Neural network for pattern association in electrical capacitance tomography

A.Y. Nooralahiyan B.S. Hoyle N.J. Bailey

Indexing terms: Neural network, Pattern association, Electrical capacitance tomography

Abstract: The paper describes the basic principles of an artificial neuron, the multilayer perceptron network and the back-propagation training algo- rithm, applied to electrical capacitance tom- ography systems for real-time, noninvasive imaging and measurement of multicomponent flows such as gas/oil and water/oil. Particular attention is given to the problems of distortion of the field ('soft field' error) and limitations on spatial resolution (imposed by the number of electrodes) in conventional image reconstruction algorithms for current systems. In addressing these issues, for the first time, an artificial neural network is employed to replace conventional image reconstruction algorithms. The system con- sists of a simulation program for a single layer multiple output network, using a variant of the back-propagation training algorithm with the principle of pattern association. The input vector consists of preprocessed capacitance measure- ments, and the output of the network directly cor- responds to the spatial image. Two similar networks are trained for gas/oil flow (small differ- ence in permittivity) and water/oil flow (large dif- ference in permittivity) with results compared. The results illustrate the feasibility of such networks for relatively accurate image reconstruction, without the difficulties associated with conven- tional algorithms.

1

Artificial neural networks (ANNs) are assemblies of many simple processors (nodes or neurons) with a large connec- tivity between them. A model of a neuron is shown in Fig. 1. The input to a neuron is either the output from other neurons or the input vector appropriately weighted by the strength of the connection.

The response of the jth neuron to an input vector (xo, xl, . . . , xJ, is an activation output level (oj) which is formed by passing a linearly weighted sum of inputs (sj), through a nonlinear sigmoid function. An extra input (x,,) is defined for the purpose of biasing the neuron. xo is always set to be on (xo = l), multiplied by a weight equal

Introduction to artificial neural network

0 IEE, 1994 Paper 119OG (ES), first received 2lst June 1993 and in revised form 25th February 1994 The authors are with the Department of Electronic & Electrical Engin- eering, University of Leeds, Leeds LS2 9JT, United Kingdom

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to minus the threshold value and added in with all the other inputs. Thus, the output of the neuron is a nonlin- ear function of the weighted sum of all the inputs (eqn. 1).

" oj =f(sj) where s - 1 w. .x . (1) j - i = o 'J '

and

where k is a positive constant that controls the spread of the function:

lim f(sj) = 1 (heaviside function) k-m

The sigmoid function limits the dynamic range and acts as an automatic gain control, the steep slope provides high gain for small signals, thus the net can accept large inputs and still remain sensitive to small changes. It is also continuously differentiable, with the first derivative as a simple function of the output (eqn. 2); an important computational consideration in the development of the training algorithm.

In general:

The most widely used neural network is the multilayer perceptron (MLP). The architecture of an MLP consists of an input layer, one or more hidden layers and an output layer. In a fully connected net (Fig. 2), each neuron in the input layer is connected to all the neurons in the first hidden layer, and each neuron in the hidden layer is connected either to all the neurons in the next hidden layer or to all the neurons in the output layer, with no intralayer connections. The function of neurons in the input layer is purely distributive with no computa- tional overhead, and that of hidden layers and output layer perform similar computation to eqn. 1.

WO+ (O=threshold value)

xo-7

I

xn

Fig. 1 Basic model of neuron

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MLP is a supervised network, trained with back- propagation training algorithm for the purpose of pattern classification or pattern association. The training

ilp layer olp layer

Fig. 2 Structure oJMLP

is often more computationally intensive than the forward evaluation of the output. The algorithm was developed in 1986 by Rumelhart, McClealland, and Williams and it is called the generalised delta rule or the back-propagation rule [I-31. This algorithm requires the units to have non- linear thresholding function that are continuously differ- entiable. The simplest form of the back-propagation algorithm is as follows:

(i) Initialise weights and thresholds to some small random values.

