Aalborg Universitet
D-bar method for electrical impedance tomography with discontinuous conductivities
Knudsen, Kim; Lassas, Matti; Mueller, Jennifer L.; Siltanen, Samuli
Publication date:
2006
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Citation for published version (APA):
Knudsen, K., Lassas, M., Mueller, J. L., & Siltanen, S. (2006). D-bar method for electrical impedance
tomography with discontinuous conductivities. Department of Mathematical Sciences, Aalborg University.
Research Report Series No. R-2006-18
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AALBORG UNIVERSITY
'
&
$
%
D-bar method for electrical impedance tomography with
discontinuous conductivities
by
Kim Knudsen, Matti Lassas, Jennifer I. Mueller and Samuli
Siltanen
R-2006-18
May 2006
Department of Mathematical Sciences
Aalborg University
Fredrik Bajers Vej 7 G DK
-
9220 Aalborg Øst Denmark
Phone:
+
45 96 35 80 80 Telefax:
+
45 98 15 81 29
URL: http://www.math.aau.dk
e
ISSN 1399–2503 On-line version ISSN 1601–7811
D-BAR METHOD FOR ELECTRICAL IMPEDANCE TOMOGRAPHY
WITH DISCONTINUOUS CONDUCTIVITIES
KIM KNUDSEN
∗
, MATTI LASSAS
†
, JENNIFER L. MUELLER
‡
, AND SAMULI SILTANEN
§
Abstract. The effects of truncating the (approximate) s cattering transform in the D-bar re-
construction method for 2-D electrical impedance tomography are studied. The method is based on
Nachman’s uniqueness proof [Ann. of Math. 143 (1996)] that applies to twice differentiable conduc-
tivities. However, the reconstruction algorithm has been successfully applied to experimental data,
which can be characterized as piecewise smooth conductivities. The truncation is shown to stabilize
the method against measurement noise and to have a smoothing effect on the reconstructed conduc-
tivity. Thus the truncation can be interpreted as regularization of the D-bar method. Numerical
reconstructions are presented demonstrating that features of discontinuous high contrast conduc-
tivities can be recovered using the D-bar method. Further, a new connection between Calder´on’s
linearization method and the D-bar method is established, and the two methods are compared nu-
merically and analytically.
Key words. inverse conductivity problem, electrical impedance tomography, exponentially
growing solution, Faddeev’s Green’s function
Abbreviated title: D-ba r method for electrical impedance tomography
1. Introduction. The 2-D inverse conductivity problem is to determine and
reconstruct an unknown conductivity distribution γ in a n open, bounded and smooth
domain Ω ⊂ R
2
from voltage-to-current measurements on the boundary ∂Ω. We
assume that there is a C > 0 such that
C
−1
< γ(x) < C, x ∈ Ω. (1)
The boundary meas urements are modeled by the Dirichlet-to-Neumann (DN) map
Λ
γ
f = γ
∂u
∂ν
|
∂Ω
,
where u is the solution to the generalized Laplace’s equation
∇ ·γ∇u = 0 in Ω, u|
∂Ω
= f. (2)
Mathematically, the problem is to show that the map γ 7→ Λ
γ
is injective and find
an algorithm for the inversion of the map. Physically, u is the electric potential in
Ω, and Λ
γ
represents knowledge of the current flux through ∂Ω resulting from the
voltage distribution f applied on ∂Ω.
The inverse conductivity problem has applications in subsurface flow monitoring
and remediation [29, 30], underground contaminant detection [10, 1 7], geophysics
[9, 26], nondestructive evalua tion [11 , 3 3, 36, 34], and a medical imaging technique
known as electrical impedance tomography (EIT) (see [8, 5] for a review article on
EIT). Conductivity dis tributions appearing in applications are typically piecewise
continuous. T his is the case for example in medical EIT, since various tissues in the
body have different conductivities, and there are discontinuities at organ boundaries.
∗
Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220
Aalborg Ø, Denmark
†
Department of mathematics, P.O. Box 1100, 02015 Helsinki University of Technology, Finland
‡
Colorado State University, Fort Collins, CO 80523, USA
§
Institute of Mathematics, Tampere U niversity of Technology, P.O. Box 553, 33101 Tampere,
Finland
1
Let us briefly outline the history of D-bar solution methods for EIT. Recently,
Astala and P¨aiv¨arinta showed [1 ] that knowledge of the DN map uniquely determines
the conductivity γ(x) ∈ L
∞
(Ω), 0 < c ≤ γ. This r esult has been generalized also
for anisotropic co nductivities in [2]. In this work we will refer to the 2-D uniquenes s
result by Nachman [24] for γ ∈ W
2,p
(Ω), p > 1 and by Brown and Uhlmann [6] for
γ ∈ W
1,p
(Ω), p > 2. The proof in [24] is constructive; that is, it outlines a direct
method for re c onstructing the conductivity γ from knowledge of Λ
γ
. This method was
realized as a numerical algorithm for C
2
conductivities in [28, 23, 15]. The uniqueness
result of Brown and Uhlmann in [6] was formulated a s a reco ns truction algorithm in
[20], which has bee n implemented in [18, 19]. There are many similarities between
the two methods. In fact it was s hown in [18] using the Brown-Uhlmann approach
that the reconstruction method of Nachman’s [24] can be extended to the class of
conductivities γ ∈ W
1+ǫ,p
(Ω), p > 2, ǫ > 0. We refer to [23, 5, 35] for discussions of
uniqueness results for γ in other spaces and Ω ⊂ R
n
, n ≥ 2.
