Electrical impedance tomography (EIT) calculates the internal conductivity distribution within a body using electrical contact measurements. The image reconstruction for EIT is an inverse problem, which is both non-linear and ill-posed. The traditional regularization method cannot avoid introducing negative values in the solution. The negativity of the solution produces artifacts in reconstructed images in presence of noise. A statistical method, namely, the expectation maximization (EM) method, is used to solve the inverse problem for EIT in this paper. The mathematical model of EIT is transformed to the non-negatively constrained likelihood minimization problem. The solution is obtained by the gradient projection-reduced Newton (GPRN) iteration method. This paper also discusses the strategies of choosing parameters. Simulation and experimental results indicate that the reconstructed images with higher quality can be obtained by the EM method, compared with the traditional Tikhonov and conjugate gradient (CG) methods, even with non-negative processing.

1. ISA Transactions 51 (2012) 808–820
Contents lists available at SciVerse ScienceDirect
ISA Transactions
journal homepage: www.elsevier.com/locate/isatrans
Reconstruction of electrical impedance tomography (EIT) images based
on the expectation maximum (EM) method
Qi Wang n, Huaxiang Wang, Ziqiang Cui, Chengyi Yang
School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 20 October 2011
Received in revised form
29 April 2012
Accepted 30 April 2012
Available online 2 June 2012
Electrical impedance tomography (EIT) calculates the internal conductivity distribution within a body
using electrical contact measurements. The image reconstruction for EIT is an inverse problem, which is
both non-linear and ill-posed. The traditional regularization method cannot avoid introducing negative
values in the solution. The negativity of the solution produces artifacts in reconstructed images in
presence of noise. A statistical method, namely, the expectation maximization (EM) method, is used to
solve the inverse problem for EIT in this paper. The mathematical model of EIT is transformed to the
non-negatively constrained likelihood minimization problem. The solution is obtained by the gradient
projection-reduced Newton (GPRN) iteration method. This paper also discusses the strategies of
choosing parameters. Simulation and experimental results indicate that the reconstructed images with
higher quality can be obtained by the EM method, compared with the traditional Tikhonov and
conjugate gradient (CG) methods, even with non-negative processing.
& 2012 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords:
Electrical impedance tomography (EIT)
Image reconstruction
Statistical method
Expectation maximization (EM) method
1. Introduction
Electrical impedance tomography (EIT) has been investigated
extensively during the past decades as a visualization and
measurement technique. Its aim is to produce images by computing electrical conductivity within the object. Sinusoidal electrical
currents are applied to volume using electrodes, and the resulting
potentials on the electrodes are measured. EIT has numerous
applications in biomedicine, industry and geology. Many potential
applications have been developed for both medical and industrial
use [1–4].
EIT has several advantages over other tomography techniques,
e.g. portability, safety, low cost, non-invasiveness and rapid
response. Thus it could provide a novel imaging solution.
However, due to the limitation of the number of sensing electrodes
and the non-linear property of the field, the imaging reconstruction
of EIT is a typical non-linear and ill-posed inverse problem, which
is unstable with respect to measurement and modeling errors [5].
Regularization is a good way to solve such a problem. Among the
regularization methods, the Tikhonov method has been generally
accepted as an important one [6]. However, the traditional
regularization methods cannot avoid introducing negative values
in the solution, i.e. the gray level of reconstructed image.
n
Corresponding author. Tel.: þ86 022 2740 5724.
E-mail addresses: wangqitju@hotmail.com (Q. Wang),
hxwang@tju.edu.cn (H. Wang), cuiziqiang@tju.edu.cn (Z. Cui),
ycysuk@tju.edu.cn (C. Yang).
The negativity of the gray level, which should be positive in real
image or conductivity distribution, produces artifacts in reconstructed images in presence of noise. Compared with these
methods, statistical techniques can obtain non-negative solution
and lower image distortion [7]. Furthermore, statistical models
provide a rigorous, effective means with which to deal with
measurement error. As a result, tomographic image reconstruction using statistical methods can provide more accurate system
models, statistical models, and physical constraints than the
conventional method [8].
As a statistical method, the expectation maximization (EM)
algorithm is often used to estimate a Poisson model from
incomplete data, i.e. data with imperfect values, or with latent
variables [9]. Furthermore, the noise level of the measurement
system can also be considered as prior information in the EM
method. Thus it is robust to measurement noise. The EM method
has been widely used for ‘‘hard-field’’ imaging, which is based on
the Poisson statistical model, e.g. gamma-ray tomography, X-ray
tomography, emission computed tomography (ECT) etc. [10–13].
The basic principle of ‘‘hard-field’’ imaging is to measure the
attenuation of the intensity of the radiation described by the
Beer–Lambert law [14]. The sensitivity field is not influenced by
the distribution of the components in the process being imaged,
i.e. the sensor field is not deformed by the process and is equally
sensitive to the process parameter in all positions throughout the
measurement volume. The sensitivity is also independent of the
process component distribution inside the measurement volume.
‘‘Hard-filed’’ sensors are typically nucleonic and optical.
0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.isatra.2012.04.011

2. Q. Wang et al. / ISA Transactions 51 (2012) 808–820
The sensitivity filed for EIT is non-linear, the sensitivity
distribution inside the field depends on the measured media, i.e.
it has the property of ‘‘soft-field’’[15]. As a result, it has low spatial
resolution although can be very fast for flow measurement. With
‘‘soft-field’’ sensors, the sensor filed is sensitive to the component
parameter distribution inside the measurement volume, in addition to the position of the component, i.e. the measured parameter, in the measurement volume. Thus, the sensor type
generates an inhomogeneous sensor field which is changed by
the phase distribution and the physical properties of the process
being imaged, meaning that the field equipotential is distorted by
variation of the electrical properties within the measurement
volume. The sensitivity distribution inside the field depends on
the parameter distribution.
Under some prior information, the mathematical models of
both the ‘‘soft-field’’ and ‘‘hard-field’’ imaging can be united. As a
result, the EM method is expected to solve the ill-posed problem
for EIT reconstruction. This paper presents the study of the EM
method for EIT reconstruction. The selection of parameters for the
EM method is also discussed. Both simulation and experimental
tests are conducted in order to prove the performance of the EM
method. The results are reported and compared with those by
using the Tikhonov and CG methods.
2. Typical reconstruction algorithms
injected current vector I. The inverse problem is also called image
reconstruction. The aim of inverse problem for EIT is to obtain the
conductivity distribution r using the boundary voltage vector U
and injected current vector I. In inverse problem, a forward model
is used to predict observations. In the specific case of EIT, a model
that predicts the spatial electric field resulting from applying a
current to a known conductivity distribution is required. The
capability to calculate the electric fields within an object also
proves an efficient method to assemble the Jacobian matrix which
is necessary to solve the inverse problem.
In order to obtain a forward model and the function of U for
EIT, the boundary conditions have to be determined. The boundary conditions arise from the current injection and voltage
measurements through the boundary electrodes. Commonly
these boundary conditions are called electrode models. In this
paper, complete electrode model is used [16].
For mathematical model, the complete electrode model
is used. The complete electrode model is defined by Laplace’s
equation
rUðsruÞ ¼ 0 in O
2.1. Forward and inverse problems
EIT is composed of forward problem and inverse problem.
The forward problem is to determine the voltage measurements,
i.e. voltage vector U for a known conductivity distribution r and
ð1Þ
and the following boundary conditions:
u þ zl s
Z
s
El
In EIT, an array of electrodes (16 electrodes in this paper) is
arranged with equispaced in a single plane around the perimeter
of the medium and a sinusoidal current are injected through these
electrodes. With the adjacent drive pattern, current is applied to
an adjacent pair of electrodes and the resultant voltages between
the remaining 13 adjacent pairs of electrodes are measured.
The three possible measurements involving one or both of the
current injecting electrodes are not used. This procedure is
repeated 16 times with current injected between successive pairs
of adjacent electrodes until all 16 possible pairs of adjacent
electrodes have been used to apply the known current [15]. This
is shown schematically in Fig. 1. This procedure produces
16 Â 13 ¼208 voltage measurements called an EIT data frame.
An estimate for the changes in cross-sectional conductivity
distribution of the object is obtained by using the voltage
measurements made on the boundary. An EIT system consists of
three parts, i.e. array electrode, data acquisition system and image
reconstruction unit, as shown in Fig. 1.
809
@u
¼ U l , on El ,
^
@n
@u
dG ¼ Il
^
@n
l ¼ 1,2,. . .,m
@u
s ^ ¼ 0 on dG [m¼ 1 El
l
@n
ð2Þ
ð3Þ
ð4Þ
In these equations s is the conductivity distribution, u is the
^
scalar potential distribution, n is the outward unit normal of the
boundary @O, zl is the contact impedance, Il is the injected current
and U l is the corresponding potentials on the electrodes, m is the
number of electrodes, El is the lth electrode, and O denotes the
object.
In addition, the following two conditions for the conservation
of charge are needed to ensure the existence and uniqueness of
the solution
m
X
Il ¼ 0
ð5Þ
Vl ¼ 0
ð6Þ
l¼1
m
X
l¼1
In order to solve the complete electrode model, numerical
techniques are preferable to analytic solution because the complexity of obtaining analytic solution usually prevents its application in the forward model. The finite element method (FEM) is
widely employed in current EIT forward model. After the FEM
discretization, the relation between the injected currents and the
measured voltages on the electrodes, i.e. the function of U can be
defined based on Eqs. (1)–(4), [17]
U ¼ Vðr; IÞ
ð7Þ
nÂ1
Fig. 1. A sketch-map of EIT sensor.
l ¼ 1,2,. . .,m
is the discrete conductivity distribution,
where vector r A R
and vector U A RnÂ1 is the discrete measured voltages. m is the
number of measurement data and n is the number of pixels in
the reconstructed image. Vðs; IÞ is the forward model mapping the
conductivity distribution s and injected current vector I to the
boundary voltage vector U.
Difference imaging is used in this paper. The aim of difference
imaging is to reconstruct the change in conductivity that occurs
over some time interval. A data set U 1 is acquired at a time t 1 and
a second data set U 2 is acquired at a later time t 2 . The algorithm
then calculates the change conductivity from time t 1 to time t 2 .

