RECOVERY OF INCLUSION SHAPE BY STATISTICAL INVERSION ...

c 2006 Institute for Scientific ° Computing and Information

INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 2, Number 4, Pages 469–483

RECOVERY OF INCLUSION SHAPE BY STATISTICAL INVERSION OF NON-STATIONARY TOMOGRAPHIC MEASUREMENT DATA DANIEL WATZENIG Abstract. The estimation of the cross-sectional material distribution inside a pipe in a non-stationary process given uncertain measurements is a challenging issue arising in various areas where flow structures composed of different material phases occur. This paper addresses the recovery of the shape and the current position of a moving material inclusion in an otherwise uniform background material from Electrical Capacitance Tomography (ECT) data. The unknown time-dependent quantities are jointly estimated applying a Particle Filter (PF) approach incorporating new measurements in each measurement update step. Consequently, the inverse ECT problem is recast as a state estimation problem consisting of a stochastic state propagation equation and a nonlinear measurement equation. The measurement update, i.e. the ECT forward problem, is solved by means of the Boundary Element method (BEM). In order to cover various physically reasonable shapes, the inclusion boundaries are modeled using a second order Fourier contour model. The novelty of this paper lies in the PF-based solution of the non-stationary inverse ECT problem using experimental data and in the treatment of potentially multi-modal probability densities of the state variables. This is particularly useful to allow for dynamic events such as splitting of inclusions. The proposed approach is validated for different test objects. Key Words. electrical capacitance tomography, statistical inversion, nonstationarity, geometric regularization.

1. Introduction Electrical Capacitance Tomography (ECT) aims at monitoring and evaluating complex dynamic industrial processes such as heterogeneous mixtures of multiple phases and components with different electric properties based on non-invasive and non-intrusive sensing principles [1]. ECT has gained importance to visualize industrial processes due to its low cost, good dynamic behavior and operability under harsh environmental conditions (i.e. high pressures and/or high temperatures) [2]. Motivated applications of ECT can be found in oil exploitation [3], pneumatic conveying [4], and in the chemical industry [5]. ECT aims at spatially resolving electrical properties of heterogeneous multi-phase flows in inaccessible pipes and vessels by measuring the capacitances between certain electrodes that are arranged around the periphery. Based on these boundary measurements the permittivity distribution inside the pipe can be estimated. Approaches to solve the ECT problem for static scenarios have been reported in the literature [6, 7, 8]. However, in Received by the editors April 14, 2006. 2000 Mathematics Subject Classification. 35R35, 49J40, 60G40. 469

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several industrial applications the physical quantities and hence the measured data are functions of time. Such an inverse problem is referred to as a non-stationary inverse problem [9]. The knowledge of the current state of a non-stationary industrial process is a necessity in order to increase process efficiency and product quality, respectively. In this paper the inverse ECT problem, i.e. the estimation of the permittivity distribution given uncertain measurements, is recast as a state estimation problem. The state equation is modeled by a stochastic differential equation which describes the state transition between consecutive time steps. The nonlinear measurement equation relates the unknown system states to measured quantities given measurement uncertainties. The estimation of the time-varying system state is a filtering problem which is commonly solved by means of Kalman Filters (KFs). KFs are restricted to uni-modal probability density functions, linear transition and measurement processes, and Gaussian process and measurement noise sources. The standard KF-based state estimation can be extended to nonlinear state transitions and measurement models leading to the Extended Kalman Filter (EKF) which linearizes the most recent state estimate. EKFs are still restricted to uni-modal state densities. Applications of linear KFs and EKFs to solve the non-stationary inverse problem have been reported in Electrical Impedance Tomography [10, 11, 12] and ECT [13]. However, the occurrence of multiple object hypothesis as frequently encountered in process tomography cannot be tackled by algorithms that are restricted to uni-modal state densities. Unlike KFs, Particle Filters (PFs) generalize the recursive Bayesian concept by using a probabilistic representation of the state transition and the measurement, respectively. Within the PF framework all the parameters included are modeled as random variables with a certain joint probability distribution. This randomness may be interpreted as parameter variability as it is related to the uncertainty of their true values. In particular, PFs are able to deal with multi-modal densities of the state variables by means of a numerical approximation. Related to the reconstruction of two-phase flows this enables to provide a physically meaningful transition model of material anomalies and their evolution in time including merge and split transitions. The applicability of PFs to solve the non-stationary inverse problems given synthetic data has already been treated for the magnetoencephalography (MEG) problem [14] and for the ECT problem [15, 16]. In this work the applicability of the PF to recover the object shape and to track the moving object from non-stationary tomographic sparse measurement data is discussed and validated for different test objects representing a perturbing material inclusion in air. In addition, the natural phenomenon that inclusions within the pipe are appearing, disappearing and tend to merge or split due to a certain flow velocity inside the pipe is treated. Compared to the KF, which solves the forward problem and the adjoint problem once in each time step, PFs suffer from increased computational cost due to the management of samples in the state-space corresponding to multiple solutions of the forward problem in each time step in order to approximate the state probability density. As a main advantage over the EKF the output of the PF is a set of samples that is used to approximate the posterior distribution rather than a single point estimate. From this sample set any point estimate, such as the Maximum A Posteriori (MAP) estimate, the conditional mean and the median of the state, can be calculated. Furthermore the posterior distribution gives information about confidence intervals of the estimates even in the

