Near-field ultrasound tomography

In this paper, a near-field tomographic solution is introduced to solve the imaging problem of fluid objects assumed to be weakly heterogeneous (Born approximation) and excited by spherical waves. The solution to the forward problem is based on the Huygens-Fresnel principle which describes the scattered field as the result of the interference scheme of all the secondary spherical waves. From the derivation of the scattered field, a new Fourier transform that has been called the elliptical Fourier transform is defined: It differs from the standard Fourier transform in that instead of a plane wave decomposition, a harmonic ellipsoidal wave decomposition is obtained. Based on this spectral analysis, a near-field Radon transform is designed that complements the "far-field tools" published in diffraction tomography literature. Then, assuming that the measuring distance is greater than one wavelength, the feasibility of reconstructing either the impedance or the velocity maps of an acoustical (perfect fluid) model is demonstrated. Numerical simulations were performed which confirmed the validity of the theory presented here; a theory which has many potential applications in future wave theory research.