Introduction to EIT
Regularized D-bar method for 2D EIT
Boundary corrected D-bar method
Reconstructing discontinuous conductivities
D-bar method with partial data
Recovering the shape of the domain from EIT data
D-bar method in dimension three
Detecting inclusions in conductivity
Recovering conductivity and its normal derivative at the boundary
This is a review of a large body of research work on electrical impedance
tomography (EIT) done in the period 1996-2013. A number of people are
involved in the research effort; their names appear below in the references
to scientific publications.
My interest in EIT goes back to the beginning of 1996 when I was a graduate
student at Helsinki University of Technology (now called Aalto University).
My PhD thesis topic was to study Adrian Nachman's seminal 1996 article on
Alberto Calderon's fundamental inverse conductivity problem, and to design
a practical reconstruction algorithm based on his proof. As explained below,
this goal has now been reached, but it was a much larger effort than just the
PhD thesis and required forming an international and multidisciplinary team.
During the last decade, I have contributed to several aspects of EIT research,
including various partial problems such as recovering inclusions in known
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 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
Applications of EIT include medical imaging (monitoring lung and heart func-
tion, detecting breast cancer at an early stage) underground prospecting (locating
water or oil reservoirs, assessing leaks) and industrial process monitoring (non-
invasive imaging of pipelines).
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 to be truncated to be 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. PDF (408 KB) Here are
some regularized reconstructions:
In the above image we see three D-bar reconstructions from simulated noisy
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 recon-
structions and their relative RMS error percentages.
Here is a picture of Matti, Kim and Samu working on the proof in 2006 at
Tampere University of Technology.
Samu, Matti and Jen in 2004:
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.
Let me present the series of studies behind the above regularization result. The
numerical implementation of Nachman's proof was the topic of my PhD thesis
Electrical Impedance Tomography and Faddeev's Green functions, Ann. Acad.
Sci. Fenn. Mathematica Dissertationes 121. PostScript (6.9 MB) I soon realized
that the task was too big to be accomplished by one person within a PhD project.
Luckily enough, my thesis supervisor Erkki Somersalo had active connections
to Rensselaer Polytechnic Institute (RPI), where he had been working together
with Margaret Cheney and David Isaacson. Jennifer Mueller was postdoccing
at the time at RPI and was interested in developing new EIT algorithms. An
international task force was formed, resulting in the article Siltanen, Mueller
and Isaacson, An implementation of the reconstruction algorithm of A. Nach-
man for the 2-D inverse conductivity problem, Inverse Problems 16 (2000),
pp. 681-699. PDF 845 KB
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
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.
Photo in 2000 at Colorado State University (Siltanen, Mueller, Isaacson):
Soon after publishing the first paper, we noticed a couple of errors in the proof
of Theorem 3.1. The theorem is true, however, and a correction was published
as Erratum, Inverse problems 17 (2001), pp. 1561-1563. PDF (2.6 MB)
Next we applied the method to high-contrast data. The promising results were
published in Siltanen, Mueller and Isaacson, Reconstruction of High Contrast
2-D Conductivities by the Algorithm of A. Nachman, Contemporary Mathematics
278 (2001), pp. 241-254. PDF 1.2 MB We continued by proving new theoretical
results about the method, applying it to a bunch of new examples, and reporting
our findings in Mueller and Siltanen, Direct reconstructions of conductivities
from boundary measurements, SIAM Journal of Scientific Computation 24 (2003),
pp. 1232-1266. PDF (617 KB) Also, we teamed up with Kim Knudsen from
Aalborg University, Denmark, and designed a new, faster solver for the D-bar
equation and explained its details in Knudsen, Mueller and Siltanen 2004,
Numerical solution method for the dbar-equation in the plane,
Journal of Computational Physics 198 (2004), pp. 500-517. PDF (311 KB)
The D-bar method was found to be useful for synthetic EIT data. But how about
real-world measurements? RPI had the ACT3 Adaptive Current Tomograph,
and we proceeded to collect data from phantoms and people. The results
suggested that the nonlinear D-bar method is capable of recovering higher
contrast deviations in the target conductivity.
The above image is from Isaacson, Mueller, Newell and Siltanen, Recon-
structions of chest phantoms by the d-bar method for electrical impedance
tomography, IEEE Transactions on Medical Imaging 23 (2004), pp. 821- 828.
PDF (664 KB) 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. PDF (1.4 MB).
Practical conductivity distributions are rarely smoothly varying; for instance
the boundaries between tissues in a human body typically correspond to jumps
in conductivity. The various theorems concerning the D-bar method assume at
least continuity of the conductivity, so what happens when the reconstruction
algorithm is applied to data from a discontinuous conductivity? This question is
examined in the papers
Knudsen, Lassas, Mueller and Siltanen,D-bar method for electrical impe-
dance tomography with discontinuous conductivities. SIAM Journal of Applied
Mathematics 67 (2007), pp. 893-913. PDF (327 KB)
Knudsen K, Lassas M, Mueller J L and Siltanen S 2008,Reconstructions of
Piecewise Constant Conductivities by the D-bar Method for Electrical Impe-
dance Tomography. Proceedings of the 4th AIP International Conference and
the 1st Congress of the IPIA, Vancouver, 2007. Journal of Physics: Conference
Series 124. PDF (241 KB)
Here is a photo of Jennifer and me in 2006 at Colorado State University.
