Electrical Capacitance Tomography Using Bayesian Inference

Colin Fox1 Daniel Watzenig2

1 Department of Physics, University of Otago, PO Box 56, Dunedin, New Zealand2 Institute of Electrical Measurement and Measurement Signal Processing,Graz University of Technology, Kopernikusgasse 24, A-8010 Graz, Austria

Email: fox@physics.otago.ac.nz & daniel.watzenig@tugraz.at

Abstract: Electrical capacitance tomography is a non-invasive imaging technique thatuses measured trans-capacitances outside a medium to recover the unknownpermittivity in the medium. We use Bayesian inference to solve the problemof recovering the unknown shape of a constant permittivity inclusion in anotherwise uniform background. Calculated statistics give accurate estimates ofinclusion area, and other properties, when using measured data.

Keywords: Electrical capacitance tomography (ECT), Bayesian inference, Inverse problem,Markov chain Monte Carlo (MCMC), Prior modelling

1. INTRODUCTION

Electrical capacitance tomography (ECT) is atechnique for recovering the spatially-varying per-mittivity of an insulating medium from measure-ment of capacitance outside the boundary of themedium. ECT is primarily used for non-invasiveimaging within inaccessible domains in applica-tions where differing materials show up as con-trasting permittivities. Electrodes are set in aninsulating material at the outside of an insulatingpipe. By applying a predefined voltage pattern toelectrodes the capacitance between pairs of elec-trodes can be directly related to measured electricpotentials, electric currents, and electric charges.

Measurements consist of all the capacitances be-tween pairs of electrodes, making up the matrix oftrans-capacitances. The interior of the pipe con-tains the material with unknown permittivity dis-tribution that is being imaged. Measured capaci-tances depend on the unknown permittivity. Theimaging problem is to ‘invert’ this relationship todetermine the unknown permittivity.

ECT has been proposed for a variety of target ap-plications such as imaging dilute as well as bulkymulti-phase flows in oil refinement, in the food in-dustry, and to observe pharmaceutical and chem-ical processes [1, 2, 3, 4]. ECT systems can beimplemented with low cost, and due to their ro-bustness and small probability of failure are suit-able for operation under harsh conditions, such asimaging in combustion chambers [5] or in the pres-ence of strong external electromagnetic fields [6].

We present a solution to a problem in ECT, of

recovering the unknown shape of a single inclu-sion with unknown constant permittivity in anotherwise uniform background material [7]. Thisproblem arose in an application with the goal ofquantifying void fraction (water/air) in oil pipelines. We apply Bayesian inference to formulatethe inverse problem as an estimation problem overthe posterior distribution for unknown permittiv-ity conditioned on measured data. Key compo-nents of the formulation are a prior model re-quiring a parametrization of the permittivity anda normalizable prior density, the likelihood func-tion that follows from a decomposition of measure-ments into deterministic and random parts, andnumerical simulation of noise-free measurements.We present the posterior distribution of inclusionarea as a check on accuracy of the method.

2. INSTRUMENTATION FOR ECT

Two measurement principles are used in ECT in-strumentation [8]:

• charge or displacement-current based mea-surements (a.c. voltage, low-impedance mea-surement)

• potential based measurements (d.c. voltage,high-impedance measurement)

For both methods each electrode is designated asa ‘transmitting’ or as a ‘receiving’ electrode witha prescribed voltage being applied to transmittingelectrodes. In the charge-based method the re-ceiving electrodes are held at virtual earth with

Proceedings of the Fifteenth Electronics New Zealand Conference ENZCon 2008

24-25 November 2008, Auckland, pp 13-18.

the displacement charge being measured, whilein the voltage-based approach the receiving elec-trodes are floating with the potential being mea-sured.

The measurements used in this paper were madewith the charge-based sensor built at Graz Univer-sity of Technology [9]. Figure 1 shows the measure-ment configuration. The front-end of the charge-

I/U

Logic HF

inclusion

I/U

Logic

grounded shielding

PVC tube

electrode

grounded front-end (1 of n)

grounded front-end (n of n)

µCimage reconstruction

algorithm

HF

Figure 1: Measurement configuration for ECT.Electrodes are placed around the pipe in the imag-ing plane. Each electrode features dedicated trans-mitting and receiving hardware.

based ECT sensor consists of an input resonantcircuit, a low-noise current-to-voltage converter, abandpass filter, a logarithmic demodulator, and a24 bit analog-to-digital converter controlled by amicroprocessor. The sensor has a carrier frequencyof 40MHz and comprises two tuneable filters ad-justed by means of variable capacitances for straycapacitance compensation.

