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

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 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

$$F_p \left( {\rho ,\theta } \right) = \sum\limits_{j = 1}^{2N + 1} {\sum\limits_{m = - 2N}^{2N} {F_{jm} } } S_j \left( \rho \right)e^{im\theta }$$

where the F_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/c₀ = 2π/λ, then the constructed solution F_l satisfies F_p(ρ,ϑ) = f(ρ,ϑ,z_p) + O(1/k²), where f denotes the exact solution to the Rytov approximation to the Helmholtz equation.

Research supported by U. S. Army Research Contract No. DAAG-29-77-G-0139.