3. 810
Q. Wang et al. / ISA Transactions 51 (2012) 808–820
The difference imaging method can improve reconstructed image
stability in the presence of problems such as unknown contact
impedance, inaccurate electrode positions, poorly known boundary shape, non-linearity and the use of 2-D approximations for
3-D electrical fields.
The calculation of the change in conductivity is performed
using a linear approximation operator. When the perturbation of
conductivity is small enough, linear algorithms are valid for
approximating the derivation of conductivity from the reference
conductivity distribution. Considering r0 is the known conductivity distribution and Vðr0 Þ is the corresponding voltages collected under the conductivity distribution r0 , r is the unknown
conductivity that does not derivate much from r0 , VðrÞ is the
corresponding voltages collected under the conductivity distribution r and applied currents I. Since VðrÞ is nonlinear with respect
to the conductivity distribution r, the Taylor series of VðrÞ at r0 ,
VðsÞ ¼ Vðs0 Þ þ JðsÀs0 Þ þ oð:sÀs0 :2 Þ
ð8Þ
mÂn
is the Jacobian matrix at r0 . It is calculated
where J ¼ Vðs0 Þ A R
column by column with the jth column describing the effect of
the change in conductivity of the jth element on the ith signal
measured between electrode pairs. For fast calculation of J, a
sensitivity method based on Geselowitz’s sensitivity theorem is
used [18].
Neglecting the high-order term of Eq. (8), then
dU ¼ J ds
ð9Þ
where vector dU ¼ VðrÞÀVðr0 Þ A RmÂ1 is the difference voltage
between two measurements; vector dr ¼ rÀr0 A RnÂ1 is the
perturbation of conductivity from the reference one.
2.2. Tikhonov regularization algorithm
where the vector Y A RmÂ1 (m is the number of projection data)
contains the measured or observed projection data, while the
vector X A RnÂ1 (n is the number of pixels of the reconstructed
image) represents an estimate of the density distribution for
measured object. The system matrix C A RmÂn models the
projection process from pixel space to projection space.
As discussed in Section 1, the EM method has been widely
used in ‘‘hard-field’’ reconstruction, which is a Poisson process
and can be expressed as [23]
X $ PoissonðkÞ
ð13Þ
where the mathematical symbol ‘‘ $’’ means probability distribution, vector X is Poisson random variable, vector k is Poisson
distributed random variable, i.e. Poisson parameter, which is
equal to the expected value of X and so is its variance and mean.
Suppose that we count along m lines. The ith element of the
m-dimensional measured projection vector, yi ðyi Z0Þ, is the number
of coincidences which are counted for the ith line during the data
collection period. The Poisson model can be expressed based on the
probability theory. If the image vector is X, the probability of the
measurement vector to be Y is
"
#
m
Y lyi
Àli
i
PL ðY9XÞ ¼
ð14Þ
e
yi !
i¼1
The reconstruction problem is to estimate the image vector X
given the data measurement Y. One approach to this problem is
the maximum likelihood method which estimates X that maximizing P L ðY9XÞ. It is equivalent to maximize the log-likelihood,
L(X) subject to nonnegative constraints on X, i.e., it finds X Z0
which maximizes
LðXÞ ¼
m
X
½yi log bi Àbi Š
ð15Þ
i¼1
Since the sensitivity matrix J in Eq. (9) is usually neither square
nor full rank, image reconstruction for EIT is a typically ill-posed
inverse problem. Small noise in the measured data can cause large
errors in the estimated conductivity. The Tikhonov regularization
is an important method to deal with the ill-posed problems
[19–21]. The usual standard-form Tikhonov approach for the
linear ill-posed problem in Eq. (9) is simply to minimize the
regularized least squares functional
The ith measurement yi can be seen as an independent Poisson
parameter with the mean given by
bi ¼ Eðyi Þ ¼
n
X
cij xj
ð16Þ
j¼1
The mathematical model for ‘‘hard-field’’ imaging can be
expressed in Eq. (12).
where cij is the ijth element of the response matrix C, bi is the
expected value of measured data. Obviously, the elements of C, X
and Y are all positive.
For the ‘‘soft-field’’ measurements of EIT, the sensitivity field is
non-linear, i.e. the sensitivity distribution inside the field depends
on the contribution of measured media. However, the inverse
problem can also be solved with sufficient accuracy by considering the linearized equation system expressed in Eq. (9), if changes
in conductivity according to the background are small.
As a result, we can see that the mathematical model for linear
EIT reconstruction is similar to the ‘‘hard-field’’ image reconstruction. In Eq. (12), the vector Y A RmÂ1 contains the changes of
measured voltage data dU, while the vector X A RnÂ1 represents
the changes of conductivity distribution dr. The matrix C A RmÂn is
the sensitivity matrix J under reference condition. However,
the positivity or negativity of the elements in vectors dU and dr
cannot be determined. In order to meet the condition of the
EM method, some prior information should be added for EIT
reconstruction.
