Skip to main content

Part of the book series: Computational Imaging and Vision ((CIVI,volume 4))

Abstract

We give the parallel sampling conditions of the 3D X-ray transform when the detector plane trajectory is a circle around the measured object. It turns out that hexagonal and interlaced sampling can be cormbined leading to an efficient scheme. In this case the detector is a rectangular grid. Numerical experiments from efficient schemes are provided.

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

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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. A.M. Cormack (1978) Sampling the Radon transform with beam of finite width. Phys. Med. Biol., 23(6):1141–1148.

    Article  PubMed  CAS  Google Scholar 

  2. L. Desbat (1993) Efficient sampling on coarse grids in tomography. Inverse lProblems, 9:251–269.

    Article  Google Scholar 

  3. L. Desbat (1995) Algebraic reconstructions from efficient sampling in tomography. In Computerized Tomography, pages 124–137. VSP. Novosibirsk 93.

    Google Scholar 

  4. L. Desbat (1995) Echantillonnage en tomographie 3D. In Quinzième colloque sur le traitement du signal et des images, GRETSI’95, pages 885–888.

    Google Scholar 

  5. L. Desbat (1996) Echantillonnage parallèle efficace en tomographie 3D. CRAS série 1. revision.

    Google Scholar 

  6. L. Desbat and D. Girard (1995) The “minimum reconstruction-error” choice of regularization parameters: some more efficient methods and their application to deconvolution problems. SIAM J. Sci. Comput., 16(6):1387–1403.

    Article  Google Scholar 

  7. A. Faridani (1990) An application of a multidimensional sampling theorem to comnputed tomography. In AMS-IMS-SIAM Conference on Integral Geometry and Tomography, volume 113, pages 65–80. Comtemporary Mathematics.

    Chapter  Google Scholar 

  8. A. Faridani (1994) A generalized sampling theorem for locally compact abelian groups. Math. Comp., 63(207):307–327.

    Article  Google Scholar 

  9. D.A. Girard (1991) Asymptotic optimality of the fast randomized versions of GCV and CL in ridge regression and regularisation. Ann. of Stat., 19(4):1950–1963.

    Article  Google Scholar 

  10. G. Golub, M. Health, and G. Wahba (1979) Generalized Cross Validation for choosing a good ridge parameter. Technometrics, 21:215–224.

    Article  Google Scholar 

  11. P. Grangeat (1993) Reconstruction d’images tridimensionnelles. INPG. Thèse d’habilitation à diriger des recherches.

    Google Scholar 

  12. A.K. Louis (1982) Optimal Sampling in Nuclear Magnetic Resonance. J. of Comput. Assist. Tomography, 6(2):334–340.

    Article  CAS  Google Scholar 

  13. F. Natterer (1977) The finite element method for ill-posed problems. RAIRO MA2N, 11(3):271–278.

    Google Scholar 

  14. F. Natterer (1986) The Mathematics of Computerized Tomography. Wiley.

    Google Scholar 

  15. F. Natterer 1993 Sampling in fan-beam tomography. SIAM J. Appl. Math., 53(2):358–380.

    Article  Google Scholar 

  16. D.P. Petersen and D. Middleton (1962). Sampling and reconstruction of wavenumber-limited functions in N-dimensional euclidean space. Inf. Control, 5:279–323.

    Article  Google Scholar 

  17. P.A. Rattey and A.G. Lindgren (1981) Sampling the 2-D Radon transform. IEEE Trans. ASSP, 29:994–1002.

    Article  Google Scholar 

  18. D. Walnut (1995) Nonperiodic sampling of bandlimited functions on unions of rectangular lattices. submitted to JFAA.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Desbat, L. (1996). Efficient Sampling in 3D Tomography. In: Grangeat, P., Amans, JL. (eds) Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine. Computational Imaging and Vision, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8749-5_7

Download citation

  • DOI: https://doi.org/10.1007/978-94-015-8749-5_7

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4723-6

  • Online ISBN: 978-94-015-8749-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics