neural network multi-criteria optimization image reconstruction

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Neural network multi-criteria optimization image reconstructiontechnique (NN-MOIRT) for linear and non-linear process

tomography

W. Warsito, L.-S. Fan *

Department of Chemical Engineering, The Ohio State University, 140 West 19th Avenue, 121 Koffolt Laboratories, Columbus, OH 43210-1180, USA

Received 20 May 2002; received in revised form 25 October 2002; accepted 25 October 2002

Abstract

In this work, an analog neural network is utilized to de velop a new image reconstruction technique for the linear as well as the

non-linear process tomography. The ultrasonic computed tomography (CT) and the electrical capacitance tomography (ECT) are

chosen to represent the linear and the non-linear tomography. The image reconstruction technique is based on a multi-criteria

optimization, namely neural network multi-criteria optimization image reconstruction technique (NN-MOIRT). The optimization

technique utilizes multi-objective functions: (a) the negative entropy function, (b) the function of the least weighted square error of

projection (integral) values between the measured data and the estimated projection data from the reconstructed image, and (c) a

smoothness function that gives a relatively small peakedness in the reconstructed image. The optimization image reconstruction

problem is then solved using the Hopfield model with dynamic neural-network computing. The technique has been tested using

simulated and measured data; this technique has shown significant improvement in accuracy and consistency compared with other

available techniques for both linear and non-linear tomography.

# 2003 Elsevier Science B.V. All rights reserved.

Keywords: Hopfield neural network; Image reconstruction; Linear tomography; Electrical capacitance tomography; NN-MOIRT

1. Introduction

In the past several years, process tomography has

emerged as a powerful technique in the study of a wide

range of multiphase flow systems. The technique

provides both local and global information on the

dynamic behavior of the flow system, which is of

considerable importance to the system design, control

and monitoring.The process tomography involves the task of recon-

structing integral (projection) measurement data from

remote sensors mounted on the periphery of the conduit

of the flow system to generate a cross-sectional image of

the vessel. The process tomography system basically

consists of three parts: (1) a sensoring system to acquire

the measurement data, (2) an electronic system for data

acquisition, and (3) a computer system for measurement

control, image reconstruction and displaying the result

[1,2]. Fig. 1 shows a schematic representation of an

electrical capacitance tomography (ECT) system.

To obtain reliable images of the flow system studied, a

successful implementation of process tomography lies in

the selection of the sensor system deployed and the

image reconstruction algorithm. Based on the sensor

physics, process tomography may be differentiated into

two types: linear tomography, also referred as to hard

field tomography, and non-linear tomography, also

referred as to soft field tomography. The first type

includes tomography techniques based on electromag-

netic radiations, such as X-ray computed tomography

(CT), gamma-ray tomography, and acoustical tomogra-

phy. This type represents the early stage of process

tomography development with a wide range of applica-

tions from medical systems to chemical processes.

Nuclear radiation based tomography techniques provide

very high spatial resolution up to 1% of column

* Corresponding author. Tel.: /1-614-292-7907; fax: /1-614-292-

1929.

E-mail address: [email protected](L.-S. Fan).

Chemical Engineering and Processing 42 (2003) 663/674

www.elsevier.com/locate/cep

0255-2701/03/$ - see front matter# 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0255-2701(02)00204-0
mailto:[email protected]:[email protected]


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diameter. However, the scanning speed is too slow (few

seconds per frame) to perform real time measurement

[3]. The non-linear tomography is represented by the

more recently developed electrical-based tomography,

such as ECT that is used to image the dielectric

properties; and electrical resistance tomography (ERT)

that is employed to image the conductivity of a multi-

phase media. The ECT and ERT have been gaining

Fig. 1. Schematic diagram of ECT system.

Fig. 2. Comparisons of reconstructed results for ultrasonic CT.

