Download - A New Forward-Problem Solver Based on a Capacitor-Mesh Model for Electrical Capacitance Tomography

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 57, NO. 5, MAY 2008 973

A New Forward-Problem Solver Based on aCapacitor-Mesh Model for Electrical

Capacitance TomographyJacek Mirkowski, Waldemar T. Smolik, Member, IEEE, M. Yang, T. Olszewski, Roman Szabatin,

Dariusz S. Radomski, and Wuqiang Q. Yang, Senior Member, IEEE

AbstractAn electrical-capacitance-tomography sensor con-sists of a set of measurement electrodes. It is difficult to calculatethe capacitance for a given sensor structure and given permit-tivity distribution, which is called solving the forward problem,theoretically. Therefore, finite-element methods (FEM) and finite-difference methods are commonly used to solve the forwardproblem and to generate a sensitivity matrix for image reconstruc-tion. This paper presents a new approach to solving the forwardproblem based on a capacitor-mesh model. It has been used foriterative image reconstruction using an updated sensitivity matrixaccording to an estimated image. In this paper, some simulationresults are presented and compared with the results obtained usingan FEM software package from Ansoft Company, showing that thenew approach is promising.

Index TermsElectrical-capacitance tomography (ECT), imagereconstruction, sensitivity matrix.

NOMENCLATURE

A Matrix in potential equation.aij Coefficient of matrix.B Vector of boundary conditions (i.e., potentials on

electrodes).Bn Applied voltage on elements.C Capacitance.C Vector of measured capacitances.E Electrical field. Distribution of permittivity. Vector of permittivity distribution.k Permittivity value.Ii,j Current in a branch of capacitor model.K Size of discrete model matrix.M Number of capacitance measurements.m Row number.N Size of sensitivity and potential vector.

Manuscript received October 1, 2006; revised October 17, 2007. This workwas supported by the British Council and the State Committee for ScientificResearch of Poland under a BritishPolish Research Partnership Programmeunder Grant WAR/341/250.

J. Mirkowski, W. T. Smolik, T. Olszewski, R. Szabatin, and D. S. Radomskiare with the Nuclear and Medical Electronics Division, Institute of Radioelec-tronics, Electronics and Information Technology Faculty, Warsaw University ofTechnology, 00-665 Warsaw, Poland.

M. Yang and W. Q. Yang are with the School of Electrical and ElectronicEngineering, The University of Manchester, M60 1QD Manchester, U.K.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIM.2007.911610

Fig. 1. Cross-sectional view of ECT sensor with 12 measurement electrodes.

nk Vector in parallel to a normal vector of kth pixel edgewith length proportional to edge size.

S Transformation matrix. Potential vector.1, 2 Potential on excitation and detection electrodes.ij Potential on a node.k Potential gradient. Surface enclosing the excitation electrode.d Directed surface element.

I. INTRODUCTION

E LECTRICAL-CAPACITANCE tomography (ECT) pro-vides a means of visualizing electrical permittivity distri-bution in a cross section. It was the first developed tomographymodality for industrial applications. The advantages of ECT arethat it is low cost, high speed, robust, nonintrusive, and non-invasive. The information provided by ECT enables improvedmonitoring and control of industrial processes.

An ECT sensor consists of a set of measurement electrodessurrounding an object. As shown in Fig. 1, the cross sectionof an ECT sensor usually has four layers: 1) an inner insulatorlayer, 2) an electrode layer including the radial screens, 3) anintermediate insulation layer, and 4) a screen layer. Typically,capacitance measurements are taken from an ECT sensor byapplying an excitation voltage sequentially to one electrodeand measuring the induced current from each of the remainingelectrodes. Having measured a set of capacitance data, an imagemay be reconstructed.

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974 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 57, NO. 5, MAY 2008

Image reconstruction is a typical inverse problem. Essen-tially, measuring capacitance is a transformation from an imag-ing space to a projection space. To obtain an image, the inversetransformation must be calculated. An analytical solution toan inverse problem, such as the inverse Radon transform, maybe applied to some hard-field tomography [6]. Because ECTis based on soft-field sensing, it is much more complicatedto solve the inverse problem with ECT than with hard-fieldtomography.

