Skip to Main Content

Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Book Review

The AMS does not provide abstracts of book reviews. You may download the entire review from the links below.


MathSciNet review: 3090425
Full text of review: PDF   This review is available free of charge.
Book Information:

Author: Sigurdur Helgason
Title: Integral geometry and Radon transforms
Additional book information: Springer, New York, 2010, xiv+301 pp., ISBN 978-1-4419-6054-2, hardcover

References [Enhancements On Off] (What's this?)

  • M. Agranovsky, P. Kuchment, and L. Kunyansky. On reconstruction formulas and algorithms for the thermoacoustic tomography. In L. Wang, editor, Photoacoustic imaging and spectroscopy, pages 89–101. CRC Press, 2009.
  • Leifur Ásgeirsson, Über eine Mittelwertseigenschaft von Lösungen homogener linearer partieller Differentialgleichungen 2. Ordnung mit konstanten Koeffizienten, Math. Ann. 113 (1937), no. 1, 321–346 (German). MR 1513094, DOI 10.1007/BF01571637
  • Robert J. Baston and Michael G. Eastwood, The Penrose transform, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1989. Its interaction with representation theory; Oxford Science Publications. MR 1038279
  • Gregory Beylkin, The inversion problem and applications of the generalized Radon transform, Comm. Pure Appl. Math. 37 (1984), no. 5, 579–599. MR 752592, DOI 10.1002/cpa.3160370503
  • H. Bockwinkel. On the propagation of light in a biaxial crystal about a midpoint of oscillation. Verh. Konink Acad. V. Wet. Wissen. Natur., 14:636, 1906.
  • Jan Boman, Helgason’s support theorem for Radon transforms—a new proof and a generalization, Mathematical methods in tomography (Oberwolfach, 1990) Lecture Notes in Math., vol. 1497, Springer, Berlin, 1991, pp. 1–5. MR 1178765, DOI 10.1007/BFb0084503
  • Jan Boman and Eric Todd Quinto, Support theorems for real-analytic Radon transforms, Duke Math. J. 55 (1987), no. 4, 943–948. MR 916130, DOI 10.1215/S0012-7094-87-05547-5
  • Shiing-shen Chern, On integral geometry in Klein spaces, Ann. of Math. (2) 43 (1942), 178–189. MR 6075, DOI 10.2307/1968888
  • A. M. Cormack. Representation of a function by its line integrals with some radiological applications II. J. Appl. Physics, 35:2908–2913, 1964.
  • H. Cramér and H. Wold, Some Theorems on Distribution Functions, J. London Math. Soc. 11 (1936), no. 4, 290–294. MR 1574927, DOI 10.1112/jlms/s1-11.4.290
  • Leon Ehrenpreis, The universality of the Radon transform, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2003. With an appendix by Peter Kuchment and Eric Todd Quinto. MR 2019604, DOI 10.1093/acprof:oso/9780198509783.001.0001
  • Adel Faridani, Erik L. Ritman, and Kennan T. Smith, Local tomography, SIAM J. Appl. Math. 52 (1992), no. 2, 459–484. MR 1154783, DOI 10.1137/0152026
  • Raluca Felea and Eric Todd Quinto, The microlocal properties of the local 3-D SPECT operator, SIAM J. Math. Anal. 43 (2011), no. 3, 1145–1157. MR 2800572, DOI 10.1137/100807703
  • David V. Finch, Uniqueness for the attenuated x-ray transform in the physical range, Inverse Problems 2 (1986), no. 2, 197–203. MR 847534
  • D. V. Finch, I.-R. Lan, and G. Uhlmann. Microlocal Analysis of the Restricted X-ray Transform with Sources on a Curve. In G. Uhlmann, editor, Inside Out, Inverse Problems and Applications, volume 47 of MSRI Publications, pages 193–218. Cambridge University Press, 2003.
