623 research outputs found
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
Forward and inverse problems in fundamental and applied magnetohydrodynamics
This Minireview summarizes the recent efforts to solve forward and inverse
problems as they occur in different branches of fundamental and applied
magnetohydrodynamics. As for the forward problem, the main focus is on the
numerical treatment of induction processes, including self-excitation of
magnetic fields in non-spherical domains and/or under the influence of
non-homogeneous material parameters. As an important application of the
developed numerical schemes, the functioning of the von-K\'{a}rm\'{a}n-sodium
(VKS) dynamo experiment is shown to depend crucially on the presence of
soft-iron impellers. As for the inverse problem, the main focus is on the
mathematical background and some first practical applications of the
Contactless Inductive Flow Tomography (CIFT), in which flow induced magnetic
field perturbations are utilized for the reconstruction of the velocity field.
The promises of CIFT for flow field monitoring in the continuous casting of
steel are substantiated by results obtained at a test rig with a low melting
liquid metal. While CIFT is presently restricted to flows with low magnetic
Reynolds numbers, some selected problems of non-linear inverse dynamo theory,
with possible application to geo- and astrophysics, are also discussed.Comment: 15 pages, 4 figures, accepted for publication in European Physical
Journal Special Topic
Jacobian of solutions to the conductivity equation in limited view
The aim of hybrid inverse problems such as Acousto-Electric Tomography or
Current Density Imaging is the reconstruction of the electrical conductivity in
a domain that can only be accessed from its exterior. In the inversion
procedure, the solutions to the conductivity equation play a central role. In
particular, it is important that the Jacobian of the solutions is
non-vanishing. In the present paper we address a two-dimensional limited view
setting, where only a part of the boundary of the domain can be controlled by a
non-zero Dirichlet condition, while on the remaining boundary there is a zero
Dirichlet condition. For this setting, we propose sufficient conditions on the
boundary functions so that the Jacobian of the corresponding solutions is
non-vanishing. In that regard we allow for discontinuous boundary functions,
which requires the use of solutions in weighted Sobolev spaces. We implement
the procedure of reconstructing a conductivity from power density data
numerically and investigate how this limited view setting affects the Jacobian
and the quality of the reconstructions
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Sampling bounds for 2-D vector field tomography
The tomographic mapping of a 2-D vector field from line-integral data in the discrete domain requires the uniform sampling of the continuous Radon domain parameter space. In this paper we use sampling theory and derive limits for the sampling steps of the Radon parameters, so that no information is lost. It is shown that if Δx is the sampling interval of the reconstruction region and xmax is the maximum value of domain parameter x, the steps one should use to sample Radon parameters ρ and θ should be: Δρ≤ Δx/√2 and Δθ≤Δx/((√2+2)|xmax|). Experiments show that when the proposed sampling bounds are violated, the reconstruction accuracy of the vector field deteriorates. We further demonstrate that the employment of a scanning geometry that satisfies the proposed sampling requirements also increases the resilience to noise
Inverse diffusion from knowledge of power densities
This paper concerns the reconstruction of a diffusion coefficient in an
elliptic equation from knowledge of several power densities. The power density
is the product of the diffusion coefficient with the square of the modulus of
the gradient of the elliptic solution. The derivation of such internal
functionals comes from perturbing the medium of interest by acoustic (plane)
waves, which results in small changes in the diffusion coefficient. After
appropriate asymptotic expansions and (Fourier) transformation, this allow us
to construct the power density of the equation point-wise inside the domain.
Such a setting finds applications in ultrasound modulated electrical impedance
tomography and ultrasound modulated optical tomography.
We show that the diffusion coefficient can be uniquely and stably
reconstructed from knowledge of a sufficient large number of power densities.
Explicit expressions for the reconstruction of the diffusion coefficient are
also provided. Such results hold for a large class of boundary conditions for
the elliptic equation in the two-dimensional setting. In three dimensions, the
results are proved for a more restrictive class of boundary conditions
constructed by means of complex geometrical optics solutions.Comment: 24 pages. Submitted to Inv. Probl. and Imaging, 201
Reconstruction of electric fields and source distributions in EEG brain imaging
In this thesis, three different approaches are developed for the estimation of focal brain activity using EEG measurements. The proposed approaches have been tested and found feasible using simulated data.
First, we develop a robust solver for the recovery of focal dipole sources. The solver uses a weighted dipole strength penalty term (also called weighted L1,2 norm) as prior information in order to ensure that the sources are sparse and focal, and that both the source orientation and depth bias are reduced. The solver is based on the truncated Newton interior point method combined with a logarithmic barrier method for the approximation of the penalty term. In addition, we use a Bayesian framework to derive the depth weights in the prior that are used to reduce the tendency of the solver to favor superficial sources.
In the second approach, vector field tomography (VFT) is used for the estimation of underlying electric fields inside the brain from external EEG measurements. The electric field is
reconstructed using a set of line integrals. This is the first time that VFT has been used for the
recovery of fields when the dipole source lies inside the domain of reconstruction. The benefit
of this approach is that we do not need a mathematical model for the sources. The test cases indicated that the approach can accurately localize the source activity.
In the last part of the thesis, we show that, by using the Bayesian approximation error approach (AEA), precise knowledge of the tissue conductivities and head geometry are not
always needed. We deliberately use a coarse head model and we take the typical variations
in the head geometry and tissue conductivities into account statistically in the inverse model.
We demonstrate that the AEA results are comparable to those obtained with an accurate head model.Open Acces
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