Min MA, Ming LI, Xiao-fang HE, Ya-nan LIU, Xi-yuan CHEN

(College of Electronic Information and Automation,Civil Aviation University of China, Tianjin 300300, China)

Abstract: Aiming at the problem of low recognition rate and large amount of data in traditional capacitive tomography, a flow pattern recognition method based on compression perceptual theory is proposed. Firstly, the original image gray scale signal is thinned by discrete cosine DCT base, and then the random Gaussian matrix is used for observation. It can use a small amount of data to accurately reconstruct the original signal, reducing the sampling time; at the same time in the imaging algorithm to avoid L1 norm regularization requires a lot of data and L0 norm optimization NP problem. The adaptive Lp (0

Key words: Capacitive tomography, Compression sensing, Cosine DCT base,The random Gaussian matrix, Adaptive Lp norm

Capacitance tomography is one of the earliest process imaging techniques that has been applied in a number of fields, such as fluidized bed drying processes, industrial scale fluidized bed internal complex gas-solid two-phase flow processes, dense phase powder, and flow measurement, nuclear reactor thermal and hydraulic characteristics of the monitoring, which, in fluidized bed monitoring to achieve a more significant application results. As a non-invasive, low-cost, non-invasive visual measurement technology, ECT technology is expected to become a new generation of industrial detection technology [1].

The ECT system consists of three parts: a sensor array unit, a data acquisition and processing unit, an image reconstruction and display unit, and a capacitive sensor array mounted around the pipeline, typically eight, twelve, or sixteen electrodes, both of which are the excitation electrode is the measuring electrode, one of which is the excitation electrode and the other electrode is the measuring electrode. The capacitance value between the electrode pairs depends on the distribution of the media in the pipeline. For the n electrodes,n(n-1)/2 sets of document [2] can be measured. The image reconstruction unit can be reconstructed according to the measured capacitance data and reconstruction algorithm. The distribution of the media in the cross section of the sensor in the pipeline could be realized, so as to realize the visualization of the medium distribution in the pipeline. The basic composition of the structure as shown in Fig.1.

At present, with reference to ECT system image reconstruction algorithm imaging speed and solution process, it can be divided into direct solution algorithm, iterative algorithm and other algorithms; direct algorithm is simple and fast, such as linear back projection method, but the image quality is poor; iterative algorithm such as Landweber Algorithm, image quality is higher, but the reconstruction speed is slow, affecting the real-time imaging. In 2006, Professor Candes of the California Institute of Technology and his team proved that in the Fourier representation of the image, if the K non-zero Fourier coefficients were included, the number of coefficients was randomly selected to reconstruct the original image only satisfyM≥2K. This discovery led to the generation of compressed sensing theory[3].

Fig.1 Electrode ECT system

In this paper, an ECT image reconstruction algorithm based on compression perceptual theory is proposed. It is found that the discrete cosine DCT transform can be used to deal with the original signal so as to satisfy the prerequisite of the compression perception theory. The random Gaussian matrix is used as the measurement matrix, and the adaptive Lp norm regularization method is used to reconstruct the gray image of the original image. The optimal threshold method is used to optimize the image, and the final image could be obtained.

1 Compressed sensing principle

1.1 The sparse representation of the signal

The sparseness and compressibility of the signal is the prerequisite for the theory of compression perception, but in practical applications, the signal is not sparse. In order to apply the compression perceptual theory, the orthogonally transformable signal can be sparse. Specific steps are as follows: Suppose a finite length one-dimensional discrete signalx; the length is N; the signal theory can be used, the signal can be linear expressed as a set of orthogonal vector basesψ=[ψ1,ψ2,ψ3,…,ψN]

(1)

Where,ψis an orthogonal matrix ofN×Ndimensions,αis a coefficient matrix ofN×1 dimensions. When there are only K nonzero elements inα, and it satisfiesK<

1.2 Design of observation matrix

According to the compression perceptual theory, the compression sensing process can be described as shown in Fig.2, where the measurement matrix can project the high-dimensional signal to low dimension and obtain the sampled signal. By designing the appropriate observing system, and the original information is contained as much as possible or more in the observed value so that the original signalxof lengthNcan be accurately reconstructed from theMobservations [5].

Fig.2 Compression sensing process

The observation process actually constructs theM×Ndimensional observation matrixfand observes the original signalxto obtainMobservationsCi=(i=1,2,3,…,M), namely:

C=fx=fψα=ACSα

(2)

Where,Cis theM×1 dimensional observation vector;fis theM×Ndimensional observation matrix. Order, known as the compression perception matrix. MakeAcs=fφ,Acsis called the compressed sensing matrix.

