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Research highlight of Finnish Centre of Excellence in Inverse Problems Research
Electrical Impedance Tomography:
- Introduction to EIT
- Regularized D-bar method for 2D EIT
- Partial Data EIT
- Recovering nonsmooth conductivities in 2D EIT
- Anisotropic EIT
- Backscattering in electrical impedance tomography
- EIT imaging of concrete structures
- Conference Presentations
- Visibility in mainstream media
Introduction to EIT
In electrical impedance tomography (EIT) an unknown physical body is probed with electric currents with the goal of revealing the inner structure of the body.
Typically, a number of electrodes is attached to the surface of the body, and electric currents are fed into the body through those electrodes. In chest imaging the electrodes might be placed like this:
The voltage potentials caused by the currents are measured at the electrodes, and this data is used to estimate the values of electric conductivity (and permittivity) in a grid of points inside the body. The result is an image of the inner structure of the body, such as above on the right.
The problem of reconstructing an image from EIT data is a nonlinear problem highly sensitive to measurement noise. Because of the sensitivity to errors in data, specially regularized methods are needed for producing meaningful EIT images.
Applications of EIT include medical imaging (monitoring lung and heart function, detecting breast cancer at an early stage) underground prospecting (locating water or oil reservoirs, assessing leaks), industrial process monitoring (non-invasive imaging of pipelines) and non-destructive testing (looking for cracks inside concrete).
Regularized D-bar method for 2D EIT
In 1996, Adrian Nachman published a breakthrough article, where he proved that infinite-precision EIT data uniquely determines a twice differentiable conductivity in a two-dimensional domain. His proof is constructive, based on the use of an intermediate object called scattering transform t(k); here k is a complex variable and t is a complex-valued function of k. The transform t(k) is first determined from EIT data via an integral equation, and the second step involves solving a D-bar equation in the auxiliary variable k, with t(k) as coefficient.
Nachman's proof can be equipped with a natural regularization step, enabling EIT imaging from finite-precision data using the D-bar approach. Namely, the scattering transform needs is set to zero outside a disc of certain radius R. Using the truncated transform as a coefficient in the D-bar equation yields a noise-robust EIT algorithm. The radius can be expressed analytically in terms of the noise level, as shown in the article Knudsen, Lassas, Mueller and Siltanen, "Regularized D-bar method for the inverse conductivity problem", Inverse Problems and Imaging 3 (2009), pp. 599-624.
Here are some regularized reconstructions for various levels of noise:
In the above image we see three D-bar reconstructions from simulated noisy EIT data calculated from a synthetic phantom. The noise levels involved are from left to right: 0.0001% corresponding to the accuracy of our Finite Element computation of the simulated measurement data, 0.01% corresponding to the accuracy of the ACT3 Impedance Imager of Rensselaer Polytechnic Institute, and 1% corresponding to the accuracy of less sophisticated EIT instruments. The three numbers in red show the truncation radius used in the nonlinear low- pass filtering of the scattering transform. The noisier data we have, the smaller radius we must choose. The bottom row of the above image shows the reconstructions and their relative RMS error percentages.
The above is the first result providing a full nonlinear regularization analysis for a global PDE coefficient reconstruction method. The work combines two traditions of inverse problems research: the school of regularization and the school of partial differential equation based analysis.
The first numerical implementation of Nachman's proof was described in the article Siltanen, Mueller and Isaacson, "An implementation of the reconstruction algorithm of A. Nachman for the 2-D inverse conductivity problem", Inverse Problems 16 (2000), pp. 681-699. See also Erratum, Inverse problems 17 (2001), pp. 1561-1563.
In the above paper, we present a numerical method where Nachman's first (ill-posed) step is simplified using a Born approximation and the D-bar equation of the second step is solved with truncated scattering data. Although we simulate EIT data only from rotationally symmetric conductivities (to reduce computational effort), the reconstruction algorithm is not restricted to such cases. This the first numerical inversion method based on complex geometrical optics solutions.
How about real-world measurements? We collected data from agar phantoms as well as human patients using the ACT3 Adaptive Current Tomograph at Rensselaer Polytechnic Institute. The results suggest that the nonlinear D-bar method is capable of recovering higher contrast deviations in the target conductivity than methods based on linearization.
The above image is from Isaacson, Mueller, Newell and Siltanen, "Reconstructions of chest phantoms by the d-bar method for electrical impedance tomography", IEEE Transactions on Medical Imaging 23 (2004), pp. 821- 828. Dynamic reconstructions from EIT data collected from a living person are documented in Isaacson, Mueller, Newell and Siltanen, "Imaging Cardiac Activity by the D-bar Method for Electrical Impedance Tomography". Physiological Measurement 27 (2006), pp. S43-S50.