The relationship between the measured capacitance and spa-tial distribution of permittivity may be derived from Maxwellsequations and is given by [12]

C =1

2 1

(x, y, z)E(x, y, z)d (1)

where C is the measured capacitance, 1 and 2 are the poten-tials on excitation and detection electrodes, is the distributionof permittivity, E is the electrical field, is the surface en-closing the excitation electrode, and d is the directed surfaceelement.

Because the electrical field is dependent on the permittivitydistribution, the relationship between capacitance and permit-tivity is nonlinear. Therefore, it is difficult to find an analyticalsolution to (1). This is called the forward problem. For agiven electrical potential on an excitation electrode, detectionelectrodes, and sensor screen (i.e., the Dirichlet boundary con-ditions) and a given medium distribution, the electrical field (orpotential) inside of the sensor is described by

(x, y, z)(x, y, z)d = 0. (2)

Equation (2) can be numerically solved, e.g., using a finite-element method (FEM) or a finite-difference method (FDM).With FEM or FDM, the region of interest is divided into manysmall elements (typically, 10 000) with constant permittivity ineach element. The potential is approximated using a polyno-mial basis to interpolate between the specified nodal values.Effectively, FEM and FDM transform the integral equation intoa large set of linear equations. The main problem with FEMis that it is time-consuming. For example, it takes more than50 h to calculate a sensitivity matrix using a FEM softwarepackage Maxwell from Ansoft Company. Some dedicated FEMsoftware has been developed by researchers, e.g., EIDORS byLionheart et al., which may be faster than Ansoft. However,the dedicated software is either not as accurate as Ansoft or istoo difficult to use. On the other hand, FDM is mostly suitablefor square or rectangular sensors. This paper presents a newapproach to solving the forward problem based on a capacitor-mesh model, which is an alternative to but is simpler than FEMand more appropriate than FDM for circular ECT sensors.

II. METHODS

A. Capacitor-Mesh Model

ECT is inherently a 3-D problem. In principle, the electricalfield should be analyzed in 3-D. However, if the length of

Fig. 2. Basic elementimage pixel. (a) Discrete square element.(b) Capacitor-mesh model.

Fig. 3. Sensitivity map for electrode pair 17 of a 12-electrode ECT sensorcalculated using the new forward-problem solver.

measurement electrodes in an ECT sensor is larger than thediameter of the sensor, the problem may be approximated by2-D for simplicity.

The first step in solving the forward problem numericallyis to divide the imaging area into many pixels, as shown inFig. 2(a). Each pixel may be characterized by its position,dimension, and electrical parameters, such as permittivity. Toassign potential and permittivity values to each discrete imagepixel, a model of a pixel is considered, as shown in Fig. 2(b).It consists of one node and four capacitance branches. If thenode in the center of the pixel is assigned a permittivity value, the four capacitors are assumed to have the same effectivepermittivity of 2. This is a basic element for building a phys-ical model for a whole ECT sensor. Somehow, this is similarto a phantom that is commonly used for electrical-resistancetomography [4].

If Kirchoffs first law is applied to the node for a current sumin this model, the following equation is obtained:

Ii+1,j + Ii,j1 + Ii1,j + Ii,j+1 = 0. (3)

From (3), the relationship between permittivity and potential,as indicated by (2) for a given node (i, j) with the considerationof the neighboring nodes, is given by

i1,ji1,j + i,j1i,j1 + i,j+1i,j+1 + i+1,ji+1,j

(i1,j + i,j1 + i,j+1 + i+1,j)i,j = 0 (4)

MIRKOWSKI et al.: NEW FORWARD-PROBLEM SOLVER BASED ON A CAPACITOR-MESH MODEL FOR ECT 975

Fig. 4. Iterative image reconstruction using the new forward-problem solver. Landweber iteration with updating sensitivity matrix.

Fig. 5. User interface.

where

i1,j1 =2i,ji1,j1i,j + i1,j1

(5)

which corresponds to the serial connection of capacitances in abranch between two adjacent nodes.

