CN101520478B

CN101520478B - Direct image reconstruction method based on capacitance tomography of round sensor

Info

Publication number: CN101520478B
Application number: CN2009100799475A
Authority: CN
Inventors: 曹章; 徐立军; 丁洁
Original assignee: Beihang University
Current assignee: Beihang University
Priority date: 2009-03-13
Filing date: 2009-03-13
Publication date: 2011-08-17
Anticipated expiration: 2029-03-13
Also published as: CN101520478A

Abstract

The present invention is a direct image reconstruction method based on electric capacitance tomography of a circular sensor, comprising the following specific operation steps: Step 1: For an ECT sensor with N electrodes on the same section, label the N electrodes counterclockwise as electrodes i(1≤i≤N), the traditional 1-by-1 excitation measurement mode is adopted, that is, N-1 actions are included in one scanning process. Step 2: Firstly, the measured capacitance value is preprocessed by linear transformation to obtain N*(N-1)/2 independently measured charge change values. Step 3: Calculate the scattering transformation t(s) of the discrete electrodes. Step 4: Reconstruct the variation of permittivity at any point. The invention is aimed at a circular measurement area, overcomes the limitation that the traditional algorithm needs to calculate the sensitivity matrix and cannot independently calculate the reconstruction results of some areas, directly realizes image reconstruction according to the capacitance measurement value, and can only directly perform image reconstruction on some areas. Due to the high real-time performance of this method, it has important practical value and application prospect to extend it to ECT.

Description

A direct approach to image reconstruction for electrical capacitance tomography based on circular sensors

(一)技术领域(1) Technical field

本发明涉及一种图像重建方法，特别是涉及一种基于圆形传感器的电容层析成像的图像重建直接方法。属于图像重建技术领域。The invention relates to an image reconstruction method, in particular to a direct method of image reconstruction based on electric capacitance tomography of a circular sensor. The invention belongs to the technical field of image reconstruction.

(二)背景技术(2) Background technology

电容层析成像技术(electrical capacitance tomography-ECT)是根据测量的电容数据来重建绝缘材料的空间介电常数的分布。ECT相比其它层析成像技术有一些优点，例如：低成本，快速响应，便携性，非侵入性和鲁棒性。在许多工业领域中具有广阔的应用前景，ECT的核心技术是，根据边界电容的一组测量值的变化量，采用图像重建方法重建内部的介电常数分布的变化量。Electrical capacitance tomography (ECT) is based on the measured capacitance data to reconstruct the distribution of the space permittivity of insulating materials. ECT has several advantages over other tomographic techniques such as: low cost, fast response, portability, non-invasiveness and robustness. It has broad application prospects in many industrial fields. The core technology of ECT is to use the image reconstruction method to reconstruct the variation of the internal permittivity distribution according to the variation of a set of measured values of boundary capacitance.

在过去，大多数图像重建算法是基于敏感度定理的，该定理由1971年Geselowitz的文章“心电图的导线理论在阻抗式测容积法的应用”电气电子工程师协会，生物医学工程学报，BME-18 38-41。(Geselowitz D B 1971 An application of electrocardiographic lead theory toimpedance plethysmography.IEEE Trans.Biomed.Eng.BME-18 38-41)和1972年Lehr的文章“一种用于阻抗式容积场计算的向量求导方法”，电气电子工程师协会，生物医学工程学报，BME-19 156-7(Lehr J 1972 A Vector Derivation Useful in Impedance Plethysmographic FieldCalculations IEEE Trans.Biomed.Eng.BME-19 156-7)中提出，是一种采用摄动原理的线性化方法。2005年Soleimani M和Lionheart W R B的文章“采用实验数据的电容层析成像的非线性图像重建”，测量科学与技术16 1987-96。(Soleimani M and Lionheart W R B 2005Nonlinear image reconstruction for electrical capacitance tomography using experimental data.Meas.Sci.Technol.16 1987-96)。In the past, most image reconstruction algorithms were based on the sensitivity theorem derived from the 1971 Geselowitz article "Application of the Wire Theory of the Electrocardiogram to Impedance Volumetrics" Institute of Electrical and Electronics Engineers, Biomedical Engineering Transactions, BME-18 38-41. (Geselowitz D B 1971 An application of electrocardiographic lead theory toimpedance plethysmography.IEEE Trans.Biomed.Eng.BME-18 38-41) and Lehr's 1972 article "A Vector Derivation Method for Impedance Volume Field Calculation" , Institute of Electrical and Electronics Engineers, Biomedical Engineering Journal, BME-19 156-7 (Lehr J 1972 A Vector Derivation Useful in Impedance Plethysmographic Field Calculations IEEE Trans.Biomed.Eng.BME-19 156-7), is a kind of A linearization method for the perturbation principle. 2005 Soleimani M and Lionheart W R B article "Nonlinear image reconstruction of electrical capacitance tomography using experimental data", Measurement Science and Technology 16 1987-96. (Soleimani M and Lionheart W R B 2005Nonlinear image reconstruction for electrical capacitance tomography using experimental data. Meas. Sci. Technol. 16 1987-96).