W. Warsito, L.-S. Fan / Chemical Engineering and Processing 42 (2003) 663 /674664


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acceptance as laboratory and industrial tools for the

purpose of multiphase flow imaging because of their

high-speed capability, low construction cost, high safety

and suitability for small or largevessels. The high-speed

capability is one of the most relevant factors in multi-

phase flow imaging for a real time monitoring. ECT is

particularly attractive to industrial systems as manychemical processes use non-conductive organic liquids

as the liquid phase[4]. Currently, commercially available

ECT systems can acquire a speed up to 100 image

frames per s.

One of the most important issues for tomography

applications in chemical processes is spatial resolution.

The process tomography systems as compared with

medical tomography systems is of relatively lower

spatial resolution due to the limited integral measure-

ment data obtained, which consequently influences the

accuracy of the reconstructed image. The relatively

smaller number of the integral measurements is commonas the result of limitations in the number of the sensors

that can be employed. The need for high-speed imaging

also prompts the number of the sensors to be limited as

few as possible. Consequently, an increase in the number

of the integral measurements consequently would in-

crease the data acquisition time, and thus would

decrease the real time capability. The issue raised is

how to acquire reconstruction results with sufficient

accuracy for process quantification from the limited

experimental integral data without sacrificing the time

resolution for reliable real time measurements.

The choice of the image reconstruction technique is

crucial as it determines the quality of the imageproduced. For linear tomography, the commonly used

filtered back projection technique (FBPT) requires a

large number of integral measurement data to obtain a

reliable accuracy[5]. In medical imaging, the number of

projection data is up to 256/256 (number of projection

directions/paths) compared with only 10/20% of that

number in chemical process applications. For multi-

phase flow imaging with limited projection data iterative

techniques based on iterative version of FBPT [5,6],

entropy maximization [7] or algebraic reconstruction

technique (ART) [8] are usually employed to obtain

more accurate reconstruction result. The iterative tech-niques implement a single criterion, i.e. the least square

error between the measured and the calculated projec-

tion data to be minimized or the entropy function to be

maximized. The iterative techniques described above are

a type of optimization technique aimed at finding a least

squared error between the measured and the estimated

projection data (capacitance data in the ECT). However,

the image reconstruction is an ill-posed problem. In flow

imaging especially, there are much fewer independent

measurements (projection data) than unknown pixel

values (field distribution). Thus, such a criterion does

not necessarily determine the image vector, and there

may be more than one possible alternative image.

Another reason why such a solution is not necessarily

good is that the least squared criterion does not

contain any information regarding the nature of a

desirable solution [9]. Therefore, more than one

objective function is required to be minimized simulta-

neously to choose the best compromise solution among

possible alternatives.

In the case of the electrical-based tomography, in

addition to the issue noted above, a non-linearity

problem due to the soft field effect of the electrical

field distribution exists for which no analytical solution

so far is available for the inverse problem. In the

electrical-based tomography, linearization using the so

called sensitivity model is usually applied before the

same algorithm as in the linear tomography is utilized

[7,8]. The commonly used image reconstruction techni-

que for the electrical tomography is based on the linear

back projection (LBP), which is essentially the same as

the back projection technique used for the linear

tomography. The technique provides only a rough

estimation of the image reconstructed. A great demand

for a high quality image generated by electrical tomo-

graphy has led many researchers to develop reconstruc-

tion techniques based on iterative algorithms. More

comprehensive reviews on the reconstruction technique

for ECT are presented by Isaksen [10]and Warsito and

Fan [11]. Most iterative techniques for electrical tomo-

graphy are of the ART type as applied for linear

tomography, in which the least square error between

the measured and the calculated integral data (current in

ERT or capacitance in ECT) is minimized. The main

problem of this criterion is the sensitivity to noise, in

which convergence is difficult to reach when the

measurement data contain noises. This is the drawback

for the imaging of highly fluctuated multiphase flow

systems.

Thus, there is a need for development of a robust

reconstruction technique for linear and non-linear

process tomography, which is capable of yielding a

reconstructed image with reliable accuracy and meeting

the required chemical process applications. In this work,

a new image reconstruction technique for linear and

non-linear process tomography is developed based on

the Hopfield analog neural network multi-criteria opti-

mization, namely the neural network multi-criteria

optimization image reconstruction technique (NN-

MOIRT). The analog neural network technique has a

major advantage over the feed forward neural network

algorithm because prior knowledge of the image pattern

to be reconstructed for network training is not required.

