62 research outputs found
Reconstruction of less regular conductivities in the plane
We study the inverse conductivity problem of how to reconstruct an isotropic
electrical conductivity distribution in an object from static
electrical measurements on the boundary of the object. We give an exact
reconstruction algorithm for the conductivity \gamma\in C^{1+\epsilon}(\ol
\Om) in the plane domain from the associated Dirichlet to Neumann map
on \partial \Om. Hence we improve earlier reconstruction results. The method
used relies on a well-known reduction to a first order system, for which the
\ol\partial-method of inverse scattering theory can be applied
Stability in Conductivity Imaging from Partial Measurements of One Interior Current
We prove a stability result in the hybrid inverse problem of recovering the
electrical conductivity from partial knowledge of one current density field
generated inside a body by an imposed boundary voltage. The region where
interior data stably reconstructs the conductivity is well defined by a
combination of the exact and perturbed data
A weighted minimum gradient problem with complete electrode model boundary conditions for conductivity imaging
We consider the inverse problem of recovering an isotropic electrical
conductivity from interior knowledge of the magnitude of one current density
field generated by applying current on a set of electrodes. The required
interior data can be obtained by means of MRI measurements. On the boundary we
only require knowledge of the electrodes, their impedances, and the
corresponding average input currents. From the mathematical point of view, this
practical question leads us to consider a new weighted minimum gradient problem
for functions satisfying the boundary conditions coming from the Complete
Electrode Model of Somersalo, Cheney and Isaacson. This variational problem has
non-unique solutions. The surprising discovery is that the physical data is
still sufficient to determine the geometry of the level sets of the minimizers.
In particular, we obtain an interesting phase retrieval result: knowledge of
the input current at the boundary allows determination of the full current
vector field from its magnitude. We characterize the non-uniqueness in the
variational problem. We also show that additional measurements of the voltage
potential along one curve joining the electrodes yield unique determination of
the conductivity. A nonlinear algorithm is proposed and implemented to
illustrate the theoretical results.Comment: 20 pages, 5 figure
Conductivity imaging from one interior measurement in the presence of perfectly conducting and insulating inclusions
We consider the problem of recovering an isotropic conductivity outside some
perfectly conducting or insulating inclusions from the interior measurement of
the magnitude of one current density field . We prove that the
conductivity outside the inclusions, and the shape and position of the
perfectly conducting and insulating inclusions are uniquely determined (except
in an exceptional case) by the magnitude of the current generated by imposing a
given boundary voltage. We have found an extension of the notion of
admissibility to the case of possible presence of perfectly conducting and
insulating inclusions. This also makes it possible to extend the results on
uniqueness of the minimizers of the least gradient problem
with to cases where
has flat regions (is constant on open sets)
On the -ray transform of planar symmetric 2-tensors
In this paper we study the attenuated -ray transform of 2-tensors
supported in strictly convex bounded subsets in the Euclidean plane. We
characterize its range and reconstruct all possible 2-tensors yielding
identical -ray data. The characterization is in terms of a Hilbert-transform
associated with -analytic maps in the sense of Bukhgeim.Comment: 1 figure. arXiv admin note: text overlap with arXiv:1411.492
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