41 research outputs found
Levenberg-Marquardt algorithm for acousto-electric tomography based on the complete electrode model
The inverse problem in Acousto-Electric tomography concerns the
reconstruction of the electric conductivity in a domain from knowledge of the
power density function in the interior of the body. This interior power density
results from currents prescribed at boundary electrodes (and can be obtained
through electro-static boundary measurements together with auxiliary acoustic
measurement. In Electrical Impedance Tomography, the complete electrode model
is known to be the most accurate model for the forward modelling. In this
paper, the reconstruction problem of Acousto-Electric tomography is posed using
the (smooth) complete electrode model, and a Levenberg-Marquardt iteration is
formulated in appropriate function spaces. This results in a system of partial
differential equations to be solved in each iteration. To increase the
computational efficiency and stability, a strategy based on both the complete
electrode model and the continuum model with Dirichlet boundary condition is
proposed. The system of equations is implemented numerically for a two
dimensional scenario and the algorithm is tested on two different numerical
phantoms, a heart and lung model and a human brain model. Several numerical
experiments are carried out confirming the feasibility, accuracy and stability
of the methods
A fully non-linear optimization approach to acousto-electric tomography
This paper considers the non-linear inverse problem of reconstructing an
electric conductivity distribution from the interior power density in a bounded
domain. Applications include the novel tomographic method known as
acousto-electric tomography, in which the measurement setup in Electrical
Impedance Tomography is modulated by ultrasonic waves thus giving rise to a
method potentially having both high contrast and high resolution. We formulate
the inverse problem as a regularized non-linear optimization problem, show the
existence of a minimizer, and derive optimality conditions. We propose a
non-linear conjugate gradient scheme for finding a minimizer based on the
optimality conditions. All our numerical experiments are done in
two-dimensions. The experiments reveal new insight into the non-linear effects
in the reconstruction. One of the interesting features we observe is that,
depending on the choice of regularization, there is a trade-off between high
resolution and high contrast in the reconstructed images. Our proposed
non-linear optimization framework can be generalized to other hybrid imaging
modalities
Limited Angle Acousto-Electrical Tomography
This paper considers the reconstruction problem in Acousto-Electrical
Tomography, i.e., the problem of estimating a spatially varying conductivity in
a bounded domain from measurements of the internal power densities resulting
from different prescribed boundary conditions. Particular emphasis is placed on
the limited angle scenario, in which the boundary conditions are supported only
on a part of the boundary. The reconstruction problem is formulated as an
optimization problem in a Hilbert space setting and solved using Landweber
iteration. The resulting algorithm is implemented numerically in two spatial
dimensions and tested on simulated data. The results quantify the intuition
that features close to the measurement boundary are stably reconstructed and
features further away are less well reconstructed. Finally, the ill-posedness
of the limited angle problem is quantified numerically using the singular value
decomposition of the corresponding linearized problem.Comment: 23 page
Acousto-Electric Tomography with Total Variation Regularization
We study the numerical reconstruction problem in acousto-electric tomography
of recovering the conductivity distribution in a bounded domain from interior
power density data. We propose a numerical method for recovering discontinuous
conductivity distributions, by reformulating it as an optimization problem with
fitting and total variation penalty subject to PDE constraints. We
establish continuity and differentiability results for the forward map, the
well-posedness of the optimization problem, and present an easy-to-implement
and robust numerical method based on successive linearization, smoothing and
iterative reweighing. Extensive numerical experiments are presented to
illustrate the feasibility of the proposed approach.Comment: 22 pages, 30 figure
Mathematics and Algorithms in Tomography
This is the eighth Oberwolfach conference on the mathematics of tomography. Modalities represented at the workshop included X-ray tomography, sonar, radar, seismic imaging, ultrasound, electron microscopy, impedance imaging, photoacoustic tomography, elastography, vector tomography, and texture analysis