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case of non-Gaussian state and noise distributions as well as summary statistics of sought parameters. The paper is structured as follows: Section 2 briefly describes the mathematical modeling of the forward map for the given sensor geometry. The proposed recursive Bayesian filtering approach to solve the non-stationary inverse ECT problem is introduced in Section 3 as well as the used state-space representation by means of truncated Fourier descriptors. The used ECT measurement setup, which is based on the measurement of displacement currents in order to acquire data for different experiments, is addressed in Section 4 followed by the evaluation of the proposed statistical inversion approach with measured data for both the static and the dynamic case in Section 5. 2. Forward modeling In ECT, the forward problem consists of determining the inter-electrode capacitances between certain electrodes at the boundary ∂Ω of an object Ω given the permittivity distribution inside, the sensor geometry, the electric excitation potential and appropriate boundary conditions. In this work only Dirichlet boundary conditions are considered since in each measurement step the driving electrode with its surface Γ0 is excited with a certain voltage u0 and the remaining electrodes with their corresponding surfaces Γe as well as the surrounding shielding are grounded. The forward problem, i.e. the determination of u given the permittivity ε of the medium inside Ω, is then given by (1)

∇ · (ε∇u) = 0 u|Γ0 = u0 u|Γe

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where ε denotes the dielectric permittivity which is given by ε = ε0 εr with ε0 is the permittivity of vacuum and εr is the relative dimensionless permittivity of the material. In the general case ε is a frequency dependent tensor quantity. However, when the medium in the object Ω is assumed to be linear and isotropic the tensor reduces to a scalar value. The forward problem can be restricted to the electrostatic case assuming that only non-conductive materials are involved. The problem defined in Equation (1) can be solved by using the BEM to compute the potential and its normal derivative at any point in the domain Ω and on its boundary ∂Ω. In this paper the permittivity is assumed to be piecewise constant, i.e. a material inclusion is a homogeneous subdomain of Ω. Within this subdomain ∇ · (ε∇u) simplifies to Laplace’s equation ∇2 u. Consequently, the BEM can be solved for each subdomain separately. Figure 1 illustrates an elliptic inclusion (εr,2 ) in an otherwise constant material with permittivity εr,2 . In order to solve the forward problem by means of the BEM, domain boundaries are approximated by N piecewise linear segments. The resultant system of equations is given by ∂u ∂n where H and G are of order N × N and are called influence matrices [17]. Using the BEM in the region of interest increases the spatial resolution, since only region boundaries are discretized instead of discretizing the entire domain Ω. However, a compromise between computational cost and quality of approximation, i.e number of used boundary elements, has to be found. A detailed description of the BEM applied to the ECT forward problem can be found in [18]. (2)

Hu = G

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Figure 1. Cross-section of the pipe. Elliptic inclusion with permittivity εr,2 in a background material εr,1 .