(ii) Present input X = xo, xl, ..., x , - ~ and target output T = t o , t , , ..., tm-l where n is the number of input nodes and rn is the number of output nodes. Set wo = -0 (bias), and xo = 1.

(iii) Calculate actual output of all the nodes in each layer (oj in eqn. l), and pass oj as input to the next layer until the output layer is reached.

(iv) In the output layer for the jth node, an error term Sj is calculated:

(3) aj = kOj(l - oj)(tj - O j )

(v) Adapt weights starting from the output layer and

(14) where wij represents the weight from node i in the layer below, to the node j in the output layer, and q is a con- stant known as the learning rate. Eqn. 4 is only valid for the output layer since it is not obvious what the target output of a hidden node should be. Thus eqn. 3 is altered and the term (ti - oj) is replaced by the summation over all the k nodes in the layer above node j.

(5 )

The propagation continues until the input layer is reached. Hence the name back-propagation. Training is easily implemented in software using computationally intensive facilities.

In this study, a single layer multioutput network is employed, with no hidden layers, thus eqn. 5 is only men- tioned here for completeness. Also, a variant of the back- propagation is used, extending the algorithm to include a momentum term, a in the weight updating equation:

working backwards :

W i j ( t + 1) = W i j ( t ) + ‘ la joj

6j = koj(1 - oj) C 6, wjk k

wij(t + 1) = w,(t) + qdjoj + a Awij(t)

where

Aw,(t) = wij(r) - w,(t - 1) 0 < a < 1

2

Electrical capacitance tomography (ECT) is a recently developed technique for imaging multicomponent pro-

Case study: electrical capacitance tomography

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cesses where traditional instrumentation cannot be used C4-q. For example, the measurement of flows such as gas/oil and water/oil in pipelines is economically impor- tant, but cannot be done reliably with turbine or venturi instruments. One such need is for ‘well testing’, where the flow is temporarily separated into its component oil, gas and water parts for measurement. Such tests reveal the state of an individual well in terms of lifetime and bore. This technique is regarded as expensive, time consuming and inefficient, since each incoming flow has to be tested in turn, and the information, although useful, can not show transient variations. ECT systems provide a pos- sible alternative which gives the advantage of straightfor- ward and continuous monitoring of several wells simultaneously. It is noninvasive, relatively low in cost, promises to be highly reliable and, importantly, does not depend upon hazardous ionizing radiation.

A typical ECT imaging system (Fig. 3) consists of a set of primary sensors, an electronic multiplexer and an image reconstruction computer. The sensing electrodes are placed on the periphery of the pipe. The electrodes are connected to a multiplexer which enables distinct measurements of capacitance for all unique combinations of pairs of electrodes. The measured data is then fed to the computer to reconstruct the time-frame cross- sectional image of the pipe.

Fig. 3 ECT imaging system

Capacitance electrode systems inherently have a non- uniform sensitivity distribution over the pipe cross section. This results in measurement errors due to elec- trode response to different permittivity distributions. This problem is particularly true for different flow regimes such as core flow, annular flow and stratified flow. Thus, a system is required with a response independent of flow regime. Mathematical modelling techniques such as the finite element method (FEM) based on electrostatic field theory are employed to model the field pattern and to investigate complex capacitor sensor configuration for design purposes [SI. The optimum design of electrodes results in minimum deviation from linearity in the pres- ence of multicomponents of flow.

It is also essential that the measuring circuit is immune from stray capacitance between the measuring electrodes and earth. One successful stray-immune technique is the use of switched-capacitor charge transfer circuit with a switching frequency of up to 2 MHz [SI. This technique offers a fast data capture rate and, due to its simple design, parallel measurement channels are used to increase the speed further without significant increase in cost and complexity.

