Reconstruction of a two-dimensional reflecting medium over a circular domain: Exact solution

Abstract

If a two-dimensional acoustic reflectivity function is defined within the interior of a circle, reflectivity data may be acquired by transmitting acoustic pulses from isotropic elements distributed around the circumference of the circle and recording the resulting backscattered sound as a function of time. If the reflectivity function fulfills the conditions of an ’’idealized’’ weakly reflecting medium, the resulting pulse–echo data may be regarded as the line integrals of this function defined over circular arcs centered at points lying on the circumference of the enclosing circle. In this paper we show that on the basis of these line integrals the unknown reflectivity in the interior of the circle can be expressed explicitly in terms of its line integrals defined over the set of paths consisting of all circular arcs whose centers lie on the circumference of the enclosing circle. We propose that the resulting reconstruction equations could provide the foundation for a new approach to reflectivity tomography. A numerical example is also given.

Inversion of Spherical Means and the Wave Equation in Even Dimensions

“…To our knowledge, the first work to tackle the problem of recovering a function from its circular means with centers on a circle was [13], whose author was interested in ultrasound reflectivity tomography. He found an inversion method based on harmonic decomposition and for each harmonic, the inversion of a Hankel transform.…”

“…Proof of (13) in Theorem 4 for n = 2. We compute (P * t∂ 2 t P f )(x) for smooth f supported in B and x ∈ B.…”

Photoacoustic Tomography

“…The hardest to come about were explicit inversion formulas, as well as some uniqueness of reconstruction results (albeit, for practical geometries, these can be considered to be resolved [3,22]). For quite a while, only series expansions had been available for the case of transducers placed along a sphere S surrounding the object [31,32]. No formulas were available for non-spherical surfaces S. The first explicit inversion formulas were obtained in [11] for the spherical geometry in odd dimensions and then extended to even dimensions in [10].…”

Mathematics of Photoacoustic and Thermoacoustic Tomography

“…The first inversion procedures for the case of closed acquisition surfaces were described in [94,95], where solutions were found for the cases of circular (in 2D) and spherical (in 3D) surfaces, respectively. These solutions were obtained by the harmonic decomposition of the measured data and of the sought function f (x), followed by equating coefficients of the corresponding Fourier series.…”

“…These solutions were obtained by the harmonic decomposition of the measured data and of the sought function f (x), followed by equating coefficients of the corresponding Fourier series. In particular, the 2D algorithm of [94] pertains to the case when the detectors are located on a circle of radius R. This method is based on the Fourier decomposition of f and g in angular variables:…”