30 research outputs found
A dissipative time reversal technique for photo-acoustic tomography in a cavity
We consider the inverse source problem arising in thermo- and photo-acoustic
tomography. It consists in reconstructing the initial pressure from the
boundary measurements of the acoustic wave. Our goal is to extend versatile
time reversal techniques to the case of perfectly reflecting boundary of the
domain. Standard time reversal works only if the solution of the direct problem
decays in time, which does not happen in the setup we consider. We thus propose
a novel time reversal technique with a non-standard boundary condition. The
error induced by this time reversal technique satisfies the wave equation with
a dissipative boundary condition and, therefore, decays in time. For larger
measurement times, this method yields a close approximation; for smaller times,
the first approximation can be iteratively refined, resulting in a convergent
Neumann series for the approximation
Theoretically exact photoacoustic reconstruction from spatially and temporally reduced data
We investigate the inverse source problem for the wave equation, arising in
photo- and thermoacoustic tomography. There exist quite a few theoretically
exact inversion formulas explicitly expressing solution of this problem in
terms of the measured data, under the assumption of constant and known speed of
sound. However, almost all of these formulas require data to be measured either
on an unbounded surface, or on a closed surface completely surrounding the
object. This is too restrictive for practical applications. The alternative
approach we present, under certain restriction on geometry, yields
theoretically exact reconstruction of the standard Radon projections of the
source from the data measured on a finite open surface. In addition, this
technique reduces the time interval where the data should be known. In general,
our method requires a pre-computation of densities of certain single-layer
potentials. However, in the case of a truncated circular or spherical
acquisition surface, these densities are easily obtained analytically, which
leads to fully explicit asymptotically fast algorithms. We test these
algorithms in a series of numerical simulations
Recommended from our members
Asymptotically Optimal High-Order Accurate Algorithms for the Solution of Certain Elliptic PDEs
The main goal of the project, "Asymptotically Optimal, High-Order Accurate Algorithms for the Solution of Certain Elliptic PDE's" (DE-FG02-03ER25577) was to develop fast, high-order algorithms for the solution of scattering problems and spectral problems of photonic crystals theory. The results we obtained lie in three areas: (1) asymptotically fast, high-order algorithms for the solution of eigenvalue problems of photonics, (2) fast, high-order algorithms for the solution of acoustic and electromagnetic scattering problems in the inhomogeneous media, and (3) inversion formulas and fast algorithms for the inverse source problem for the acoustic wave equation, with applications to thermo- and opto- acoustic tomography
Microlocally accurate solution of the inverse source problem of thermoacoustic tomography
We consider the inverse source problem of thermo- and photoacoustic
tomography, with data registered on an open surface partially surrounding the
source of acoustic waves. Under the assumption of constant speed of sound we
develop an explicit non-iterative reconstruction procedure that recovers the
Radon transform of the sought source, up to an infinitely smooth additive error
term. The source then can be found by inverting the Radon transform.
Our analysis is microlocal in nature and does not provide a norm estimate on
the error in the so obtained image. However, numerical simulations show that
this error is quite small in practical terms. We also present an asymptotically
fast implementation of this procedure for the case when the data are given on a
circular arc in 2D.Comment: 6 figure
Fast, high-order solution of surface scattering problems
We present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three-dimensional space. This algorithm evaluates the scattered field, through fast, high-order solution of the boundary integral equation. The high-order of the solver is achieved through use of partition of unity together with analytical resolution of kernel singularities. The acceleration in turn, results from a novel approach which, based on high-order "two-face" equivalent source approximations, reduces the evaluation of far interactions to evaluation of 3-D FFTs. We demonstrate its performance with a variety of numerical results. In particular, we show that the present algorithm can evaluate accurately in a personal computer, scattering from bodies of acoustical sizes of several hundreds
A mathematical model and inversion procedure for Magneto-Acousto-Electric Tomography (MAET)
Magneto-Acousto-Electric Tomography (MAET), also known as the Lorentz force
or Hall effect tomography, is a novel hybrid modality designed to be a
high-resolution alternative to the unstable Electrical Impedance Tomography. In
the present paper we analyze existing mathematical models of this method, and
propose a general procedure for solving the inverse problem associated with
MAET. It consists in applying to the data one of the algorithms of
Thermo-Acoustic tomography, followed by solving the Neumann problem for the
Laplace equation and the Poisson equation.
For the particular case when the region of interest is a cube, we present an
explicit series solution resulting in a fast reconstruction algorithm. As we
show, both analytically and numerically, MAET is a stable technique yilelding
high-resolution images even in the presence of significant noise in the data