The above reduction can be performed approximately by computer. Here is one
example. On the left: original conductivity. Middle: reconstruction assuming that
the conductivity is constant near the boundary (although this is not the case).
Right: boundary corrected reconstruction.
For more examples and details, see the submitted manuscript Siltanen and Tamminen,
Reconstructing conductivities with boundary corrected D-bar method.PDF (572 KB)
Here are a couple of very first examples of reconstructions based on the mu-Hilbert
transform and the transport matrix:
More details are available in
Astala K, Mueller J L, Paivarinta L, Peramaki A and Siltanen S 2011,
Direct electrical impedance tomography for nonsmooth conductivities.
Inverse Problems and Imaging 5(3), pp. 531-549. PDF (284 KB)
It turned 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
A solution for two-dimensional EIT is offered by Teichmüller space theory,
as shown in
Kolehmainen V, Lassas M, Ola P and Siltanen S 2013,
Recovering boundary shape and conductivity in electrical impedance tomography.
Inverse Problems and Imaging 7(1), pp. 217-242. PDF (1.9 MB)
Here is an example image. On the left there is the original conductivity.
On the right you see a reconstruction computed solely from the EIT data,
with no information whatsoever on the shape of the domain on the left.
Aspects of practical D-bar method for dimension three are discussed in the article
Cornean, Knudsen and Siltanen, Towards a d-bar reconstruction method for
three-dimensional EIT. Journal of Inverse and Ill-Posed Problems 14(2006),
pp. 111-134.PDF (231 KB)
Here is a picture taken at Aalborg University in 2001.
Left: Horia Cornean, right: Kim Knudsen.
Recently, numerical reconstruction results have appeared as well, due to Jutta
Bikowski, Fabrice Delbary, Per Christian Hansen, Kim Knudsen, Jennifer Mueller
and the team of David Isaacson.
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The work was started in 1999 when I visited Professor Masaru Ikehata in Gunma
University, Japan. He explained to me his enclosure method that reveals the convex
hull of a set of inclusions. We decided to design together a numerical implementation
of his theoretical method, resulting in the publication Ikehata and Siltanen,
Numerical method for finding the convex hull of an inclusion in conductivity from
boundary measurements, Inverse Problems 16(2000), pp. 1043-1052. PDF 259 KB
(We remark that simultaneously and independently, Martin Hanke and
implemented the enclosure method numerically as well.)
Here are a couple of simple reconstructions from the above paper:
In the enclosure method one chooses a half-plane and uses the boundary data
to decide whether the inclusion intersects the half-plane. It is also possible to use
cones for probing for inclusions, so that more than just the convex hull of inclusions
can be recovered. Such method, based on the Mittag-Leffler function, is described
in the article Ikehata and Siltanen,Electrical impedance tomography and Mittag-
Leffler's function, Inverse Problems 20(2004), pp. 1325-1348. PDF (476 KB)
It actually turned out that it is not easy to decide form EIT data whether
intersects an inclusion or not. However, we came up with a computational ad hoc
strategy that gives useful results. Here is a sample reconstruction of three inclusions
from noisy EIT data:
Here is a picture taken during my visit to Gunma University in 2006. Ikehata
is on the left.
Instead of using cones, it is possible to use hyperbolic transformation
to derive a
probing method based on discs. This is described for dimensions two and three in
the articleIde, Isozaki, Nakata, Siltanen and Uhlmann,Probing for electrical
inclusions with complex spherical waves. Communications on Pure and Applied
Mathematics 60(2007), pp. 1415-1442. PDF (351 KB) Here is a reconstruction
from noisy EIT data in dimension two:
The hyperbolic probing approach was extended numerically to 3D in Ide, Isozaki,
Nakata and Siltanen, Local detection of three-dimensional inclusions in electrical
impedance tomography. Inverse Problems 26 (2010), 035001. PDF (500 KB)
Here is a reconstruction from noisy EIT data in dimension three. Left image shows
the best possible recovery using infinite-precision data, and the right image shows
reconstruction from noisy EIT data.
Here are Nakata, Ide and Isozaki at the excellent Futomaru sushi restaurant in
Tsukuba, Japan, in 2008.
This is me with Gunther Uhlmann in Seattle in 2008:
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Numerical solution for the two-dimensional problem was described in Nakamura,
Siltanen, Tanuma and Wang,Numerical recovery of conductivity at the boundary
from the localized Dirichlet to Neumann map, Computing 75(2005), pp. 197-213.
PDF (400 KB). Here are reconstructed trace (top) and normal derivative (bottom)
using 48 electrodes:
The work has been generalized to dimension three as well and reported in the
article Nakamura, Ronkanen, Siltanen and Tanuma, Recovering conductivity
at the boundary in three-dimensional electrical impedance tomography,
PDF (1 MB), to appear in Inverse Problems and Imaging.
Here is a picture of our team in 2007:
In the above paper, we introduce a calibration step allowing approximate
recovery of the trace and the normal derivative in dimension three. Here original
(top) and recovered (bottom) trace on the lateral boundary of a 3D cylinder:
Here original (top) and recovered (bottom) normal derivative:
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