Reconstruction of electrical capacitance tomography images based on fast linearized alternating direction method of multipliers for two-phase flow system☆

Chongkun Xia,Chengli Su*,Jiangtao Cao,Ping Li

School of Information and Control Engineering,Liaoning Shihua University,Fushun 113001,China

1.Introduction

Two-phase flow is a mixed- flow pattern widely found in nature,especially in chemical,petroleum,electricity,nuclear power and metallurgical industries[1,2].Two-phase flow image reconstruction plays an increasing important role in the automation of energy industry.In recent years,process tomography(PT)technology has been developed rapidly,and it has great potential and broad prospects in solving the two-phase flow detection problems for industrial applications.Among various tomography techniques such as X-ray,electrical,ultrasonic,nuclear magnetic resonance,and microwaves,the tomography technique based on measurement of electrical properties has received significant attention[3,4].Electrical Resistance Tomography(ERT)[5],Electromagnetic Tomography(EMT)[6],and Electrical capacitance tomography(ECT)[7]constitute the three main types of electrical-based tomography techniques.ECT provides the permittivity distribution across the two-phase flow by measuring the capacitances between all pairs of electrodes surrounding this flow in a relatively short time.Compared with other tomography techniques,ECT is a relatively new and has some distinct advantages:being safe(i.e.non-radioactive),non-invasive,lower cost,portable and much faster.Thus,it can be adopted for applications requiring real-time performance such as visualization of oil-water pipeline flows and gas-solid flows.Since the ECT system is a typical nonlinear system,the measurements from independent capacitance(projection data)are limited and rare,far less than the number of pixels of the reconstructed image.Besides,the ECT inversion problem has no analytical solution.At the same time,since ECT has some essential properties such as nonlinear and“soft field”,the stability of solution of ECT systems is bad and there is a serious morbidness[8-10].

The successful application of ECT measurement is largely dependent on the speed and accuracy of imaging reconstruction algorithms.Now there are more commonly used methods for ECT image reconstruction,such as the linear back projection algorithm(LBP)[11],the Tikhonov regularization method[12],the Landweber iteration method[13],the conjugate gradient method(CG)[14,15],the neural network algorithm[16]and so on.The LBP algorithm is simple and has fast reconstruction speed.Due to the relatively poor image quality,the LBP is only a qualitative algorithm.The Tikhonov regularization method is an effective method to solve the inverse problems,in which the regularization parameters have a great impact on image quality,but the selection of these parameters generally adopts the empiric values and has no enforceable rules.The Landweber iteration algorithm has in recent years been widely adopted.However,it is only the steepest descental gorithm from the viewpoint of numerical optimization,and the rate of convergence is relatively slow.When the coefficient matrix is symmetric positive definite,the CG algorithm has a short imaging time and fast convergence,but the CG has a bad performance on the complex flow patterns.For ECT image reconstruction,the neural network algorithm in essence is a pattern recognition method,and the successful application of the method completely depends on the rational network structure and training samples.However,it is difficult to obtain complete training samples owing to the randomness and complexity of the multiphase flow mid-stream change.And there are some difficulties in determining the network structure in the actual application.

Compressed sensing(CS)is a mathematical framework with several powerful theorems that provide insight into how a high resolution image can be inferred from a relatively small number of measurements using sophisticated computational methods[17-19].Recently,CS has witnessed an increased high demand for fast,efficient and in-expensive signal processing algorithms,applications and devices[20,21].Besides,the CS is growing rapidly in application of the electron tomography(ET),such as magnetic resonance imaging(MRI)[22]and X-ray computed tomography[23].ECT image reconstruction is to obtain the material distribution from small number of known capacitance measurements.Hence,ECT inverse problem also can be approached as a CS problem[24].The alternating direction method of multiphier(ADMM)is a new and effective method to solve the convex optimization problem,especially in the area of CS[25].So far,no study has not been found that a combination of ADMM and CS is used to solve ECT inverse problem.

In this paper,inspired by the mathematical framework of CS,the ECT data acquisition is regarded as a linear measurement process of permittivity distribution signals over a pipe section.Then we design a new measurement matrix and use L1 regularization to convert ECT inverse problem to a convex relaxation problem.Finally,a new fast alternating direction method of multipliers which contained linearized idea(FLADMM)is employed to solve the objective function.The simulation and experimental results show that the proposed method improves the accuracy and the quality of the reconstructed images.

