173 research outputs found
Imaging polarizable dipoles
We present a method for imaging the polarization vector of an electric dipole
distribution in a homogeneous medium from measurements of the electric field
made at a passive array. We study an electromagnetic version of Kirchhoff
imaging and prove, in the Fraunhofer asymptotic regime, that range and
cross-range resolution estimates are identical to those in acoustics. Our
asymptotic analysis provides error estimates for the cross-range dipole
orientation reconstruction and shows that the range component of the dipole
orientation is lost in this regime. A naive generalization of the Kirchhoff
imaging function is afflicted by oscillatory artifacts in range, that we
characterize and correct. We also consider the active imaging problem which
consists in imaging both the position and polarizability tensors of small
scatterers in the medium using an array of collocated sources and receivers. As
in the passive array case, we provide resolution estimates that are consistent
with the acoustic case and give error estimates for the cross-range entries of
the polarizability tensor. Our theoretical results are illustrated by numerical
experiments.Comment: 35 pages, 18 figure
Exterior cloaking with active sources in two dimensional acoustics
We cloak a region from a known incident wave by surrounding the region with
three or more devices that cancel out the field in the cloaked region without
significantly radiating waves. Since very little waves reach scatterers within
the cloaked region, the scattered field is small and the scatterers are for all
practical purposes undetectable. The devices are multipolar point sources that
can be determined from Green's formula and an addition theorem for Hankel
functions. The cloaking devices are exterior to the cloaked region
Study of noise effects in electrical impedance tomography with resistor networks
We present a study of the numerical solution of the two dimensional
electrical impedance tomography problem, with noisy measurements of the
Dirichlet to Neumann map. The inversion uses parametrizations of the
conductivity on optimal grids. The grids are optimal in the sense that finite
volume discretizations on them give spectrally accurate approximations of the
Dirichlet to Neumann map. The approximations are Dirichlet to Neumann maps of
special resistor networks, that are uniquely recoverable from the measurements.
Inversion on optimal grids has been proposed and analyzed recently, but the
study of noise effects on the inversion has not been carried out. In this paper
we present a numerical study of both the linearized and the nonlinear inverse
problem. We take three different parametrizations of the unknown conductivity,
with the same number of degrees of freedom. We obtain that the parametrization
induced by the inversion on optimal grids is the most efficient of the three,
because it gives the smallest standard deviation of the maximum a posteriori
estimates of the conductivity, uniformly in the domain. For the nonlinear
problem we compute the mean and variance of the maximum a posteriori estimates
of the conductivity, on optimal grids. For small noise, we obtain that the
estimates are unbiased and their variance is very close to the optimal one,
given by the Cramer-Rao bound. For larger noise we use regularization and
quantify the trade-off between reducing the variance and introducing bias in
the solution. Both the full and partial measurement setups are considered.Comment: submitted to Inverse Problems and Imagin
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