Since difference imaging is executed between two different
conductivity distributions in EIT, a conductivity change can be
seen relatively non-negative to the medium with lower conductivity. Thus dU needs to be computed with some prior information in order to keep dr non-negative [24],
Y ¼ CX
dU ¼ ðÀ1Þq ðU mea ÀU ref Þ
2
2
F Tikh ðdrÞ ¼ :J drÀdU:2 þ a:Ldr:2
ð10Þ
without any constraints. Then the minimization of Eq. (10) can be
given by
drTikh ¼ ðJ T J þ aLÞÀ1 J T dU
ð11Þ
where L is the regularization matrix, which is selected as a unity
matrix in the standard form. a Is the regularization parameter,
which can be selected by the L-curve, generalized cross-validation, or the quasi-optimality method [20].
However, the Tikhonov regularization method introduces
negative values in the solution i.e. negative gray level, which
produces artifacts in reconstructed images in presence of noise,
especially in the center where the sensitivity is much lower [22].
3. The expectation maximization (EM) algorithm
3.1. Suitability of the EM method for EIT reconstruction
ð12Þ
ð17Þ

4. Q. Wang et al. / ISA Transactions 51 (2012) 808–820
where U ref is the reference measurement of the background, U mea
is the measurement of the tested medium, q is index determined
by some prior information, that is, q¼ 2 when U ref is the
measurement of lower conductivity compared with U mea , q¼1
when U ref is of higher one.
According to Eq. (17), the vectors dU and ds in Eq. (9) can be
transformed to be positive. Additionally, the elements of the
sensitivity matrix J can be normalized to the range of 0 and
1 based on the normalization method [25]. As a result, the nonnegative constraint mathematical model of EIT has the same
condition with the mathematical model of ‘‘hard’’ field tomography. That means dU is assumed to be the projection data, ds is
assumed to be the density distribution, and J is assumed to be the
system matrix. Thus, the ith measurement dU i can be seen as an
independent Poisson distributed random variable. Then the EM
method can be used to the non-negative constraint mathematical
model of EIT.
Under the linear approximation in Eq. (9) and the prior
information of non-negativity in Eq. (17), the statistical model
of EIT can be assumed as Poisson model, as shown in Formula
(13), i.e. the EM method is suitable to solve the ill-posed problem
for EIT. We assume that the vector dU is Poisson distributed with
the Poisson parameter k ¼ Jdr, i.e. let the ith measurement dU i be
an independent Poisson distributed random variable with the
mean given by
bi ¼ EðdU i Þ ¼
n
X
Jij dsj
ð18Þ
j¼1
Obviously, it is equal to the expected value of measurement,
B ¼ EðdUÞ ¼ J dr
the problem as prior information. Furthermore, the likelihood
maximization is transformed to the minimization problem,
min WðdrÞ ¼
n
XÈ
É
2
ððJ drÞi þ o2 ÞÀðdU i þ o2 ÞlogððJ drÞi þ o2 Þ þ g:Ldr:2
i¼1
ð23Þ
where dr Z 0, U i ¼ maxU i ,0, the parameter o2 and the elimination of negative values of dri have a stabilizing effect on the
minimization. In practice, o2 is the noise level of the measurement system. L is the regularization matrix. WðdrÞ is proved to be
strictly convex, so Problem (23) has the unique global minimum
solution.
In order to obtain the non-negative solution, we define the
projection operator
(
dsi if dsi Z0
½PðdrÞŠi ¼ maxðdsi ,0Þ ¼
ð24Þ
0
if dsi o0
Then Problem (23) is transformed to
min fðdrÞ : ¼ min WðPðdrÞÞ
ð19Þ
ð20Þ
Fig. 2. Typical EIT system with a 16-electrode sensor.
3.2. Solving the EM method
The maximum likelihood method, which estimates ds that
maximizing the log-likelihood, is used to solve the EM method [9].
The original likelihood function LðdrÞ is established,
LðdrÞ ¼
m
X
½dU i log li Àli Š
ð21Þ
i¼1
where li ¼ ðJdrÞi is the expected value of measurement. In order
to accelerate convergence, the maximum likelihood function is
combined with the least squares merit function, which takes the
form of
max Gðdr, gÞ ¼ LðdrÞÀgPðdrÞ
m
X
½dU i log li Àli ŠÀgPðdrÞ
¼
ð22Þ
i¼1
The penalty function PðdrÞ should be symmetric and twice
2
differentiable [26], which is selected as :Ldr:2 in this paper.
The parameter g is a positive weighted parameter, which controls
the tradeoff between the likelihood function and the penalty form
to obtain better results.
Since the true solution in this paper is known to have only
non-negative entries, the non-negativity constraint is added to
ð25Þ
Most statistical methods for image reconstruction require
minimizing an objective function related to measurement statistics. For realistic image sizes, direct minimization methods are
computationally intractable, so iterative methods are required
[27]. A gradient projection-reduced Newton (GPRN) method is
used in this paper [26]. We perform reduced Newton (RN)
iteration with an initial guess, which is obtained by the gradient
As a result, the non-negative constraint mathematical model
of EIT in Eq. (9) has the same condition with the mathematical
model of ‘‘hard-field’’ tomography in Eq. (12). That means the
non-negative image vector X is the non-negative constrained
perturbation of conductivity dr. The measured projection vector
Y is the non-negative difference voltage dU. According to Eq. (13),
B is the expected value of measurement and can be given by
B ¼ EðdUÞ ¼ J dr
811
Fig. 3. Simulated conductivity distributions.

5. 812
Q. Wang et al. / ISA Transactions 51 (2012) 808–820
Fig. 4. Profiles of image error versus the number of iterations based on the two methods. The subfigures (a)–(d) are associated with simulated conductivity distribution
(a)–(d) in Fig. 3, respectively.
projection method to solve the non-negative problem. Thus the
GPRN method consists of two stages:
(1) The gradient projection (GP) method. The projected gradient is
defined based on the non-negative constraint,
8
< @@Wi ðdrÞ
ds
n
o
½rfðdrÞŠi ¼
: min 0, @W ðdrÞ
@dsi
if
dsi Z 0,
if dsi ¼ 0:
ð26Þ
(
½rR fðdrÞŠi ¼
The iterative step for the GP method is
pðkÞ ’ÀrWðdrðkÞ Þ
dr
ðk þ 1Þ
k
where h:,:i represents the inner product of two vectors,
0 o c1 o c2 o 1.
(2) The RN method. The GP method results in an asymptotically
linear convergence rate, which is very slow. To improve the
convergence rate, second order derivative information, e.g.
the Newton method, is incorporated based on the results of
the GP method. The gradient vector and the Hessian matrix
are simplified according to the non-negative constraint,
@W
@dsi
ð27Þ
k k
’Pðdr þ t p Þ
ð28Þ
where t 40 is the step length, which is obtained by the Wolfe
inexact line search strategy [28],
D
fðdrðkÞ þ tðkÞ pðkÞ Þ r fðdrðkÞ Þ þ c1 rWðdrðkÞ Þ,PðdrðkÞ þ tpðkÞ ÞÀdrðkÞ
E
ð29Þ
D
rðdrðkÞ þ tðkÞ pðkÞ Þ Z c2 rWðdrðkÞ Þ,PðdrðkÞ þ tpðkÞ ÞÀdrðkÞ
E
HR ¼ ½r
if dsi ¼ 0
0
2
R fðd
rÞŠij ¼
ðdrÞ
8
<1
if dsi ¼ 0 or dsj ¼ 0
2
: @d@ i W sj
s @d
ðdrÞ
otherwise
ð32Þ
2
where rR fðdrÞ and rR fðdrÞ are simplified or reduced
gradient vector and Hessian matrix of fðdrÞ, respectively.
If H denotes the ordinary, unreduced Hessian of W, then
HR ¼ DI HDI þDA
(
nÂ1
ð30Þ
ð31Þ
otherwise
where DA ¼ diagðli Þ, l A R
, li ¼
dsi ¼ 0
,D ¼ IÀDA .
0 dsi a 0 I
1
ð33Þ

6. Q. Wang et al. / ISA Transactions 51 (2012) 808–820
813
Fig. 5. Profiles of image error versus the number of iterations with different regularization parameters for the GPRN method. The subfigures (a)–(d) are associated with
simulated conductivity distribution (a)–(d) in Fig. 3, respectively.
The iterative step for RN method is
p ’rWðdr Þ
ð34Þ
sðkÞ ’ÀHÀ1 pðkÞ
R
ð35Þ
drðk þ 1Þ ’PðdrðkÞ þ tðkÞ sðkÞ Þ
ð36Þ
ðkÞ
ðkÞ
The step length t 4 0 is according to Formulas (29) and (30).
It should be noted that more than one iteration step can be taken
in the GP stage for the initial guess about drNewton .
0
4. Results and discussions
4.1. Simulation phantoms
A simulation study of a two-dimensional EIT problem was
performed using the measurement setup depicted in Fig. 2.
In simulation, the measured voltages were simulated using the
complete electrode model and adjacent current patterns, which
consist of 16 current excitation configurations and 13 corresponding voltage measurement configurations for each current excitation, as shown in Fig. 1. An adjacent detection strategy was used
for producing simulated voltage data. The excitation current
is 71 mA. The lower conductivity (background) and the higher
conductivity (objects) are 1 S/m and 3 S/m, respectively. The
forward problem is solved using a finite element method. A mesh
of adaptive fist-order triangular elements produced in COMSOLs
is used for the forward calculations. In order to simulate the
typical noise levels in real measurement systems, the Gaussian,
zero mean random noise is added to the simulated voltages.
The amplitude of the noise is 71% of simulated voltage amplitude.
4.2. Choices of parameters for the EM algorithm
Selection of parameters is very important because an inappropriate parameter can lead to a useless result. In this section,
the strategies of choosing parameters for the EM algorithm based
on numerical simulation are discussed. All of the algorithms were
implemented using MATLAB on a PC with a 2.8 GHz CPU and 1GB
memory.
4.2.1. Iteration number
According to Section 3.2, iterative methods are used to solve
the EM method. Two methods were compared in this paper.
The first one is the GP method; the second one is the GPRN
method with initial value of two GP iterations. Four conductivity

7. 814
Q. Wang et al. / ISA Transactions 51 (2012) 808–820
distributions in Fig. 3 were reconstructed by the two methods
with 100 iterations. The plots of relative image errors (RE) defined
in Eq. (37) versus iterative steps are calculated and shown in
Fig. 4. For both of the two methods, the regularization parameter
g and the parameter o2 are selected as 0.35 and 0.01, respectively.
.
2
2
RE ¼ :rÀrn :2 :rn :2
ð37Þ
where r is the calculated conductivity vector and rn is the real
one in the simulation model.
It can be seen from Fig. 3 that the GPRN method is more
oscillatory than the GP method, i.e. the GP method is more stable.
However, the GPRN method can reach the least RE below 0.2 at an
early iteration step. It can effectively improve the quality of
reconstruction, compared with the GP method. This also demonstrates that the reduced Newton method based on the GP initial
guess can dramatically reduce the number of iteration steps.
an appropriated parameter g can effectively improve the accuracy
of the reconstructed images.
The existing methods of choosing a regularization matrix, i.e.
weighting operator, L in the Tikhonov system are still effective in
the GPRN method. L is a matrix that defines a suitable smoothing
norm for the reconstruction problem. The matrix L is of size p  n,
where p r n. Typically, L is the identity matrix,
0
1
1
B
C
1
B
C
nÂn
i:e: L ¼ B
ð38Þ
CAR
@
A
&
1
or a banded matrix approximation to the ðnÀpÞth derivative. For
example, an approximation to the first derivative is given by the
matrix
0
1
À1 1
B
C
À1 1
B
C
ðnÀ1ÞÂn
L1 ¼ B
ð39Þ
CAR
@
A
& &
À1
4.2.2. Regularization parameter and regularization matrix
In order to solve EM problem based on the GPRN method, the
regularization parameter g and the regularization matrix L should
be carefully selected, according to Eq. (9).
The regularization parameter g is important to determine the
weight of the regularization. With too little regularization, reconstructions have highly oscillatory artifacts due to noise amplification. With too much regularization, the reconstructions are too
smooth; some of detail information, like sharp transitions in
conductivity is not able to be presented [24]. Fig. 5 plots REs
versus 100 iterations with different regularization parameters.
For the initial guess, the number of GP iterations is 2. The results
demonstrate that reconstruction with too large or too small
regularization parameters will seriously reduce the image quality.
The REs of different methods against g with 5 iteration steps
are calculated and shown in Fig. 6, which also demonstrates that
1
while an approximation to the
0
1 À2 1
B
1 À2 1
B
L2 ¼ B
@
& & &
1
À2
second derivative is
1
C
C
ðnÀ2ÞÂn
CAR
A
ð40Þ
1
The identity matrix L is to minimize the norm of the solution
only, while L1 and L2 can result in smooth solution. In order to
investigate the relationship between the weighting matrixes and
the quality of the reconstructed images, both the iteration
number and regularization parameter for the EM methods with
different weighting matrixes were studied. In Fig. 7, two methods
with regularization matrixes L, L1 and L2 are compared to
determine the iteration number. The first one is the GP method,
the second one is the GPRN method with initial value of two GP
Fig. 6. Profiles of image error versus the parameter g with 5 iterations based on the GPRN method. The subfigures (a)–(d) are associated with simulated conductivity
distribution (a)–(d) in Fig. 3, respectively.

8. Q. Wang et al. / ISA Transactions 51 (2012) 808–820
815
Fig. 7. Profiles of image error versus the number of iterations based on the GP and GPRN methods with different regularization matrixes. (a) L regularization matrix.
(b) L1 regularization matrix. (c) L2 regularization matrix.
iterations, as discussed in Section 3.2. The conductivity distribution in Fig. 2(a) is reconstructed by the two methods with
regularization matrixes L, L1 and L2 , respectively. The iteration
number for both of the two methods is 100. The plots of the RE
versus iterative steps are calculated and shown in Fig. 7. For both
of the two methods, the regularization parameter g and the
parameter o2 are selected as 0.35 and 0.01, respectively.
It can be seen from Fig. 7 that the GPRN method can reach the
least RE at an early iteration step for all of the three regularization
matrixes. The GP and GPRN methods with the regularization
matrix L1 give more stable RE plots than L, however, the RE of
reconstructed images also dramatically increase because of the
over-smoothing effect of the weighting operator L1 . The regularization matrix L2 has even stronger smoothing effect than L1 and
leads to the distortion solution. As a result, the regularization
matrix L defined in Eq. (38) is selected.
Fig. 8 plots the REs versus 100 iterations with different
regularization parameters. Similar to the results in Fig. 5, the
reconstruction with too large or too small regularization parameters will reduce the image quality. Although the GPRN method
with the regularization matrix L2 can obtain stable solution, the
GPRN method with the regularization matrix L can give the best
quality of the reconstructed images.
The REs of different methods against g with 5 iteration steps
are calculated and shown in Fig. 9, which also demonstrates that
regularization matrix L is the most appropriated for GPRN
method, compared with other two regularization matrixes.
Computational time is another factor which must be considered in assessing the proposed method. Table 1 shows the
computational time of the GPRN method with different regularization parameters and matrixes versus the iteration numbers.
The GPRN method with the weighting matrix L has the least
computational time, compared with L1 and L2 . For the same
regularization matrix, the reconstruction speed is not affected
by the variation of the regularization parameters.
4.2.3. Selection of parameter o
The real measured voltages contains systematic and random
errors, which can be modeled as a Gaussian random variable with
mean zero and variances o2 [9]. Then the statistical model for
measured voltage data of EIT system can be expressed as
dU i $ PoissonðJ drÞi þ Gaussianð0, o2 Þ
ð41Þ
The variance of the Gaussian is selected according to the
measurement accuracy of the system. In this paper, o2 is selected

9. 816
Q. Wang et al. / ISA Transactions 51 (2012) 808–820
Fig. 8. Profiles of image error versus the number of iterations with different regularization parameters for the GPRN method with different regularization matrixes.
(a) L regularization matrix. (b) L1 regularization matrix. (c) L2 regularization matrix.
as 0.01, which is almost equal to the typical noise level in real EIT
system with 16 electrodes.
In practice, as the conductivity distribution is unknown
beforehand, it is difficult to determine the cut off criterion based
on the RE. Under the same experimental conditions, the REs
against iteration for different objects based on the same algorithm
have similar tendency. Thus, some typical distributions can be
tested for choosing the parameters g, L and iteration number with
a stopping criterion. All these procedures can be done before the
real experiments. Then the object can be reconstructed based on
the selected parameters. Although only approximate parameters
are used for experiments, both simulation and experimental
results indicate that the EM method with the GPRN iteration
can give better reconstruction results than conventional methods.
In the following simulations and experiments, the performance of
the EM method using these strategies of choosing parameters will
be shown.
the regularization parameter g is 0.35. o2 is 0.01. The number
of initial GP iterations is 2. For the Tikhonov method, the
regularization parameter a is selected by the L-curve method.
The regularization matrix for both the Tikhonov and the EM
methods are identity matrix. The iteration numbers for the CG
and the EM methods are 60 and 5, respectively.
The reconstructed images are shown in Fig. 10. For the
convenience of comparison, the gray levels are normalized to
the range from 0 to 1. We can see that, the artifacts of reconstructed images can be effectively decreased with non-negative
processing, i.e. the negative parts of the gray level in dr are set to
be zero. However, compared with the Tikhonov and CG methods,
the EM method can give the best results. The quality of reconstruction is satisfied in both size and position, even in the center
of the sensing field. The RE defined in Eq. (37) is calculated and
displayed in Fig. 11, which also demonstrates that the EM method
performs the best.
4.3. Numerical simulation results
4.4. Experimental results
In order to evaluate the proposed method and the strategies of
choosing parameters, conductivity distribution in Fig. 3 is reconstructed based on the simulated voltage data. For the EM method,
An experimental study was performed using a measurement
setup similar to the above simulation study, as shown in
Fig. 12(a). Here, the two-dimensional imaging domain was set

10. Q. Wang et al. / ISA Transactions 51 (2012) 808–820
817
Fig. 9. Profiles of image error versus the parameter g with 5 iterations based on the GPRN method with different regularization matrixes. (a) L regularization matrix. (b) L1
regularization matrix. (c) L2 regularization matrix.
Table 1
Computational time of the GPRN method with different regularization parameters
and weighting matrixes.
Computational time
Regularization matrix
g
1
5
10
L
g ¼ 0:1
g ¼ 0:35
g ¼ 0:8
0.829392
0.842526
0.837984
4.524379
4.674582
4.852321
12.437544
12.239005
12.596712
L1
g ¼ 0:1
g ¼ 0:35
g ¼ 0:8
0.938076
0.933124
0.937772
5.336022
5.304658
5.322889
13.619874
13.707223
13.447516
L2
g ¼ 0:1
g ¼ 0:35
g ¼ 0:8
0.977821
0.988632
0.973597
5.784235
5.674882
5.798545
14.105478
14.35564
14.09089
up using a cylindrical container of inner diameter d ¼20 cm, filled
with tap water to a height of 10 cm. Sixteen composite electrodes
evenly distributed on the inner surface of the container.
The composite electrode is composed of the outer current electrode,
i.e. excitation electrode, and the inner voltage electrode, i.e. the
measurement electrode, as shown in Fig. 12(b). Adjacent currents
injection from a single current source and adjacent voltage
measurement strategies are used. The structure of the EIT
imaging system is shown in Fig. 12(c). In the data acquisition
and control system, the AC-based sensing electronics mainly
consists of the resistance-to-voltage (R/V) converter and the ac
programmable gain amplifier (AC-PGA). The digital quadrature
demodulation is implemented in the digital platform. The digital
signal from the digital-to-analog converter (DAC) is obtained and
processed by the low-cost, high-capacity FPGA (Xilinx Spartan-3
XC3S400), where the modules of micro-control unit (MCU), digital
phase sensitive demodulation (digital PSD), and first in, first
out (FIFO) are contained. All measurements were made at the
frequency of 500 kHz [29].
The experiments have been conducted for plastic rods positioned in tap water. The plastic rods with diameter d¼ 2 cm,
placed at different positions inside the imaging domain, corresponding to four different measurement cases. The true conductivity distribution and reconstructed images are shown in Fig. 13.
The parameters for the three reconstruction methods are selected
according to the simulation results, as discussed in Section 4.3.
The REs are calculated and shown in Fig. 14. For the convenience
of comparison, the gray levels are normalized to the range from
0 to 1. From Figs. 13 and 14, we can observe that the EM method
not only keeps images accuracy but also suppresses images noise,
compared with the Tikhonov and CG methods
The calculation time of the three methods are also compared
in Table 2. It is obvious that the Tikhonov method, which is a
direct method, has the best real-time performance. Although the
iteration number of the CG method is larger than the EM method,
the computational time of each iterative step for the CG method is
much shorter. Furthermore, the EM method needs additive two
GP iterations for initial guess. Thus the EM method obtains good
image quality at the cost of relatively long computational time.
However, the proposed method is suitable for the situation where
the image accuracy is the most importance, and more over, the
demand for the real-time performance is relatively low. For a
better understanding of a dynamic conductivity distribution, e.g.
identification of dynamic multi-phase flow regimes, monitoring
of lung ventilation, there is obviously a current need for improving the calculation time while keeping the high resolution for
instantaneous observation.
4.5. Discussion about the 3-D imaging based on the EM method
Due to computational complexity, reconstructions have
usually been over 2-D FEM in this paper, i.e. electrode

11. 818
Q. Wang et al. / ISA Transactions 51 (2012) 808–820
Fig. 10. Reconstructed images with simulated voltage data using five methods.
Instead of producing fully 3-D reconstruction it is also possible
to exploit 3-D aspects with 2-D reconstruction on each layer.
In the forward problem, the Jacobian matrix of 3-D EIT problem is
established based on the complete electrode model. The 3-D EIT
forward problem is discretized and solved by the FEM. Then the
elements in sensitivity matrix of each plane can be approximated
within each element by an interpolating function. In the inverse
problem, the linear approximation of 3-D EIT problem takes the
following form
dU 3D ¼ J3D dr3D
Fig. 11. Relative image errors versus simulated conductivity distributions
obtained by using the five methods.
placement for 2-D reconstruction algorithms is confined to
planar arrangements that match the 2-D reconstruction geometry. However, most industrial and medical electrical imaging
problems are fundamentally 3-D, the electric current is not
confined to the measurement plane. Therefore, the EIT problem
is inherently 3-D. Off-plane structures will have effect on the
reconstructed images. 3-D reconstruction algorithms with
multi-plane electrode arrangements can be used to more
accurately reconstruct impedance distributions. Since both
the measurements on and off the drive plane can be obtained,
the number of measurements increases, compared with the 2-D
problem, while the number of pixels in reconstructed image for
each layer is not changed. As a result, the ill-condition of the
EIT reconstruction problem is weakened. Furthermore, the
measurements off the drive plane may be used to correct for
off-drive-plane resistivity changes.
ð42Þ
where dU 3D is measured data vector, including on-plane and
off-plane driving, J 3D is the sensitivity matrix based on the 3-D
EIT complete electrode model, dr3D is the conductivity vector
of one plane, which is to be solved. From Eqs. (9) and (42), we
can see that, the linear approximation of 2-D and 3-D EIT
problems are similar, so the EM method, which is used to solve
2-D EIT problem in this paper, can also be used to solve 3-D
problem. The EM method is expected to have the potential to
expand to 3-D EIT reconstruction and improve the quality of
reconstructed images, according to the 2-D reconstruction
results discussed in this paper. To improve the resolution of
reconstructed images it will be necessary to use an increased
number of electrode layers and increase the number of
electrodes on the layers.
5. Conclusion
In this paper, the EM method is presented for EIT image
reconstruction. The solution is obtained by the GPRN method.
The strategies of choosing the parameters are discussed. Both
simulation and experimental results have shown that the EM
method presents obvious superiority in both the quality of
reconstructed images and accuracy of the recovered conductivity
values, compared with the common Tikhonov regularization and
CG methods.

12. Q. Wang et al. / ISA Transactions 51 (2012) 808–820
Fig. 12. Experimental setup. (a) Experimental system developed by Tianjin University. (b) Composite electrode. (c) Schematic diagram of the EIT system.
Fig. 13. Reconstructed images of plastic rods by using the three methods.
819

13. 820
Q. Wang et al. / ISA Transactions 51 (2012) 808–820
Fig. 14. Relative image errors versus true conductivity distributions obtained by
using the three methods.
Table 2
Comparison of the three methods in terms of computational time for experiment.
Index of true
distribution
Computational time (s)
Tikhonov
method
Distribution
Distribution
Distribution
Distribution
(a)
(b)
(c)
(d)
CG method
(60 iterations)
EM method (2 GP iterations
and 5 GPRN iterations)
0.0493
0.0492
0.0564
0.0614
0.9482
1.1218
0.9634
1.2661
2.7986
2.8873
2.8138
2.8895
Acknowledgments
This work is supported by the National Natural Science
Foundation of China (50937005, 61001135, and 60820106002)
and by the Natural Science Foundation of Tianjin Municipal
Science and Technology Commission under Grant 11JCYBJC06900.
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