The performance of the image reconstruction technique

of this study is compared with other commonly used

techniques.

W. Warsito, L.-S. Fan / Chemical Engineering and Processing 42 (2003) 663 /674 665


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2. Theory

2.1. Forward problem in integral measurement

The tomography technique can be divided into two

steps: the integral measurement and the image recon-struction. The integral measurement is a forward

problem which transforms the field property X(x , y )

into the measured integral data, y , which is easily

measurable, e.g. time-of-flight of an ultrasonic wave or

intensity of an electrical signal. In the linear tomography

the relationship between the field property and the

measured integral data becomes linear and can be

expressed as:

Y(s; u) gL(s; u)

X(x; y)dl (1)

where L (s , u ) is the projection line as a function of thedistance from the origin, s and the angle, u . In the case

of the ECT, the field property distribution corresponds

to the permittivity, o(x , y), while the measured para-

meter is the capacitance data. The forward problem is

based on the following Poissons equation[12,13]:

9o(x; y)9f(x; y)0 (2)

where o(x , y ) is the permittivity distribution, f (x , y ) is

the electrical field distribution. This is second-order,

elliptic-type partial differential equations with the typi-

cal boundary conditions of Dirichlet or Neumann type.

The measured capacitance C(s , u) is obtained by

integration ofEq. (2):

C(s; u) 1

DV(s; u) GG(s; u)

o(x; y)9f(x; y)dl (3)

where DV(s , u ) is the voltage difference between the

source and the detector electrodes. G (s , u) is the curve

enclosing the detector electrode. Eq. (3) relates the

dielectric constant (permittivity) distribution, o(x , y),

to the measured capacitances C(s , u). That is, for a

given medium distribution o(x , y ) and the boundary

conditions, the capacitances can be calculated using, for

example, the finite element method (FEM).

2.2. Inverse problem in image reconstruction

The image reconstruction process is an inverse

problem involving the estimation of the field property

distribution X(x , y ) from the measured parameter Y(s ,

u). For linear tomography, the commonly used image

reconstruction technique is the FBPT and ART. In the

FBPT, the field property distribution X(x , y) is

obtained analytically by integrating the convolution of

the measured integral data, Y(s , u), and a filter

function, g(s ):

X(x; y)gp

0

[Y(s; u)g(s)]sx cos uy sin udu (4)

The major problem of the technique is that it requires a

large number of projection data, which is not suitable

for process applications, because in most cases thenumber of collectable projection data is limited. An

iterative technique is usually applied for this case. Here,

the reconstructed image (field distribution) is used to

predict the projection data, which is then compared with

the measured projection data to correct the recon-

structed image iteratively.

The ART type is based on the series expansion ofEq.

(1).In matrix formulation it can be written as:

YAX (5)

where Y is the measurement vector, X is the image

vector (field distribution), and A is the projection

matrix. The image reconstruction using the ART is theproblem of estimating image vector X such that the

estimated integral projection data Y*(X)5/Y, given a

measurement vector Y. The ART can be expressed by

the following form

X(k1)

X(k)a(k)AT(YY(X(k))) (6)

Here, AT is the transpose matrix of A, X(k) is the

estimated image vector in the k-th iteration, a is a

relaxation factor, also called as gain factor or weighting

factor. The ART requires less number of projection data

than IFBPT; it is, however, generally sensitive to noise.