B. Cell Model

The idea of the pixel model is based on a cell method [2]. Inthis approach, the pixel is a cell with constant parameters and aboundary. The electric field on the boundary can be calculatedby assuming that the electric field is constant on each side

of a square. On this assumption, (2), for a square element, issimplified to

4k=1

kknk = 0 (6)

where k is the permittivity value, k is the potential gradientat the kth pixel edge, and nk is a normal vector to the kthpixel edge.

Using an arithmetic mean value for the potential and permit-tivity values at the pixel edge, the similar relationship to (4) isobtained but with coefficients given by

i1,j1 =i,j + i1,j1

2. (7)


Fig. 6. Potential distribution and normalized sensitivity map for different numerical phantoms calculated by three methods. The maps for the electrode pairs 17for a 12-electrode sensor are presented.

C. Calculation of Potential DistributionLet us assume that the potential values i,j : i, j 1,K,

where K is the size of the discrete model matrix, are the ele-ments of a 1-D vector of size N = KK using an assignmentn = (i 1)K + j. The potential for the given node n, as givenin (4), may be written as

anKnK + an1n1 + ann+ an+1n+1 + an+Kn+K = 0 (8)

with the following coefficients:

an,n = (i1,j + i,j1 + i,j+1 + i+1,j)an,nN =i1,jan,n1 =i,j1an,n+1 =i,j+1an,n+N =i+1,j . (9)

For one node, an equation can then be written as

A(m) = 0 (10)

where m 1,M is the row number of matrix A.

For those nodes, which are assigned to the electrodes orscreens, (10) can be expressed in a simple form as

an,nn = Bn (11)

where an,n = 1 due to the applied voltage Bn on these ele-ments. All equations for potential in mesh nodes constitute asystem equation

A = B (12)

where B represents the boundary conditions, i.e., potentials onthe electrodes and the screen elements, and A is a large butwell-defined matrix. In our tests, the potential equation is solvedby the Gauss elimination method. In one matrix decomposition,it is possible to find a potential distribution for all applicationelectrodes.

D. Calculation of Sensitivity MatrixA sensitivity matrix, which describes the physical model

of a given sensor, must be obtained for image reconstruction.A number of endeavors have been made for the calculation


Fig. 7. Horizontal-cut profiles of potential distribution and normalized sensitivity maps for the electrode pairs 17. The pixel number is on the x-axis. Thevoltage or normalized sensitivity are on the y-axis. (o) Capacitor-mesh model. () Cell method. () Ansoft.

of sensitivity matrices [9] and for the optimization of sensordesign [10][14].

While various image-reconstruction algorithms have beendeveloped for ECT [16], almost all of them are based on thediscrete linear approximation of (1), which can be expressed as

C = S (13)

where C is a vector of measured capacitances, is a vector ofpermittivity distribution, and S is a transformation matrix.

This transformation matrix is called the linearized sensitivitymatrix, because its elements represent the sensitivity of capaci-tance to a small change of permittivity in a given image pixel

smn =Cmn

. (14)

The sensitivity matrix may be calculated from the potentialdistributions using the Geselowitz formula [5]

s(l1,l2),(i1)N+j =

El1 [i, j] El2 [i, j]Vl1Vl2

d (15)


where l1 and l2 are the electrode numbers, and i, j are nodecoordinates.

From (2), it can be seen that sensitivity coefficients stronglydepend on the potential distribution for a given sensor. Fig. 3shows a typical sensitivity distribution, which is called a sensi-tivity map for electrodes 1 and 7 of a 12-electrode ECT sensorand is calculated using the new forward-problem solver.

E. Image Reconstruction With Updated Sensitivity Matrix

An efficient forward-problem solver is particularly importantto iterative image reconstruction, because the forward problemmust be solved many times. With iterative image reconstruc-tion, it is necessary to update the sensitivity matrix accordingto the estimated permittivity distribution, i.e., the reconstructedimage, because of the nonlinearity feature of ECT. Whilemany researchers have realized the importance of updating thesensitivity matrix during an iterative process [3], this has notbeen done.