3. Statistical inversion using particle filtering The reconstruction of the spatial permittivity distribution inside the sensor is an ill-posed problem subjected to sensor noise. In this work the main focus is on the reconstruction of objects with sharp boundaries. The boundary between two distinct permittivity regions represents a closed contour C in R2 , which can be described by means of a vector-valued function R → R×R : s 7→ c(s) parameterized by s ∈ [0, 1]. The Cartesian coordinates (x, y) of a point on the contour are given by the elements of c(s) = (x(s), y(s))T . The representation of closed contours by both B-splines and Fourier descriptors with application to ECT have been investigated in [15]. In contrast to B-spline models Fourier descriptors allow to recover both local contour properties like corners and indentations as well as global properties like translation of the entire contour. The Fourier contour model is given by (3)

x ˆ(s) =

N −1 ax,0 1 X + (ax,i cos(2πis) + bx,i sin(2πis)) 2 2π i=1

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N −1 1 X ay,0 + (ay,i cos(2πis) + by,i sin(2πis)) . 2 2π i=1

where ax,i , ay,i , bx,i and by,i for i = 0 . . . N -1 are the system states which describe the boundary of a material inclusion within the domain Ω. The estimation of the system states at different time instants given uncertain measurements can be solved by a recursive Bayesian filtering approach such as the PF algorithm which incorporates all available information by means of measurements zk into an recursively updated state estimate x ˆk . PFs have gained importance in the last decade and consequently various fields of application can be found in the literature including visual localization, ballistic tracking, phase tracking and detection of signals in noise. The PF numerically approximates the potentially multi-modal probability density function (pdf) of the state vector using the principle of stochastic sampling the required pdf is represented by a scatter of particles which propagate through the state-space. The set of N particles x(m) , which are randomly chosen from the state-space and their respective weights w(m) , are used to approximate the pdf


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according to Equation (5). (5)

fx (x) ≈ {x(m) , w(m) }m=1,...,N

Assuming that the underlying process represents a first order Markov chain, the state transition in Equation (12) can be reformulated as conditional probability density p(xk |xk−1 ). Furthermore, the measurement model in Equation (13) is expressed by p(zk |xk ). The PF keeps track of the current state estimate represented by p(xk |Zk ), where Zk = {z1 , . . . , zk } denotes the history of measurements acquired up to time step k. In the following a brief introduction to PF-based dynamic state estimation is given. For a general introduction to the topic the reader is referred to e.g. [19, 20, 21]. The basic processing steps involved in the estimation process are: Prediction: At each iteration of the PF the transition model is used to obtain a state prediction p(xk |Zk−1 ) using the Chapman-Kolmogorov equation Z p(xk |xk−1 )p(xk−1 |Zk−1 )dxk−1 , (6) p(xk |Zk−1 ) = Ω T

where Zk−1 = (z0 , . . . , zk−1 ) denotes the history of all measurements acquired up to time k − 1. Numerically the conditional density p(xk |Zk ) is represented using (m) (m) the sample set S = {xk , wk } where m = 1, . . . , N . The number of samples used to represent the state density is crucial for real-time operation. In general a trade-off between the quality of approximation and processing time is required. In this work a residual resampling step of O(N ) is applied to ensure a good approximation using as few particles as possible [19, 22]. A comparison of different resampling strategies including residual resampling, minimum variance resampling and sampling-importance resampling can be found in Douc et al. [23]. Measurement Update: In the second step within the PF algorithm, the measurement model p(zk |xk ) is used to estimate the posterior density p(xk |Zk ) by straight forward application of Bayes’ theorem: (7)

p(xk |Zk ) =

p(zk |xk )p(xk |Zk−1 ) p(Zk )

The measurement likelihood function is modeled by a multivariate Gaussian distribution (cf. Section 4), ½ ¾ 1 T −1 (8) p(zk |xk ) ∝ exp − (Qm − zk ) Σ (Qm − zk ) , 2 where Qm denotes the vector of electric charges, which is directly proportional to the vector of measured displacement currents. The set of weighted samples generated by the PF is used to approximate the posterior distribution. From these samples, any estimate of the system state can be calculated. In this work the expected value of the state given by (9)