For the ECT, as seen from Eq. (3), it is difficult to

obtain an explicit expression that relates the measured

capacitances to the fraction and position of the dielectric

components inside the measured domain. Since there is

no general method of solution even for the forward

problem, in which the equation becomes linear, approx-

imation methods are usually used. The most common

method is the use of the so-called sensitivity model

[12,14]. Based on this model, Eq. (3) can be written in

matrix expression as:

CSG (7)

where C is the measured capacitance matrix, G is the

image vector (permittivity distribution) and S is the so-called sensitivity map. The image vector G is obtained

from the following matrix multiplication:

GSTC (8)

where ST is the matrix transpose ofS. The reconstruc-

tion technique is referred to as the LBP. The recon-

structed image using the LBP is blurred, showing a

smoothing effect on the sharp transitions between the

different dielectric constants. To obtain more accurate

reconstruction results, an iterative procedure as in the

linear tomography is usually applied. Eq. (7) corre-

sponds toEq. (5)for the linear tomography. Therefore,



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an iterative technique used for the linear tomography is

also applicable to this problem.

2.3. Multi-criterion optimization in image reconstruction

problem

Considering the error due to the measurement in-

accuracy and the truncation in the series expansion and

linearization, bothEqs. (5) and (7) can be written as

YAXe (9)

where e is an M-dimensional error vector. A corre-

sponds to the projection matrix for the linear tomogra-

phy, and the sensitivity matrix for the ECT. Then, the

reconstruction problem is to find methods for estimating

the imagevector (field property distribution) X from the

measurement vector Y, and minimize the error e. Since

we obviously do not know e , the problem is to find acompromise solution of the system under certain

conditions (criteria), such that

AX5Y (10)

The criteria that have been used for the image recon-

struction problem are usually of the form: choose as the

solution of Eq. (10) an image vector X for which the

value of some function fi(X) is minimal, and if there is

more than one X which minimizes fi(X) choose among

these one for which the value of some other function

fj(X) is minimal [9]. The multi-criteria optimization

based solution method is usually employed to find an

image which (a) has the largest entropy; (b) has the least

weighted square error between the measured data set

and the estimated value calculated from the recon-

structed image; (c) is smooth and has a relatively small

peakedness. The corresponding objective functions are,

respectively:

f1(X)g1

XNj1

Xj lnXj: (11)

f2(X)1

2g2AXY

2

1

2g2

XMi1

XNj1

AijXjYi

2(12)

f3(X)1

2g3(X

TNXX

TX) (13)

where g1, g2 and g3 are normalized constants between 0

and 1. N is N/N non-uniformity matrix. Again, the

multi-criteria optimization for the reconstruction pro-

blem is to choose an imagevector for which thevalue of

the multi-objective function F(X)/[f1(X), f2(X), f3(X)]T

is minimized simultaneously. This can be realized by the

weighted sum technique stated mathematically as:

minimizeXP

F(X)X

i

wifi(X); (i1; 2; 3)

so that AXY50:

( (14)

where wiis a weight operating on the objective function

fi(X) and can be interpreted as the relative weight or

worth of the objective compared with the other

objectives, i.e.:

X3i1

wi1 : (15)

P is the feasible set of constrained decisions defined by

linear constraints:

PfX RNAX5Y; X]0; Y RMg (16)

2.4. Solution using Hopfield dynamic neural networks

Here, we use an analog neural network based on the

Hopfield model to solve the optimization problem (Eq.

(14)). More details of the reconstruction technique are

described elsewhere [11]. To solve the optimization

problem using the neural network technique, the gray

level (attenuation coefficient distribution for X-ray

tomography, sound velocity distribution for acoustical

tomography, or permittivity map for ECT) Xj in the

image vector is mapped into the output variable vj for

neuron j bounded by 0 and 1. The output variable is a

continuous and monotonic increasing function of the

internal state of the neuron:

XjvjfS(uj); (17)

where uj is the internal state variable of the neuron j.

Function fS is called activation function with typical

choice of the form as:

fS(uj) 1

1 exp(buj); (18)

where b is a steepness gain factor that determines the

vertical slope and the horizontal spread of the sigmoid-

shape function. The non-linear sigmoid function forces

the neuron output to converge to binary output 1 or 0.