Fig. 4 shows the iterative algorithm with sensitivity matrixupdating, which is based on the new forward-problem solverusing the capacitor-mesh model and conventional Landweberiteration [15].

First, the sensitivity matrix for uniform permittivity distribu-tion is calculated. The linear back-projection (LBP) algorithmis used to obtain the initial solution for the iterative procedureusing

= STC (16)where C is the normalized capacitance data, and S is thenormalized sensitivity matrix [8].

From the current image, a vector C(t) of capacitance iscalculated. The differences between the measured capacitanceCm and the estimated capacitance are used to compute the re-laxation factor (t). The Landweber step is performed to obtainthe new estimate (t+1) of the solution. If the Ttot iteration isreached, the algorithm will stop. If the Tup iteration is achieved,the forward solver will execute. The new sensitivity matrix iscalculated according to the current estimate of an image.

F. Implementation of SoftwareThe software is implemented in Pascal language in a Delphi

3.0 environment. The software package is named CAPTOMand has the following main functions:

1) configuring the geometrical and electrical parameters ofa sensor, such as the number and size of electrodes, radialscreens geometry, and insulator permittivity;

2) simulating the electric-field distribution and calculating asensitivity matrix;

3) measuring a set of capacitances from the definedsensor;

4) implementing the LBP, the Landweber iteration, andother iterative algorithms;

5) evaluating the quality of reconstructed images.The elaborated software calculates the sensitivity matrix in

12 s using a 1-GHz Intel processor. Fig. 5 shows a screen shotof the user interface.

Fig. 8. Sequence of reconstructed images. First image: Numerical phantom.Second image: Result of LBP. The next ones: Results of sensitivity matrixupdating and 100 successive Landweber iterations.

III. SIMULATION AND EVALUATION

The sensor configuration shown in Fig. 1 with 12 measure-ment electrodes was used in the following simulation. The threemethods of discretization were used: 1) the capacitor-meshmodel, 2) the cell method, and 3) FEM. The first two methodswere implemented in the CAPTOM software. FEM modelingwas performed using the Ansoft software. Four permittivitydistributions were used for tests: 1) uniform distribution = 1,2) stratified low = 1 and high = 3, 3) bar phantom low = 1and high = 3, and 4) rod phantom = 80. The potentialdistributions and sensitivity maps were calculated using theabove three methods, as shown in Fig. 6. From the results,the differences between profiles of sensitivity maps generatedby the different methods can be seen. In addition, it can beseen that dimensions of objects (rods or bars) are closer withthe capacitor model and the cell model than with the Ansoftsoftware. The differences in the sensitivity maps are small. Theimages reconstructed using the proposed method for calculatingsensitivity maps are comparable to those using the Ansoftsoftware. Fig. 7 shows more comparisons between the threemethods.

The iterative algorithm with sensitivity matrix updating de-scribed earlier was used for image reconstruction based onsimulated capacitance measurements. After each Landweberiteration, the distance between the measured and estimatedcapacitances was calculated and used to adjust the relaxationfactor . Fig. 8 shows the result of the reconstruction of thenumerical phantom of seven rods. The first image (in the leftupper corner) is the numerical phantom. The first reconstructedimage is the result of the LBP procedure. The next ones arethe results of sensitivity matrix updating and 100 successiveLandweber iterations. The results show that the reconstructedimage is clear, and all phantom rods are visible.


IV. CONCLUSION

A new forward-problem solver for ECT has been developedbased on a capacitor-mesh model. It can be used to calculatethe electrical-field distribution for any ECT sensor and any per-mittivity distribution. To make a comparison, the cell methodhas been introduced. The two methods produce similar formsof potential equations but different coefficients. To validate theproposed forward model, a comparison of sensitivity maps andimage reconstruction was performed. As a reference, the resultsobtained using an FEM software package from Ansoft wereused. It is shown that the proposed forward-problem solver pro-duces comparable results to FEM modeling but at much lowercomputational cost. Because of its fast computation speed, itcan be used in iterative image reconstruction with an updatedsensitivity matrix. The capacitor-mesh model presented in thispaper can be easily extended to 3-D cases.

