75 research outputs found
Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography
The paper contains a simple approach to reconstruction in Thermoacoustic and
Photoacoustic Tomography. The technique works for any geometry of point
detectors placement and for variable sound speed satisfying a non-trapping
condition. A uniqueness of reconstruction result is also obtained
Time reversal in thermoacoustic tomography - an error estimate
The time reversal method in thermoacoustic tomography is used for
approximating the initial pressure inside a biological object using
measurements of the pressure wave made on a surface surrounding the object.
This article presents error estimates for the time reversal method in the cases
of variable, non-trapping sound speeds.Comment: 16 pages, 6 figures, expanded "Remarks and Conclusions" section,
added one figure, added reference
On the injectivity of the circular Radon transform arising in thermoacoustic tomography
The circular Radon transform integrates a function over the set of all
spheres with a given set of centers. The problem of injectivity of this
transform (as well as inversion formulas, range descriptions, etc.) arises in
many fields from approximation theory to integral geometry, to inverse problems
for PDEs, and recently to newly developing types of tomography. The article
discusses known and provides new results that one can obtain by methods that
essentially involve only the finite speed of propagation and domain dependence
for the wave equation.Comment: To appear in Inverse Problem
A series solution and a fast algorithm for the inversion of the spherical mean Radon transform
An explicit series solution is proposed for the inversion of the spherical
mean Radon transform. Such an inversion is required in problems of thermo- and
photo- acoustic tomography. Closed-form inversion formulae are currently known
only for the case when the centers of the integration spheres lie on a sphere
surrounding the support of the unknown function, or on certain unbounded
surfaces. Our approach results in an explicit series solution for any closed
measuring surface surrounding a region for which the eigenfunctions of the
Dirichlet Laplacian are explicitly known - such as, for example, cube, finite
cylinder, half-sphere etc. In addition, we present a fast reconstruction
algorithm applicable in the case when the detectors (the centers of the
integration spheres) lie on a surface of a cube. This algorithm reconsrtucts
3-D images thousands times faster than backprojection-type methods
Extremal discs and the holomorphic extension from convex hypersurfaces
Let D be a convex domain with smooth boundary in complex space and let f be a
continuous function on the boundary of D. Suppose that f holomorphically
extends to the extremal discs tangent to a convex subdomain of D. We prove that
f holomorphically extends to D. The result partially answers a conjecture by
Globevnik and Stout of 1991
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
Thermoacoustic tomography with detectors on an open curve: an efficient reconstruction algorithm
Practical applications of thermoacoustic tomography require numerical
inversion of the spherical mean Radon transform with the centers of integration
spheres occupying an open surface. Solution of this problem is needed (both in
2-D and 3-D) because frequently the region of interest cannot be completely
surrounded by the detectors, as it happens, for example, in breast imaging. We
present an efficient numerical algorithm for solving this problem in 2-D
(similar methods are applicable in the 3-D case). Our method is based on the
numerical approximation of plane waves by certain single layer potentials
related to the acquisition geometry. After the densities of these potentials
have been precomputed, each subsequent image reconstruction has the complexity
of the regular filtration backprojection algorithm for the classical Radon
transform. The peformance of the method is demonstrated in several numerical
examples: one can see that the algorithm produces very accurate reconstructions
if the data are accurate and sufficiently well sampled, on the other hand, it
is sufficiently stable with respect to noise in the data
Thermoacoustic tomography with an arbitrary elliptic operator
Thermoacoustic tomography is a term for the inverse problem of determining of
one of initial conditions of a hyperbolic equation from boundary measurements.
In the past publications both stability estimates and convergent numerical
methods for this problem were obtained only under some restrictive conditions
imposed on the principal part of the elliptic operator. In this paper
logarithmic stability estimates are obatined for an arbitrary variable
principal part of that operator. Convergence of the Quasi-Reversibility Method
to the exact solution is also established for this case. Both complete and
incomplete data collection cases are considered.Comment: 16 page
Reconstruction of a function from its spherical (circular) means with the centers lying on the surface of certain polygons and polyhedra
We present explicit filtration/backprojection-type formulae for the inversion
of the spherical (circular) mean transform with the centers lying on the
boundary of some polyhedra (or polygons, in 2D). The formulae are derived using
the double layer potentials for the wave equation, for the domains with certain
symmetries. The formulae are valid for a rectangle and certain triangles in 2D,
and for a cuboid, certain right prisms and a certain pyramid in 3D. All the
present inversion formulae yield exact reconstruction within the domain
surrounded by the acquisition surface even in the presence of exterior sources.Comment: 9 figure
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