However, when an output value between 0 and 1 isequally expected as the extreme values, as in the

application of tomography for concentration measure-

ment, we use the following linear activation function:

fS(uj)

0 if uj5j

b

bujj if j

bBujB1

j

b

1 if 1j

b5uj

8>>>>>>>>>>>>>>>:

(19)

whereb andj become the slope and the intercept of the



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line, respectively. In the Hopfield model dynamics

neural network the neural, activities of the output vjas well as the input ujdepend on time t . The evolution

of the network is determined by the N-coupled ordinary

differential equations[15,16]:

C0j dujdt@E(

v)

@vj(20)

whereC0jis an associated capacitance in thejth neuron

in the Hopfield nets with time constant t (t/R0jC0j,

R0j is the associated resistance), and E is the system

energy of the Hopfield networks. The corresponding

neural networks energy function to the optimization

problem inEq. (14) can be written as:

E(X)v1g1

XNj1

Xj lnXj1

2v2g2AXY

2

12v3g3(X

TNXX

TX)

XM

i1

C(zi)X

N

j1

1

Rj

gXj

0

f1S (X)dX (21)

where Nis the number of neurons in the Hopfield nets,

which is equal to the number of pixels in the digitized

image. Mis the number of integral data measurements.

The first three terms in Eq. (21) are the interactive

energy among neurons based on the objective functions.

The forth term is related to the constraint violation,

which must be minimized. The fifth term encourages thenetwork to operate in the interior of the N-dimensional

unit cube {05/Xj5/1} that forms the state space of the

system. The constraint function C(zi) is defined as:

dC

dzid(zi)

0 if zi50

azi if zi0

(22)

where,

ziXNj1

AijXjYi (23)

Herea is a large penalty parameter. The function C(zi)provides for the output of the neuron j to be a large

positive value when the corresponding constraint equa-

tion is being violated. Since the Hopfield network is a

monotonic descent technique as described above, it will

converge to the first local minimum it encounters. By

introducing the penalty function, temporary increases in

the descending evolution of the Hopfield network

energy is permitted in order to allow escape from local

minima. The form of penalty parameter a is chosen as:

a(t)a0zexp(ht) (24)

Here a0, z and h are positive constants. This modified

Hopfield network provides a mechanism for escaping

local minima by varying the direction of motion of the

neurons. In the initial steps, the ascent step is taken

largely by the penalty function, but less likely as the

algorithm proceeds.

For simplicity, choosing R0j/R0 and C0j/C0, and

redefining R0C0, g1v1/C0, g2v2/C0and g3v3/C0as, t , g1,g2and g3, respectively, the time evolution of the internal

state variable of neurons in the networks becomes:

u?(t)u(t)

t

[g1(1ln X(t))g2ATz(t)g3(NX(t)X(t))

ATd(z(t))] (25)

where,

u(t) [u1(t); u2(t); . . . ; uN(t)]T;

X(t) [X1(t); X2(t); . . . ; XN(t)]T

;z(t) [z1(t); z2(t); . . . ; zN(t)]

T:

The image vector Xj becomes the output of jth

neuron, which is calculated from the sigmoid function

(Eq. (18)orEq. (19)):

Xj(t)fS(uj(t)); j1; 2; . . . ; N (26)

Unlike in the multi-layer feed forward (MLF) neural

network, the parameter uj is calculated here by solving

Eq. (25) using, for example, Eulers method. Here, we

set C0/1.0, until the system converges to a stationary

state. Take t/1.0 so that time is measured in units t ,

and time step length Dt/0.01. The iteration process ofthe internal state variable of jth neuron and the

corresponding pixel value of the image are, respectively,

given by:

uj(tDt)uj(t)u?j(t)Dt (27)

Xj(tDt)fS(uj(tDt))Xj(t)f?S(u)u?j(t)Dt (28)

where fS? (u )/dfS(uj)/duj and uj?(t )/duj(t )/dt . The

stopping rule is used when the changes in the firing

rates become insignificant, i.e. for all pixelsjGj(t/Dt)/Gj(t)j/1.

