The aim of electrical impedance tomography is to form an image of the conductivity distribution inside an unknown body using electric boundary measurements. The computation of the image from measurement data is a non-linear ill-posed inverse problem and calls for a special regularized algorithm. One such algorithm, the so-called D-bar method, is improved in this work by introducing new computational steps that remove the so far necessary requirement that the conductivity should be constant near the boundary. The numerical experiments presented suggest two conclusions. First, for most conductivities arising in medical imaging, it seems the previous approach of using a best possible constant near the boundary is sufficient. Second, for conductivities that have high contrast features at the boundary, the new approach produces reconstructions with smaller quantitative error and with better visual quality.
International audienceThis work deals with an inverse boundary value problem arising from the equation of heat conduction. Mathematical theory and algorithm is described in dimensions 1–3 for probing the discontinuous part of the conductivity from local temperature and heat flow measurements at the boundary. The approach is based on the use of complex spherical waves, and no knowledge is needed about the initial temperature distribution. In dimension two we show how conformal transformations can be used for probing deeper than is possible with discs. Results from numerical experiments in the one-dimensional case are reported, suggesting that the method is capable of recovering locations of discontinuities approximately from noisy data
We present a few ways of using conformal maps in the reconstruction of two-dimensional conductivities in electrical impedance tomography. First, by utilizing the Riemann mapping theorem, we can transform any simply connected domain of interest to the unit disk where the D-bar method can be implemented most efficiently. In particular, this applies to the open upper half-plane. Second, in the unit disk we may choose a region of interest that is magnified using a suitable Möbius transform. To facilitate the efficient use of conformal maps, we introduce input current patterns that are named conformally transformed truncated Fourier basis; in practice, their use corresponds to positioning the available electrodes close to the region of interest. These ideas are numerically tested using simulated continuum data in bounded domains and simulated point electrode data in the half-plane. The connections to practical electrode measurements are also discussed.2010 Mathematics Subject Classification. 65N21, 35R30, 65N15.
Abstract. We consider an inverse boundary value problem for the heat equation ∂tu = div (γ∇xu) in (0, T ) × Ω, u = f on (0, T ) × ∂Ω, u t=0 = u 0 , in a bounded domain Ω ⊂ R n , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time :. Fix a direction e * ∈ S n−1 arbitrarily. Assuming that ∂D(t) is strictly convex for 0 ≤ t ≤ T , we show that k and sup{e * ·x ; x ∈ D(t)} (0 ≤ t ≤ T ), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ∂ν u(t, x) (0,T )×∂Ω . The knowledge of the initial data u 0 is not used in the proof. If we know min 0≤t≤T sup x∈D(t) x · e * , we have the same conclusion from the local Dirichlet-to-Neumann map. The results have applications to nondestructive testing. Consider a physical body consisiting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking.
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