Fig.1.Principles of ECT system.

Fig.2.Layout of ECT sensors.

2.ECT System and Its Mathematical Model

2.1.Structure of ECT system

ECT system mainly consists of three parts:capacitive sensor,data acquisition and signal process,and image reconstruction,as shown in Fig.1.Pairs of metal electrode plates are mounted uniformly around the outside of an insulted pipe and shielding cover.Fig.2 shows the layout of ECT sensors.In Fig.2,R1 is the pipe diameter;R2 is the distance between the center of pipeline and plate;R3 is the distance between the center of pipeline and the shield cover;and θ is the plate angle.The basic principle of ECT is to use the multi-phase flow media with different dielectric constant to obtain capacitance value of each pair of electrodes installed in insulated pipeline outer wall as the capacitance sensor.The capacitance values which reflect the distribution of dielectric constant across the pipeline is used to retrieve the two-phase flow concentration distribution map on the pipeline cross section by the computer.Moreover,with some certain image reconstruction algorithms,we can use the values to restore the physical phase image in the pipeline,namely intuitive access to distribution information of two-phase flow.

Fig.3.Measurement model of ECT system.

2.2.Measurement principle

There are two major computational aspects in the ECT image reconstruction:the forward problem and the inverse problem[26].The forward problem of ECT system determines the inter-electrode capacitance from the permittivity distribution.The inverse problem aims to determine the permittivity distribution from the measured capacitance data.The result is usually presented as a visual image,hence the process is called image reconstruction.Fig.3 shows the measurement model of ECT system.

The electrical field inside an ECT sensor can be calculated using the Poisson equation:

where ε(x,y)and φ(x,y)are respectively dielectric constant and electrical potential distributions.The relationship between the capacitance and the permittivity distribution is governed by

where Q is the charge,V is the potential difference between two electrodes forming the capacitance and Ω is the electrode surface.In ECT application,the image reconstruction model can be simplified as

where x is an m×1 dimensional vector indicating the normalized capacitance values;Θ1is an n×1 dimensional vector standing for the normalized permittivity distribution,and in the reconstructed image it denotes the gray level value;Ψ is a matrix of dimension m×n,and it is called the sensitivity field matrix.

The purpose of the ECT inverse problems is finding Ψ rapidly and effectively from the known Θ1and x parameters.There are three major difficulties with image reconstruction in ECT:

(1)The relationship between the permittivity distribution and capacitance is nonlinear and the electric field is distorted by the material that is present;this is the so-called“soft field”effect;

(2)The number of independent measurement data is less than that of the pixels of the reconstructed image;

(3)The inverse problem is ill-posed and ill-conditioned.

Since the ECT image reconstruction is sensitive to noise in the input data,its solution is unstable.As for ECT image reconstruction,some prior information or constraints on the solution should be imposed to obtain a meaningful reconstruction result.

3.Algorithm Analysis

ECT image reconstruction is a typical ill-posed problem and its solution is numerically unstable.Therefore,the methods which ensure a stable numerical solution and enhance the quality of the reconstructed images should be employed.Compressed sensing(CS)is based on the idea that a signal can be reconstructed from a very small number of measurements while the signal is sparse.In this section,we utilize the image restoration theory of the CS framework to solve the ill-posed problem.

3.1.Measurement matrix and sparse representation

The well-known Shannon's sampling theorem states that to recover a signal exactly,the sampling rate must be as least the Nyquist rate,which is twice the maximum frequency of the signal.In contrast,using CS,far fewer samples or measurements at far below the Nyquist rate are required to recover the signal,as long as the signal is sparse and the measurement is incoherent[18,19].The sparse representation of signal X is also a very important part of the CS frame.For two-phase flow system,the original signal of ECT(i.e.the pipeline permittivity distribution)is approximate sparse.However,the sparsity is determined by the medium in the pipeline,and for most flow patterns,the sparsity are unable to meet the requirement of CS.To meet the sparsity request of CS,the original signal of ECT has to be converted to be sparse by orthogonal transformation.Firstly,we give

where Θ1is an n×1 dimensional vector standing for the normalized permittivity distribution;θ is an n×n orthogonal matrix(i.e.sparse basis of Θ1);α is an n×1 sparse coefficient(i.e.α is the projection of original signal Θ1in the sparse basis θ).So,the mathematical model of an ECT system can be described as follows:

Besides,with the zero expansion operation and random recombination,the sensitivity matrix can be used as the measurement matrix.The central challenge of ECT is to determine the permittivity distribution from capacitance measurement,namely the image reconstruction.The change of capacitance in response to the perturbation of the permittivity distribution is nonlinear.Because the permittivity perturbation is usually small,Problem(5)can be linearized as

where A is the sensitivity matrix of the capacitance change y to permittivity change x,Φis the reconstructed image and y is the so-called“compressed sensing data”.The image is reconstructed by solving this linear equation.For a 16-electrode system,we can obtain 256 measurements which contains 120 independent measurements because the capacitance of all electrode pairs are measured.The reconstructed dimension of the permittivity distribution depends on the grid size in discretizing this area.For example,a 61×61 grid as we use later to generate an image of 3721 pixels,which is much larger than the number of measurements.Due to the small number of measurements and large image dimension,the equation is ill-posed.

We propose to fi nd a sparse vector by solving the augmented l1-regularized problem as follows:

where τ,y∈(0,∞)are the variables.

3.2.Alternating direction method of multipliers

The alternating direction method of multipliers(ADMM)is based on a variable-splitting technique and a method of multipliers framework.Eckstein and Bertsekas[27]combined the method of multipliers(MM)and the alternating direction method(ADM)to develop the ADMM,which thus has the virtue of both methods.Besides,it is preferable to other methods,because it decouples many difficult problems into simple sub-problems that can be easily solved.Consider a constrained optimization problem as described by

where f(u)and g(v)are the convex functions,C and D are the given matrices,b is a given vector,and u and v are the unknown variables.The augmented Lagrangian function based on the constrained optimization Problem(7)is given as follows:

However,an accurate and joint optimization for u and v can be costly.In contrast,the ADMM utilizes a separable structure in optimization Problem(7)and replaces the joint least squares by two simpler sub-problems.Specifically,the ADMM minimizes u and v of Γ(u,v,λk)separately via a nonlinear Gauss-Seidel type iteration.After just one alternating least square for u and v,the multiplier λ is updated immediately.In short,when given(vk,λk),the ADMM iterates as follows:

The above ADMM proposed to solve the constrained optimization Problem(7)can be summarized as follows.

Algorithm1:ADMM algorithm(1)Initialization:Given μ ＞0,starting points v0 and λ0,and iteration index k=0;(2)Update u:u k+1=arg min uΓ（u,v k,λk）(3)Update v:v k+1=arg min Γ（u k+1,v k,λk）(4)Update λ:λk+1=λk-μ(Cu k+1+Dv k+1-b)(5)Iteration is terminated if the termination condition is satisfied;otherwise,set k=k+1 and return to step(2).v

We terminate the ADMM algorithm when the relative change in the u vector between two consecutive iterations becomes small enough.

Due to the strict mathematical foundation,the ADMM algorithm has been successfully applied to reconstruct images and get good reconstructed quality[28].But when the observation matrix is not the specified matrix,the ADMMis highly complex in computation,that leads to a long image reconstruction time.And this is unacceptable for an ECT system that has a high speed performance demand.

3.3.Fast linearized alternating direction method of multipliers

Obviously,it is a problem that how we can speed up execution while maintaining the reconstruction quality for the ADMM algorithm.Yang et al.[28]studied the use of alternating direction algorithms for the l1-regularized problem arising from sparse solution reconstruction in CS.Inspired by this work[28],we present the fast linearized alternating direction method of multipliers algorithm(FLADMM)for solving the augmented l1-regularized problem in this section.

