar 

  105. Yu, H., Wang, G.: Compressed sensing based interior tomography. Phys. Med. Biol. 54, 2791–2805 (2009)

    Article  Google Scholar 

  106. Zalcman, L.: Offbeat integral geometry. Am. Math. Mon. 87, 161–175 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  107. Zalcman, L.: Uniqueness and nonuniqueness for the Radon transform. Bull. Lond. Math. Soc. 14(3), 241–245 (1982). doi:10.1112/blms/14.3.241

    Article  MATH  MathSciNet  Google Scholar 

  108. Zampighi, G., Zampighi, L., Fain, N., Wright, E., Cantelle, F., Lanzavecchia, S.: Conical tomography II: a method for the study of cellular organelles in thin sections. J. Struct. Biol. 151(3), 263–274 (2005)

    Article  Google Scholar 

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Acknowledgements

Both authors thank the American Institute of Mathematics and their colleaguesGaik Ambartsoumian, Raluca Felea, and Clifford Nolan in an American Instituteof Mathematics (AIM) SQuaREs program for discussions at AIM on microlocalanalysis and radar imaging that informed this work. They appreciate JürgenFrikel’s careful reading of the chapter. The authors thank the MittagLeffler Institute for the congenial atmosphere as they worked on some ofthe research presented in this chapter. The second named author thanksJan Boman, Alfred Louis, Frank Natterer, and many other friends andcolleagues for important discussions about tomography and microlocalanalysis over the years. Both authors thank Birsen Yazıcı for interestingdiscussions.

The first named author was partially supported by NSF grant DMS 1109417. Additionally, he benefited from the support of Airbus Group Corporate Foundation Chair in “Mathematics of Complex Systems” established at TIFR CAM and ICTS TIFR, Bangalore, India, and from a German DAAD Research Stays grant. The second named author was partially supported by NSF grant DMS 1311558.

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  1. Centre for Applicable Mathematics, Tata Institute for Fundamental Research, Sharada Nagar, Chikkabommasandra Post Bag 6503, GKVK Post Office, 560065, Bangalore, Karnataka, India

    Venkateswaran P. Krishnan

  2. Department of Mathematics, Tufts University, 503 Boston Ave., 02155, Medford, MA, USA

    Eric Todd Quinto

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  1. Venkateswaran P. Krishnan
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  2. Eric Todd Quinto
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Correspondence to Venkateswaran P. Krishnan or Eric Todd Quinto .

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  1. Computational Science Center, University of Vienna, Vienna, Austria

    Otmar Scherzer

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Krishnan, V.P., Quinto, E.T. (2015). Microlocal Analysis in Tomography. In: Scherzer, O. (eds) Handbook of Mathematical Methods in Imaging. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0790-8_36

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