Nachman’s D-bar approach in [24] is based on the evalua tion of the scattering
transform t(k) by the formula
t(k) =
Z
∂Ω
e
ik¯x
(Λ
γ
− Λ
1
)ψ(·, k)dσ(x), k ∈ C, x = x
1
+ ix
2
, (3)
where Λ
1
denotes the DN map cor responding to the homogeneous conductivity 1.
Then γ can be recovered by solving a D-bar equation containing t(k). The functions
ψ(·, k) in (3) are traces of certain exponentially growing solutions to (2), i.e. solutions
that behave like e
ikx
asymptotically as either |x| or |k| tends to infinity. These traces
can in principle be found by solving a particular boundary integ ral equation. However,
as solving such an equation is quite sensitive to measurement noise, the following
approximation to t(k) wa s introduced in [28]:
t
exp
(k) =
Z
∂Ω
e
i
k¯x
(Λ
γ
− Λ
1
)e
ikx
dσ(x). (4)
For smooth high-contrast conductivities, approximating t by truncated t
exp
yields
good reconstructions, see [2 8, 23]. Truncation is necessary for stabilizing the method
against measurement noise.
Formula (4) allows the evaluation of t
exp
(k) for L
∞
conductivities, and the D-
bar method is found to be effective even when the conductivity does not sa tisfy the
assumptions of the original reconstruction theorem. In [15], quite accurate reconstr uc-
tions are computed from experimental data collected on a phantom chest consisting
of agar heart and lungs in a saline-filled tank. They are the first reconstructions
using the D-bar method on a disco ntinuous conductivity and on measure d data. In
[16] the D-bar algorithm with a differencing t
exp
approximation is used to reconstruct
conductivity changes in a human chest, particularly pulmonary perfusion.
Our aim is to better understand the reconstruction o f realistic co nductivities from
noisy EIT data using the D-bar method by studying its application to piecewise
smooth conductivities. Section 2 gives necessary background on the method and
its variants. In Section 3 we prove that reconstructions from any truncated scat-
tering data are smooth. In Section 4 we show that the reconstructions from noisy
data using truncated t
exp
are stable. We remark that previous work [22, 3] shows
that the exact reconstructio n algorithm is stable in a restricted sens e, i.e. as a map
defined on the range of the forward oper ator Λ : γ 7→ Λ
γ
. In contrast, we show
that the approximate reconstruction is continuously defined on the entire data space
2
L(H
1/2
(∂Ω), H
−1/2
(∂Ω)). As an application of the stability we consider in Section 5
mollified versions γ
λ
of a piecewise continuous conductivity distribution γ, and show
that reconstructions of γ
λ
converg e to reconstructions of γ as λ → 0. This means that
no systematic artifacts are introduced when the reconstruction method is applied to
conductivities outside the assumptions of the theory.
In Section 6 a connection between the linearization method of Calder´on [7] and
the D-bar method is established. Calder´on’s method is written in terms of t
exp
and is
revealed to be a low-order approximation to the D-bar method. The simple example
of the unit disk containing one concentric ring of constant conductivity with a dis-
continuity at the interface is studied in depth in Section 7. We write t
exp
as a ser ies
showing the asymptotic growth rate. Reco nstructions by Calder´on’s method and the
D-bar method with the t
exp
approximation are expressed in explicit formulas.
In Section 8 we illustrate our theoretical findings by numerical examples. We
find that both the D-bar method and Calder´on’s method can approximately recover
the location of a discontinuity. Also, both metho ds yield good reconstructions of
low-contrast conductivities, but have difficulties in recovering the actual conductivity
values in the presence of high contrast features near the boundary.
2. The D-bar reconstruction method. In this section we briefly review the
reconstruction method based on the proof by Nachman [24]. We will describe both
the exact mathematical algorithm and an approximate numerical algorithm.
2.1. Exact reconstruction from infinite precision data. The r econstruc-
tion method uses exponentially growing solutions to the conductivity equation. Sup-
pose γ − 1 ∈ W
1+ǫ,p
(R
2
) with p > 2 and γ ≡ 1 in R
2
\ Ω. Then the equation
∇ · γ∇u = 0 in R
2
(5)
has a unique exponentially growing solution ψ that behaves like e
ixk
, where x is un-
derstood as x = x
1
+ ix
2
∈ C and the parameter k = k
1
+ ik
2
∈ C. More precisely
(e
−ixk
ψ(x, k) − 1) ∈ W
1,p
(R
2
) with p > 2. The construction of expo nentially grow-
ing solutions is done by reducing the conductivity equation either to a Schr¨odinger
equation (requires two derivatives on the conductivity) or to a first order system (re-
quires one derivative). The intermediate object in the reconstruction method is the
scattering transform defined in terms of the DN map by (3).
The rec onstruction algorithm consists of the two steps
Λ
γ
→ t → γ. (6)
In order to compute t from Λ
γ
by (3) one needs to find the trace of ψ(·, k) on ∂Ω. It
turns out that ψ|
∂Ω
satisfies
ψ(·, k)|
∂Ω
= e
ikx
− S
k
(Λ
γ
− Λ
1
)ψ(·, k). (7)
Here S
k
is the single-layer operator
(S
k
φ)(x) :=
Z
∂Ω
G
k
(x − y)φ(y)dσ(y), k ∈ C \ 0 , (8)
where the Faddeev’s Green’s function G
k
is defined by
G
k
(x) :=
e
ikx
(2π)
2
Z
R
2
e
ix·ξ
|ξ|
2
+ 2k(ξ
1
+ iξ
2
)
dξ, −∆G
k
= δ. (9)
3