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R. Szabatin, Multichannel capacitance tomograph for dynamic processimaging, Opto-Electron. Rev., vol. 11, no. 3, pp. 175180, 2003.

[2] M. Bullo, F. Dughiero, M. Guarnieri, and E. Tittonel, Isotropic andanisotropic electrostatic field computation by means of the cell method,IEEE Trans. Magn., vol. 40, no. 2, pp. 10131016, Mar. 2004.

[3] Q. Chen, B. S. Hoyle, and H. J. Strangeways, Electric field interac-tion and an enhanced reconstruction algorithm in capacitance processtomography, in Tomographic Techniques for Process Design and Op-eration, M. S. Beck, E. Campogrande, M. Morris, R. A. Williams, andR. C. Waterfall, Eds. Southampton, U.K.: Comput. Mech., 1993,pp. 205212.

[4] A. R. Daniels, R. G. Green, and I. Basarab-Horwath, Modelling of three-dimensional resistive discontinuities using HSPICE, Meas. Sci. Technol.,vol. 7, no. 3, pp. 338342, Mar. 1996.

[5] D. Geselowitz, An application of electrocardiographic lead theory toimpedance plethysmography, IEEE Trans. Biomed. Eng., vol. BME-18,no. 1, pp. 3841, Jan. 1971.

[6] G. T. Herman, Image Reconstruction From Projections: The Fundamen-tals of Computerized Tomography. New York: Academic, 1980.

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[11] M. Soleimani and W. R. B. Lionheart, Non-linear image reconstructionfor electrical capacitance tomography using experimental data, 2005.Private communication.

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[15] W. Q. Yang, D. M. Spink, T. A. York, and H. McCann, An image-reconstruction algorithm based on Landwebers iteration method forelectrical-capacitance tomography, Meas. Sci. Technol., vol. 10, no. 11,pp. 10651069, Nov. 1999.

[16] W. Q. Yang and L. H. Peng, Image reconstruction algorithms forelectrical capacitance tomography, Meas. Sci. Technol., vol. 14, no. 1,pp. R1R13, Jan. 2003 (Review Article).

[17] [Online]. Available: http://eidors3d.sourceforge.net

Jacek Mirkowski was born in Warsaw, Poland, in1948. He received the M.Sc. degree in electronicsapparatus and the Ph.D. degree from Warsaw Univer-sity of Technology, in 1971 and 1981, respectively.

From 1971 to 2001, he was with the Facultyof Electronics, Warsaw University of Technology,where he was an Associate, Senior Associate, andSenior Scientist. Since 2003, he has collaboratedwith the Process Tomography Group at the Nuclearand Medical Electronics Division, Institute of Ra-dioelectronics. His main research interests include

industrial-process tomography, particularly electrical-capacitance tomography,image-reconstruction algorithms, and measurement-control apparatus and sys-tems. He has published nearly 100 scientific papers.

Waldemar T. Smolik (M06) received the M.Sc.degree in electronics engineering and the Ph.D. de-gree in electronic apparatus from Warsaw Universityof Technology, Warsaw, Poland, in 1991 and 1997,respectively.

Since 1997, he has been an Assistant Professorwith the Nuclear and Medical Electronics Division,Institute of Radioelectronics, Electronics and Infor-mation Technology Faculty, Warsaw University ofTechnology. Since 1998, he has also been headingthe Computer Tomography Laboratory at this In-

stitute. His main research interests are computer tomography and medicalimaging. He has published over 50 scientific papers.

M. Yang, photograph and biography not available at the time of publication.

T. Olszewski was born in Warsaw, Poland, in 1957.He received the M.Sc. degree in electronic apparatusfrom Warsaw University of Technology in 1982.

Since 1989, he has been with the Division ofNuclear and Medical Electronics, Institute of Ra-dioelectronics, Faculty of Electronics and Informa-tion Technology, Warsaw University of Technology,where he is currently a Senior Lecturer. His main re-search interests include digital electronics, program-mable logic devices, and impedance tomography.