2.1 Conventional image reconstruction technique In electrical capacitance tomography, a forward problem is initially solved to determine capacitance values for known dielectric distributions within the cross section. The theoretical analysis of a given capacitance system is characterised by Laplace’s equation (eqn. 6), and the associated (Dirichlet) boundary conditions.

v c E ( x , Y ) V 4 ( X > Y)l = 0 (6)

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where $(x, y) is the spatial potential distribution, and E(x, y) is the spatial permittivity distribution. Eqn. 6 is initially solved to determine the spatial potential distribu- tion $(x, y). However, due to irregularity of the flow dis- tribution E(X , y), eqn. 6 has no general analytical solution. Thus, a numerical method based on FEM is used to divide the cross-section into P quadrilateral elements cor- responding to Q nodes [4]. The resultant nodal poten- tials are used to determine the capacitance between an electrode pair i-j, by performing the following integration numerically [4] :

1 ci = - 7 E(& Y)V$(X , y ) lii (7) ii

where V is the voltage applied to the source electrode and Li is the curve encompassing the detecting electrodes. The potential distribution $(x, y) is dependent upon the dielectric distribution (eqn. 6). This interdependence is referred to as the 'soft field' characteristic of capacitance sensors.

The image reconstruction aims to solve the inverse problem to determine the distribution of dielectric con- stant E(x, y) within the cross-section, from a limited number of measurements. For an eight-electrode system the number of independent two-electrode measurements (N) is 28, N = n(n - 1)/2. Once again, due to the irregu- larity of the two-component flow, there is no general ana- lytical solution for the inverse of eqn. 7, thus approximations are often made for a numerical solution. The FEM represents the pipe cross-section as an array of sensitivity values, each value corresponding to the magni- tude of capacitance change for variation in permittivity at that pixel (p). This precalculated 'sensitivity matrix' Si, j(p) and the normalised capacitance measurements Ai, are used to obtain a grey level image G(p) (eqn. 8), approximating the cross-sectional dielectric distribution E ( X , Y ) ~41 .

"-1 " 1 Ai, js, j(P)

c i si,j@) i = l j = i + l

(8) G @ ) = i = l j = i + l

"-1

where n is the number of electrodes, and Si,j(p) is the capacitance sensitivity of pixel p for electrode i-j:

Ci, j(P) - Ci, j(emPtY pipe) Ci, j(fu11 pipe) - Ci, j(emPtY pipe) Si, j ( P ) =

3 Description of the problem

There are two major issues concerning the image recon- struction process; namely, the problem of the soft field error, and the dependency of spatial resolution on the design and the number of the electrodes in the system.

in the flow. For two-component flows with similar per- mittivity, such as gas and oil with relative permittivities of 1 and 3, respectively, the above assumption is con- sidered reasonable and conventional reconstruction algo- rithms cope well. However, for flows with relatively large difference in permittivity, such as water and oil (with rela- tive permittivities of about 80 and 3, respectively), the field is distorted and the resulting images contain signifi- cant errors.

Iterative algorithms have been investigated to over- come such distortions [8, 91. The iteration starts with the initial estimation of permittivity distribution and the forward solution produces a sensitivity distribution. The inverse solution then projects this distribution back, to obtain the permittivity distribution. Typically, the algo- rithm tries to minimise the error (difference between the calculated and the estimated permittivities) by updating the permittivities. The process is computationally inten- sive, and the rate of convergence is critical for a real-time system. In some cases, where the initial estimate is poor, convergence may not be achieved. Although the theoreti- cal basis for iterative methods are well defined, in general they have not proved to be very successful, to date.

3.2 Restrictions on spatial resolution The grey level of an image with relatively poor resolution can provide very useful data, especially when no other nonintrusive methods can be applied other than tom- ography. With current ECT systems, spatial resolution is dependent on the number and design of electrodes. Thus, the spatial resolution of a 1Zelectrode system (projection N = 66) is better than eightelectrode system (projection N = 28). However, the increase in the number of elec- trodes is restricted by the capability of sensor hardware to resolve smaller capacitance values. This means that increasing the number of electrodes to, say, 16 or 24 elec- trodes would, at best, result in diminishing returns. Also, as the number of electrodes increases, so does the compu- tational overhead; the complexity of measurement cir- cuitry; and particularly the sensitivity specification of the sensor electrode, whereas, in an eight-electrode system with an ANN approach, the resolution of the image is only restricted by the capability (sensitivity specification) of the electrodes to detect capacitance changes. If these changes are significant enough to form the basis for detectable features in the data set, the network may be trained to extract and associate these features with corre- sponding output patterns.