Variable splitting is a very simple procedure that involves creating a new auxiliary variable Ξ∈RNto serve as the argument ofunder the constraint.Then,the augmented l1-regularized Problem(7)is clearly equivalent to the constrained optimization problem as follows:

In Problem(10),is strongly convex and has the Lipschitz continuous gradient;τ||Ξ||1is convex and non-smooth.Note that if γ=0,thenmay not be strongly convex if the matrix A does not have full column rank.In many applications,this is indeed the case since the number of measurement y is usually smaller than their dimensions(i.e.,M＜N).However,the parameter γ＞0 guarantees the strong convexity ofand hence the global linear convergence of FLADMM algorithm when applied to Problem(10).The augmented Lagrangian function of Problem(10)is given as follows:

where soft(·,Th)=sgn(·)max{|·|-Th,0}is the soft thresholding function with threshold Th,and sgn(·)is the sign function.A key ingredient of Eq.(13)is the so-called soft thresholding function,which is the Moreau proximal mapping of the regularization||Θ||1.

In order to accelerate the convergence of the above iteration,the fast method of Beck et al.[29]is applied,and Ξ is updated again as follows:

Second,for the givenof Problem(10)with respect to Θ is given by

Since A is a random Gaussian matrix and(I being an identity matrix),it is complicated to solve this problem[28].Inspired by the work of Yang et al.[30],we use a linearized strategy for the quadratic termto accelerate the convergence of the FLADMM.With this linearization,the quadratic term can be approximated by

The above FLADMM algorithm which is proposed to solve the optimization Problem(10)can be summarized as follows.

Algorithm 2:FLADMM algorithm(1)Initialization:Given μ＞0,γ＞0,α＞0,τ＞0,starting points d0,Θ0,Ξ0,t0=1 and iteration index k=0;(2)Compute ω:ω=1+α(2γ+μ);Compute P:P=I-αA T A;Compute q:q=A T y;(3)Update t:tk+1=2;(4)Update Ξ:1+■■■■■■■■■■■1+4t2 k Ξk）;(5)Update Θ:τ μ）;Update Ξ again:Ξk+1=Ξk+1=soft（Θk-d k,Ξk+1+（tk-1 tk+1）（Ξk+1-Θk）;(6)Update d:d k+1=d k-(Θk+1-Ξk+1);(7)Iteration is terminated if the termination condition is satisfied;otherwise,set k=k+1 and return to step(3).1 ω（PΘk+α（q+μ（Ξk+1+d k）））;Update Θ again:Θk+1=Θk+1=Θk+1+（tk-1 tk+1）（Θk+1-

We terminate the FLADMM algorithm when the relative change in the sparse coefficient vector between two consecutive iterations becomes small enough,i.e.

From the above algorithm description,we can find that the FLADMM only requires one soft thresholding projection and matrix vector multiplications at each iteration.And the FLAMDD is substantially accelerated owing to the fast linearization idea from[29].In addition,the computational complexity of the ADMMis O(MN2).In contrast,the computational complexity of steps 4,5 at each iteration for the FLADMM algorithm is O(N2),whereas the computational complexities of steps 3 and 6 are both O(N),and others are only O(1).Therefore,the FLADMM has a better image reconstruction performance,more robust and a faster convergence rate.

4.Results and Discussion

In this section,some static and dynamic experimental results are presented to evaluate the performance of the proposed FLADMM.The hardware platform running these algorithms consists of a PC computer with a Core 2 Duo 2.8 GHz CPU and 4 GB of RAM.

4.1.Numerical simulation

For the static image reconstruction,the difference between the numerical simulation and the static experiment is very small,and the simulation can allow for flexibility in testing the performance of the FLADMM algorithm.Hence,we adopt the numerical simulation to verify the effectiveness and compare the reconstruction results with other algorithms.All algorithms are implemented using the MATLAB 2014a software.Four typical flow patterns are chosen for the simulations,which are presented in Fig.4.The potential distribution data of four typical patterns are derived from the Finite Element Analysis(FEA)tool in COMSOL MultiphysicsR.The structure of a 3D simulation model of ECT is shown in Fig.5.Moreover,the permittivity data acquisition process can be also regarded as an ECT positive problem.Its boundary condition is the Dirichlet type.When the electrode i(i=1,2,…N-1,N is the plate number)is source electrode,the boundary condition is described as follows:

Fig.4.Simulated test objects.