ˆ k = E{xk |Zk } ≈ x

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4. ECT measurement setup Two measurement methods to tackle the ECT problem have been developed in the literature – measuring electric charges (cf. [6]) and measuring electric voltages [18]. A comparison of the two different measurement principles is given in [24]. For both methods electrodes are excited with a prescribed voltage in turn. While in the charge-based method the other electrodes are grounded, i.e. set to zero potential, the remaining electrodes in the voltage-based approach are floating. The sensor used in this work is based on the measurement of displacement currents [25]. By measuring the displacement currents, which depend on the materials inside the corresponding charges at the electrodes, can be computed based on the mathematical model described in Section 2. For the i-th electrode the electric charge Qi over the electrode surface Γe is given by Z ∂u (10) Qi = ε ds ∂n Γe where n denotes the inward normal vector. The objective in ECT is to minimize the difference between the vector of measured charges Qm and the vector of computed charges Qc . Figure 2 illustrates the schematic of the principal hardware components and the measurement configuration. The cross-section of a PVC pipe with a diameter of 100 mm is used as measurement plane. The 16 measurement electrodes are equally spaced around the surface of the pipe. The electrodes are 16 mm wide with a spacing of 21 mm and have a length of 40 mm. The setup is protected from electromagnetic interference by a grounded outer shield. Every single electrode can be alternately used as a transmitter and a receiver. The front-end electronics of an electrode consists of a transmitting amplifier and an input stage comprising a current-tovoltage converter, a bandpass filter, and a high frequency peak rectifier. Due to the electronically tunable input filter a high immunity against stray capacitances is achieved. A excitation frequency of 40 MHz is used for the displacement current measurements. Every electrode of the ECT sensor is individually controlled by a microprocessor and the 16 front-ends are connected to the signal preprocessor unit via a serial link. To allow for efficient image reconstruction, a PC is connected to the setup via local area network.

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Figure 2. Measurement configuration and schematic of the used ECT sensor.


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A single measurement frame consists of 16 projections, according to the 16 available transmitting electrodes. For one projection a specific electrode acts as transmitter while all the others sense the displacement current. A measurement frame consequently consists of 16 · 15 = 240 entries. The first 90 displacement current values of a measured frame are shown in Figure 3. The curve is crown-shaped because the coupling between the transmitting and the adjacent receiving electrodes is much higher than between transmitting and opposite electrodes. The mutual capacitances between the electrodes are in the range between 1 fF to 5 pF. This means that at the same time very small capacitances need to be measured and a high dynamic range is necessary, posing high demands on the sensor hardware. The existing setup features frame rates of 100 frames/s and a linear measurement range of 60 dB. Figure 4(a) depicts the relative error between mathematical model and data acquired from the measurement system for an empty pipe. According

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to the normalized quantile plot in Figure 4(b), the model error is Gaussian distributed which justifies the multivariate Gaussian measurement model introduced in Section 3. Subsequently, the offset error is corrected for the empty pipe. By comparing simulated and measured data for a well-defined target – a centered PVC rod – the gain error between model and measurement is corrected. 5. Measurement results In order to validate the proposed approach the following experiments have been performed: • Static reconstruction of a PVC rod of a diameter of 20 mm in an air-filled pipe. • Estimation of the position and the shape of a moving PVC rod with a diameter of 50 mm • Estimation of the position and the shape of a moving PVC rod with a diameter of 20 mm • Reconstruction of two targets simultaneously – each of it has a diameter of 20 mm The reconstructions were carried out using a second order Fourier contour model. This restriction of the model complexity results in a geometric regularization of the contour. According to the general Fourier contour model represented by equations (3) and (4) introduced in Section 3, the state vector xk is comprised of (11)

xk = [ax,0 ay,0 ax,1 ay,1 bx,1 by,1 ax,2 ay,2 bx,2 by,2 ]Tk .

Taking further into account that the contour evolves over time, the state-space representation of a contour including its dynamics is defined by (12) (13)

xk zk

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where f (·) represents the state transition of the state x from time k − 1 to time k subjected to process noise v. Assuming a random walk transition model Equation (12) simplifies to (14)

xk = xk−1 + vk−1 .

The nonlinear measurement equation h(·) relates the observed boundary quantities zk to the state vector xk and the measurement noise w. The nonlinear measurement equation is given by the forward problem defined in Section 2. In order to improve the convergence behavior of the PF the state vector xk is segmented into the translation xT = [ax,0 ay,0 ]Tk and the deformation part xD = [ax,1 ay,1 . . . bx,2 by,2 ]Tk where different process noise levels for each part are assumed, (15)

xk = (xTT , xTD )Tk .

A standard deviation of 2× 10−3 is assumed for the process noise of the translational part. A standard deviation of 1× 10−4 is prescribed for the noise level which is associated with the deformation part of the dynamic contour. 5.1. Shape estimation of a static target. In this experiment 30 particles are used to reconstruct a semi-circle-shaped inclusion (PVC, εr = 3.5) in air (εr = 1.0). The PF is initialized with circular contours. After 10 iterations the PF has converged to the right position and shape. Figure 5(a) depicts different samples of the posterior distribution whereas the different gray levels correspond to the


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Figure 5. Static reconstruction of a PVC rod with a diameter of 20 mm. (a) Estimation result. (b) Convergence of the measurement log-likelihood for different initial states of the PF. (c) Histogram of sample areas at final iteration. (d) Histogram of sample circumference at final iteration. associated weights. The bold black contour denotes one single estimate – the sample mean. Figure 5(b) shows the evolution of the measurement log-likelihood for three different initial states of circular shapes with diameters of 1 cm, 1.2 cm and 0.5 cm. As a result, the PF needs more iterations if the initial state is far from the shape to be reconstructed. For all three initial states the right position was estimated after about 5 iterations. The shape estimation depends strongly on the initial state of the PF. Figure 5(c) depicts the estimated areas for all involved particles at the final iteration, Figure 5(d) illustrates the estimated circumferences for the samples at the final iteration. The posterior variability for both parameters - area and circumference - is summarized in Table 1. Table 1. Posterior variability of included particles at final iteration. Quantities Area [m2 ] Circumference [m]

true values 3.14×10−4 6.28×10−2

mean 3.08×10−4 6.23×10−2

standard deviation 3.51×10−6 3.54×10−4

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5.2. Single object tracking. This section presents reconstruction results for a single moving target using measured data acquired from our ECT sensor in order to validate the Bayesian filtering approach for object tracking. Figure 6 depicts the measurement setup for dynamic measurements. Two stepping motors, which are controlled via a PC, are used to move the PVC rod within the pipe.

Figure 6. Measurement setup for single object tracking. The two stepping motors are mounted at the top of the pipe. A circular inclusion with a diameter of 50 mm is translated from the left to the right using a stepper motor (cf. Figure 7(a)). Within the performed move measurements are acquired at eight different positions spaced along the x-axis. In each position 10 measurement frames are collected and provided to the reconstruction algorithm. By using a smaller step size the PF requires less than 10 iterations to converge to the new position. In Figure 7(b) the evolution of the shape center of the estimated object position is shown. The references of the start and the stop

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Figure 8. Sample variability at different positions. (a) Small sample variance at the boundary of the pipe. (b) Larger sample variance in the center of the pipe where the sensitivity is typically small compared to regions close to the electrodes. The bold black contours denote the sample mean while the gray-shaded contours depict the involved particles associated with their respective weights. positions are known. Deviations in x-direction and y-direction in each position are below 5% with respect to the diameter of the pipe. This is a quite remarkable result since the spatial resolution of tomographic sensor systems is usually between 5-10% [3]. Figure 8(a)and Figure 8(b) emphasize the typical overall sensitivity of soft-field tomographic sensors. The estimated sample variance of a target at the boundary of the pipe is small compared to the sample variance of the reconstructed target in the center of the pipe. The progression of the measurement log-likelihood is depicted in Figure 9(a). Iterations, during which the object was repositioned,

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are indicated by triangular markers. Position changes implicate distinct changes in the log-likelihood function. In accordance to the typical overall sensitivity of tomographic measurement systems the deterioration in the log-likelihood is significant at positions close to the boundary. The last experiment shows the tracking result of a circular inclusion that is moved 25 mm in y-direction (cf. Figure 9(b)). This experiment indicates an accurate reconstruction even for small objects. 5.3. Identification of two targets. The last experiment shows the possibility to track multiple targets due to the proposed multi-model state-space representation. To verify the multi-modality, one PF was applied to reconstruct two different targets. Figure 10 clearly illustrates the multi-modal posterior state density for the entry in the state vector which corresponds to the y-position of the contour over iterations k. This result indicates the presence of two targets inside the pipe. Consequently, a second PF has to be initialized. First results to use a filter bank to track multiple targets have been published recently by the authors for an electric potential based ECT sensor [16].