2.1 Measurement Uncertainty

The sensor front-end and the subsequent instru-mentation introduce noise sources to the measure-ment process. Applied voltage also has error, butthis error is not significant and we don’t considerthat here (see [10] for an analysis that does). To in-vestigate the robustness and repeatability of dataacquisition, the distribution of the measured dis-placement currents was examined over multiplemeasurement with a given a fixed permittivity dis-tribution [7]. Figure 2 shows the normalized quan-tile plot and the histogram for 2000 measurementsat one electrode. The measured electrode displace-ment currents exhibit noise properties that can bewell modelled as additive zero mean Gaussian withstandard deviation σ ≈ 0.07µA. The matrix ofsample correlation coefficients for all electrodes isshown in Figure 3. As can be seen, off-diagonalelements are plausibly zero, so the measurementerror covariance matrix is modelled as Σ = σ2I

where I is the identity matrix. Accordingly, thedensity for measuring d given permittivity definedby θ is the multivariate Gaussian,

π(d|θ) ∝ exp

−1

2(qm − d)T Σ−1(qm − d)

(1)

−4 −3 −2 −1 0 1 2 3 4−2.14

−2.12

−2.1

−2.08

−2.06x 10

−7

standard normal quantiles

qu

an

tile

s o

f in

pu

t sa

mp

le

receiving electrode

−2.13 −2.12 −2.11 −2.1 −2.09 −2.08 −2.07

x 10−7

0

50

100

150

200

250

displacement current [µ A]

num

ber

of sam

ple

s

Figure 2: Distribution of 2000 measured displace-ment currents on one electrode. The distributioncan be well modelled as Gaussian distribution ac-cording to the normalized quantile plot (top) andthe histogram of the displacement currents (bot-tom).

1

5

9

13

16

1

5

9

1316

0

0.2

0.4

0.6

0.8

1

corr

ela

tion c

oeffic

ient ρ

xy

Figure 3: Matrix of correlation coefficients. Off-diagonal elements are almost zero.

where qm denotes the vector of simulated displace-ment currents (or charges).

3. DATA SIMULATION

We denote the measurement region by Ω, beingthe region bounded by the electrodes and outershield. Let Γi, i = 1, 2, . . . , Ne denote the bound-ary of electrode i when there are Ne electrodes, andlet Γs be the inner boundary of the outer shield.Then the boundary to Ω is ∂Ω = ∪Ne

i=1Γi ∪ Γs.In the absence of internal charges, the electric po-tential u satisfies the generalized Laplace equationin which the permittivity ε appears as a coeffi-cient, along with Dirichlet boundary value condi-tions corresponding to the voltage asserted at elec-trodes.

For the case when electrode i is held at potential v0

while all others are held at virtual earth, then thepotential ui satisfies the Dirichlet boundary value

problem (BVP),

∇ · (ε∇ui) = 0, in Ω,

ui|Γi= v0, (2)

ui|Γj= 0, j 6= i,

ui|Γe= 0.

For brevity we have not written ε (r) and ui (r)showing the functional dependence on positionr ∈ Ω, but take that spatial variation to be im-plicit. The charge at the sensing electrode j canbe determined by integration of the electric dis-placement over the electrode boundary,

qi,j = −

∮

Γj

ε∂ui

∂ndr (3)

where n is the inward normal vector.

Our computer solution of the BVP uses a coupledfinite element method (FEM) and boundary ele-ment method (BEM) scheme [11, 7], with the re-gion being imaged using a BEM formulation cou-pled to a FEM discretization of the insulating pipeand region outside the electrodes.

4. BAYESIAN INFERENCE

In the Bayesian formulation, inference about θ isbased on the posterior density

π(θ|d) =π(d|θ)π(θ)

π(d). (4)

where π(d|θ) is the likelihood function (for fixedd) in equation 1, and π(θ) denotes the prior den-sity expressing the information of θ prior to themeasurement of d. The denominator π(d) =∫

θ π(d|θ)π(θ)dθ is a finite normalizing constant,since our formulation for ECT is fixed, that doesnot need to be calculated as we can work with thenon-normalized posterior distribution.

