Contents:

Background

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

You can find more details on EIT research in the textbook I wrote with Jennifer Mueller:

.

The book is available for ordering at the SIAM bookstore.

For computational resources (Matlab files) related to the book, see this page.

## Background

## 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.

## 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 with t(k) as coefficient.

## Boundary corrected D-bar method

The above D-bar reconstructions assume that the conductivity is constant near

the boundary. This assumption is used in Nachman's original article for simplifying

the structure of the proof; More general cases are reduced to the "constant near the

boundary" situation by a special boundary correction technique:

1. Recover the trace and normal derivative of the conductivity from EIT data.

2. Continue the conductivity outside the domain so that it becomes one near the

boundary of a big disc.

3. Express EIT data at the boundary of the big disc in terms of the measured data.

4. Reconstruct on the big disc.

## Reconstructing discontinuous conductivities

A breakthrough article by Kari Astala and Lassi Päivärinta in 2006 shows that

a wide class of discontinuous conductivities is uniquely determined from infinite-

precision EIT data. Numerical reconstruction algorithms based on that result are

under construction by a five-member research team: Kari Astala, Jennifer Mueller,

Allan Perämäki, Lassi Päivärinta and myself.

## D-bar method with partial data

In practice it is often possible to cover the patient only partly by electrodes.

But the original D-bar method assumes measurements known on the whole boundary.

What can one do? Here is one possible answer:

.

The book is available for ordering at the SIAM bookstore.

For computational resources (Matlab files) related to the book, see this page.

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

background conductivity.

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 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 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).

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

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 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 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**,

Inverse Problems and Imaging

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**,

man for the 2-D inverse conductivity problem

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

results about the method, applying it to a bunch of new examples, and reporting

our findings in

from boundary measurements

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

Journal of Computational Physics

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

PDF (664 KB) Dynamic reconstructions from EIT data collected from a living

person are documented in

Cardiac Activity by the D-bar Method for Electrical Impedance Tomography.

Physiological Measurement

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

Piecewise Constant Conductivities by the D-bar Method for Electrical Impe-

dance Tomography.

the 1st Congress of the IPIA, Vancouver, 2007. Journal of Physics: Conference

Series

Here is a photo of Jennifer and me in 2006 at Colorado State University.

the boundary. This assumption is used in Nachman's original article for simplifying

the structure of the proof; More general cases are reduced to the "constant near the

boundary" situation by a special boundary correction technique:

1. Recover the trace and normal derivative of the conductivity from EIT data.

2. Continue the conductivity outside the domain so that it becomes one near the

boundary of a big disc.

3. Express EIT data at the boundary of the big disc in terms of the measured data.

4. Reconstruct on the big disc.

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)

a wide class of discontinuous conductivities is uniquely determined from infinite-

precision EIT data. Numerical reconstruction algorithms based on that result are

under construction by a five-member research team: Kari Astala, Jennifer Mueller,

Allan Perämäki, Lassi Päivärinta and myself.

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)

But the original D-bar method assumes measurements known on the whole boundary.

What can one do? Here is one possible answer:

Computational Experiments.

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

partial-boundary data.

available. A CT or MRI scan would give it, but there are cost, scheduling and

radiation dose issues related to that solution. Also, the shape of the patient

changes in time due to breathing and position.

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.

possible to derive a constructive D-bar method also in dimension three, at least

for smooth conductivities and infinite-precision data. However, compared to the

two-dimensional case there are many technical difficulties in modifying the method

for finite-precision data, arising especially from so-called exceptional points leading

to non-uniqueness of the exponentially behaving solutions.

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

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.

background conductivity (intact machine part or healthy tissue) and looks for

possible deviations from the background (air bubble or cancerous tumor), called

from the knowledge of electrical measurements at the boundary.

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

(We remark that simultaneously and independently, Martin Hanke and
Martin Brühl

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

It actually turned out that it is not easy to decide form EIT data whether
a cone

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 article**Ide, Isozaki, Nakata, Siltanen and Uhlmann**,*Probing for electrical
inclusions with complex spherical waves.*
Communications on Pure and Applied

Mathematics

from noisy EIT data in dimension two:

The hyperbolic probing approach was extended numerically to 3D in **Ide, Isozaki,
Nakata and Siltanen**,

impedance tomography.

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:

at the boundary

conductivity

whole conductivity inside the domain, but sometimes the restricted problem is

useful as an intermediate step. For example, the knowledge of the conductivity

at the boundary helps the design of a good initial guess for iterative EIT methods

such as regularized least squares. Also, the boundary corrected D-bar method

described above needs boundary reconstruction.

Numerical solution for the two-dimensional problem was described in **Nakamura,
Siltanen, Tanuma and Wang**,

from the localized Dirichlet to Neumann map,

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:

Research home page of Samuli Siltanen