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Figure 10. Evolution of posterior distribution p(xk |zk ) of the (2) second state parameter xk (y-coordinate of the target) over time updates k. The multi-modal distribution indicates the presence of two or more inclusions. Figure 11 depicts the reconstruction result of two targets. Two PF are used to estimate the targets. In order to achieve repeatable conditions for the splitting algorithms two PVC rods have been used in the following two-stage experiment. In the first step one PVC rod was placed into the pipe. Once the PF has converged to a steady state, a second PVC rod of the same size is placed into the pipe. Rather than spreading particle equally within the state-space, 30 particles that represent circular contours of a constant diameter are deliberately placed across the cross-section of the pipe. This physically meaningful strategy helps to decrease the number of particles considerably that is required to initialize the filter. The gray circles depict the true positions of the targets to be reconstructed. After 30 iterations (cf. Figure 11(d)) both position and shape of both targets are reconstructed reasonably well. The splitting procedure over time evolution, i.e. the required initialization


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Figure 11. Multi-modal state transition. The tracking results for (a) 2 (b) 4 (c) 8 and (d) 30 iterations are plotted. The gray contours denote the true targets while the black bold shapes correspond to the estimated contours. of a second PF, is based on the information of the measurement likelihood function p(zk |xk ). A certain threshold level is defined (log(p(zk |xk ))=-40) to enable or disable the second PF. Once the second PF is activated, the number of particles is increased to 50 to ensure filter convergence. The number of iterations has also to be extended due to the higher complexity since the state-space variables are doubled. This approach is even feasible in the presence of more than two targets since log(p(zk |xk )) remains at small values for a bad data fit. Table 2 summarizes the estimation results for both targets. Estimated areas and circumferences where chosen to be significant parameters for the reconstruction quality. As a result the sample means of the estimated parameters are quite close to their true values and the corresponding sample variance is small. Table 2. Variability in estimated areas and circumferences. Quantities Area of upper circle [m2 ] Circumference of upper circle [m] Area of lower circle [m2 ] Circumference of lower circle [m]

true values 3.14×10−4 6.28×10−2 3.14×10−4 6.28×10−2

mean 2.91×10−4 6.22×10−2 2.81×10−4 6.11×10−2

standard deviation 5.23×10−6 5.87×10−4 8.91×10−6 9.73×10−4

482

D. WATZENIG

6. Conclusion This contribution addresses a recursive Bayesian filtering approach to solve the non-stationary inverse ECT problem. A PF based reconstruction algorithm is applied to measured data provided by an ECT sensor measuring displacement currents in order to reconstruct static as well as non-stationary material inclusions in an otherwise uniform background material. Assuming a random walk motion model simple convex contours are tracked over time. The multi-modal state-space representation of the PF is particularly useful to model split and merge transitions, which typically occur in industrial processes. The applicability of the proposed PF algorithm to reconstruct single time-varying and multiple circular inclusions is demonstrated using experimental data. Sample mean and sample variance of meaningful parameters are given in order to show the reliability of the estimation results. The performed experiments using known objects and known positions indicate that the reconstruction performance is suitable for tomographic purposes. Beside the advantage of PFs over KFs to deal with any non-linearities in the the state transition and the measurement equation, PFs produce a set of samples that can be used to compute summary statistics of the sought parameters. Furthermore, single point estimates, their reliability as well as certain confidence intervals can be calculated. Future work will concentrate on reconstructing non-stationary inclusions with more complex shapes. Since the forward problem in each iteration has to be solved several times – depending on the number of particles – special focus will be put on fast solver for the BEM.

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