Recovering nonsmooth conductivities in 2D EIT
Practical conductivity distributions are rarely smoothly varying; for instance the boundaries between tissues in a human body typically correspond to jumps in conductivity. The class of conductivities originally considered by Calderon in 1980 was the space of essentially bounded (and strictly positive) functions. This problem was solved in the breakthrough article Astala and Päivärinta, Calderon's inverse conductivity problem in the plane, Ann. of Math. 163 (2006), pp. 265-299, in the affirmative: infinite-precision EIT measurements at the boundary do determine an essentially bounded conductivity uniquely.
Numerical reconstruction algorithms based on the above result are under construction. Partial results are available in the article Astala, Mueller, Päivärinta and Siltanen, Numerical computation of complex geometrical optics solutions to the conductivity equation, to appear in Applied and Computational Harmonic Analysis.
Partial Data EIT
Hamilton S and Siltanen S, Nonlinear Inversion from Partial EIT Data:
Computational Experiments 2013. (to appear) Arxiv PDF (516 KB)
It turns out that the boundary integral equation, which is the starting point of any D-bar style algorithm, can be solved in a basis of localized functions (Haar
wavelets in our case).
Here is a result.
Left: original conductivity,
middle: D-bar reconstruction from full-boundary data,
right: D-bar reconstruction from partial-boundary data.
Add work by Astala, Lassas, Päivärinta, as well as future work.
Backscattering in electrical impedance tomography
Suppose that the measurement probe used for gathering EIT data is (infinitesimally) small, consists of two electrodes and can be moved along the object boundary. Under such circumstances, the available data is of backscattering nature. In 1 it is shown that in two dimensions such measurements uniquely determine an insulating simply connected cavity in constant background. Moreover, it is possible to use the backscatter data of EIT to localize more general compactly supported inclusions by reconstructing the so-called convex backscattering support, which is a nonempty subset of the inhomogeneity under only mild assumptions on the measurements 2.
1. M. Hanke, N. Hyvönen, and S. Reusswig, An inverse backscatter problem for electric impedance tomography, SIAM Journal on Mathematical Analysis, 41, 1948-1966 (2009).
2. M. Hanke, N. Hyvönen, and S. Reusswig, Convex backscattering support in electric impedance tomography, accepted to Numerische Mathematik.
EIT imaging of concrete structures
Concrete is distinctly the most extensively used construction material in the world - twice as much concrete is used in construction around the world than the total of all other building materials, including wood, steel, plastic and aluminum. During the life-cycle of concrete, there are several situations necessitating measurement and analysis of the structural integrity and other properties of concrete structures. Many of the existing methods for assessment of concrete conditions are destructive in nature. Non-destructive modalities exist but most of them are inaccurate and prone to uncertainties of the models.
The aims of non-destructive testing (NDT) include:
- Localization of reinforcing bars
- Detection of rebar corrosion rate
- Crack evalution
- Evaluation of concrete cover-zone properties (chlorides, moisture, etc)
In this project, we apply Electrical Impedance Tomography (EIT) to non-destructive testing of concrete. In concrete, metals (reinforcing bars and fibers), cracks, air voids, moisture and chlorides carry contrast with respect to conductivity, and hence EIT might serve as a tool to get information of those properties/structures. Advantages of EIT are that it is a relatively inexpensive, quick and safe modality.
Left: Measurement setup in the case of a cylindrical specimen. Right: Measurements of a concrete slab.
Left: Photo of a cracked concrete specimen. Right: EIT reconstruction. The colors represent the electrical conductivity, indicating that the conductivity of the crack is close to zero.
Reconstructed conductivity distribution in a concrete slab. Blue color indicates a large delamination inside the slab.
K. Karhunen, A. Seppänen, A. Lehikoinen, P.J.M. Monteiro, J.P. Kaipio: "Electrical resistance tomography imaging of concrete", Cement and Concrete Research,40: 137-145,doi:10.1016/j.cemconres.2009.08.023,2010.
K. Karhunen, A. Seppänen, A. Lehikoinen, J. Blunt,J.P. Kaipio, P.J.M. Monteiro: "Electrical Resistance Tomography for Assessment of Cracks in Concrete", ACI Materials Journal 2010 (In Review).
Mueller J L and Siltanen S,
Linear and Nonlinear Inverse Problems with Practical Applications. SIAM 2012.
The book is available for ordering at the SIAM bookstore.
For computational resources (Matlab files) related to the book, see this page.
Visibility in mainstream media