A Superresolution Image Reconstruction Algorithm Based on Landweber in Electrical Capacitance Tomography

According to the image reconstruction accuracy influenced by the “soft field” nature and ill-conditioned problems in electrical capacitance tomography, a superresolution image reconstruction algorithm based on Landweber is proposed in the paper, which is based on the working principle of the electrical capacitance tomography system. The method uses the algorithm which is derived by regularization of solutions derived and derives closed solution by fast Fourier transform of the convolution kernel. So, it ensures the certainty of the solution and improves the stability and quality of image reconstruction results. Simulation results show that the imaging precision and real-time imaging of the algorithm are better than Landweber algorithm, and this algorithm proposes a new method for the electrical capacitance tomography image reconstruction algorithm.


Introduction
Electrical capacitance tomography (ECT) is a high-tech cross-section detection technology in recent years, and it can achieve the visualization measurement of the multiphase media component parameter in the closed pipe and process container equipment.The ECT technology could be able to obtain the permittivity distribution inside the pipeline through measuring the capacitance values between the different plates of the capacitive sensing electrode array around the pipeline.It has the advantages of noninvasive, simple structure, low cost, fast response, good safety performance, wide application range, and so forth.So, it can be used for measuring the two-phase flow in the petroleum, chemical and electric power industries, and so forth.
There are three principal factors with image reconstruction for the electrical capacitance tomography system.Firstly, it is the "soft field" effect; the sensitivity performance is in homogeneity affected by the medium distribution.Secondly, the measurements are less than the pixels of the image, and the underdetermined equation has numerous solutions.Thirdly, the solution stability is poor; even small perturbation in the measured capacitance data can cause great error.Due to the nonlinear and the "soft field" effect, the stability of the ECT system is poor and seriously pathological.So, it is difficult for image reconstruction [1].At present, there are many methods about the ECT image reconstruction, such as the linear back projection (LBP), regularization method, Landweber iterative method [2], genetic algorithms [3], projection Landweber iterative method, conjugate gradient method [4], improved Gauss-Newton method [5], and acceleration algorithm based on polynomial [6].The characteristic of the LBP is simple and fast reconstruction.But because its imaging quality is relatively poor, strictly speaking, the algorithm is only a qualitative algorithm.Regular methods commonly use empirical values, because the regular selection of parameters influences the imaging quality.It is specially worth mentioning that the online preiterative method (OIOR) based on Landweber iterative method has the same imaging quality with several Landweber iterations under the same speed with LBP, but its spatial resolution has yet to exceed Landweber method.

Basic Principle of ECT System
Electrical capacitance tomography system mainly consists of three parts: capacitive sensor, data acquisition and signal processing, and image reconstruction, as shown in Figure 1.These data will be sent to the computer after processing.These measurement values reflect the distribution of the dielectric constant in the pipeline.The distribution information of the fluid medium in the pipeline is obtained through some image reconstruction algorithm by the computer. is the total number of independent electrode pairs, which is available for the ECT system with the -electrode plate.Consider In this paper, the typical 8-electrode capacitance sensor is the research object, we can get 28 independent measured values as  1 ,  2 ,  3 , . . .,  28 .The capacitance value of degree of influence on the same plate pair has different effects on the medium of the different locations in the pipeline.That is the soft field characteristic.On the other hand, the same point of the medium within the pipeline has different effects on the capacitance values of different plate pairs.The capacitance values between a pair of capacitor plates can actually be approximately regarded as the varying degrees superposition by the capacitance of all the points within the pipeline.The capacitance measurement values can be seen as the projection data of the multiphase fluid on a direction within the pipeline.The electrical capacitance tomography obtains each phase parameter of the multiphase flow from the projection data; then reconstruct the image.According to the electrical principles, ignoring the impact of the distribution changes of each subphase of the pipe radial multiphase fluid and the shield, the measure value   which is for the capacitance between any two electrodes in the array type of the 8-electrode capacitance sensor is where  = 1, 2, . . ., 28,  is the pipeline cross-sections, (, ) represents the distribution function of the dielectric in the pipeline cross-section, and   (, , (, )) is the sensitivity distribution function of the measuring capacitance   , namely, the sensitivity of the capacitance   to the media on point (, ).
The sensitivity distribution data is the priori data required for the image reconstruction.The sensitivity distribution which is also known as the sensitive field can reduce the soft field error of the sensitive field and improve the quality of the image reconstruction.It is extremely necessary to carry out in-depth analysis on the sensitivity distribution.
The calculation formula of the sensitivity function is where  (oil)    is empty-pipe capacitance, that is, the capacitance when the pipeline is filled with the discrete phase (oil), and it is a constant, where  = 1, 2, . . ., 28.When the dielectric constant of the th microelement is  2 in the pipeline and the dielectric constant of other microelements is  1 ,   () is the capacitance value.

