Nuutti Hyvönen

Complete electrode model of electrical impedance tomography: Approximation properties and characterization of inclusions

In electrical impedance tomography one tries to recover the spatial admittance distribution inside a body from boundary measurements. In theoretical considerations it is usually assumed that the boundary data consists of the Neumann-to-Dirichlet map; when conducting real-world measurements, the obtainable data is a linear finite-dimensional operator mapping electrode currents onto electrode potentials. In this paper it is shown that when using the complete electrode model to handle electrode measurements, the corresponding current-to-voltage map can be seen as a discrete approximation of the traditional Neumann-to-Dirichlet operator. This approximating link is utilized further in the special case of constant background conductivity with inhomogeneities: It is demonstrated how inclusions with strictly higher or lower conductivities can be characterized by the limit behavior of the range of a boundary operator, determined through electrode measurements, when the electrodes get infinitely small and cover all...

The factorization method applied to the complete electrode model of impedance tomography

The factorization method is a tool for recovering inclusions inside a body when the Neumann-to-Dirichlet operator, which maps applied currents to measured voltages, is known. In practice this information is never at hand due to the discreteness and physical properties of the measurement devices. The complete electrode model of impedance tomography includes these physical characteristics but leads to a finite-dimensional data set, called the resistivity matrix. The main result of this work is an approximation link relating the resistivity matrix to the Neumann-to-Dirichlet operator in the

Factorization method and irregular inclusions in electrical impedance tomography

L^2

APPROXIMATING IDEALIZED BOUNDARY DATA OF ELECTRIC IMPEDANCE TOMOGRAPHY BY ELECTRODE MEASUREMENTS

-operator norm. This result allows us to extend the factorization method to the framework of real-life electrode measurements using a regularized series criterion which is easy to implement in practice. The truncation index of the sequence criterion, which represents the stopping index of the regularization scheme, can be computed solely from the measured, perturbed, and finite-dimensional data. The functionality of ...

Simultaneous recovery of admittivity and body shape in electrical impedance tomography: an experimental evaluation

In electrical impedance tomography, one tries to recover the conductivity inside a physical body from boundary measurements of current and voltage. In many practically important situations, the investigated object has known background conductivity but it is contaminated by inhomogeneities. The factorization method of Andreas Kirsch provides a tool for locating such inclusions. Earlier, it was shown that under suitable regularity conditions positive (or negative) inhomogeneities can be characterized by the factorization technique if the conductivity or one of its higher normal derivatives jumps on the boundaries of the inclusions. In this work, we use a monotonicity argument to generalize these results: we show that the factorization method provides a characterization of an open inclusion (modulo its boundary) if each point inside the inhomogeneity has an open neighbourhood where the perturbation of the conductivity is strictly positive (or negative) definite. In particular, we do not assume any regularity of the inclusion boundary or set any conditions on the behaviour of the perturbed conductivity at the inclusion boundary. Our theoretical findings are verified by two-dimensional numerical experiments.

Simultaneous Reconstruction of Outer Boundary Shape and Admittivity Distribution in Electrical Impedance Tomography

In electric impedance tomography, one tries to recover the spatial admittance distribution inside a body from boundary measurements of current and voltage. In theoretical considerations, it is usually assumed that the available data is the infinite-dimensional Neumann-to-Dirichlet map, i.e. one assumes to be able to use any boundary current and measure the corresponding potential everywhere on the object boundary. However, in practice, the data consists of a finite-dimensional operator mapping the electrode currents onto the corresponding electrode potentials. What is more, the measurements are affected by the contact impedance at the electrode-object interfaces. In this paper, we show that the introduction of a suitable nonorthogonal projection operator makes it possible to approximate the Neumann-to-Dirichlet map by its electrode counterpart in the topology of bounded linear operators on square integrable functions, with the discrepancy depending linearly on the distance between centers of adjacent electrodes. In particular, convergence is proved without assuming that the electrodes cover all of the object boundary. The theoretical results are complemented by two-dimensional numerical experiments.

Convex Source Support and Its Application to Electric Impedance Tomography

In this paper, the simultaneous retrieval of the exterior boundary shape and the interior admittivity distribution of an examined body in electrical impedance tomography is considered. The reconstruction method is built for the complete electrode model and it is based on the Frechet derivative of the corresponding current-to-voltage map with respect to the body shape. The reconstruction problem is cast into the Bayesian framework, and maximum a posteriori estimates for the admittivity and the boundary geometry are computed. The feasibility of the approach is evaluated by experimental data from water tank measurements. The results demonstrate that the proposed method has potential for handling an unknown body shape in a practical setting.

Characterizing inclusions in optical tomography

The aim of electrical impedance tomography is to reconstruct the admittivity distribution inside a physical body from boundary measurements of current and voltage. Due to the severe ill-posedness of the underlying inverse problem, the functionality of impedance tomography relies heavily on accurate modelling of the measurement geometry. In particular, almost all reconstruction algorithms require the precise shape of the imaged body as an input. In this work, the need for prior geometric information is relaxed by introducing a Newton-type output least squares algorithm that reconstructs the admittivity distribution and the object shape simultaneously. The method is built in the framework of the complete electrode model and is based on the Frechet derivative of the corresponding current-to-voltage map with respect to the object boundary shape. The functionality of the technique is demonstrated via numerical experiments with simulated measurement data.

Convex backscattering support in electric impedance tomography

The aim in electric impedance tomography is to recover the conductivity inside a physical body from boundary measurements of current and voltage. In many situations of practical importance, the investigated object has known background conductivity but is contaminated by inhomogeneities. In this work, we try to extract all possible information about the support of such inclusions inside a two-dimensional object from only one pair of measurements of impedance tomography. Our noniterative and computationally cheap method is based on the concept of the convex source support, which stems from earlier works of Kusiak, Sylvester, and the authors. The functionality of our algorithm is demonstrated by various numerical experiments.

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In optical tomography, one tries to determine the spatial absorption and scattering distributions inside a body by using measured pairs of inward and outward fluxes of near-infrared light on the object boundary. In many practically important situations, the scatter and the absorption inside the object are smooth apart from inclusions where at least one of the two optical parameters jumps to a higher or lower value. In this work, we investigate the possibility of characterizing these inhomogeneities in the framework of the diffusion approximation of the radiative transfer equation using the factorization method: for purely scattering inclusions, or if the scattering and absorption coefficients interplay in a correct way, the outcoming flux corresponding to a point source belongs to the range of an operator, determined through boundary measurements, if and only if the point source lies inside one of the inclusions.