Skip to main content
Top

2011 | OriginalPaper | Chapter

Tomography

Author : Gabor T. Herman

Published in: Handbook of Mathematical Methods in Imaging

Publisher: Springer New York

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We define tomography as the process of producing an image of a distribution (of some physical property) from estimates of its line integrals along a finite number of lines of known locations. We touch upon the computational and mathematical procedures underlying the data collection, image reconstruction, and image display in the practice of tomography. The emphasis is on reconstruction methods, especially the so-called series expansion reconstruction algorithms.
We illustrate the use of tomography (including three-dimensional displays based on reconstructions) both in electron microscopy and in x-ray computerized tomography (CT), but concentrate on the latter. This is followed by a classification and discussion of reconstruction algorithms. In particular, we discuss how to evaluate and compare the practical efficacy of such algorithms.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literature
1.
go back to reference Artzy E, Frieder G, Herman GT (1981) The theory, design, implementation and evaluation of a three-dimensional surface detection algorithm. Comput Graph Image Process 15: 1–24CrossRef Artzy E, Frieder G, Herman GT (1981) The theory, design, implementation and evaluation of a three-dimensional surface detection algorithm. Comput Graph Image Process 15: 1–24CrossRef
2.
go back to reference Banhart J (2008) Advanced tomographic methods in materials research and engineering. Oxford University Press, OxfordCrossRef Banhart J (2008) Advanced tomographic methods in materials research and engineering. Oxford University Press, OxfordCrossRef
4.
go back to reference Browne JA, De Pierro AR (1996) A row-action alternative to the EM algorithm for maximizing likelihood in emission tomography. IEEE Trans Med Imaging 15:687–6996CrossRef Browne JA, De Pierro AR (1996) A row-action alternative to the EM algorithm for maximizing likelihood in emission tomography. IEEE Trans Med Imaging 15:687–6996CrossRef
5.
go back to reference Censor Y, Zenios SA (1998) Parallel optimization: theory, algorithms and applications. Oxford University Press, New York Censor Y, Zenios SA (1998) Parallel optimization: theory, algorithms and applications. Oxford University Press, New York
6.
go back to reference Censor Y, Altschuler MD, Powlis WD (1988) On the use of Cimmino’s simultaneous projections method for computing a solution of the inverse problem in radiation therapy treatment planning. Inverse Probl 4:607–623MathSciNetMATHCrossRef Censor Y, Altschuler MD, Powlis WD (1988) On the use of Cimmino’s simultaneous projections method for computing a solution of the inverse problem in radiation therapy treatment planning. Inverse Probl 4:607–623MathSciNetMATHCrossRef
7.
go back to reference Chen LS, Herman GT, Reynolds RA, Udupa JK (1985) Surface shading in the cuberille environment (erratum appeared in 6(2):67–69, 1986). IEEE Comput Graph Appl 5(12):33–43MATHCrossRef Chen LS, Herman GT, Reynolds RA, Udupa JK (1985) Surface shading in the cuberille environment (erratum appeared in 6(2):67–69, 1986). IEEE Comput Graph Appl 5(12):33–43MATHCrossRef
8.
go back to reference Cormack AM (1963) Representation of a function by its line integrals, with some radiological applications. J Appl Phys 34:2722–2727MATHCrossRef Cormack AM (1963) Representation of a function by its line integrals, with some radiological applications. J Appl Phys 34:2722–2727MATHCrossRef
9.
go back to reference Crawford CR, King KF (1990) Computed-tomography scanning with simultaneous patient motion. Med Phys 17:967–982CrossRef Crawford CR, King KF (1990) Computed-tomography scanning with simultaneous patient motion. Med Phys 17:967–982CrossRef
10.
go back to reference Crowther RA, DeRosier DJ, Klug A (1970) The reconstruction of a threedimensional structure from projections and its application to electron microscopy. Proc R Soc Lon Ser-A A317: 319–340CrossRef Crowther RA, DeRosier DJ, Klug A (1970) The reconstruction of a threedimensional structure from projections and its application to electron microscopy. Proc R Soc Lon Ser-A A317: 319–340CrossRef
12.