The Principle of the Landweber Iterative Algorithms
The ECT image reconstruction is a typical ill-posed problem, and its solution is unstable.In order to obtain meaningful results of reconstruction, we should use the method that can improve the quality and stability of the solution.Regularization technique is an effective method to deal with pathological problems, and Landweber iteration method is a typical regularization techniques.The Landweber iterative format can be expressed as [7] The iterative method actually uses the steepest decent method with the step size as  to solve the minimum of the quadratic functional ‖ − ‖ 2 .Actually, if () = ( −   ) (−1) +    is the steepest descent iterative sequence with the step size as .On the other hand, constant  in the iterative scheme can also be seen as a relaxation factor, and it is a modified form from  () =  (−1) − ( (−1) − ).
The  () is expressed as  () =   ; the operator   :  →  is Use the singular system (  ,   ,   ) of compact operator  to expand.The expression of   is as follows: where the filtering function may be expressed as follows: When constructing the regularization solution sequence   by Landweber iteration, the number of iterations  = () should be given, and it is as the iteration stopping criteria.One common method is that iteration stops when the first  which meets formula ( 9) is disappeared: The so-called semiconvergence phenomenon will disappear when solves the inverse problem with Landweber iterative method.That is, in the early stages of the iteration approximate solution can be stably improved, and the effect of self-regularization is shown, but the iterations will tend to diverge when iterations exceed a certain threshold [8].So, the key question is to find a suitable termination principle in order to match the number of iterations and the error level of original data.However, in practical applications, the error level of the original data is difficult to obtain.So, there is a common method to determine the number of iterations by empirical methods.
Landweber iterative method has various modifications and most improvements concentrate on accelerating the problem.From the numerical optimization point, Landweber iterative method belongs to the most descent methods, and its convergence rate is relatively slow.V-method is an accelerated method of Landweber algorithm.The method by Brakhage is a polynomial acceleration method whose applications are broader than the method by Chebyshev.V-method can be expressed as the following iterative scheme [9]: where  1 = 0,  1 = (4V + 2)/(4V + 1); when  ≥ 1, iterative scheme is as follows: In fact, standard Landweber iterative method can be seen as a special kind of steepest descent method.Standard landweber iterative method steepest descent method have the same iteration format, and the only difference is the selected aspect of the iteration step.The steepest descent method requires searching from each of iteration by the 1-D to obtain the optimal step size.Landweber iteration method makes the step size in the interval [0, ‖‖ −2 ].

