264 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
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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