go back to reference DeRosier DJ, Klug A (1968) Reconstruction of three-dimensional structures from electron micrographs. Nature 217:130–134CrossRef DeRosier DJ, Klug A (1968) Reconstruction of three-dimensional structures from electron micrographs. Nature 217:130–134CrossRef
13.
go back to reference Edholm P, Herman GT, Roberts DA (1988) Image reconstruction from linograms: implementation and evaluation. IEEE Trans Med Imaging 7: 239–246CrossRef Edholm P, Herman GT, Roberts DA (1988) Image reconstruction from linograms: implementation and evaluation. IEEE Trans Med Imaging 7: 239–246CrossRef
14.
go back to reference Edholm PR, Herman GT (1987) Linograms in image reconstruction from projections. IEEE Trans Med Imaging 6:301–307CrossRef Edholm PR, Herman GT (1987) Linograms in image reconstruction from projections. IEEE Trans Med Imaging 6:301–307CrossRef
15.
go back to reference Eggermont PPB, Herman GT, Lent A (1981) Iterative algorithms for large partitioned linear systems, with applications to image reconstruction. Linear Algebra Appl 40:37–67MathSciNetMATHCrossRef Eggermont PPB, Herman GT, Lent A (1981) Iterative algorithms for large partitioned linear systems, with applications to image reconstruction. Linear Algebra Appl 40:37–67MathSciNetMATHCrossRef
16.
go back to reference Epstein CS (2007) Introduction to the mathematics of medical imaging, 2nd edn. SIAM, Philadelphia Epstein CS (2007) Introduction to the mathematics of medical imaging, 2nd edn. SIAM, Philadelphia
17.
go back to reference Frank J (2006a) Electron tomography: methods for three-dimensional visualization of structures in the cell, 2nd edn. Springer, New York Frank J (2006a) Electron tomography: methods for three-dimensional visualization of structures in the cell, 2nd edn. Springer, New York
18.
go back to reference Frank J (2006b) Three-dimensional electron microscopy of macromolecular assemblies: visualization of biological molecules in their native state. Oxford University Press, New YorkCrossRef Frank J (2006b) Three-dimensional electron microscopy of macromolecular assemblies: visualization of biological molecules in their native state. Oxford University Press, New YorkCrossRef
19.
go back to reference Gordon R, Bender R, Herman GT (1970) Algebraic Reconstruction Techniques (ART) for three-dimensional electron microscopy and x-ray photography. J Theor Biol 29:471–481CrossRef Gordon R, Bender R, Herman GT (1970) Algebraic Reconstruction Techniques (ART) for three-dimensional electron microscopy and x-ray photography. J Theor Biol 29:471–481CrossRef
20.
go back to reference Hanson KM (1990) Method of evaluating image-recovery algorithms based on task performance. J Opt Soc Am A 7:1294–1304CrossRef Hanson KM (1990) Method of evaluating image-recovery algorithms based on task performance. J Opt Soc Am A 7:1294–1304CrossRef
21.
go back to reference Herman GT (1981) Advanced principles of reconstruction algorithms. In: Newton TH, Potts DG (eds) Radiology of skull and brain, vol 5: Technical aspects of computed tomography. C.V. Mosby, St. Louis, pp 3888–3903 Herman GT (1981) Advanced principles of reconstruction algorithms. In: Newton TH, Potts DG (eds) Radiology of skull and brain, vol 5: Technical aspects of computed tomography. C.V. Mosby, St. Louis, pp 3888–3903
22.
go back to reference Herman GT (2009) Fundamentals of computerized tomography: image reconstruction from projections, 2nd edn. Springer, London Herman GT (2009) Fundamentals of computerized tomography: image reconstruction from projections, 2nd edn. Springer, London
23.
go back to reference Herman GT, Kuba A (2007) Advances in discrete tomography and its applications. Birkhäuser, BostonMATHCrossRef Herman GT, Kuba A (2007) Advances in discrete tomography and its applications. Birkhäuser, BostonMATHCrossRef
24.
go back to reference Herman GT, Lent A (1976) Iterative reconstruction algorithms. Comput Biol Med 6:273–294CrossRef Herman GT, Lent A (1976) Iterative reconstruction algorithms. Comput Biol Med 6:273–294CrossRef
25.
go back to reference Herman GT, Liu HK (1979) Three-dimensional display of human organs from computed tomograms. Comput Graph Image Process 9:1–21CrossRef Herman GT, Liu HK (1979) Three-dimensional display of human organs from computed tomograms. Comput Graph Image Process 9:1–21CrossRef
26.
go back to reference Herman GT, Meyer LB (1993) Algebraic reconstruction techniques can be made computationally efficient. IEEE Trans Med Imaging 12:600–609CrossRef Herman GT, Meyer LB (1993) Algebraic reconstruction techniques can be made computationally efficient. IEEE Trans Med Imaging 12:600–609CrossRef
27.
go back to reference Herman GT, Naparstek A (1977) Fast image reconstruction based on a Radon inversion formula appropriate for rapidly collected data. SIAM J Appl Math 33:511–533MathSciNetMATHCrossRef Herman GT, Naparstek A (1977) Fast image reconstruction based on a Radon inversion formula appropriate for rapidly collected data. SIAM J Appl Math 33:511–533MathSciNetMATHCrossRef
28.
go back to reference Herman GT, Tuy HK, Langenberg KJ, Sabatier PC (1988) Basic methods of tomography and inverse problems. Institute of Physics Publishing, Bristol Herman GT, Tuy HK, Langenberg KJ, Sabatier PC (1988) Basic methods of tomography and inverse problems. Institute of Physics Publishing, Bristol
29.
go back to reference Hounsfield GN (1973) Computerized transverse axial scanning tomography: Part I, description of the system. Br J Radiol 46:1016–1022CrossRef Hounsfield GN (1973) Computerized transverse axial scanning tomography: Part I, description of the system. Br J Radiol 46:1016–1022CrossRef
30.
go back to reference Hudson HM, Larkin RS (1994) Accelerated image reconstruction using ordered subsets of projection data. IEEE Trans Med Imaging 13:601–609CrossRef Hudson HM, Larkin RS (1994) Accelerated image reconstruction using ordered subsets of projection data. IEEE Trans Med Imaging 13:601–609CrossRef
31.
go back to reference Kalender WA (2006) Computed tomography: fundamentals, system technology, image quality, applications, 2nd edn. Wiley-VCH, Munich Kalender WA (2006) Computed tomography: fundamentals, system technology, image quality, applications, 2nd edn. Wiley-VCH, Munich
32.
go back to reference Kalender WA, Seissler W, Klotz E, Vock P (1990) Spiral volumetric CT with single-breath-hold technique, continuous transport, and continuous scanner rotation. Radiology 176:181–183 Kalender WA, Seissler W, Klotz E, Vock P (1990) Spiral volumetric CT with single-breath-hold technique, continuous transport, and continuous scanner rotation. Radiology 176:181–183
33.
go back to reference Katsevich A (2002) Theoretically exact filtered backprojection-type inversion algorithm for spiral CT. SIAM J Appl Math 62:2012–2026MathSciNetMATHCrossRef Katsevich A (2002) Theoretically exact filtered backprojection-type inversion algorithm for spiral CT. SIAM J Appl Math 62:2012–2026MathSciNetMATHCrossRef
34.
go back to reference Kinahan PE, Matej S, Karp JP, Herman GT, Lewitt RM (1995) A comparison of transform and iterative reconstruction techniques for a volume-imaging PET scanner with a large axial acceptance angle. IEEE Trans Nucl Sci 42:2181–2287CrossRef Kinahan PE, Matej S, Karp JP, Herman GT, Lewitt RM (1995) A comparison of transform and iterative reconstruction techniques for a volume-imaging PET scanner with a large axial acceptance angle. IEEE Trans Nucl Sci 42:2181–2287CrossRef
35.
go back to reference Lauterbur PC (1979) Medical imaging by nuclear magnetic resonance zeugmatography. IEEE Trans Nucl Sci 26:2808–2811CrossRef Lauterbur PC (1979) Medical imaging by nuclear magnetic resonance zeugmatography. IEEE Trans Nucl Sci 26:2808–2811CrossRef
36.
go back to reference Levitan E, Herman GT (1987) A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography. IEEE Trans Med Imaging 6: 185–192CrossRef Levitan E, Herman GT (1987) A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography. IEEE Trans Med Imaging 6: 185–192CrossRef
37.
go back to reference Lewitt RM (1990) Multidimensional digital image representation using generalized Kaiser-Bessel window functions. J Opt Soc Am A 7:1834–1846CrossRef Lewitt RM (1990) Multidimensional digital image representation using generalized Kaiser-Bessel window functions. J Opt Soc Am A 7:1834–1846CrossRef
38.
go back to reference Lewitt RM (92) Alternatives to voxels for image representation in iterative reconstruction algorithms. Phys Med Biol 37:705–716 Lewitt RM (92) Alternatives to voxels for image representation in iterative reconstruction algorithms. Phys Med Biol 37:705–716
39.
go back to reference Lorensen W, Cline H (1987) Marching cubes: a high-resolution 3D surface reconstruction algorithm. Comput Graph 21(4):163–169CrossRef Lorensen W, Cline H (1987) Marching cubes: a high-resolution 3D surface reconstruction algorithm. Comput Graph 21(4):163–169CrossRef
40.
go back to reference Maki DD, Birnbaum BA, Chakraborty DP, Jacobs JE, Carvalho BM, Herman GT (1999) Renal cyst pseudo-enhancement: Beam hardening effects on CT numbers. Radiology 213:468–472 Maki DD, Birnbaum BA, Chakraborty DP, Jacobs JE, Carvalho BM, Herman GT (1999) Renal cyst pseudo-enhancement: Beam hardening effects on CT numbers. Radiology 213:468–472
41.
go back to reference Marabini R, Rietzel E, Schroeder R, Herman GT, Carazo JM (1997) Threedimensional reconstruction from reduced sets of very noisy images acquired following a single-axis tilt schema: application of a new three-dimensional reconstruction algorithm and objective comparison with weighted backprojection. J Struct Biol 120:363–371CrossRef Marabini R, Rietzel E, Schroeder R, Herman GT, Carazo JM (1997) Threedimensional reconstruction from reduced sets of very noisy images acquired following a single-axis tilt schema: application of a new three-dimensional reconstruction algorithm and objective comparison with weighted backprojection. J Struct Biol 120:363–371CrossRef
42.
go back to reference Marabini R, Herman GT, Carazo J-M (1998) 3D reconstruction in electron microscopy using ART with smooth spherically symmetric volume elements (blobs). Ultramicroscopy 72:53–65CrossRef Marabini R, Herman GT, Carazo J-M (1998) 3D reconstruction in electron microscopy using ART with smooth spherically symmetric volume elements (blobs). Ultramicroscopy 72:53–65CrossRef
43.
go back to reference Matej S, Lewitt RM (1996) Practical consideration for 3D image-reconstruction using spherically-symmetrical volume elements. IEEE Trans Med Imaging 15:68–78CrossRef Matej S, Lewitt RM (1996) Practical consideration for 3D image-reconstruction using spherically-symmetrical volume elements. IEEE Trans Med Imaging 15:68–78CrossRef
44.
go back to reference Matej S, Herman GT, Narayan TK, Furuie SS, Lewitt RM, Kinahan PE (1994) Evaluation of task-oriented performance of several fully 3D PET reconstruction algorithms. Phys Med Biol 39:355–367CrossRef Matej S, Herman GT, Narayan TK, Furuie SS, Lewitt RM, Kinahan PE (1994) Evaluation of task-oriented performance of several fully 3D PET reconstruction algorithms. Phys Med Biol 39:355–367CrossRef
45.
go back to reference Matej S, Furuie SS, Herman GT (1996) Relevance of statistically significant differences between reconstruction algorithms. IEEE Trans Image Process 5:554–556CrossRef Matej S, Furuie SS, Herman GT (1996) Relevance of statistically significant differences between reconstruction algorithms. IEEE Trans Image Process 5:554–556CrossRef
46.
