Ultrasound Tomography by Galerkin or Moment Methods

  • Steven A. Johnson
  • Frank Stenger
Part of the Lecture Notes in Medical Informatics book series (LNMED, volume 23)


Ultrasound B-scan imaging is now a well established and valuable clinical tool. Improvements in transducer arrays and microprocessor controls have led to the development of real-time linear and sector scanners which produce images of remarkable clarity and resolution compared with B-scanners of only a few years ago. Further improvements in B-scan images are predicted to occur as larger transducers, apertures and improved dynamic focusing methods are employed. The use of Doppler ultrasound alone or in combination with real-time B-scan imaging is expected to increase in importance as the clinical significance of high resolution Doppler images is appreciated.


Inverse Scattering Scattered Field Moment Method Incident Field Acoustical Image 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Morse PM and Ingard KV (1968). Theoretical Acoustics. McGraw-Hill Book Company, New York.Google Scholar
  2. 2.
    Greenleaf JF, Johnson SA, Lee SL, Herman GT and Wood EH (1974). Algebraic reconstruction of spatial distributions of acoustic absorption within tissue from their two-dimensional acoustic projections. (In) Acoustical Holography, (Green PS, ed), Plenum Press, New York, Vol. 5, pp 591–603.Google Scholar
  3. 3.
    Greenleaf JF, Johnson SA, Samayoa WF and Duck FA (1975). Algebraic reconstruction of spatial distributions of acoustic velocities in tissue from their time-of-flight profiles. (In) Acoustical Holography, (Booth, Newell ed), Plenum Press, New York, Vol. 6, pp 71–90.Google Scholar
  4. 4.
    Wolf, E (1969). Three-dimensional structure determination of semi-transparent objects from holographic data. Optics Communicator 1 (4): 153–156.CrossRefGoogle Scholar
  5. 5.
    Mueller RK, Kaveh M and Wade G (1979). Reconstructive tomography and applications to ultrasonics. Proc IEEE 67: 567.CrossRefGoogle Scholar
  6. 6.
    Kaveh M, Soumekh M, Lu ZQ, Mueller RK (1982). Further results on diffraction tomography. (In) Acoustical Imaging, Plenum Press, New York, Vol. 12, pp 599–608.Google Scholar
  7. 7.
    Norton SJ and Linzer M (1982). Correcting for ray refraction in velocity and attenuation tomography: A perturbation approach. Ultrasound Imaging 4: 201–233.CrossRefGoogle Scholar
  8. 8.
    Stenger F (1982). Asymptotic ultrasonic inversion based on using more than one frequency. (In) Acoustical Imaging, Plenum Press, New York, Vol. 11, pp 425–444.Google Scholar
  9. 9.
    Stenger F, Berggren MJ, Johnson SA and Wilcox CH (January 1983). Rational function frequency extrapolation in ultrasonic tomography. Submitted to Ultrasonic Imaging.Google Scholar
  10. 10.
    Harrington RF. Field Computation by Moment Methods. The Macmillan Company, Inc., New York; copies may also be obtained from the author in the Electrical Engineering Department, Syracuse University, New York.Google Scholar
  11. 11.
    Richmond J (May 1965). Scattering by a dielectric cylinder of arbitrary cross-sectional shape. IEEE Transactions on Antennas and Propagation, pp 334–341.Google Scholar
  12. 12.
    Hagmann MJ, Gandhi OP and Ghodgaonkar DK (1981). Application of moment methods to electromagnetic biological imaging. MITfs International Microwave Symposium Digest, p 432.Google Scholar
  13. 13.
    Yoon TH, Kim SY and Ra JW (June 1982). Reconstruction of distributed dielectric objects using low-frequency waves. Conference Proceedings of the 1982 International GeoScience and Remote Sensing Symposium, Munich, Germany; IEEE Transactions on GeoScience and Remote Sensing, IEEE Symposium Digest 3: 3. 1–4.Google Scholar
  14. 14.
    Johnson SA, Yoon TH and Ra JW (Feb 1983) Inverse scattering solu-tions of the scalar Helmholtz wave equation by a multiple source moment method. Electronics Letters 19 (4): 130.CrossRefGoogle Scholar
  15. 15.
    Stenger F (April 1981). Numerical methods on Whittaker cardinal, or sine functions. SIAM Review 23 (2).Google Scholar
  16. 16.
    Johnson SA and Tracy ML. Inverse scattering solutions by a sine basic, multiple source, moment method-Part I: Theory. Submitted to Acoustical Imaging.Google Scholar
  17. 17.
    Tracy ML and Johnson SA (May 1983). Inverse scattering solutions by a sine basic, multiple source, moment method — Part I I: Numerical evaluations. Submitted to Acoustical Imaging.Google Scholar
  18. 18.
    Censor Y (Oct 1981). Raw-action methods for huge and sparce systems and their applications. SIAM Review 23 (4): 446–466.CrossRefGoogle Scholar
  19. 19.
    Fletcher R (1980). Practical methods of optimization. (In) Unconstrained Optimization, Vol. 1, John Wiley & Sons, New York.Google Scholar
  20. 20.
    Gill PE, Murray W and Wright MH (1981). Practical Optimization. Academic Press, New York.Google Scholar
  21. 21.
    Kaczmarz S (1935). Angenäherte Auflosung von System Linearer Gleichungen. Bull Akad Polon Science Lett, A. 35, pp 355–357.Google Scholar
  22. 22.
    Herman GT (1980). Image Reconstruction from Projections, the Fun-damentals of Computerized Tomography. Academic Press, New York.Google Scholar
  23. 23.
    Raison A and Rabinowitz P (1978). A First Course in Numerical Analysis, McGraw-Hill Book Company, New York.Google Scholar
  24. 24.
    Fast imaging algorithms based upon Fast Fourier Transform (FFT) implementation of the back projection algorithm \({{\gamma }_{\ell }} {\text{FF}}{{{\text{T}}}^{{ - 1}}} \{ {\text{FFT[}}\gamma _{\ell }^{\prime }]/{\text{FFT[sin}}{{{\text{c}}}^{2}}({{k}_{0}}|{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}}|)]\}\) where \(\begin{array}{*{20}{c}} {\sum\limits_{i}^{N} {{{\gamma }_{\ell }}\sin {{c}^{2}}({{k}_{0}}|{{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}}}}_{\ell }} - {{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}}}}_{i}}|)} } & {\gamma _{\ell }^{\prime }} \\ \end{array} = \frac{{2\pi }}{\Phi }\sum\limits_{\emptyset }^{\Phi } {(1/{{f}_{{\emptyset \ell }}})\sum\limits_{m}^{M} {S_{{\Phi m}}^{{(s)}}D_{{m\ell }}^{*}} }\) for solving Equation (39) and Fast Fourier Transform implementation of conjugate gradient methods for solving Equation (38) look promising for reducing computational time dependence from N5/2 of this paper to N3/2 log N. These suggestions were made by S.A. Johnson and D.T. Borup, respectively, Department of Electrical Engineeering, University of Utah, prior to April 1, 1983.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Steven A. Johnson
    • 1
    • 2
    • 3
  • Frank Stenger
    • 1
    • 2
    • 3
  1. 1.Department of BioengineeringUniversity of UtahSalt Lake CityUSA
  2. 2.Department of Electrical EngineeringUniversity of UtahSalt Lake CityUSA
  3. 3.Department of MathematicsUniversity of UtahSalt Lake CityUSA

Personalised recommendations