由于ECT具有‘软’场特性，即空间灵敏度的分布随着空间介电常数分布的变化而变化。在过去的几年中，有研究人员采用更新的灵敏度矩阵进行迭代的图像重建，但非常耗时，且该类迭代的收敛性尚未得到证实。详见2004年Fang W的文章“一种电容层析成像图像重建的非线性算法”，测量科学与技术15 2124-32。Li Y和Yang W Q2008年的文章“对于复杂分布的非线性Landweber迭代法进行图像重建”，测量科学与技术19 94014。Smolik W等人在2006年的文章“更新灵敏度矩阵的电容层析成像图像重建算法的实际数据验证”，第四届国际过程层析成像会议(华沙，波兰)第85-89页(Fang W 2004 A nonlinear image reconstructionalgorithm for electrical capacitance tomography Meas.Sci.Technol.15 2124-32.Li Y and YangW Q 2008 Image reconstruction by nonlinear Landweber iteration for complicated distributionsMeas.Sci.Technol.19 94014。Smolik W，Mirkowski J，Olszewski T and Szabatin R 2006Verification of image reconstruction algorithm with sensitivity matrix updating for real data inelectrical capacitance tomography.In：Proc.4th Int.Symp.On Process Tomography，(Warsaw，Poland)p 85-9)。Since ECT has a 'soft' field characteristic, that is, the distribution of spatial sensitivity changes with the distribution of spatial permittivity. In the past few years, some researchers used an updated sensitivity matrix for iterative image reconstruction, but it was very time-consuming, and the convergence of this type of iteration has not been confirmed. See Fang W's 2004 article "A Nonlinear Algorithm for Image Reconstruction in Electrical Capacitance Tomography" for details, Measurement Science and Technology 15 2124-32. Li Y and Yang W Q2008 article "Image reconstruction with nonlinear Landweber iterative method for complex distributions", Measurement Science and Technology 19 94014. Smolik W et al. 2006 "Real data validation of an electrical capacitance tomography image reconstruction algorithm with an updated sensitivity matrix", 4th International Conference on Process Tomography (Warsaw, Poland) pp. 85-89 (Fang W 2004 A nonlinear image reconstructionalgorithm for electrical capacitance tomography Meas.Sci.Technol.15 2124-32.Li Y and YangW Q 2008 Image reconstruction by nonlinear Landweber iteration for complicated distributionsMeas.Sci.Technol.19 94014。Smolik W，Mirkowski J，Olszewski T and Szabatin R 2006 Verification of image reconstruction algorithm with sensitivity matrix updating for real data electrical capacitance tomography. In: Proc. 4th Int. Symp. On Process Tomography, (Warsaw, Poland) p 85-9).

1980年，Calderon提出了一个解决二维逆问题的新的线性化方法。Allers和Santosa实施Calderon的算法通过把逆问题简化成一个矩问题，在电阻层析成像中将电导率展成Zernike多项式的形式求解。详见A.Allers和F.Santosa的文章”电阻层析成像中一种线性化问题的稳定性和分辨力分析”，逆问题7，515-533(1991)(A.Allers，and F.Santosa，″Stability andresolution analysis of a linearized problem in electrical impedance tomography，″Inverse Probl 7，515-533(1991))。Bikowski和Mueller把Calderon方法应用于二维ERT来重建电导率的分布。详见J.Bikowski，和J.L.Mueller的文章“采用Calderon方法的二维EIT图像重建”，反问题与图像，2，43-61(2008)(J.Bikowski，and J.L.Mueller，″2D EIT reconstructions using Calderon′smethod，″Inverse Problems and Imaging 2，43-61(2008))。In 1980, Calderon proposed a new linearization method for solving the two-dimensional inverse problem. Allers and Santosa implemented Calderon's algorithm by reducing the inverse problem to a moment problem and solving the electrical conductivity as Zernike polynomials in electrical resistance tomography. See A.Allers and F.Santosa's article "Stability and resolution analysis of a linearization problem in electrical resistance tomography", Inverse Problem 7, 515-533 (1991) (A.Allers, and F.Santosa , "Stability and resolution analysis of a linearized problem in electrical impedance tomography," Inverse Probl 7, 515-533 (1991)). Bikowski and Mueller applied the Calderon method to two-dimensional ERT to reconstruct the conductivity distribution. See J.Bikowski, and J.L.Mueller's article "2D EIT reconstructions using the Calderon method", Inverse Problems and Images, 2, 43-61 (2008) (J.Bikowski, and J.L.Mueller, "2D EIT reconstructions using Calderon's method, "Inverse Problems and Imaging 2, 43-61 (2008)).