3. Simulations and experiments

The image reconstruction algorithm proposed in this

work is tested by reconstructing simulated and actual

projection measurement data based on ultrasonic CT

and ECT. The simulated projection data for ultrasonic

CT is calculated based on Eq. (1) with a known field

property distribution. The projections are conducted

from 18 directions and eight paths with the same spacing

for each direction. The simulated capacitance data for

ECT is calculated based on Eq. (3) with a given



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permittivity distribution using the FEM. A 12-electrode

capacitance sensor model is used for the simulation. The

images are reconstructed on 32/32 digitized image

pixels from 66 measured capacitance data.

An experiment to test the reconstruction technique for

real capacitance data is carried out in 5 cm I.D. column

using a 12-electrode sensor with a dielectric material inthe column. The schematic diagram of the ECT is given

in Fig. 1. The data acquisition system used is from

Process Tomography Limited (UK) and is capable of

capturing image data up to 100 frames per s. Sixty-six

measured capacitance values from the 12 electrodes for

one frame are used to reconstruct the image on 32/32

digitized image pixels using the algorithm described

above, i.e. the resolution (pixel size) is about 3% of the

column diameter. The reconstruction results using the

new algorithm are also compared with those using other

techniques based on the same experimental data. No

experiment is performed to test the reconstructiontechnique for the linear tomography.

An experiment to test the reconstruction technique for

flow imaging of gas/liquid/solid flows in bubble

column is also performed. The experiment is carried

out in a 10 cm diameter column using the same ECT

system as described above. The gas distributor is a single

nozzle with a diameter of 0.5 cm. Air (dielectric

constant/1), Norpar 15 (paraffin, density/773 kg/

m3, viscosity/0.253 mPa/s, dielectric constant/2.2)

and glass beads (density/2470 kg/m3, average dia-

meter/200 mm, dielectric constant/3.8) are used as

the gas, liquid, and solid phases, respectively. The

experiments are conducted in a semi-batch mode with-out inlet liquid flow with the liquid levels maintained at

60/80 cm during fluidization and the total solids

loading of 40% by volume. A twin-plane capacitance

sensor with time resolution of 50 frames per s per plane

is used to perform a real time imaging of the three-phase

flow system. The NN-MOIRT reconstructs the permit-

tivity map from the collected measurement capacitance

data off-line. A three-phase capacitance model com-

bined with a two-region model solids concentration in

the emulsion phase of the three-phase system is used to

attain the gas and the solids concentration distributions.

The two-region model assumes a uniform solids con-centration distribution in the solid/liquid emulsion

phase. More details on the models are described else-

where[17].

For linear tomography, the data obtained by Warsito

et al. [6] for air/water/glass beads system based on

ultrasonic tomography is used. The column used is 14

cm I.D. and 140 cm height. The gas distributor is six

nozzles located at the center of the column. The average

diameter of the glass beads is 260 mm with density of

2470 kg/m3 with the solids loading less than 2% and the

gas superficial velocity less than 2 cm/s. The ultrasonic

tomography technique could simultaneously differenti-

ate the gas and the solids concentrations in the three-

phase system based on measurements of the time-of-

flight and the attenuation of the ultrasonic wave

transmitted through the three-phase media. It is, how-

ever, only the time average measurement data available,

as the technique is too slow to perform a real time

imaging. The NN-MOIRT reconstructs the time-of-flight and the attenuation data to obtain the time

average cross-sectional distributions of the gas and the

solids concentration distributions.

4. Results and discussion

Fig. 2shows the reconstruction results from simulated

projection data based on the ultrasonic tomography

using the NN-MOIRT as well as commonly used

reconstruction techniques for linear tomography: ART

and iterative filtered back-projection technique(IFBPT). No noise is added to the simulated projection

data. Clearly, the reconstruction results from the limited

projection data by the IFBPT are severely distorted. The

ART gives better results than IFBPT; it is, however,

difficult to reach convergence especially when noise

exists (data not shown). Over 200 iterations are needed

for the ART for the noise-free data as compared with

less than ten iterations for the IFBPT. More iteration is

needed for the ART depending on the magnitude of the

noise added. The NN-MOIRT gives the best results

among the three reconstruction techniques and con-

verges in less than half of the iterations required for the

ART.Fig. 3 shows the reconstruction results for the ECT

using the NN-MOIRT in comparison with commonly

used techniques, including the LBP and the iterative

version of LBP (ILBP). No noise is added in the

simulated capacitance data for these results. From Fig.

