773 research outputs found
An Analysis of Finite Element Approximation in Electrical Impedance Tomography
We present a finite element analysis of electrical impedance tomography for
reconstructing the conductivity distribution from electrode voltage
measurements by means of Tikhonov regularization. Two popular choices of the
penalty term, i.e., -norm smoothness penalty and total variation
seminorm penalty, are considered. A piecewise linear finite element method is
employed for discretizing the forward model, i.e., the complete electrode
model, the conductivity, and the penalty functional. The convergence of the
finite element approximations for the Tikhonov model on both polyhedral and
smooth curved domains is established. This provides rigorous justifications for
the ad hoc discretization procedures in the literature.Comment: 20 page
Adaptive Reconstruction for Electrical Impedance Tomography with a Piecewise Constant Conductivity
In this work we propose and analyze a numerical method for electrical
impedance tomography of recovering a piecewise constant conductivity from
boundary voltage measurements. It is based on standard Tikhonov regularization
with a Modica-Mortola penalty functional and adaptive mesh refinement using
suitable a posteriori error estimators of residual type that involve the state,
adjoint and variational inequality in the necessary optimality condition and a
separate marking strategy. We prove the convergence of the adaptive algorithm
in the following sense: the sequence of discrete solutions contains a
subsequence convergent to a solution of the continuous necessary optimality
system. Several numerical examples are presented to illustrate the convergence
behavior of the algorithm.Comment: 26 pages, 12 figure
Reconstruction of a piecewise constant conductivity on a polygonal partition via shape optimization in EIT
In this paper, we develop a shape optimization-based algorithm for the
electrical impedance tomography (EIT) problem of determining a piecewise
constant conductivity on a polygonal partition from boundary measurements. The
key tool is to use a distributed shape derivative of a suitable cost functional
with respect to movements of the partition. Numerical simulations showing the
robustness and accuracy of the method are presented for simulated test cases in
two dimensions
Parametric Level Set Methods for Inverse Problems
In this paper, a parametric level set method for reconstruction of obstacles
in general inverse problems is considered. General evolution equations for the
reconstruction of unknown obstacles are derived in terms of the underlying
level set parameters. We show that using the appropriate form of parameterizing
the level set function results a significantly lower dimensional problem, which
bypasses many difficulties with traditional level set methods, such as
regularization, re-initialization and use of signed distance function.
Moreover, we show that from a computational point of view, low order
representation of the problem paves the path for easier use of Newton and
quasi-Newton methods. Specifically for the purposes of this paper, we
parameterize the level set function in terms of adaptive compactly supported
radial basis functions, which used in the proposed manner provides flexibility
in presenting a larger class of shapes with fewer terms. Also they provide a
"narrow-banding" advantage which can further reduce the number of active
unknowns at each step of the evolution. The performance of the proposed
approach is examined in three examples of inverse problems, i.e., electrical
resistance tomography, X-ray computed tomography and diffuse optical
tomography
Electrical Impedance Tomography: A Fair Comparative Study on Deep Learning and Analytic-based Approaches
Electrical Impedance Tomography (EIT) is a powerful imaging technique with
diverse applications, e.g., medical diagnosis, industrial monitoring, and
environmental studies. The EIT inverse problem is about inferring the internal
conductivity distribution of an object from measurements taken on its boundary.
It is severely ill-posed, necessitating advanced computational methods for
accurate image reconstructions. Recent years have witnessed significant
progress, driven by innovations in analytic-based approaches and deep learning.
This review explores techniques for solving the EIT inverse problem, focusing
on the interplay between contemporary deep learning-based strategies and
classical analytic-based methods. Four state-of-the-art deep learning
algorithms are rigorously examined, harnessing the representational
capabilities of deep neural networks to reconstruct intricate conductivity
distributions. In parallel, two analytic-based methods, rooted in mathematical
formulations and regularisation techniques, are dissected for their strengths
and limitations. These methodologies are evaluated through various numerical
experiments, encompassing diverse scenarios that reflect real-world
complexities. A suite of performance metrics is employed to assess the efficacy
of these methods. These metrics collectively provide a nuanced understanding of
the methods' ability to capture essential features and delineate complex
conductivity patterns. One novel feature of the study is the incorporation of
variable conductivity scenarios, introducing a level of heterogeneity that
mimics textured inclusions. This departure from uniform conductivity
assumptions mimics realistic scenarios where tissues or materials exhibit
spatially varying electrical properties. Exploring how each method responds to
such variable conductivity scenarios opens avenues for understanding their
robustness and adaptability
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