Leonid Kunyansky

Abstract:

We present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three-dimensional space. This algorithm evaluates scattered fields through fast, high-order, accurate solution of the corresponding boundary integral equation. The high-order accuracy of our solver is achieved through use of partitions of unity together with analytical resolution of kernel singularities. The acceleration, in turn, results from use of high-order equivalent source approximations, which allow for fast evaluation of non-adjacent interactions by means of the three-dimensional fast Fourier transform (FFT). Our acceleration scheme has dramatically lower memory requirements and yields much higher accuracy than existing FFT-accelerated techniques. The present algorithm computes one matrix-vector multiply in O(N^6/5 log N) to O(N^4/3 log N) operations (depending on the geometric characteristics of the scattering surface), it exhibits super-algebraic convergence, and it does not suffer from accuracy breakdowns of any kind. We demonstrate the efficiency of our method through a variety of examples. In particular, we show that the present algorithm can evaluate accurately, on a personal computer, scattering from bodies of acoustical sizes (ka) of several hundreds.

Abstract:

The inversion problem for the 3D parallel-beam exponential ray transform is solved through inversion of a set of the 2D exponential Radon transforms with complex-valued angle-dependent attenuation. An inversion formula for the latter 2D transform is derived; it generalizes the known Kuchment-Shneiberg formula valid for real angle-dependent attenuation. We derive an explicit theoretically exact solution of the 3D problem which is valid for arbitrary closed trajectory that does not intersect itself. A simple reconstruction algorithm is described, applicable for certain sets of trajectories satisfying Orlov's condition. In the latter case, our inversion technique is as stable as the Tretiak-Metz inversion formula. Possibilities of further reduction of noise sensitivity are briefly discussed in the paper. The work of our algorithm is illustrated by an example of image reconstruction from two circular orbits.

Abstract:

An exact formula approach to nonuniform attenuation correction in single-photon emission computed tomography (SPECT) was discussed. The formula admits a numerical implementation via a direct generalization of the filtered backprojection (FBP) algorithm. The formula can be used for fast computing of an efficient first approximation for more complicated SPECT reconstruction techniques.

Abstract:

We derive explicit formulae for the reconstruction of a function from its integrals over a family of spheres, or for the inversion of the spherical mean Radon transform. Such formulae are important for problems of thermo- and photo-acoustic tomography. A closed-form inversion formula of a filtration-backprojection type is found for the case when the centres of the integration spheres lie on a sphere in surrounding the support of the unknown function. © 2007 IOP Publishing Ltd.

Abstract:

Studying classical wave propagation in periodic high contrast photonic and acoustic media naturally leads to the following spectral problem: -Δu = λεu, where ε(x) (the dielectric constant) is a periodic function that assumes a large value ε near a periodic graph Σ in ℝ2 and is equal to 1 otherwise. High contrast regimes lead to appearence of pseudo-differential operators of the Dirichlet-to-Neumann type on graphs. The paper contains a technique of approximating these pseudo-differential spectral problems by much simpler differential ones that can sometimes be resolved analytically. Numerical experiments show amazing agreement between the spectra of the pseudo-differential and differential problems.