Fig.5.Structure of ECT model using COMSOL.

where the Γ1,Γ2,…ΓNis the location of measure electrodes in the sensor system,Γsis the location of outer shield,Γpqis the location of radial electrodes and Vcis the driving voltage.

Table 1 Comparison of reconstructed tomogram

The COMSOL software options are:Homepage＞Space Dimension＞3D;Add Physics AD/DC＞Electrostatics;Select Study Type＞Preset Studies＞Stationary.The capacitances across different electrodes for these permittivity distributions are calculated using the finite-element method(FEM)according to Eqs.(1)and(2).The Delaunay triangular mesh generation based on linear interpolation is adopted.In order to reduce the influence of various kinds of interference on the capacitance,the original capacitances are normalized as follows:

whereis the capacitance between plate i and plate j when the pipe is filled with dielectric 1(water),is the capacitance between plates i and plate j when the pipe is filled with dielectric 2(oil).

The black color part stands for the high permittivity materials with a value of2.1(water),and the white color part represents the low permittivity materials with a value of 1.0(oil).The dielectric constants of pipeline wall and insulating filler materials are both 5.Besides,it is very important to select appropriate parameters for obtaining permittivity distribution data.From our experience,R1 is 71.3 mm,R2 is 82.5 mm,R3 is 92.7 mm,θ is 22.5°and the electrode size is 10 mm×20 mm.And a 12-electrode ECT is simulated,and these sensors are distributedevenly and the size of the reconstructed region is 96 mm×96 mm for the circular cross section.An image using 32×32 pixels is presented and each pixel takes a value between 0 and 225 on the gray scale.The research object,respectively,uses the typical core flow,annular flow,four bubbles and stratified flow.The image reconstruction results are as shown in Table 1(black area is water,and white area for oil).

Table 2 Mean square error σ2

In order to evaluate the imaging performance of the super resolution image reconstruction algorithm in Electrical Capacitance Tomography,the Landweber algorithm,LBP method and CG method are used for comparative evaluation.Three objective evaluation parameters(mean square error,signal to noise ratio,and peak signal to noise ratio)are used to index the performance of algorithms.The details are as follows.

4.1.1.Mean square error

Mean square error indicates the proximity of two images of the same size.As a very intuitive and effective evaluation index,mean square error rate can clearly show the effect of image reconstruction.Usually,in the super resolution reconstruction experiment,the reconstructed images are compared with the known original images in terms of the mean square error:

Table 3 Signal to noise ratio

Table 4 Peak signal to noise ratio

where M,N are the length and width of the image.f(i,j)represents the grayness value of pixels of the original image,and g(i,j)for the pixels of the reconstructed image.

The exact figures of the evaluations simulated by Matlab are shown in Table 2.

4.1.2.Signal to noise ratio

The noise in the origin data has an important impact on image reconstruction for ECT.Therefore,the image signal to noise ratio SNRshould not be ignored.Signal to noise ratio is used to measure the ratio between the reconstructed image and the original image.The larger the ratio is,the more information the image contains.On the contrary,the smaller the ratio is,the less information the image contains.Consider

the evaluations are shown in Table 3.

4.1.3.Peak signal to noise ratio

Peak signal to noise ratio is also often used as an evaluation index of image compression:

It indicates the ratio between the maximum possible power signal and the destructive noise power which affects the representation accuracy.The larger the ratio is,the less the distortion is.The exact evaluations are shown in Table 4.

From Tables 1 and 2,the quality of the reconstructed images for the four flow patterns has been improved by using the algorithm proposed in this paper.For the flow patterns(a),(b)and(d),the improvement of image reconstruction quality is quite obvious,but for the four bubbles pattern(c),compared with other 3 methods,the σ2of the proposed algorithm has improved very significantly.In Table 3,the clarity of reconstructed images also improves by using the proposed method.From Table 4,the image distortion has been reduced significantly,which indicates that the proposed FLADMM has a better tolerance for noise signals.In conclusion,the resolution in the periphery and center of the imaging area is better than other reconstruction algorithm.

4.2.Experimental results and analysis

The above simulation results show that the proposed method is suitable for static images reconstruction of ECT systems.To inspect and verify actual functions of this method further,a dynamic experiment of gas-solid two-phase flow measurement in a CFB(circulating fluidized bed)system is tested.The permittivity of the solid particle is about 3.0,and the permittivity of the air is about 1.0.

