Special Functions and Their Applications. By N. N. Lebedev, translated by R. A. Silverman. Pp. xii, 308. 96s. 1965. (Prentice-Hall)

We survey mathematical developments in the inverse method of electrical impedance tomography which consists in determining the electrical properties of a medium by making voltage and current measurements at the boundary of the medium. In the mathematical literature, this is also known as Calderon's problem from Calderon's pioneer contribution (Calderon 1980 Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980) p 65 (Soc. Brasil. Mat.)). We concentrate this review around the topic of complex geometrical optics solutions that have led to many advances in the field. In the last section, we review some counterexamples to Calderon's problems that have attracted a lot of interest because of connections with cloaking and invisibility.

http://www.dim.uchile.cl/~axosses/calderoniprevised.pdf

Electrical impedance tomography and Calderón's problem

A strategy for regularizing the inversion procedure for the two-dimensional D-bar reconstruction algorithm
based on the global uniqueness proof of Nachman [Ann. Math. 143 (1996)] for the ill-posed inverse conductivity problem is presented. The strategy utilizes truncation of the boundary integral equation and the scattering transform. It is shown that this leads to a bound on the error in the scattering transform and a stable reconstruction of the conductivity; an explicit rate of convergence in appropriate Banach spaces is derived as well. Numerical results are also included, demonstrating the convergence of the reconstructed conductivity to the true conductivity as the noise level tends to zero. The results provide a link between two traditions of inverse problems research: theory of regularization and inversion methods based on complex geometrical optics. Also, the procedure is a novel regularized imaging method for electrical impedance tomography.

Regularized d-bar method for the inverse conductivity problem

This survey article deals mainly with two inverse problems and the relation between them. The first inverse problem we consider is whether one can determine the electrical conductivity of a medium by making voltage and current measurements at the boundary. This is called electrical impedance tomography and also Calderon’s problem since the famous analyst proposed it in the mathematical literature (Calderon in On an inverse boundary value problem. Seminar on numerical analysis and its applications to continuum physics (Rio de Janeiro, 1980), Soc Brasil Mat. Rio de Janeiro, pp. 65–73, 1980). The second is on travel time tomography. The question is whether one can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as a geometry problem, the boundary rigidity problem. Can we determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points? These two inverse problems concern visibility, that is whether we can determine the internal properties of a medium by making measurements at the boundary. The last topic of this paper considers the opposite issue: invisibility: Can one make objects invisible to different types of waves, including light?

/pdf/inverse-problems-seeing-the-unseen-48aitkxoj9.pdf

Inverse problems: seeing the unseen

Current-voltage measurements in electrical impedance tomography (EIT) can be partially ordered with respect to definiteness of the associated self-adjoint Neumann-to-Dirichlet operators. With this ordering, a pointwise larger conductivity leads to smaller current-voltage measurements, and smaller conductivities lead to larger measurements. We present a converse of this simple monotonicity relation and use it to solve the shape reconstruction (a.k.a. inclusion detection) problem in EIT. The outer shape of a region where the conductivity differs from a known background conductivity can be found by simply comparing the measurements to that of smaller or larger test regions.

Monotonicity-based shape reconstruction in electrical impedance tomography ∗

Let a physical body Ω in ℝ2 or ℝ3 be given. Assume that the electric conductivity distribution inside Ω consists of conductive inclusions in a known smooth background. Further, assume that a subset Γ ⊂ ∂Ω is available for boundary measurements. It is proved using hyperbolic geometry that certain information about the location of the inclusions can be exactly recovered from static electric measurements on Γ. More precisely: given a ball B with center outside the convex hull of Ω and satisfying (B ∩ ∂Ω) ⊂ Γ, boundary measurements on Γ with explicitly given Dirichlet data are enough to determine whether B intersects the inclusion. An approximate detection algorithm is introduced based on the theory. Numerical experiments in dimension two with simulated noisy data suggest that the algorithm finds the inclusion-free domain near Γ and is robust against measurement noise. © 2007 Wiley Periodicals, Inc.

/pdf/probing-for-electrical-inclusions-with-complex-spherical-2ik18m9rdw.pdf

Probing for electrical inclusions with complex spherical waves

Particle-Based Transparent Rendering of Implicit Surfaces and its Application to Fused Visualization

Assume one is given a three-dimensional bounded domain with an unknown conductivity distribution inside. Further, suppose that the conductivity consists of a known background and unknown anomalous regions (inclusions) where conductivity values are unknown and different from the background. A method is introduced in Ide et al (2007 Commun. Pure Appl. Math. 60 1415–42) for locating inclusions approximately from noisy localized voltage-to-current measurements performed at the boundary of the body. The method is based on the use of complex geometrical optics solutions and hyperbolic geometry; numerical testing is presented in the aforementioned paper for the two-dimensional case. This work reports the results of computational implementation of the method in dimension three, where both the simulation of data and the computerized inversion algorithm are more complicated than in dimension two. Three new regularizing steps are added to the algorithm, resulting in significantly better robustness against noise. Numerical experiments are reported, suggesting that the approximate location of the inclusions can be reliably recovered from the data with a realistic level of measurement noise. Potential applications of the results include early diagnosis of breast cancer, underground contaminant detection and nondestructive testing.

/pdf/local-detection-of-three-dimensional-inclusions-in-3fd9tdn6u8.pdf

Local detection of three-dimensional inclusions in electrical impedance tomography

This paper explores a method for creating large-scale urban 3D models using Historical GIS data. The method is capable of automatically generating realistic VR models based on GIS data at a low cost. 3D models of houses are created from polygon data, fences from line data, and pedestrians and trees from point data. The method is applied to the Virtual Kyoto Project in which the landscape of the whole city of Kyoto of the early Edo era (ca 17C) is reconstructed.

Reconstruction of Kyoto of the Edo era based on arts and historical documents: 3D urban model based on historical GIS data

The large-scale simulation of the electromagnetic wave propagation using meshless time domain method (MTDM) is numerically investigated. Moreover, compute unified device architecture (CUDA) and OpenMP is adopted for parallelization technique to reduce the computation time. The results of computation show that the execution time of the time evolution calculation on GPU is 8.8 time faster than that of CPU. In addition, the execution time of the shape function generation procedure can be speedup about 7842 times by proposed scheme and OpenMP.

Susumu Nakata

Papers

Probing for electrical inclusions with complex spherical waves

Particle-Based Transparent Rendering of Implicit Surfaces and its Application to Fused Visualization

Local detection of three-dimensional inclusions in electrical impedance tomography

Reconstruction of Kyoto of the Edo era based on arts and historical documents: 3D urban model based on historical GIS data

Large-Scale Simulation of Electromagnetic Wave Propagation Using Meshless Time Domain Method With Parallel Processing