Numerical Solution of Nonlinear Equations pp 371-406 | Cite as

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

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## Abstract

This paper describes a numerical method for reconstructing the function f(\(\bar r\)) = ω where the F

^{2}[c(r)^{−2}−c_{0}^{−2}], where \(c\left( {\bar r} \right)\) denotes the speed of sound in a bounded body, and c_{0}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 cosϑ_{j}, a sinϑ_{j}, z_{p}), where ϑ_{j}= jπ/(2N+1), z_{p}= ph, p = 1,2,…,2N+5. We then describe the reconstruction of f(x,y,z_{p}) = F_{p}(ρ,ϑ) in the form$$F_p \left( {\rho ,\theta } \right) = \sum\limits_{j = 1}^{2N + 1} {\sum\limits_{m = - 2N}^{2N} {F_{jm} } } S_j \left( \rho \right)e^{im\theta }$$

_{jm}are complex numbers and the S_{j}(ρ) are "Chapeau" splines on a nonequi-spaced mesh. If h and aπ/(2N+1) are of the order of 1/k^{1/3}, where k = ω/c_{0}= 2π/λ, then the constructed solution F_{l}satisfies F_{p}(ρ,ϑ) = f(ρ,ϑ,z_{p}) + O(1/k^{2}), 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

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© Springer-Verlag 1981