In order to ensure that the original structure can still be maintained after the signal is projected and the projection mode is linear, the compression perception theory requires that the measurement matrix must satisfy the Restricted Isometry property (RIP) condition [6] or the transform domain matrix is not relevant for all Has a K-sparse form of the signalx. The limit equidistant constantσkthat defines the measurement matrixfsatisfies the following formula:

(3)

If the constantσk∈(0,1), then the measurement matrixfsatisfies the K-order limit equidistant condition. In document [6], it is pointed out that when the observation matrixfand the sparse baseψare not correlated, whenM≥C·Kln(N/K)≈4×K，Acscan satisfy RIP conditions at a high probability. In this paper, the Gaussian random matrix is chosen as the observation matrix and the discrete cosine transform base as the sparse base. The unique solution of the equation (2) could be obtained.

1.3 Signal reconstruction algorithm

For the compression sensing method, the essence of signal reconstruction is to find the most sparse solution vectorx, apply the precondition ofKterm sparse to (2), convert the problem of incursion into the minimum L0 norm problem, and solve the coefficient Vectorα, to reconstruct the original signalx, namely:

αopt=argmin||α||0

s.t.C=fψα,xopt=ψαopt

(4)

However, the L0 norm has a nonconvexity, which makes the problem NP difficult problem. Donoho and chen et al. proposed that the problem can be transformed into convex optimization problem by L1 norm approximation L0 norm, and then the linear programming (BP) [8], sparse gradient projection algorithm (GPSR) [9], original dual interior point method [10], and so on. The method of solving the problem of compression sensing [7] and the mathematical model of the ECT system is as follows:

C=Sg

(5)

Where,Sis the normalizedM×Ndimensional sensitivity coefficient matrix,gis the normalizedN×1 dimensional image gray matrix, andCis the measuredM×1 dimensional capacitance matrix.

Based on the compression perception theory, the ECT system needs to reconstruct the image gray signal of the original signal. In order to meet the preconditions, it is necessary to do the thinning of the image gray scale signal. In this paper, the cosine DCT base is used to trinity the image signal,

g=ψDCTα

(6)

Where,αis the K-sparse coefficient vector forN×1 dimensional ;ψDCTis the cosine DCT base, and its expression is as follows:

Where,m,n= 1, 2, 3, …,N

Substitute the formula (6) into formula (5), the ECT mathematical model could be obtained based on the theory of compressed sensing:

C=Sg=SψDCTα=Acsα

(7)

Where,Acsis the system compression perception matrix andCis the projection result obtained by observing the original image grayscale signalgthrough the observation matrixSfor the sampling system.

The theory of compression sensing requires that the measurement matrixSsatisfy the RIP condition. The equivalent condition is that the measurement matrix is not related to the sparse base matrix. In litarature [11], the stochastic selection electrode is used to excite and follow the random sequence sampling method in combination with the ECT system. Improve the randomness of the observation matrix, while controlling the number of independent measurements, reducing the consumption of the time in the sample. In the literature[3], the data of the measurement capacitance gets increased or decreased by using the zero-padding method, that is, it is assumed that some fictitious capacitive plates are added, and the sensitivity of all the mesh nodes relative to these fictitious capacitive plates is zero, which increases the sampling rate. Then the sensitivity matrix after adding fictitious capacitance is arranged in a random arrangement to form a new sensitivity matrix in order to improve the randomness of the observation matrix.

2 ECT image reconstruction algorithms based on adaptive Lp norm

In literature[12], it is proved that the reconstruction algorithm based on Lp (0

Rao is proposed IRLS (Iteratively Reweighted Least Square, IRLS [13]) based on Lp (0

(8)

The weightwiis:

(9)

The iteration formula for the sparse coefficientαis:

(10)

Q=diag(1./w)

However, in the actual case, the final result obtained depends on the norm valuep,p=0 is not the best choice for all cases, here is an improved scheme, so that the norm value p can choose the optimal value according to the different input. An iterative weighted least squares method is called as adaptive Lp norm [13-14].

In this paper, by inputting different adaptive p values, in order to make the p value converge sensitive, definite：

The gradient change is approximately:

▽rRMSEk=rRMSEk-rRMSEk-1

The algorithm is as follows:

(1) Set the initial valuek=0，pk=1，ε=1， Iter=1 000，α(0)=f+y.

(2) Fork=1:Iter

elsek=k+1, go to(3).

(6) Use the IRLS method to solve:εk+1

(7) if it satisfy ||fa-C||≤η, end, if notα(0)=αrepeat (2)-(6).

