Skip to main content
SpringerLink
Log in

X-ray Tomography

  • Chapter
Inverse Problems and Imaging

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1943))

We give a survey on the mathematics of computerized tomography. We start with a short introduction to integral geometry, concentrating on inversion formulas, stability, and ranges. We then go over to inversion algorithms. We give a detailed analysis of the filtered backprojection algorithm in the light of the sampling theorem. We also describe the convergence properties of iterative algorithms.We shortly mention Fourier based algorithms and the recent progresses made in their accurate implementation. We conclude with the basics of algorithms for cone beam scanning which is the standard scanning mode in present days clinical practice.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bal G and Ryzhik L 2003 Time reversal and refocusing in random media SIAM J. Appl. Math. 63 1475–98

    Article  MATH  MathSciNet  Google Scholar 

  2. Bardos C and Fink M 2002 Mathematical foundations of the time reversal mirror Asymptotic Analysis 29 157–182

    MATH  MathSciNet  Google Scholar 

  3. Berryman J G, Borcea L, Papanicolaou G C and Tsogka C 2002 Statistically stable ultrasonic imaging in random media Journal of the Acoustic Society of America 112 1509–1522

    Article  Google Scholar 

  4. Bleistein N, Cohen J K, and Cohen J W 2001 Mathematics of multidimensional seismic imaging, migration, and inversion (Springer: New York)

    MATH  Google Scholar 

  5. Blomgren P, Papanicolaou G, and Zhao H 2002 Super-Resolution in Time-Reversal Acoustics, Journal of the Acoustical Society of America 111 238–248

    Article  Google Scholar 

  6. Borcea L, Papanicolaou G and Tsogka C 2003 Theory and applications of time reversal and interferometric imaging Inverse Problems 19 5139–5164

    MathSciNet  Google Scholar 

  7. Cheney M, Isaacson D and Lassas M 2001 Optimal Acoustic Measurements SIAM J Appl Math, 61 1628–1647

    Article  MATH  MathSciNet  Google Scholar 

  8. Claerbout J 1985 Fundamentals of geophysical data processing: with applications to petroleum prospecting, (CA: Blackwell Scientific Publications, Palo Alto)

    Google Scholar 

  9. Collins M D and Kuperman W A 1991 Focalization: Environmental focusing and source localization J. Acoust. Soc. Am. 90 (3) 1410–1422

    Article  Google Scholar 

  10. Collins M D and Kuperman W A 1994 Inverse problems in ocean acoustics Inverse Problems 10 1023–1040

    Article  MATH  MathSciNet  Google Scholar 

  11. Dautray R and Lions J L Mathematical Analysis and Numerical Methods for Science and Technology Vol. 1–6, Springer.

    Google Scholar 

  12. Devaney A 2004 Superresolution processing of multi-static data using time reversal and MUSIC to appear in J. Acoust. Soc. Am.

    Google Scholar 

  13. Dorn O 1998 A transport-backtransport method for optical tomography Inverse Problems 14 1107–1130

    Article  MATH  MathSciNet  Google Scholar 

  14. Dorn O, Bertete-Aguirre H, Berryman J G and Papanicolaou G C 1999 A nonlinear inversion method for 3D electromagnetic imaging using adjoint fields Inverse Problems 15 1523-1558

    Article  MATH  MathSciNet  Google Scholar 

  15. Dowling D R and Jackson D R 1992 Narrow band performance of phase conjugate arrays in dynamic random media J. Acoust. Soc. Am. 91 3257-3277

    Article  Google Scholar 

  16. Dowling D R 1993 Acoustic pulse compression using passive phase-conjugate processing J. Acoust. Soc. Am. 95 (3) 1450–1458

    Article  Google Scholar 

  17. Draeger C and Fink M 1997 One-Channel Time Reversal of Elastic Waves in a Chaotic 2D-Silicon Cavity Physical Review Letters 79 (3) 407–410

    Article  Google Scholar 

  18. Edelmann G F, Akal T, Hodgkiss W S, Kim S, Kuperman W A and Chung H C 2002 An initial demonstration of underwater acoustic communication using time reversal IEEE J. Ocean. Eng. 27 (3) 602–609

    Article  Google Scholar 

  19. Emami S M, Hansen J, Kim A D, Papanicolaou G, Paulraj A J, Cheung D and Prettie C 2004 Predicted time reversal performance in wireless communications using channel measurements submitted

    Google Scholar 

  20. Fink M 1997 Time reversed acoustics Phys. Today 50 34-40

    Article  Google Scholar 

  21. Fink M and Prada C 2001 Acoustic time-reversal mirrors Inverse Problems 17 R1–R38

    Article  MATH  Google Scholar 

  22. Foschini G J 1996 Layered space-time architecture for wireless communications in a fading environment Bell Labs Technical Journal 1 (2) 41–59

    Article  Google Scholar 

  23. Fouque J P, Garnier J and Nachbin A 2004 Time reversal for dispersive waves in random media to appear in SIAM J. Appl. Math.

