·Table of Contents ·Computer Processing and Simulation | Research of the Image Reconstruction Algorithm for Electrical Resistance Tomography Based on Neural NetworkWei YingPO Box 321, School of Information Science and Engineering, Northeastern University, Shenyang, 110006, P. R. China E-mail:MCGMZWZC@pub.ln.cninfo.net Tel & Fax: +86 24 23891977 Wu Xinjie Zhao Jinchuang Wang Shi Contact |
In formula (1),Ñ is gradient operator, f(x,y) is the function of voltage distribution in the sensing field, W is region of the sensing field, ¶Wis the boundage of W , s(x,y) is the function of unkown conductance, J is the excite current density which is exerted on the sensor electrode, U_{J}^{0} is the function of the voltage distribution boundage, ¶f /¶n is the direction derivative along the boundage.
When s (x,y) is known, the problem of solving the f(x,y)in any point in the field is called forward problem of ERT, which is from (1a) (1c)or (1a) (1b) to solve f(x,y), it is problems all known that Dirichlet problem and Neumann problem of Laplace equation, the solution is suitable to meaning of Hadamard. Authors adopted finite element method (FEM) and have set up ERT simulateion software, with it got the solution of forward problem of ERT conveniently. From the known boundage condition of U_{J}^{0} to reverse compute s(x,y) is inverse problem of ERT, it is the finally aim of ERT, but the solution is unsuitable and nonlineary. To conquer the unsuitability and nonlinearility is the key and difficulty of ERT image reconstruct, so must adopt suitable reconstruct algorithm to gain high precision solution.
Fig 1: typical ERT system structure |
An ERT system can be divided into three basic parts as shown in Fig.1 : the sensing array, the data acquire & process system and reconstructing unit. A set of rectangular electrodes are mounted flushly inside surface of a process pipe to form an annular sensing array. These electrodes are excited by a current source one pair by one pair in turn, and all the voltages values between any other two eletrodes are measured. These data reflect the interaction between the conductivity distribution in the region of interest interaction complyes with Laplace equation. The total number of independent measurements is N(N-3)/2, where N is the number of electrodes, being 104 for 16 electrodes, these data are fed into a computer to reconstruct the image of conductivity distribution using qualitative or quantitative algorithm. The last step is to extract useful flow parameters from the reconstructed image by the computer, such as concentration, velocity etc.
(2) |
Fig 2: Structure of RBF neural network | Fig 3: The sensing field is divided into 800 pixels in the pipe |
Fig 4: The even square error-epoch curve in the training in the RBF neural network |
Before training the network, the 104 measurement value had been normalized between 0-1 in order to make the neural network distinguishing those resemble training sample and the learning apt to convergence. The learning method of the neural network adopt teacher guide selecting RBF center. During each time of learning produced a radical basis function neuron, such again, the number of neurons increased continuously until get the expected error or get the maximum epoch (detail of the procedure of the training-learning of RBF neural network is in the reference [2]).
60 typical flow pattern samples had been trained for the network, there are 104 measurement voltages for each flow pattern, so the input of the network is a matrix of 104´60,the pipe is divided into 800 pixels, therefore the output of the network is a matrix of 800´60. The convergence speed of the RBF network is very fast, when trained for 58 times, the error of network output is 10^{-10},the curve of error-training is shown in Fig.4.
Tab 1 :Results of different flow-pattern by three kinds of ERT reconstruct alogrithms |
(3) |
In formula (3), consider area of different pixel element may not be the same, so in formula (3) A(p) express the area of the p^{th} element. The grey value distribution of set up model is g_{s}(p), the grey value distribution of the reconstructed image is g_{B}(p), g_{B}(p) is numerical as follows:
(4) |
Select SIE_{n} as the image quality evaluate factor of the section. The smaller SIE_{n }is, the higher quality image is. The SIE_{n} of back-project and sensitive coefficient algorithm is about 11% and 8% respectively, the SIE_{n} of RBF neural network is not more than 5%
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