4.1 Representation and Prior Modelling

Our solution uses a polygonal representation of theboundary, so the permittivity is defined by the pa-rameter θ = ((x1, y1), (x2, y2), . . . , (xn, yn)) givingthe vertexes of an n-gon for some fixed n typi-cally in the range 8 to 32. We also include thetwo permittivity values, but we will omit that con-sideration here for clarity. A basic prior densityover this representation is to sample each vertex(xk, yk) uniformly in area from the allowable do-main Ω and restrict to simple polygons, i.e. notself crossing. This prior density has the form

π(θ) ∝ I (θ) (5)

where I is the indicator function for θ represent-ing a feasible polygon. That is, the prior densityis constant over allowable polygons. The changeof variables relation for probability distributionsshows that a uniform density in vertex positiongives a density over area that scales as (area)−1/2.Hence for large polygons, and inclusions, wheremost polygons are simple, this prior puts greaterweight on small areas resulting in estimated ar-eas that will always be smaller than the true area.The constraint that the polygon be simple compli-cates this picture for small area inclusions, since agreater proportion of small polygons are self cross-ing. The overall effect is that the area of largeinclusions will be underestimated while the areaof small inclusions will be overestimated, with thedivision between ‘small’ and ‘large’ depending onthe number of vertexes n. This effect necessar-ily occurs in regularized or least-squares fittingof contour-based models, since effectively the con-stant prior model is used.

Since we are primarily interested in area of inclu-sions, we explicitly specify a prior in terms of area,given in equation (6) [12]. Denote the circumfer-ence of the inclusion by c(θ) and the area of thepolygon by Γ(θ). The prior density is

π(θ) ∝ exp

−1

2σ2pr

(

c(θ)

2√

Γ(θ)π− 1

)

I (θ) (6)

in which the variance σ2pr is chosen to penalize

small and large areas.

4.2 Posterior Exploration

The product of equations 1 and 6 gives the non-normalized posterior distribution distribution thatwe explore to learn about the unknown permit-tivity. Exploration of the posterior distributionis performed using Markov chain Monte Carlo(MCMC) sampling that generates a Markov chainwith equilibrium distribution π(·|d) by simulatingan appropriate transition kernel [13].

The standard Metropolis-Hastings (MH) algo-rithm has been extended to deal with transitionsin state space with differing dimension [13], allow-ing insertion and deletion of parameters. Eventhough we do not use variable-dimension states inthe example we prefer this ‘reversible jump’, orMetropolis-Hastings-Green (MHG), formalism asit greatly simplifies calculation of acceptance prob-abilities for the subspace moves that we employ.

The reversible jump formalism considers the com-posite parameter (θ, γ) where θ is the usual statevector, and γ is the vector of random numbers usedto compute the proposal θ′. Similarly, (θ′, γ′) isthe composite parameter for the reverse proposal.

Then the MCMC sampling algorithm with MH dy-namics can be written as:Let the chain be in state θn = θ, then θn+1 is de-termined in the following way:

• Propose a new candidate state θ′ from θ withsome proposal density q(θ, θ′)

• Calculate the MH acceptance ratio

α(θ, θ′) = min

(

1,π(θ′|d)q(γ′)

π(θ|d)q(γ)

∣

∣

∣

∣

∂(θ′, γ′)

∂(θ, γ)

∣

∣

∣

∣

)

(7)

• Set θn+1 = θ′ with probability α(θ, θ′), i.e. accept the proposed state, otherwise setθn+1 = θ, i.e. reject.

• Repeat

The last factor in equation 7 denotes the Jacobiandeterminant of the transformation from (θ, γ) to(θ′, γ′).

We find a combination of M = 4 moves givesa suitably efficient MCMC in this example [7].These are translation, rotation, and scaling of thepolygon, and moving the position of one vertex ofthe polygon. The vertex move ensures irreducibil-ity but, by itself, would lead to a very slow algo-rithm. The remaining moves are designed to givean efficient algorithm. A new candidate θ′ is pro-posed from θ by randomly choosing one of thesefour moves and using a random step size λi tunedfor each move. The Jacobian term in the MHGalgorithm for the moves translation, rotation, andvertex move is 1. For the scaling move the Jaco-bian term is λ−2n+1 [7].

5. RESULTS

In the following different shaped material inclu-sions are recovered from simulated and measureddata using MCMC sampling. For all experimentsinclusions with different shapes and a permittiv-ity of εr = 3.5 in an air-filled pipe (εr = 1.0)are considered. For the experiments using syn-thetic data 2,000,000 samples were drawn fromthe posterior distribution, using a simulated dataset corrupted by noise. The data was createdusing 10,000 boundary elements in the forwardmap. For the experiment using measured data1,000,000 samples were drawn from a posterior dis-tribution. Noise standard deviation in both caseswas σ = 6.7 × 10−4.