但是由于ECT传感器存在屏蔽层，同时文献中电阻层析成像采用的是电流激励、测量电压的激励测量策略，而ECT中采用的是电压激励、测量电流的激励测量模式，Calderon的算法无法直接用于ECT。由于这种方法属于一种直接方法，且实时性高，将其推广至ECT，即可以只对部分区域直接进行图像重建，具有重要的应用价值。However, due to the existence of a shielding layer in the ECT sensor, and the electrical resistance tomography in the literature uses an excitation measurement strategy of current excitation and voltage measurement, while ECT uses an excitation measurement mode of voltage excitation and current measurement, Calderon’s algorithm cannot be used directly. in ECT. Since this method is a direct method and has high real-time performance, it is of great application value to extend it to ECT, that is, to directly perform image reconstruction on only a part of the region.

(三)发明内容(3) Contents of the invention

1、目的：本发明的目的是提供一种基于圆形传感器的电容层析成像的图像重建直接方法，它克服了现有技术的不足，可以实现快速图象重建。1. Purpose: The purpose of the present invention is to provide a direct method for image reconstruction based on electrical capacitance tomography of a circular sensor, which overcomes the deficiencies in the prior art and can realize fast image reconstruction.

2、技术方案：本发明是一种基于圆形传感器的电容层析成像的图像重建直接方法，包括下列具体操作步骤：2. Technical solution: The present invention is a direct method for image reconstruction based on electrical capacitance tomography of a circular sensor, comprising the following specific steps:

步骤一：对于同一截面上具有N个电极的ECT传感器，将N个电极逆时针标号为电极i(1≤i≤N)，采用传统的1-by-1激励测量模式，即，一次测量过程中，包括N-1个动作。第1步，电极1上施加幅值为V的交流电压，其余N-1个电极均接地或与地保持同电位，测量得到电极1分别与电极2到N共N-1个电容值；第2步，电极2上施加幅值为V的交流电压，其余N-1个电极均接地或与地保持同电位，测量得到电极2分别与电极3到N共N-2个电容值；第3步，电极3上施加幅值为V的交流电压，其余N-1个电极均接地或与地保持同电位，测量得到电极3分别与电极4到N共N-3个电容值；以此类推，第N-1步，电极N-1上施加幅值为V的交流电压，其余N-1个电极均接地或与地保持同电位，测量得到电极N-1与电极N共1个电容值。共测量得到N*(N-1)/2个独立测量电容变化值。如ΔC_i，j是电极对i-j(i≠j，1≤j≤N)之间的电容变化量。Step 1: For an ECT sensor with N electrodes on the same section, label the N electrodes counterclockwise as electrode i (1≤i≤N), and adopt the traditional 1-by-1 excitation measurement mode, that is, one measurement process , including N-1 actions. In the first step, an AC voltage with an amplitude of V is applied to electrode 1, and the other N-1 electrodes are grounded or kept at the same potential as the ground, and N-1 capacitance values are obtained from electrode 1 and electrodes 2 to N respectively; Step 2, apply an AC voltage with an amplitude of V on electrode 2, and the other N-1 electrodes are all grounded or kept at the same potential as the ground, and N-2 capacitance values are obtained from electrode 2 and electrodes 3 to N respectively; the third Step 1: Apply an AC voltage with an amplitude of V on the electrode 3, and the other N-1 electrodes are all grounded or kept at the same potential as the ground, and the measured electrode 3 and electrodes 4 to N have a total of N-3 capacitance values; and so on , step N-1, apply an AC voltage with an amplitude of V on the electrode N-1, and the other N-1 electrodes are all grounded or kept at the same potential as the ground, and a total capacitance of electrode N-1 and electrode N is measured . A total of N*(N-1)/2 independently measured capacitance change values are obtained. For example, ΔC _i,j is the capacitance variation between the electrode pair ij (i≠j, 1≤j≤N).

步骤二：首先通过线性变换将测量的电容值进行预处理，得到N*(N-1)/2个独立测量电荷变化值。如Δq_j ^k为第k次测量时第j个电极上的电荷变化量。Step 2: Firstly, the measured capacitance value is preprocessed by linear transformation to obtain N*(N-1)/2 independently measured charge change values. For example, Δq _j ^k is the amount of charge change on the jth electrode during the kth measurement.