  • Bela Frigyik, Plamen Stefanov, and Gunther Uhlmann, The X-ray transform for a generic family of curves and weights, J. Geom. Anal. 18 (2008), no. 1, 89–108. MR 2365669, DOI 10.1007/s12220-007-9007-6
  • Paul Funk, Über eine geometrische Anwendung der Abelschen Integralgleichung, Math. Ann. 77 (1915), no. 1, 129–135 (German). MR 1511851, DOI 10.1007/BF01456824
  • Richard J. Gardner, Geometric tomography, Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, Cambridge, 1995. MR 1356221
  • I. Gelfand, S. Gindikin, and M. Graev. Integral geometry in affine and projective spaces. J. Sov. Math., 18:39–167, 1982.
  • I. M. Gel′fand, S. G. Gindikin, and M. I. Graev, Izbrannye zadachi integral′noĭ geometrii, Dobrosvet, Moscow, 2000 (Russian, with Russian summary). MR 1795833
  • I. M. Gelfand, S. G. Gindikin, and M. I. Graev, Selected topics in integral geometry, Translations of Mathematical Monographs, vol. 220, American Mathematical Society, Providence, RI, 2003. Translated from the 2000 Russian original by A. Shtern. MR 2000133, DOI 10.1090/mmono/220
  • I. M. Gelfand and M. I. Graev. Geometry of homogeneous spaces, representations of groups in homogeneous spaces and related questions of integral geometry. Trudy Moskov. Mat. Obshch., 8:321–390, 1959.
  • I. M. Gel′fand, M. I. Graev, and Z. Ja. Šapiro, Differential forms and integral geometry, Funkcional. Anal. i Priložen. 3 (1969), no. 2, 24–40 (Russian). MR 0244919
  • I. M. Gel′fand, M. I. Graev, and N. Ja. Vilenkin, Obobshchennye funktsii, Vyp. 5. Integral′naya geometriya i svyazannye s neĭ voprosy teorii predstavleniĭ, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1962 (Russian). MR 0160110
  • I. M. Gel′fand, M. I. Graev, and N. Ya. Vilenkin, Generalized functions. Vol. 5, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1966 [1977]. Integral geometry and representation theory; Translated from the Russian by Eugene Saletan. MR 0435835
  • Simon Gindikin, Real integral geometry and complex analysis, Integral geometry, Radon transforms and complex analysis (Venice, 1996) Lecture Notes in Math., vol. 1684, Springer, Berlin, 1998, pp. 70–98. MR 1635612, DOI 10.1007/BFb0096091
  • R. Godement, The decomposition of $L^{2}\,(G/\Gamma )$ for $\Gamma =\textrm {SL}(2,\,Z)$, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 211–224. MR 0210827
  • Fulton B. Gonzalez and Tomoyuki Kakehi, Pfaffian systems and Radon transforms on affine Grassmann manifolds, Math. Ann. 326 (2003), no. 2, 237–273. MR 1990910, DOI 10.1007/s00208-002-0398-1
  • Fulton B. Gonzalez and Tomoyuki Kakehi, Moment conditions and support theorems for Radon transforms on affine Grassmann manifolds, Adv. Math. 201 (2006), no. 2, 516–548. MR 2211536, DOI 10.1016/j.aim.2005.02.009
  • Allan Greenleaf and Gunther Uhlmann, Nonlocal inversion formulas for the X-ray transform, Duke Math. J. 58 (1989), no. 1, 205–240. MR 1016420, DOI 10.1215/S0012-7094-89-05811-0
  • Eric L. Grinberg, On images of Radon transforms, Duke Math. J. 52 (1985), no. 4, 939–972. MR 816395, DOI 10.1215/S0012-7094-85-05251-2
  • V. Guillemin. Some remarks on integral geometry. Technical report, MIT, 1975.
  • Victor Guillemin, On some results of Gel′fand in integral geometry, Pseudodifferential operators and applications (Notre Dame, Ind., 1984) Proc. Sympos. Pure Math., vol. 43, Amer. Math. Soc., Providence, RI, 1985, pp. 149–155. MR 812288, DOI 10.1090/pspum/043/812288
