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# An algorithm for ultrasonic tomography based on inversion of the Helmholtz equation

• F. Stenger
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 878)

## Abstract

This paper describes a numerical method for reconstructing the function f($$\bar r$$) = ω2[c(r)−2−c 0 −2 ], 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
$$F_p \left( {\rho ,\theta } \right) = \sum\limits_{j = 1}^{2N + 1} {\sum\limits_{m = - 2N}^{2N} {F_{jm} } } S_j \left( \rho \right)e^{im\theta }$$
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

Line Segment Helmholtz Equation Spherical Wave Wave Source Cardinal Function
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.

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## References

1. 
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).Google Scholar
2. 
Bleistein, N. and R. A. Handelsman, Asymptotic Expansion of Integrals, Holt, Rinehart and Winston (1975).Google Scholar
3. 
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.Google Scholar
4. 
McNamee, J., F. Stenger and E. L. Whitney, Whittaker's Cardinal Function in Retrospect, Math. Comp. 25 (1971), pp. 141–154.
5. 
Morse, P. M. and K. U. Ingard, Theoretical Acoustics, McGraw-Hill, N.Y. (1965).Google Scholar
6. 
Mueller, R. K., M. Kaveh and G. Wade, Reconstructive Tomography and Applications to Ultrasonics, Proc. IEEE 67 (1979) pp. 567–587.
7. 
Olver, F. W. J., Asymptotics and Special Functions, Academic Press, N.Y. (1974).
8. 
Rawson, E. G., Vibrating Verifocal Mirrors for 3-D Imaging. IEEE, Spectrum 6 (1969) pp. 37–43.
9. 
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.

## Copyright information

© Springer-Verlag 1981

## Authors and Affiliations

• F. Stenger
• 1
1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA

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