3, it is seen that LBP only gives rough estimations of the

original images. Clearly, the ILBP and the NN-MOIRT

give much better results than the LBP, and the NN-

MOIRT gives the best accuracy and consistency. When

a Gaussian noise (up to 20 dB) is added, the ILBP no

longer reconstructs the original images accurately (Fig.

4). Again, the LBP only reconstructs the original imagesvery roughly. The reconstructed results by the ILBP are

severely distorted from the original images. Meanwhile,

the NN-MOIRT shows its robustness to the noise and

still gives a good accuracy and consistency despite of the

existence of the noise. The robustness of a reconstruc-

tion technique with respect to noise is crucial as

measurement data always contains noise varying in

magnitude.

Reconstruction results from actual measurement data

using the ECT are shown in Fig. 5(a and b). Fig. 5(a)

shows the reconstruction results of a static tube placed

in a column and measured by the capacitance sensor.

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The 3-dimensional image is obtained by stacking 50

image frames taken during 1 sec in the same position.

Fig. 5(b) shows the reconstruction results of a tube

circling with the period of one circle per s in the column

measured by the capacitance sensor. The 3-dimensional

image is obtained by stacking 125 image frames taken

during 2.5 s in the same position. Clearly, both results

reveal the accuracy of the NN-MOIRT in reconstructing

the actual experimental data.

Fig. 6 shows time average distributions of gas and

solids concentrations in a dilute gas/liquid/solid flow

system at a different level of the columns based on the

Fig. 3. Comparisons of reconstructed results for ECT from noise free simulated capacitance data.

Fig. 4. Comparisons of reconstructed results for ECT from noise contented simulated capacitance data.

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ultrasonic tomography measurement. In the gas/liquid

system, there is no remarkable change in the gas holdup

profile with increasing the column level. In the gas/

liquid/solid system, the time averaged gas holdup

profile changes from a doughnut-shape distribution in

the lower region to a center-peak distribution in the

upper region. Meanwhile, the solids concentration

profile turns a doughnut-shape distribution with a

peak in the center region to a well-defined wall-peak

distribution in the upper region. Warsito et al. [6]

described that the doughnut-shape distribution in the

lower region for the gas holdup is due to the spiral

motion of the gas bubbles, and the solids concentration

profile is a result of bubble/particle interaction and

interaction between large scale liquid vortices and

particles.

Fig. 7 shows the quasi-3D real time distributions of

the gas bubbles and the solids concentrations in the two-

and the three-phase systems at gas velocity of 5 cm/s and

column levels of 20 and 55 cm above the gas distributor

Fig. 5. Reconstruction results of dielectric material using ECT.

Fig. 6. Time averaged cross-sectional distributions of gas and solids holdup in G/L and G/L/S systems based on ultrasonic tomography (UG/2

cm/s).



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based on ECT. The quasi-3D image is constructed by

stacking 200 tomography images captured in 4 s for the

same plane but different in time. The vertical coordinate

is taken as time by choosing the length scale for the time

coordinate (Z-coordinate) as the bubble rise velocity/

time. The quasi-3D images show the flow structures of

the gas/liquid flow along with the effect of the gas

velocity at levels 20 and 55 cm above the distributor. A

cutoff boundary for the gas holdup of 0.2 is chosen from

the images to determine the bubble surface and yields

reasonable bubble images, i.e. only sufficiently large

bubbles are imaged.

From the figure, for the gas/liquid system, at a lower

level (z/20 cm) the bubble swarms rise in spiral motion

while rocking back and forth from the sides of the wall

with period of 2/3 s. At a higher level (z/55 cm), the

spiral motion becomes insignificant, and the bubbles are

more concentrated in the center region of the column.

