Abstract
This paper describes a numerical method for reconstructing the function f(\(\bar r\)) = ω2[c(r)−2−c −20 ], where \(c\left( {\bar r} \right)\) denotes the speed of sound in a bounded body, and c0 denotes the speed of sound in the medium surrounding the body, for both the case of plane wave excitation, \(e^{i\left( {\bar k \cdot \bar r - \omega t} \right)}\), and spherical wave excitation, \({{e^{ik\left| {\bar r - \bar r} \right.} s^{\left| { - i\omega t} \right.} } \mathord{\left/{\vphantom {{e^{ik\left| {\bar r - \bar r} \right.} s^{\left| { - i\omega t} \right.} } {\left[ {4\pi \left| {\bar r - \bar r_s } \right|} \right]}}} \right.\kern-\nulldelimiterspace} {\left[ {4\pi \left| {\bar r - \bar r_s } \right|} \right]}}\). It is assumed that the body is located in the interior of a cylinder of radius a, having the z axis as its axis of symmetry, that the ultrasonic sound pressure is measured on the surface of this cylinder at the points (a cosϑj, a sinϑj, zp), where ϑj = jπ/(2N+1), zp = ph, p = 1,2,…,2N+5. We then describe the reconstruction of f(x,y,zp) = Fp(ρ,ϑ) in the form
where the Fjm are complex numbers and the Sj(ρ) are "Chapeau" splines on a nonequi-spaced mesh. If h and aπ/(2N+1) are of the order of 1/k1/3, where k = ω/c0 = 2π/λ, then the constructed solution Fl satisfies Fp(ρ,ϑ) = f(ρ,ϑ,zp) + O(1/k2), where f denotes the exact solution to the Rytov approximation to the Helmholtz equation.
Keywords
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.
Research supported by U. S. Army Research Contract No. DAAG-29-77-G-0139.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Ball, J., S. A. Johnson and F. Stenger, Explicit Inversion of the Helmholtz Equation for Ultrasound Insonification and Spherical Detection, to appear in Proc. of Houston Conference on Acoustical Imaging 9 (1980).
Bleistein, N. and R. A. Handelsman, Asymptotic Expansion of Integrals, Holt, Rinehart and Winston (1975).
Greenleaf, J. F., S. A. Johnson, W. F. Samoyoa and F. A. Duck, Reconstruction of Spacial Distributions of Retractive Indices in Tissue from Time of Flight Profiles, Image Processing for 2D and and 3D Reconstruction from Projections: Theory and Practice in Medicine and the Physical Sciences. A Digest of Technical Papers. August 4–7 (1975), Stanford, Cal., pp. MA2-1–MA2-4.
McNamee, J., F. Stenger and E. L. Whitney, Whittaker's Cardinal Function in Retrospect, Math. Comp. 25 (1971), pp. 141–154.
Morse, P. M. and K. U. Ingard, Theoretical Acoustics, McGraw-Hill, N.Y. (1965).
Mueller, R. K., M. Kaveh and G. Wade, Reconstructive Tomography and Applications to Ultrasonics, Proc. IEEE 67 (1979) pp. 567–587.
Olver, F. W. J., Asymptotics and Special Functions, Academic Press, N.Y. (1974).
Rawson, E. G., Vibrating Verifocal Mirrors for 3-D Imaging. IEEE, Spectrum 6 (1969) pp. 37–43.
Stenger, F. and S. Johnson, Ultrasonic Transmission Tomography Based on the Inversion of the Helmholtz Wave Equation for Plane and Spherical Wave Insonification, Appl. Math. Notes 4 (1979) pp. 102–127.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1981 Springer-Verlag
About this paper
Cite this paper
Stenger, F. (1981). An algorithm for ultrasonic tomography based on inversion of the Helmholtz equation. In: Allgower, E.L., Glashoff, K., Peitgen, HO. (eds) Numerical Solution of Nonlinear Equations. Lecture Notes in Mathematics, vol 878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090689
Download citation
DOI: https://doi.org/10.1007/BFb0090689
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-10871-9
Online ISBN: 978-3-540-38781-7
eBook Packages: Springer Book Archive