4 Neural network based solution

In Fig. 4 the conventional reconstruction algorithm is replaced with a trained neural network. A single layer multioutput network (SLMON) is used, with 28 neurons in the input layer (corresponding to 28 distinct capac-

3.1 The soft field error Image reconstruction algorithms for capacitance tom- ography assume that the effect of the dielectric configu- ration on the sensitivity distribution of the sensors is insignificant (i.e. the sensitivity distribution is calculated with respect to uniform (single component) permittivity distribution). Due to this assumption, the resulting image contains an error referred to as the 'soft field' error (see Section 2.1). This error is proportional to the difference between the relative permittivities of the two components

IEE Proc.-Circuits Devices Sysf., Vol. 141, No. 6, December 1994

0000303000 ooOo3oooOO ,0000000000,

SLMON Fig. 4 Neurol net model for ECT

519

itance measurements for an eight-electrode system), fully connected to a grid of 100 neurons in the output layer (corresponding to the spatial image), with no intralayer connections. The input vector is preprocessed to reduce the dynamic range of the capacitance data set. Each node in the output layer represents a fragment of the overall image. The function of neurons in the input layer is purely distributive with no computational overhead, and that of output layer is similar to the basic model (Fig. 1).

SLMON is a supervised network trained with a variant of the back-propagation algorithm for the purpose of pattern association. The network was imple- mented in C and trained on a general purpose work- station to recognise 39 different sets of simulated data (patterns). The training sets include data for an empty pipe (filled with gas), a full pipe (filled with oil) and 37 sets of data consisting of the smallest identifiable gas components (bubbles) in oil. The bubbles are placed in different positions within the oil which together cover the entire cross section of the pipe (Fig. 4).

SLMON learns to associate different features (corresponding to different individual bubbles in the training sets), to their respective output pattern (i.e. the correct firing of neurons in the output layer). The tomog- raphic images are then created by extracting features in a given capacitance measurement set, and superimposing the associated output patterns, hence the name 'multi output' network.

&electrode ECI

F.E. phantom gas-oil vater-oil

m e f l o w talc T=16.66ms 000ms void-F = 0.042 fideTity-99.65'/. 99.65'/0

5 Results and discussions

The tests carried out include different flow regimes such as core flow, annular flow and stratified flow. Tests were also carried out on various bubble flow regimes, consist- ing of different bubble sizes and multiple bubbles of similar size. Two such networks were constructed in soft- ware: one trained with simulation result for gas/oil, and the other trained for water/oil. The results illustrate the ability of a trained SLMON to superimpose the output of several training sets to produce a relatively accurate result corresponding to a given finite element generated simulation, or 'phantom'. This is possible due to the feature extraction principle, enabling the net to make optimum use of variations in a measured data set, and associating a set of features with a given output. Image results are presented in Fig. 5, with simulation of each flow pattern on the left, and SLMON-estimated images of gas/oil and water/oil to its right. The relative permit- tivities of components for simulated measurements are: gas = 1, oil = 3, and water = 80. Two thresholds of 0.25 and 0.75 are used to distinguish three components in the image. In the case of gas/oil flow these are: oil (0 to 0.25), oil-gas mixture (foam) (0.25 to 0.75) and gas (0.75 to 1). For oil/water flow, the component values correspond to oil, oil/water mixture (emulsion) and water, respectively. The finite element phantom displays the void fraction of gas (shown in white) in oil (the shaded area within the

8-electrode ECT

F.E. phantom gas-oil water-oil

st railfled mlc-T- 0.00 ms O.OOmS void-F=0.500 f 1 d e l i t y ~ 9 5 . 4 5 ~ 1 . 93.18.1.