Superresolution Image Reconstruction Algorithm
4.1.Superresolution Reconstruction Model.In the ECT image reconstruction, small changes of capacitance value which is measured can cause the fluctuation of the solution.This can cause the deterioration of the imaging quality.Sequence images about the same object with different angles can make up such issues to a certain extent.The destination of image restoration is to recover image that has been degraded and make sure that the processed image is close to the original as possible Figure 2 simulated the degradation processing of sequence images.
In order to establish mathematical model of superresolution image reconstruction, suppose that   is th lowresolution image,  is high-resolution image which needs to be reconstruct,   is the geometry transformation matrix,   is the fuzzy matrix,  is the downsampling matrix, and   is the additional noise.We can obtain superresolution reconstruction model which is as follows [10]: where  is the  × 1 vector, which is super-resolution image vectors permutation that we need to solve. shows a low-resolution image with a total of  frames.  is the     × 1 vector, which is said to get the low-resolution image whose size is   ×   from the -frame./  = /  =  is the multiple of the corresponding superresolution reconstruction.  is the motion compensation matrix of  ×  which does pixel registration.And it is obtained by general multilevel block-matching registration.
is blurring function matrix ×. is downsampling matrix     × .  is the Gaussian noise obtained by the low-resolution image of the th frame of the image.Array formula (12) by a column vector of the image and as follows: where  is the smooth operator for solving ill-posed problem and it usually uses the Laplacian in super-resolution reconstruction. is the regularization parameter and it can be obtained by experiences in engineering approximation or other methods.Transformation is shown as follows: Formula (15) becomes Minimizing the previous equation is equivalent to solving the following equations: Since the matrix  cannot be diagonalized, the solution of the previously mentioned equation is computationally intensive part in the super-resolution image reconstruction.Approximate the registration results among the images by iterative solution algorithm.And solve the previous equation by preconditioned conjugate gradient method.The computational complexity of the algorithm is high, and the computing complexity with the number of low-resolution images increases exponentially.Then, analyze the previous equation, and derive a direct calculation algorithm, so that the complexity of computation increases linearly with the number of low-resolution image.

The Export of Closed Solution.
The solutions of (19) can be directly expressed as follows: According to the matrix inversion lemma, The equation obtained by formula (20) is shown as follows: The formula obtained by solution is shown as follows: According to formula, ( 14) and ( 17),   is a  ×  block matrix, and the (, ) block is      −1  −         whose size is     ×     .The calculation uses the calculation model of cyclic matrix, so   ,   , and  are all the block cyclic matrixes.According to the nature of the block cyclic matrix,      −1  −       still is a block cyclic matrix whose size is ×.Since  is the next sampling matrix, the super resolution multiples on the horizontal and vertical directions are the same.     −1  −         is still a block cyclic matrix whose size is     ×     .Since block cyclic matrix plus the unit matrix is still a faster cycle matrix, matrix   + −1  is a  ×  partitioned matrix; each block of the matrix is still the block cyclic matrix whose size is     ×     .According to the nature that the block cyclic matrix can be diagonalized, let  , be the (, ) block cyclic matrix of   +  −1 : where Λ , is the diagonal matrix which is diagonalized by the corresponding block circulant matrix. is the corresponding orthogonal matrix, so the following formula can be obtained: The corresponding calculation mainly converts to seek the inverse of the corresponding block diagonalization matrix: Let  be an element permutation matrix of block matrix [11].   = , [  Λ] ,;, = [Λ] ,;, .The function is to substitute the (, ) element of the first (, ) block matrix for the (, ) element of the (, ) block matrix. So, where every Ã is a  ×  matrix.The inverse of formula ( 27) is shown as follows: According to the nature and inverse of the block diagonal matrix in formula (29), its inverse is also the block diagonalization of the corresponding matrix inverse.That is, Thus, the inversion of the matrix whose size is ( ×  × ) × ( ×  × ) transforms into the inversion of  ×   ×  matrix.According to the nature of the permutation matrix, formula (31) is shown as follows: where every Λ , is ( × ) × ( × ) diagonal matrix and substituted into formula (26): According to formula (14), ( 16), ( 17), (23), and (32), formula (33) can be obtained.
So, the superresolution image reconstruction computable closed formula is derived.In the derivation process, there are many methods to be used, such as the diagonalization of the block circulation diagonal matrix, fast Fourier transform, downsampling matrix transpose multiplying, small matrix inversion, and vector product.The block circulant matrix can be diagonalized by fast Fourier transform of the convolution kernel.Down sampling matrix transpose multiplication is equivalent to add zero in the corresponding position.For the small size of the matrix inversion, it can be calculated directly.reconstruction algorithm based on Landweber iterative can significantly improve imaging quality, stability, and so on.So, it proposes a new method for ECT image reconstruction algorithm.

Figure 2 :
Figure 2: Simulated degradation processing of sequence images.