Fast inverse scattering solutions using the distorted Born iterative method and the multilevel fast multipole algorithm

Abstract

The distorted Born iterative method ͑DBIM͒ computes iterative solutions to nonlinear inverse scattering problems through successive linear approximations. By decomposing the scattered field into a superposition of scattering by an inhomogeneous background and by a material perturbation, large or high-contrast variations in medium properties can be imaged through iterations that are each subject to the distorted Born approximation. However, the need to repeatedly compute forward solutions still imposes a very heavy computational burden. To ameliorate this problem, the multilevel fast multipole algorithm ͑MLFMA͒ has been applied as a forward solver within the DBIM. The MLFMA computes forward solutions in linear time for volumetric scatterers. The typically regular distribution and shape of scattering elements in the inverse scattering problem allow the method to take advantage of data redundancy and reduce the computational demands of the normally expensive MLFMA setup. Additional benefits are gained by employing Kaczmarz-like iterations, where partial measurements are used to accelerate convergence. Numerical results demonstrate both the efficiency of the forward solver and the successful application of the inverse method to imaging problems with dimensions in the neighborhood of ten wavelengths.

An Inverse Fast Multipole Method for Geometry Reconstruction Using Scattered Field Information

“…Previous work in which the FMM is applied in inverse problems are mostly focused on the source reconstruction method, an integral equation technique for antenna diagnostics [24], [25]. The FMM has also been applied to rough surface profile reconstruction [26] and constitutive parameters retrieval when combined with a distorted Born iterative method for acoustic problems [27]. The contribution of this work is not only VSRM speed-up, but also the study of the FMM properties when applied to an inverse electromagnetic problem.…”

Ultrasound tomography for simultaneous reconstruction of acoustic density, attenuation, and compressibility profiles

“…13, the multilevel fast multipole algorithm incorporated in the DBIM as developed in Ref. 14, as well as in the time-domain and frequencydomain eigenfunction methods described in Refs. 15 and 16. It is well known that the density profile in many biomedical applications such as breast imaging is not constant.…”

The fast multipole method and Fourier convolution for the solution of acoustic scattering on regular volumetric grids

“…For example, estimation and correction of aberration requires simulation of propagation of ultrasound pulses through aberrating media [1–3]. Iterative inverse scattering algorithms such as the distorted Born iterative method [4–7] and the eigenfunction method [8–10] require repeated computation of the forward scattering problem for successive guesses of a reconstructed medium. The arbitrary nature of the scattering media, the desire for automatic satisfaction of the radiation condition, and the embarrassingly parallel nature of solutions at multiple frequencies make frequency-domain integral equation solvers particularly attractive where repeated calculations are required.…”