go back to reference Narayan TK, Herman GT (1999) Prediction of human observer performance by numerical observers: an experimental study. J Opt Soc Am A 16:679–693CrossRef Narayan TK, Herman GT (1999) Prediction of human observer performance by numerical observers: an experimental study. J Opt Soc Am A 16:679–693CrossRef
47.
go back to reference Natterer F, Wübbeling F (2001) Mathematical methods in image reconstruction. SIAM, PhiladelphiaMATHCrossRef Natterer F, Wübbeling F (2001) Mathematical methods in image reconstruction. SIAM, PhiladelphiaMATHCrossRef
48.
go back to reference Poulsen HF (2004) Three-dimensional x-ray diffraction microscopy: mapping polycrystals and their dynamics. Springer, BerlinCrossRef Poulsen HF (2004) Three-dimensional x-ray diffraction microscopy: mapping polycrystals and their dynamics. Springer, BerlinCrossRef
49.
go back to reference Radon J (1917) Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber Verh Sächs Akad Wiss, Leipzig, Math Phys Kl 69:262–277 Radon J (1917) Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber Verh Sächs Akad Wiss, Leipzig, Math Phys Kl 69:262–277
50.
go back to reference Ramachandran GN, Lakshminarayanan AV (1971) Three-dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms. Proc Natl Acad Sci USA 68:2236–2240MathSciNetCrossRef Ramachandran GN, Lakshminarayanan AV (1971) Three-dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms. Proc Natl Acad Sci USA 68:2236–2240MathSciNetCrossRef
51.
go back to reference Scheres SHW, Gao H, Valle M, Herman GT, Eggermont PPB, Frank J, Carazo J-M (2007) Disentangling conformational states of macromolecules in 3D-EM through likelihood optimization. Nat Methods 4:27–29CrossRef Scheres SHW, Gao H, Valle M, Herman GT, Eggermont PPB, Frank J, Carazo J-M (2007) Disentangling conformational states of macromolecules in 3D-EM through likelihood optimization. Nat Methods 4:27–29CrossRef
52.
go back to reference Scheres SHW, Nuñez-Ramirez R, Sorzano COS, Carazo JM, Marabini R (2008) Image processing for electron microscopy single-particle analysis using XMIPP. Nat Protocols 3:977–990CrossRef Scheres SHW, Nuñez-Ramirez R, Sorzano COS, Carazo JM, Marabini R (2008) Image processing for electron microscopy single-particle analysis using XMIPP. Nat Protocols 3:977–990CrossRef
53.
go back to reference Shepp LA, Logan BF (1974) The Fourier reconstruction of a head section. IEEE Trans Nucl Sci 21:21–43CrossRef Shepp LA, Logan BF (1974) The Fourier reconstruction of a head section. IEEE Trans Nucl Sci 21:21–43CrossRef
54.
go back to reference Shepp LA, Vardi Y (1982) Maximum likelihood reconstruction for emission tomography. IEEE Trans Med Imaging 1:113–122CrossRef Shepp LA, Vardi Y (1982) Maximum likelihood reconstruction for emission tomography. IEEE Trans Med Imaging 1:113–122CrossRef
55.
go back to reference Sorzano COS, Marabini R, Boisset N, Rietzel E, Schröder R, Herman GT, Carazo JM (2001) The effect of overabundant projection directions on 3D reconstruction algorithms. J Struct Biol 133:108–118CrossRef Sorzano COS, Marabini R, Boisset N, Rietzel E, Schröder R, Herman GT, Carazo JM (2001) The effect of overabundant projection directions on 3D reconstruction algorithms. J Struct Biol 133:108–118CrossRef
56.
go back to reference Udupa JK, Herman GT (1999) 3D imaging in medicine, 2nd edn. CRC Press, Boca Raton Udupa JK, Herman GT (1999) 3D imaging in medicine, 2nd edn. CRC Press, Boca Raton
Metadata
Title
Tomography
Author
Gabor T. Herman
Copyright Year
2011
Publisher
Springer New York
DOI
https://doi.org/10.1007/978-0-387-92920-0_16

Premium Partner