3 Experimental results and analysis

In this paper, four typical simulation flow patterns are simulated by gas-solid and gas-liquid two-phase flow. The COMSOL 5.2 finite element software is used to draw the model and measure the capacitance value. COMSOL5.2 automatic meshing. The split unit is triangular, the pattern reconstruction is calculated by using 3228 rectangular grids, and the ECT image reconstruction is performed using MATLAB R2014a. Through the COMSOL software, the core flow, double bubble flow, four bubble flow and laminar flow simulation are used to solve the ECT problem. The imaging quality of the model is compared with the image of the above algorithm. The image error and image of the correlation coefficient of quantitative are analyzed.

(1) Image error (IME), the smaller the image error, the higher the imaging accuracy, the greater the error, the lower the accuracy. Defined as follows:

(11)

(2)Image correlation coefficient (CORR), the greater the image correlation coefficient, the higher the spatial resolution of the image, the smaller the correlation coefficient, the lower the image resolution, defined as follows:

(12)

In the simulation experiments, the information about the size and shape of the object to be reconstructed and the distribution of the relative permittivity values of each pixel is known. The image quality is evaluated by comparing the correlation and error of the reconstructed image with the image to be reconstructed.

In order to reflect the adaptive effect of the algorithm used in this paper, Table 5.13 gives the imaging results based on the four bubble flows, the norm value P and the penalty term F, and the adaptive values.

According to the above analysis, we can see that the images with fixed parameters have more artifacts and relatively low resolution. The adaptive method proposed in this paper is superior to the fixed parameters. Therefore, the selection of parameters affects the reconstruction quality of the image. The method proposed in this paper can avoid the problem well and adaptively select the optimal parameters, so that the image reconstruction quality is better than the fixed parameter method.

In order to better compare the effect of image reconstruction, Table 3 shows the comparison of imaging results based on LBP, regularization, SVD and the proposed algorithm for different flow patterns.

Table 1 Reconstruction quality for variouspandλ

Table 2 Reconstruction data for variouspandλ

Simulation prototypeP=1,λ=0.1P=0.1,λ=1P=0.3,λ=1自適應(yīng)p,λIME0.286 50.277 40.270 60.265 9CORR0.711 80.745 60.757 20.789 7

Table 3 Different algorithm reconstruction results

Table 4 Image reconstruction speed comparison(s)

Flow patternCore flowDouble bubble flowFour bubbles flowLaminar flowLBP0.0460.0520.0760.087Tikhonov2.5833.2353.3533.064SVD3.0353.4354.0523.484Adaptive Lp norm2.7693.1312.9552.982

Table 5 Image relative error(IME)

Flow patternCore flowDouble bubble flowFour bubbles flowLaminar flowLBP0.301 20.649 50.634 30.626 8Tikhonov0.271 80.543 10.286 50.453 1SVD0.321 60.564 20.591 20.512 4Adaptive Lp norm0.286 90.385 60.265 90.332 9

Table 6 Image correlation coefficient(CORR)

Flow patternCore flowDouble bubble flowFour bubbles flowLaminar flowLBP0.635 60.405 40.398 20.642 1Tikhonov0.778 00.682 10.701 70.561 3SVD0.512 00.430 60.462 70.514 5Adaptive Lp norm0.685 20.752 10.729 70.806 7

According to the comparison of the results of different algorithms and the analysis of the relevant data, we can see that the LBP algorithm is faster than the other methods in terms of imaging speed due to the reversal of the matrix inverse by the transposition of the sensitivity matrix. Analysis of Table 5.17 and Table 5.18 image correlation error shows that the regularization algorithm image error, capacitance residual are relatively large, resulting in less image correlation coefficient (CORR). Singular value decomposition method, because the sensitivity of the matrix number is too large, the process of collecting data in the slight noise, will lead to serious image distortion, and make the final image error is very large, the correlation coefficient is very small. Three algorithms for the core flow and other simple model of the imaging accuracy is relatively optimistic, for complex flow patterns, imaging results are not ideal. The method of adaptive Lp norm based on compression perception proposed in this paper makes good use of the sparseness of the data obtained by the compression perceptual theory, reduces the redundancy between the data, less artifacts and better image quality than the other three Algorithm, positioning more accurate, with lower error and higher correlation.

4 Conclusion

This paper proposed an ECT image reconstruction algorithm based on the compression perceptual theory. The compression perceptual theory reconstructs the image accurately in the case of less measurement data. Therefore, it can solve the problem of poorly generated ECT system measurement data. In this paper, the adaptive Lp norm is used to replace the L1 regularization model, and the optimal parameters can be selected by inputting the data. The simulation results show that the method has high imaging accuracy. Saving the sampling time to improve the speed of reconstruction, has a good application prospects.

Acknowledgements

This work is supported by the Joint funding project of the National Natural Science Foundation of China and the China Civil Aviation Administration (No.U1733119); civil aviation science and Technology project (No.20150220).