A burn-in period of 50000 samples was found suit-able after testing the algorithm with different ini-tial states and different ratios of moves. Duringthe burn-in-period, which strongly depends on the

initial state of the Markov chain, the sampling dis-tribution is not the equilibrium distribution. Oncethe chain is in equilibrium only every 100th sampleis stored. For the results presented, the four pro-posal moves were chosen with equal probability,giving an acceptance rate of about 2%.

5.1 Experiment 1 – elliptic and fancy-shaped con-tour (simulated data)

Figure 4(a) and Figure 4(b) illustrate the resul-tant posterior variability in inclusion shape for anelliptical inclusion and a more fancy shape, respec-tively. The figures show scatter plots, constructedby plotting one point randomly from each state,uniformly in boundary length, and gives a graphi-cal display of the probability density that a bound-ary passes through any element of area. Points inthe scatter plot are clustered around the true con-tour plotted in dashed gray. Due to the decreasedsensitivity in the center of the pipe, the margin ofdeviation of scattered points for the elliptic con-tour is greater towards the center than in the re-gion close to the electrodes. For the fancy contour

increased marginof deviation

(a)

deviation

(b)

Figure 4: Scatter plots. (a) Entire domain Ω withelliptic-shaped inclusion. (b) fancy-shaped con-tour.

in Figure 4(b) deviations from the true contour inregions of low sensor sensitivity are clearly visi-ble. Furthermore, there are significant outliers in

the right part of the estimation result despite thispart of the contour being close to the boundary.The reason is that the distinct corner in the trueboundary which is not well modelled by our prior,which rejects candidate states with sharp anglesbetween two boundary elements. Note, however,that this mis-modelling in the prior is evident fromthe scatter plot in the region near the sharp corner,and indicates the significant benefit of posterior er-ror estimates available in a Bayesian analysis.

5.2 Experiment 2 – circular contour (measureddata)

Figure 5(a) shows the posterior variability in inclu-sion shape and position for measured data. Sam-ples from the posterior distribution are consistentwith the circular shape of the PVC rod. Due tothe lack of a reference measurement system, thetrue shape is not depicted. However, knowing thegeometry of the rod allows validation of the re-construction results by comparing estimates to thetrue area and circumference. The centered graycircle-shaped contour represents the initial state ofthe Markov chain. MAP state and CM state esti-mates are presented in Figure 5(b). In accordancewith the first example (same shape and proper-ties but synthetic data) MAP and CM estimatesalmost coincide indicating that the posterior dis-tribution has a well-defined single mode. Table1 summarizes the inference. Mean, standard de-viation and the integrated auto-correlation time(IACT) are evaluated for inclusion area Γ, circum-ference c and center coordinates (x, y) of the cir-cular inclusion. The integrated auto-correlationtime is, roughly, the number of correlated sampleswith the same variance reducing power as one in-dependent sample. Hence a small IACT indicatesa statistically efficient Markov chain.

6. DISCUSSION AND CONCLUSION

The predominant approach to inverse problemsin ECT is based on various deterministic itera-tive (e.g. regularized least squares) or non-iterative(e.g. linear back projection) approximations forimage reconstruction.

Inferential solutions to inverse problems providesubstantial advantages over deterministic meth-ods, such as the ability to treat arbitrary forwardmaps and error distributions, and to use a widerange of representations of the unknown systemincluding parameter spaces that are discrete, dis-continuous, or even variable dimension.

Markov chain Monte Carlo sampling (MCMC) hasrevolutionized computational Bayesian inferenceand is currently the best available technology for

(a)

Pipe

MAP

CM

(b)

Figure 5: Reconstruction results. (a) Scatter plot.Randomly chosen points of the posterior distri-bution are plotted. (b) Detail plot of point es-timates. MAP estimate (gray) and the CM esti-mate (dashed black) calculated from the posteriordistribution.

a comprehensive analysis of inverse problems, al-lowing quantitative estimates and exploration ofhigh-dimensional posterior distributions withoutspecial mathematical structure. An impedimentto the application of Bayesian analyses to practicalinverse problems lies in the computational cost ofMCMC sampling. In recent years several promis-ing algorithms and advances have been suggestedthat give substantial speedup for computationallyintensive problems including capacitance tomog-raphy. There is some hope that eventually thecomputational cost of sampling will not be sub-stantially greater than that of optimization.

