An FFT-based algorithm for reconstructing inhomogeneous circular cylindrical shells from noisy data

Abstract

The inverse problem of reconstructing an inhomogeneous, circular, cylindrical shell, from the knowledge of scattering data, is of importance in diverse fields, including the imaging of thin veins and arteries in medicine, of co-axial cables in non-destructive testing, and of volcanic pipes in volcanology. Optical fibres provide another important example of shells evaluated by such non-invasive means.

In as much as the problem has such diverse applications, it will be useful to have an algorithm for this purpose, which (i) is computationally efficient; (ii) provides good reconstructions when the data is corrupted with additive noise which is often assumed to be Gaussian; (iii) does not employ a small-perturbations approximation, such as Born’s or Rytov’s, which restricts their scope of application.

One such algorithm is proposed in this paper. Here, computational efficiency is achieved by recognising that the circular geometry of the problem translates itself into circulant matrices that are easily inverted by invoking the Fast Fourier Transform (FFT); while the desensitization of the method to corrupting noise is carried out by employing the least-squares method.

Illustrative computer simulations verify the validity of the technique proposed.