Roman Szabatin was born in Warsaw, Poland, in1946. He received the M.Sc. degree in electronicsapparatus and the Ph.D. degree from Warsaw Univer-sity of Technology, in 1972 and 1983, respectively.

Since 1973, he has been with the Nuclear andMedical Electronics Division, Institute of Radio-electronics, Electronics and Information TechnologyFaculty, Warsaw University of Technology, as anAssociate, Senior Associate, and Senior Scientist.Since 2000, he has been the Head of the ProcessTomography Group and the Vice Dean of the Elec-

tronics and Information Technology Faculty, Warsaw University of Technology.His main research interests include nuclear and medical instrumentation andmedical imaging (gamma cameras, single photon emission computer tomog-raphy (SPECT), and PET tomography). He has published over 50 scientificpapers.

Dr. Szabatin was the recipient of the Ministry of Higher Education Prizein 1987 and 1993. He was the main Organizer of the Fourth InternationalSymposium on Process Tomography, Poland, in 2006.


Dariusz S. Radomski received the M.Sc. degree incontrol engineering and the Ph.D. degree in controlscience from the Technical University of Warsaw,Warsaw, Poland, in 1995 and 2001, respectively,and the Ph.D. degree in medical biology from theMedical University of Warsaw in 2006.

Since 2001, he has been an Assistant Professorwith the Nuclear and Medical Division, WarsawUniversity of Technology. He also conducts researchprojects on the sexuality of physically disabled per-sons. His main research interests are in the mathe-

matical modeling of the reproductive and endocrine systems, epidemiologicalmethods applied to reproductive medicine, diagnostic supporting systems, andthe application of an inverse problem in the medical and technical areas.

Dr. Radomski is a member of the International Society for ClinicalBiostatistics and the Polish Society of Gynecology and Obstetrics.

Wuqiang Q. Yang (SM05) was born in ShandongProvince, China, in 1956. He received the B.Eng.degree (with distinction) in instrumentation andprocess control and the M.Sc. and Ph.D. degrees(with distinction) in navigation and control fromTsinghua University, Beijing, China, in 1982, 1985,and 1988, respectively.

From 1988 to 1991, he was a Lecturer withTsinghua University. Since 1991, he has been withthe School of Electrical and Electronic Engineering,University of Manchester Institute of Science and

Technology (currently, The University of Manchester), Manchester, U.K., asa Research Associate, Research Fellow, Lecturer, Senior Lecturer, and, finally,Professor. In 2006, he took a sabbatical leave with the Massachusetts Instituteof Technology, Cambridge, as a Visiting Professor. He has published nearly200 scientific papers, including review articles. He has been a Guest Professorwith Tianjin University, Tianjin, China, and a Visiting Professor with ShangQiuNormal University, ShangQiu, China. His main research interests includeindustrial-process tomography, particularly electrical-capacitance tomography,image-reconstruction algorithms, sensing and data-acquisition systems, circuitdesign, instrumentation, multiphase measurement, and image-based control.

Prof. Yang has been member and a Fellow of the Institution of ElectricalEngineers since 1997 and 2004, respectively. He is a Chartered Engineer, anOfficial Referee for over 20 professional journals in the U.K., the U.S., Canada,Holland, and China, Science Advisor to the Chinese Academy of Sciences, Ed-itorial Board member of the Sensor Review Journal and Sensors and Transduc-ers Journal, Guest Editor of the Measurement Science and Technology Journal,Qualified Expert in Intota, U.S., and a Panel member of the Natural ScienceFoundation of China. He was the recipient of the 1997 IEE Measurement Prize,the 1997 Honeywell Prize from the Institute of Measurement and Control, the2000 IEE Ayrton Premium, and the Global Research Award from the RoyalAcademy of Engineering in 2006. His biography has been included in WhosWho in the World, Whos Who in Science and Engineering, and the Dictionaryof International Biography (Cambridge) since 2002.

Download - A New Forward-Problem Solver Based on a Capacitor-Mesh Model for Electrical Capacitance Tomography

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