We demonstrated sample-based recovery of a sin-gle inclusion with unknown shape and unknownconstant permittivity from measured and simu-lated electrical capacitance data. The accuracyof results speak for themselves.

There is now a range of well-developed tools forstochastic modelling and Bayesian inference forinverse problems. Notwithstanding application-specific modelling difficulties, applying those toolsis now a well-defined procedure. The interested

Table 1: Output statistics for circular inclusion estimated from measured dataQuantities true values mean standard deviation IACTx-coordinate of center [m] – 3.71×10−2 2.32×10−5 5.89×102

y-coordinate of center [m] – -1.14×10−2 3.02×10−5 4.65×102

Area Γ [m2] 3.14×10−4 3.13×10−4 6.88×10−6 1.10×103

Circumference c [m] 6.28×10−2 6.24×10−2 1.57×10−4 1.88×103

Log-likelihood – -46.10 1.72×10−1 3.99×102

reader can learn more from two local sources: Thispaper is a (very) abridged version of a tutorialreview that the authors recently completed [14].The modelling and computational methods forBayesian inference that we apply in this paperhave been taught for more than a decade in thegraduate course ‘Physics 707 Inverse Problems’ atAuckland University. In 2009 that course will betaught both in Auckland and Otago (under themoniker ELEC 404) with an on-line text availablevia the latter course.

7. REFERENCES

[1] D. S. Holder, Electrical impedance tomogra-phy: methods, history and applications (Se-ries in Medical Physics and Biomedical Engi-neering, IOP Publishing Ltd) 2005.

[2] T. York, “Status of electrical tomographyin industrial applications,” Jnl. of ElectronicImaging 10 pp 608-619, 2001.

[3] H. S. Tapp, A. J. Peyton, E. K. Kemsley andR. H. Wilson, “Chemical engineering applica-tions of electrical process tomography,” Jnl.of Sensors and Actuators B, 92, pp 17-24,2003.

[4] J. C. Gamio, J. Castro, L. Rivera, J. Alamil-lia, F. Garcia-Nocetti and L. Aguilar, “Vi-sualisation of gas-oil twophase flow in pres-surised pipes using electrical capacitance to-mography,” Flow Meas. and Instr., 16, pp129-134 2005.

[5] R. C. Waterfall, “Imaging combustion usingelectrical capacitance tomography,” AdvancedSensors and Instr. Systems for CombustionProcesses, pp 12/1-12/4, 2000.

[6] T. Dyakowski, “Application of electrical ca-pacitance tomography for imaging industrialprocesses,” Jnl. of Zhejiang University Sci-ence, 12, pp 1374-1378, 2005.

[7] D. Watzenig Bayesian inference for processtomography from measured electrical capaci-tance data, (PhD thesis, Graz University ofTechnology) 2006.

[8] B. Kortschak, H. Wegleiter and B.Brandstatter, “Formulation of cost func-tionals for different measurement principles

in nonlinear capacitance tomography,” Meas.Sci. and Techn., 18, pp 71-78, 2007.

[9] H. Wegleiter, A. Fuchs, G. Holler and B. Ko-rtschak, “Development of a displacement cur-rent based sensor for electrical capacitance to-mography applications,” Flow Measurementand Instrumentation, 19, pp 241-250, 2008.

[10] C. Fox, G. Nicholls and M. Palm, “Efficientsolution of boundary-value problems for im-age reconstruction via sampling,” Jnl. of Elec-tronic Imaging, 9, pp 251-259, 2000.

[11] B. Kortschak and B. Brandstatter, “A FEM-BEM approach using level-sets in tomogra-phy,” Jnl. for Comp. and Math. in Electricaland Electronic Eng., 24, pp 591-605, 2004.

[12] D. Watzenig and C. Fox, “Posterior variabil-ity of inclusion shape based on tomographicmeasurement data,” Jnl. of Phys.: Conf. Se-ries, 6th Int. Conf. on Inverse Problems inEng.: Theory and Practice, Paris, France,June 15 - 19, 2008 (in press).

[13] P. J. Green, “Reversible jump Markov chainMonte Carlo computation and Bayesianmodel determination,” Jnl. Biometrika, 82,711-732, 1995.

[14] D. Watzenig and C. Fox, “A review of sta-tistical modelling and inference for electri-cal capacitance tomography,” Meas. Sci. andTechn., 2008 (invited review, submitted).