    Google Scholar 

  24. Gesbert D, Shafi M, Shiu D, Smith P J and Naguib A 2003 From Theory to Practice: An Overview of MIMO Space-Time Coded Wireless Systems IEEE J. Select. Areas Commun. 21 (3) (tutorial paper) 281–301

    Article  Google Scholar 

  25. Haber E, Ascher U and Oldenburg D 2000 On optimization techniques for solving nonlinear inverse problems Inverse Problems 16 1263–1280

    Article  MATH  MathSciNet  Google Scholar 

  26. Hodgkiss W S, Song H C, Kuperman W A, Akal T, Ferla C and Jackson D R 1999 A long range and variable focus phase conjugation experiment in shallow water J. Acoust. Soc. Am. 105 1597–1604

    Article  Google Scholar 

  27. Hursky P, Porter M B, Cornuelle B D, Hodgkiss W S, and Kuperman W A 2004 Adjoint modeling for acoustic inversion J. Acoust. Soc. Am. 115 (2) 607–619

    Article  Google Scholar 

  28. Jackson D R and Dowling D R 1991 Phase conjugation in underwater acoustics J. Acoust. Soc. Am. 89 (1) 171–181

    Article  Google Scholar 

  29. Kim A D, Blomgren P and Papanicolaou G 2004 Spatial focusing and intersymbol interference in time reversal submitted

    Google Scholar 

  30. Klibanov M V and Timonov A 2003 On the mathematical treatment of time reversal Inverse Problems 19 1299–1318

    Article  MATH  MathSciNet  Google Scholar 

  31. Kuperman W A, Hodgkiss W S, Song H C, Akal T, Ferla C, and Jackson D R 1998 Phase conjugation in the ocean: Experimental demonstration of an acoustic time reversal mirror J. Acoust. Soc. Am. 103 25-40

    Article  Google Scholar 

  32. Kyritsi P, Papanicolaou G, Eggers P and Oprea A 2004 MISO time reversal and delay spread compression for FWA channels at 5GHz preprint

    Google Scholar 

  33. Marsden J E and Hughes T J R Mathematical Foundations of Elasticity (Prentice Hall, 1968)

    Google Scholar 

  34. Montaldo G, Lerosey G, Derode A, Tourin A, de Rosny J and Fink M 2004 Telecommunication in a disordered environment with iterative time reversal Waves Random Media 14 287–302

    Article  Google Scholar 

  35. Montaldo G, Tanter M and Fink M 2004 Real time inverse filter focusing through iterative time reversal to appear in J. Acoust. Soc. Am. 115 (2)

    Google Scholar 

  36. Natterer F and Wübbeling F 1995 A propagation-backpropagation method for ultrasound tomography Inverse Problems 11 1225-1232

    Article  MATH  MathSciNet  Google Scholar 

  37. Oestges C, Kim A D, Papanicolaou G and Paulraj A J 2004 Characterization of space-time focusing in time reversal random fields submitted to IEEE Trans. Antennas Prop.

    Google Scholar 

  38. Papanicolaou G, Ryzhik L and Solna K 2004 Statistical stability in time-reversal SIAM J. Appl. Math. 64 1133–1155

    Article  MATH  MathSciNet  Google Scholar 

  39. Prada C, Manneville S, Spoliansky D and Fink M 1996 Decomposition of the time reversal operator: Detection and selective focusing of two scatterers J. Acoust. Soc. Am. 96 (4) 2067–2076

    Article  Google Scholar 

  40. Prada C, Kerbrat E, Cassereau D and Fink M 2002 Time reversal techniques in ultrasonic nondestructive testing of scattering media Inverse Problems 18 1761–1773

    Article  MATH  MathSciNet  Google Scholar 

  41. Ryzhik L, Papanicolaou G and Keller J B 1996 Transport Equations for Elastic Waves and Other Waves in Random Media Wave Motion 24 327–370

    Article  MATH  MathSciNet  Google Scholar 

  42. Simon H S, Moustakas A L, Stoytchev M and Safar H 2001 Communication in a disordered world Physics Today 54 (9) (online at http://www.physicstoday.org/pt/vol-54/iss-9/p38.html)

  43. Snieder R K and Scales J A 1998 Time-reversed imaging as a diagnostic of wave and particle chaos Phys. Rev. E 58 (5) 5668–5675

    Article  Google Scholar 

  44. Smith K B, Abrantes A A M and Larraza A 2003 Examination of time-reversal acoustics in shallow water and applications to non-coherent underwater communications J. Acoust. Soc. Am. 113 (6) 3095–110

    Article  Google Scholar 

  45. Solna K 2002 Focusing of time-reversed reflections Waves Random Media 12 1–21

    Article  MathSciNet  Google Scholar 

  46. Song H, Kuperman W A, Hodgkiss W S, Akal T and Guerrini P 2003 Demonstration of a High-Frequency Acoustic Barrier With a Time-Reversal Mirror IEEE J. Acoust. Eng. 28 (2) 246–249

    Google Scholar 

  47. Tsogka C and Papanicolaou G C 2002 Time reversal through a solid-liquid interface and super-resolution Inverse Problems 18 1639-1657

    Article  MATH  MathSciNet  Google Scholar 

  48. Vögeler M 2003 Reconstruction of the three-dimensional refractive index in electromagnetic scattering by using a propagation-backpropagation method Inverse Problems 19 739–753

    Article  MATH  MathSciNet  Google Scholar 

  49. Yang T C 2003 Temporal Resolutions of Time-Reversal and Passive-Phase Conjugation for Underwater Acoustic Communications IEEE J. Ocean. Eng. 28 (2) 229–245

    Article  Google Scholar 

  50. Zhao H 2004 Analysis of the Response Matrix for an Extended Target SIAM J. Appl. Math. 64 (3) 725–745

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Numerische und Angewandte Mathematik, University of Münster, Einsteinstraße 62, 48149, Münster, Germany

    Frank Natterer

Authors
  1. Frank Natterer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Gregorio Millán Institute of Fluid Dynamics Nanoscience and Industrial Mathematics, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911, Madrid, Spain

    Luis L. Bonilla

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Natterer, F. (2008). X-ray Tomography. In: Bonilla, L.L. (eds) Inverse Problems and Imaging. Lecture Notes in Mathematics, vol 1943. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78547-7_2

Download citation

Publish with us

Policies and ethics

Search

Navigation