The schematic diagram of the experimental platform is shown in Fig.6.The experimental platform consists of the material circulating system,riser tube,loop seal system,bag filter,cyclone separator,and compressed air system.In particular,the ECT measurement system is included in the platform.The riser tube is 0.4 m in diameter and 3.94 m high,and the top-hat chamber is 0.2 m high.Moreover,the butter fly valve is installed to meter the solids re-circulating flow-rate of the loop seal system.The nozzle structure of riser tube is a bed-base with a round hole(0.25 m in diameter).The gas-solid device needs to be kept working at a minus pressure.

Data acquisition is very important for ECT system in the practical application.However,the measured capacitances of an ECT system are very small so that the measurement mostly is disturbed by the stray capacitance and the high-drift of the instrument base-line.

Fig.6.The experimental rig of ECT system for the gas-solid flow measurement.

Fig.7.The micro-capacitance gauging circuit diagram of data acquisition based on the differential sampling principle.

Fig.8.The signal sequence diagram.

Table 5 The operational parameters setting

Hence,the experiment adopts the micro-capacitance gauging circuit based on the differential sampling principle to collect the capacitances quickly and accurately.The micro-capacitance gauging circuit diagram is shown in Fig.7.The sequence diagram is shown in Fig.8.In Fig.7,Cxis the capacitance to be measured.Viis the excitation source.U1,Crand S1compose the charge amplifier.The switch(S2,S3),voltage follower(U1,U2)and capacitors(Ch1,Ch2)compose the sample hold device(S/H).Casand Cbsare the equivalent capacitances of the stray capacitances.Casis driven by excitation source Viand the voltage difference between two ends of the capacitor Cbsis zero.So the stray capacitances have no effect on capacitance measurement.Moreover,the proposed micro-capacitance gauging circuit can acquire the original images at 800 frames per second[31].

This experiment adopts 12-electrode square sensor system.And the operation condition of this experiment is shown in Table 5.The dynamic images reconstructed by the FLADMM algorithm are shown in Table 6.

Table 6 shows the reconstructed images by the FLADMM algorithm for a series of time instants with an interval about 0.417 s.As can be expected,the FLADMM algorithm shows satisfactory dynamic imaging performance,and it can fast reconstruct the distributions of solid particles over the cross section in a riser tube.The typical core annulus structure and the particle clusters can be observed.From Table 6,we can also find that the proposed algorithm has clear reconstructed images.Under the same experimental conditions,other methods such as LBP and CG,are used to offer some contrast.The results indicate that the proposed method has a higher image reconstruction quality and a faster operation.In particular,due to the round nozzle structure of riser tube,the particle concentration inthe vicinity of the wall is relatively high,and in the center area the particle concentration is relative low.Table 6 also verifies this fact clearly.This phenomenon also indicates the images reconstructed by the FLADMM fit the real application.In addition,it can be found that the shape of the core-annulus structure changes with time owing to the intense interaction between gas and solid phases.These results indicate that the proposed algorithm is successful in reconstructing the dynamic objects.

Table 6 Time series of reconstructed images by the FLADMM algorithm

5.Conclusions

In this paper,a generalized objective functional,which has been developed using a new fast linearized alternating direction method of multipliers,is proposed.This objective functional transforms the ECT image reconstruction problem into a convex relaxation problem.The FLADMM algorithm is employed to solve the objective functional.The algorithm is tested by the simulation and the dynamic experiment.The evaluation indexes of the simulation,such as σ2,RSNand PSNR,show remarkable improvement in the accuracy and spatial resolution of the reconstructed images,and the artifacts in the reconstructed images can be eliminated effectively,which indicate that the proposed algorithm is successful in solving ECT inverse problem.The dynamic experiment results show that the proposed algorithm is successful in reconstructing the dynamic objects in the lab-scale fluidized bed equipment.And the proposed method also fulfills the real-time requirement of ECT system in the application.

In addition,to enable ECT technology usable in a real industry environment,more work on the hardware and software of ECT systems should be done and the image reconstruction algorithms should be further developed.

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