  • Victor Guillemin and Shlomo Sternberg, Geometric asymptotics, Mathematical Surveys, No. 14, American Mathematical Society, Providence, R.I., 1977. MR 0516965
  • V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols, Duke Math. J. 48 (1981), no. 1, 251–267. MR 610185
  • Harish-Chandra, Spherical functions on a semisimple Lie group. I, Amer. J. Math. 80 (1958), 241–310. MR 94407, DOI 10.2307/2372786
  • Harish-Chandra, Spherical functions on a semisimple Lie group. II, Amer. J. Math. 80 (1958), 553–613. MR 101279, DOI 10.2307/2372772
  • S. Helgason, Radon-Fourier transforms on symmetric spaces and related group representations, Bull. Amer. Math. Soc. 71 (1965), 757–763. MR 179295, DOI 10.1090/S0002-9904-1965-11380-5
  • Sigurđur Helgason, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds, Acta Math. 113 (1965), 153–180. MR 172311, DOI 10.1007/BF02391776
  • Sigurđur Helgason, A duality in integral geometry on symmetric spaces, Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965) Nippon Hyoronsha, Tokyo, 1966, pp. 37–56. MR 0229191
  • Sigurđur Helgason, A duality for symmetric spaces with applications to group representations, Advances in Math. 5 (1970), 1–154 (1970). MR 263988, DOI 10.1016/0001-8708(70)90037-X
  • Sigurdur Helgason, The Radon transform, Progress in Mathematics, vol. 5, Birkhäuser, Boston, Mass., 1980. MR 573446
  • Sigurdur Helgason, Integral geometry and multitemporal wave equations, Comm. Pure Appl. Math. 51 (1998), no. 9-10, 1035–1071. Dedicated to the memory of Fritz John. MR 1632583, DOI 10.1002/(SICI)1097-0312(199809/10)51:9/10<1035::AID-CPA5>3.3.CO;2-H
  • Sigurdur Helgason, The Radon transform, 2nd ed., Progress in Mathematics, vol. 5, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1723736, DOI 10.1007/978-1-4757-1463-0
  • Sigurdur Helgason, Geometric analysis on symmetric spaces, 2nd ed., Mathematical Surveys and Monographs, vol. 39, American Mathematical Society, Providence, RI, 2008. MR 2463854, DOI 10.1090/surv/039
  • Fritz John, The ultrahyperbolic differential equation with four independent variables, Duke Math. J. 4 (1938), no. 2, 300–322. MR 1546052, DOI 10.1215/S0012-7094-38-00423-5
  • Fritz John, Plane waves and spherical means applied to partial differential equations, Interscience Publishers, New York-London, 1955. MR 0075429
  • Tomoyuki Kakehi, Integral geometry on Grassmann manifolds and calculus of invariant differential operators, J. Funct. Anal. 168 (1999), no. 1, 1–45. MR 1717855, DOI 10.1006/jfan.1999.3459
  • M. Kashiwara, A. Kowata, K. Minemura, K. Okamoto, T. Ōshima, and M. Tanaka, Eigenfunctions of invariant differential operators on a symmetric space, Ann. of Math. (2) 107 (1978), no. 1, 1–39. MR 485861, DOI 10.2307/1971253
  • Alexander Katsevich, Improved cone beam local tomography, Inverse Problems 22 (2006), no. 2, 627–643. MR 2216419, DOI 10.1088/0266-5611/22/2/015
  • Venkateswaran P. Krishnan, A support theorem for the geodesic ray transform on functions, J. Fourier Anal. Appl. 15 (2009), no. 4, 515–520. MR 2549942, DOI 10.1007/s00041-009-9061-5
  • Peter Kuchment, Generalized transforms of Radon type and their applications, The Radon transform, inverse problems, and tomography, Proc. Sympos. Appl. Math., vol. 63, Amer. Math. Soc., Providence, RI, 2006, pp. 67–91. MR 2208237, DOI 10.1090/psapm/063/2208237