~

F.E. phantom gas-oil water-ail F.E.phantorn gas-oll water-oil

bubble flow calc-T= 16.67ms O.OOrns vOid-F - 0.202 f ideliiy=95.20% 99.75%

annular flow calc-T=O.OOrns 16.67ms void-F =0.205 fidelity=91.5Zgl. 93.23'/.

F.E. phantom gas-ail water_oil F.E.phantom gas-oil water-oil

bubble f low colc-1=16.67ms 16.67rns void-F= 0.158 f idel i ty = 97.62% 97.05%

calc-T= 0.000ms 16.67rns vaid-F= 0.796 f idel i ty= 92.O9V0 97.68'h care flow

Fig. 5

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SLMON-generated images for different bubbleflows, and variousflow regimes

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cross section of the pipe). The water fraction in the image is outlined, and the emulsion and the foam are shown in fuzzy shades. Fidelity is a measure of accuracy of result- ant void fraction (generated by SLMON), compared to the void fraction of the simulated phantom.

The network is first loaded with gas/oil weight matrix for forward evaluation. The matrix of weights is then replaced with the corresponding water/oil matrix, and the same forward evaluation is carried out for water fraction in oil. The output of SLMON is mapped to the cross- section of the pipe, and the images indicate the firing of individual neurons correlating to the area of void fraction displayed by the phantom. In Fig. 5, with up to 50% water (stratified flow), and 80% water (core flow), in the pipe, SLMON maintains its high fidelity, as well as rela- tively correct position of firing neurons. It produces an accurate output for a small bubble in the centre of the pipe with void fraction of 0.042 (resolution of 4.2%), and is able to cope with different bubble sizes and similar bubble positioned at the edge of the pipe. The compari- son of annular flow and core flow images, shows the complementary firing of neurons to produce the opposite image.

The important point to note is that the net has only been trained to recognise single fixed sized bubbles in dif- ferent positions, with no training for any particular flow regime. However, it is able to associate and interpret a given data set to produce multiple outputs effectively from just one set of measurements. As illustrated, the net can be trained to recognise two-component flows and produce comparable quality of output, regardless of the relative differences in component permittivities. This alle- viates the problem of distortion of the field- (soft-field effect), and the consequential problems in conventional image reconstruction algorithms. Thus a SLMON trained to identify say gas/oil flow is just as easily trained for water/oil flow without detriment to its performance.

ECT systems, in general, do not produce high resolution images; resolutions of about 10% in the centre of the pipe are regarded as acceptable, improving towards the edge of the pipe. In comparison, the resolution of a SLMON generated image is significantly improved to 4.2% in the centre of the pipe (Fig. 5). Also, for a given number of electrodes (say 8 or 12), the resolution may be improved further by adding more neurons to SLMON and training the net for smaller identifiable elements. Thus, for a given design, the resolution is limited only by the electrode capability to detect smaller objects (Section 3.2). In contrast, conven- tional image reconstruction algorithms are governed by

the number of boundaries formed by the sensing elec- trode pairs (eqn. 8). Thus the resolution of the image is dependent on the sensitivity of electrodes and the number of electrodes in the design.

The exact time for forward evaluation of 100 nodes fully connected to 28 inputs can easily be calculated for a given processor (eqn. 1). The reported calculation time shown in these results is obtained by reading the clock on IPX SUN microsystems which includes system overhead. In the tests carried out, the calculation time would either be zero or 16 ms (resolution of the clock), ensuring the real-time performance of over 50 frames per second.

In a horizontal oil pipe, in which components could be assumed to travel at the same rate (due to viscosity with no pressure gradient), the measured data from two closely positioned SLMON systems could be correlated to determine the mass flow rate. Correlation may also be performed on image fragment (individual neurons), should the materials be travelling at different velocities. This situation may occur in a vertical oil pipe in which gas travels upwards rather quicker than the oil (the buoy- ancy effect of gas in oil). The spatial distribution of the two components is determined and the velocity is mea- sured by correlating the two images neuron by neuron. Thus, the flow rate of both components can be obtained by dividing the distance separating the two systems by the corresponding correlation delay.

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