Comparing the images at the lower and the higher levels

shows that there are more relatively large bubbles (up to

Fig. 7. Quasi-3D gas and solids holdup distributions in G/L and G/L/S systems based on ECT (UG/5 cm/s).



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5 cm diameter, indicated by dark red color) mixed with

small bubbles (indicated with light blue clouds) at the

upper level than that at the lower level. Meanwhile, for

the gas/liquid/solid system, a significant difference in

the gas bubble flow structures in the upper region as

compared with those in the lower region is clearly

observed. In the upper region, the bubble size is moreuniform, the bubbles are more concentrated in the

center region, and there is no relatively large bubbles

observed. This indicates that the interaction between the

bubbles and the solid particles causes significant bubble

break-ups, resulting in smaller size and more uniform

bubbles in the upper region. For the solids concentra-

tion distribution, the solids concentration in the region

where the gas holdup is nearly zero (almost no gas

bubble, indicated by bright red color) is almost uniform

throughout the column. This supports thevalidity of the

assumption made in the two-region model employed in

this work. The solids concentration profile is in opposi-tion to the gas holdup profile.

Fig. 8shows the cross-sectional profiles of the gas and

the solid holdups in the gas/liquid and the gas/liquid/

solid systems at a different level of the columns and

various gasvelocities averaged over a period of 4 s. The

time averaged gas holdup profiles in both the gas/liquid

and the gas/liquid/solid systems have a similar trend,

i.e. a doughnut-shape distribution in the lower region

and a center-peak distribution in the upper region. This

result indicates that the spiral motion becomes less

significant as the level of the column increases.

This phenomenon is the same as seen in Fig. 6 based

on ultrasonic tomography measurement regardless of

the types of liquid used, the gas distributor, the solids

loading and the column size. By increasing the column

level, the solids concentration profile turns from a

double-ring distribution or a single ring

(wall-peak distribution) with some amount of solid

particles in the center in the lower region to a well-

defined wall-peak distribution in the upper region. This

is also similar to those observed by the ultrasonictomography.

5. Concluding remarks

A new image reconstruction technique based on

analog neural network optimization, namely NN-

MOIRT is developed in this work for linear as well as

non-linear tomography. The reconstruction technique

implements multi-criteria optimization solved by a

modified Hopfield neural network. Comparisons withthe other commonly used reconstruction techniques for

both linear and non-linear tomography revealed much

improvement of the technique in the accuracy, the

consistency and the robustness to noise. The technique

is also applicable for any linear or non-linear tomogra-

phy using the appropriate projection (or sensitivity)

matrix. The technique also has the advantage of much

reduced computation time due to the possibility for

hardware implementation and inherent parallelism in

the computation. Some examples of the use of the

reconstruction technique to image multiphase flows in

three-phase bubble column based on time average modeultrasonic tomography and real time ECT have also

been presented.

Fig. 8. Time averaged cross-sectional distributions of gas and solids holdup in G/L and G/L/S systems based on ECT atUG/5 cm/s (left) and 15

cm/s (right) (color map is the same as in Fig. 1).

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Appendix A: Notations

A projection matrix

C capacitance

C0 associated coefficient in Hopfield neuron

E network energy

e error matrixg(s ) filter function

k number of iteration

L , l projection line

M number of integral measurement

N smoothness matrix

N number of image pixel

R0 associated resistance in Hopfield neuron

S sensitivity matrix

t time

Dt time step length

s projection distance from origin

u neuron internal statevariableuG gas superficial velocity

DV voltage difference

v, v neuron output vector, neuron output variable

w1,2,3 weight coefficient

X, X image matrix, gray level

x variable

y variable

Y, Y projection matrix, projectionvalue

z parameter defined inEq. (23)

Symbols

a penalty factor

a0 initial penalty factorb steepness gain factor

o permittivity

f electrical field distribution

G curve enclosing the detector electrode

g1, g2, g3 normalized coefficients

h coefficient inEq. (24)

P set of constraint decision defined inEq. (16)

u projection angle

t unit time

j coefficient inEq. (19)

z coefficient inEq. (24)

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