  • Serguei Lissianoi, On stability estimates in the exterior problem for the Radon transform, Tomography, impedance imaging, and integral geometry (South Hadley, MA, 1993) Lectures in Appl. Math., vol. 30, Amer. Math. Soc., Providence, RI, 1994, pp. 143–147. MR 1297571
  • H. Minkowski. About bodies of constant width. Mathematics Sbornik, 25:505–508, 1904.
  • F. Natterer, The mathematics of computerized tomography, Classics in Applied Mathematics, vol. 32, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. Reprint of the 1986 original. MR 1847845, DOI 10.1137/1.9780898719284
  • Clifford J. Nolan and Margaret Cheney, Microlocal analysis of synthetic aperture radar imaging, J. Fourier Anal. Appl. 10 (2004), no. 2, 133–148. MR 2054305, DOI 10.1007/s00041-004-8008-0
  • Eric Todd Quinto, The dependence of the generalized Radon transform on defining measures, Trans. Amer. Math. Soc. 257 (1980), no. 2, 331–346. MR 552261, DOI 10.1090/S0002-9947-1980-0552261-8
  • Eric Todd Quinto, Singularities of the X-ray transform and limited data tomography in $\textbf {R}^2$ and $\textbf {R}^3$, SIAM J. Math. Anal. 24 (1993), no. 5, 1215–1225. MR 1234012, DOI 10.1137/0524069
  • Eric Todd Quinto, Andreas Rieder, and Thomas Schuster, Local inversion of the sonar transform regularized by the approximate inverse, Inverse Problems 27 (2011), no. 3, 035006, 18. MR 2772525, DOI 10.1088/0266-5611/27/3/035006
  • J. Radon. Über die Bestimmung von Funktionen durch ihre Integralwerte langs gewisser Mannigfaltigkeiten. Ber. Verh. Sach. Akad., 69:262–277, 1917.
  • Luis A. Santaló, Integral geometry and geometric probability, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. With a foreword by Mark Kac. MR 2162874, DOI 10.1017/CBO9780511617331
  • L. A. Shepp and J. B. Kruskal, Computerized Tomography: The New Medical X-Ray Technology, Amer. Math. Monthly 85 (1978), no. 6, 420–439. MR 1538734, DOI 10.2307/2320062
  • Kennan T. Smith, Donald C. Solmon, and Sheldon L. Wagner, Practical and mathematical aspects of the problem of reconstructing objects from radiographs, Bull. Amer. Math. Soc. 83 (1977), no. 6, 1227–1270. MR 490032, DOI 10.1090/S0002-9904-1977-14406-6
  • Robert S. Strichartz, Harmonic analysis on Grassmannian bundles, Trans. Amer. Math. Soc. 296 (1986), no. 1, 387–409. MR 837819, DOI 10.1090/S0002-9947-1986-0837819-6
  • E. Vainberg, I. A. Kazak, and V. P. Kurozaev. Reconstruction of the internal three-dimensional structure of objects based on real-time integral projections. Soviet Journal of Nondestructive Testing, 17:415–423, 1981.
  • Yangbo Ye, Hengyong Yu, and Ge Wang, Cone-beam pseudo-lambda tomography, Inverse Problems 23 (2007), no. 1, 203–215. MR 2302969, DOI 10.1088/0266-5611/23/1/010
    •
    Review Information:

    Reviewer: Fulton Gonzalez
    Affiliation: Department of Mathematics, Tufts University, Medford, Massachusetts 02155
    Email: fulton.gonzalez@tufts.edu
    Reviewer: Eric Todd Quinto
    Affiliation: Department of Mathematics, Tufts University, Medford, Massachusetts 02155
    Email: todd.quinto@tufts.edu
    Journal: Bull. Amer. Math. Soc. 50 (2013), 663-674
    DOI: https://doi.org/10.1090/S0273-0979-2012-01391-5
    Published electronically: December 6, 2012
    Review copyright: © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.