Skip to main content
SpringerLink
Log in

Some mathematical problems motivated by medical imaging

  • Conference paper
  • First Online:
Inverse Problems

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1225))

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 69.99
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. Bockwinkel, "On the propagation of light in a two-axial crystal around a center of vibration," Versl. Kon Akad. v. Wet. XIV 2 (1906), 636–651.

    Google Scholar 

  2. Radon, J., "Uber die Bestimmung von Funktionen durch ihre Integral werte längs gewisser Mannifaltigkeiten," Berichte Saechsische Akademic der Wissenschaften 69 (1917), 262–277.

    Google Scholar 

  3. Herman, G.T., Image Reconstruction from Projections, (Academic Press, New York, 1980).

    MATH  Google Scholar 

  4. Herman,G.T., editor, Proceedings of the IEEE, Special Issue on Computerized Tomography, vol. 71, no. 3 (March 1983).

    Google Scholar 

  5. Davison, M.E. and F.A. Grűnbaum, "Tomographic reconstruction with arbitrary directions," Communications on Pure and Applied Mathematics 34 (1981), 77–120.

    Article  MATH  MathSciNet  Google Scholar 

  6. Smith, K.T., D.C. Solmon and S.L. Wagner, "Practical and mathematical aspects of the problem of reconstructing objects from radiographs," Bulletin of the American Mathematical Society, 83 (1977), 1227–1270.

    Article  MATH  MathSciNet  Google Scholar 

  7. Gerchberg, R.W. and W.O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik, 35, no. 2 (1972), 237–246.

    Google Scholar 

  8. Tuy. H and A. Lent, "An iterative method for the extrpolation of band-limited functions," J. Math. Analysis & Appl. 83, 2, (1981), 554–565.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Papoulis, "A new algorithm in spectral analysis and band limited extrapolation," IEEE Trans. on Circuits and Systems, vol. CAS-22, no. 9 (1975), 735–742.

    Article  MathSciNet  Google Scholar 

  10. Slepian, D. and H.P. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty, I." Bell Systems Technical Journal, 40, no. 1 (1961), 43–64.

    MATH  MathSciNet  Google Scholar 

  11. Landau, H.J. and Pollak, H.P., "Prolate spheroidal wave functions, Fourier analysis and uncertainty, II." Bell Systems Technical Journal, 40, no. 1 (1961), 65–84.

    MATH  MathSciNet  Google Scholar 

  12. Landau, H.J. and Pollak, H.P., "Prolate spheroidal wave functions, Fourier analysis and uncertainty, III." Bell Systems Technical Journal, 41, no. 4 (1962), 1295–1336.

    MATH  MathSciNet  Google Scholar 

  13. Slepian, D., "Prolate spheroidal wave functions, Fourier analysis and uncertainty, IV," Bell Systems Technical Journal, 43, no. 6 (1964), 3009–3058.

    MATH  MathSciNet  Google Scholar 

  14. Slepian, D., "Prolate spheroidal wave functions, Fourier analysis and uncertainty," Bell Systems Technical Journal, 57, no. 5 (1978), 1430–2371.

    Google Scholar 

  15. Slepian, D., "Some comments on Fourier analysis, uncertainty and modelling," SIAM Review, 25 (1983), 379–394.

    Article  MATH  MathSciNet  Google Scholar 

  16. Grűnbaum, F.A., "A new property for reproducing kernels of classical orthogonal polynomials," J. of Math. Analysis and Appl., 95 (1983), 491–500.

    Article  Google Scholar 

  17. Perlstadt, M., "Chopped orthogonal polynomial expansions — some discrete cases," SIAM J. Alg. Disc. Meth. 4 (1), (1983), 94–100.

    Article  MATH  MathSciNet  Google Scholar 

  18. Perlstadt, M., "A property of orthogonal polynomial families with polynomial duals," SIAM J. Math. Anal. 15 (5), (1984), 1043–1054.

    Article  MATH  MathSciNet  Google Scholar 

  19. Perline,R., "Discrete time-band limiting operators and commuting tridiagonal matrices," to appear in SIAM J. Alge. Disc. Meth.

    Google Scholar 

  20. Duistermaat, J. and F.A. Grűnbaum, "Differential equations in the eigenvalue parameter," Comm. Math. Physics, 103 (1986), 177–240.

    Article  MATH  Google Scholar 

  21. Grűnbaum, F.A., "A remark on Hilbert's matrix," Linear Algebra and its Applications, 43 (1982), 119–124.

    Article  MathSciNet  Google Scholar 

  22. Bertero, M. and F.A. Grűnbaum, "Commuting differential operators for the finite Laplace transform," Inverse Problems, 1 (1985), 181–192.

    Article  MATH  MathSciNet  Google Scholar 

  23. Coddington,E. and N.Levinson, Theory of Ordinary Differential Equations, (McGraw-Hill Book Co., 1955).

    Google Scholar 

  24. Grűnbaum,F.A., "The Bloch equations and two nonlinear extensions of the Fourier transform," Proceedings, Third Annual Meeting, Society of Magnetic Resonance in Medicine, New York (1984), 281–282.

    Google Scholar 

  25. Grűnbaum, F.A., "An inverse problem related to the Bloch equations and the nonlinear Fourier transform," Inverse Problems, 1 (1985), L25–L28.

    Article  Google Scholar 

  26. Hoult, D., "The solution of the Bloch equations in the presence of a varying B1 field," J. Magnetic Resonance, 35 (1979), 69.

    Google Scholar 

  27. Grűnbaum, F.A. and Hasenfeld, A., "An exploration of the invertibility of the Bloch transform," Inverse Problems, 2 (1986), 75–81.

    Article  MathSciNet  Google Scholar 

  28. Hasenfeld, A., "An inverse problem for in vivo NMR spatial localization," Ph.D. Thesis, University of California, Berkeley (1985).

    Book  Google Scholar 

  29. Flaschka, H., "The Toda lattice, I: Existence of integrals," Phys. Rev. B., 9, (1974), 1924–1925.

    Article  MathSciNet  Google Scholar 

  30. Flaschka, H., "On the Toda lattice, II: Inverse scattering solution," Prog. Theor. Physics, 51 (1974), 703–716.

    Article  MathSciNet  Google Scholar 

  31. Deift, P., T. Nanda and C. Tomei, "Ordinary differential equations and the symmetric eigenvalue problem," SIAM J. Numer. Analysis, 20, 1 (1983), 1–22.

    Article  MATH  MathSciNet  Google Scholar 

  32. Feig, E. and F.A. Grűnbaum, "Tomographic methods in range-Doppler radar," to appear in Inverse Problems, vol. 2, 2 (1986).

    Google Scholar 

  33. Grűnbaum, F.A., "A remark on the radar ambiguity function," Transactions on Information Theory, IEEE, vol. IT-30, no. 1, (January 1984), 126–127.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, 94720, Berkeley, California

    F. Alberto Grünbaum

Authors
  1. F. Alberto Grünbaum
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Istituto Matematico “U.DINI”, Università di Firenze, Viale Morgagni 67/a, 50134, Firenze, Italy

    Giorgio Talenti

Rights and permissions

Reprints and permissions

Copyright information

© 1986 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Grünbaum, F.A. (1986). Some mathematical problems motivated by medical imaging. In: Talenti, G. (eds) Inverse Problems. Lecture Notes in Mathematics, vol 1225. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072661

Download citation

  • DOI: https://doi.org/10.1007/BFb0072661

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17193-5

  • Online ISBN: 978-3-540-47353-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation