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- 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 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 ½PðdrÞi 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 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 ÀgPðdrÞ 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 ½rfðdrÞi ¼ : min 0, @W ðdrÞ @dsi if dsi Z 0, if dsi ¼ 0: ð26Þ ( ½rR fðdrÞi 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 rÞij 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 Â 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. References [1] Webster JG. Electrical impedance tomography. Bristol, UK: Adam Hilger; 1990 ). [2] Dolgin M, Einziger PD. 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