$[\begin{matrix} {Δq Δq}_{11}^{k k} \\ {Δq Δq}_{22}^{k k} \\ {Δq Δq}_{33}^{k k} \\ . . \\ . . \\ . . \\ {Δq Δq}_{N N}^{k k} \end{matrix}] = = [\begin{matrix} {ΔC ΔC}_{1,1 1,1} & - - {ΔC ΔC}_{1,2 1,2} & - - {ΔC ΔC}_{1,3 1,3} & . . . . . . & {- - ΔC ΔC}_{11,, N N} \\ - - {ΔC ΔC}_{2,1 2,1} & {ΔC ΔC}_{2,2 2,2} & {- - ΔC ΔC}_{2,3 2,3} & . . . . . . & {- - ΔC ΔC}_{22,, N N} \\ - - {ΔC ΔC}_{3,1 3,1} & {- - ΔC ΔC}_{3,2 3,2} & {C C}_{3,3 3,3} & . . . . . . & {- - ΔC ΔC}_{33,, N N} \\ . . & . . & . . & . . & . . \\ . . & . . & . . & . . & . . \\ . . & . . & . . & . . & . . \\ {- - ΔC ΔC}_{N N,, 11} & {- - ΔC ΔC}_{N N,, 22} & {- - ΔC ΔC}_{N N,, 33} & . . . . . . & {ΔC ΔC}_{N N,, N N} \end{matrix}] [\begin{matrix} {V V}_{11}^{k k} \\ {V V}_{22}^{k k} \\ {V V}_{33}^{k k} \\ . . \\ . . \\ . . \\ {V V}_{N N}^{k k} \end{matrix}] - - - - - - ((11))$

其中， $Δ C_{i, i} = Σ_{j = 1, j &NotEqual; i}^{N} Δ C_{i, j}$ 是第i个电极上的自身电容变化量，ΔC_i，j是电极对i-j(i≠j)之间的电容变化量。V_j ^k为第k次测量时第j(1≤j≤N)个电极上虚拟施加的电压，满足in, $Δ C_{i, i} = Σ_{j = 1, j &NotEqual; i}^{N} Δ C_{i, j}$ is the capacitance variation on the i-th electrode, and ΔC _i,j is the capacitance variation between the electrode pair ij (i≠j). V _j ^k is the virtual applied voltage on the jth (1≤j≤N) electrode during the kth measurement, satisfying

${V V}_{j j}^{k k} = = \{\begin{matrix} cos cos ((k k \frac{j j}{N N} 22 π π)),, & k k = = 11 . . . . . . N N / / 22 \\ sin sin ((((k k - - N N / / 22)) \frac{j j}{N N} 22 π π)),, & k k = = N N / / 22 + + 11 . . . . . . N N - - 11 \end{matrix} - - - - - - ((22))$

步骤三：计算离散电极的散射变换t(s)Step 3: Calculate the scattering transformation t(s) of the discrete electrodes

$t t ((s the s)) = = \frac{Δθ Δθ}{A A} {Σ Σ}_{m m = = 11}^{\frac{N N}{22}} {a a}_{m m} ((\overset{&OverBar; &OverBar;}{s the s})) {Σ Σ}_{n no = = 11}^{\frac{N N}{22}} {a a}_{n no} ((s the s)) {Σ Σ}_{j j = = 11}^{N N} [[{e e}^{- - im im {θ θ}_{j j}} ((Δ Δ {q q}_{j j}^{n no} + + IΔ IΔ {q q}_{j j}^{n no + + N N / / 22}))]] - - - - - - ((33))$

其中，A表示电极的面积， ${Δq}_{j}^{n} = A (Λ_{ϵ} - Λ_{1}) V_{j}^{n}$ 表示第k次测量时第j个电极上产生的电荷变化量。 $a_{n} (s) = \frac{{(is)}^{n}}{n!},$ $a_{m} (\overset{&OverBar;}{s}) = \frac{{(i \overset{&OverBar;}{s})}^{m}}{m!},$ s表示复数s＝s₁+Is₂的共轭函数， $I = \sqrt{- 1},$ s₁和s₂均为实数。Among them, A represents the area of the electrode, ${Δq}_{j}^{no} = A (Λ_{ϵ} - Λ_{1}) V_{j}^{no}$ Indicates the amount of charge change generated on the jth electrode during the kth measurement. $a_{no} (the s) = \frac{{(is)}^{no}}{no!},$ $a_{m} (\overset{&OverBar;}{the s}) = \frac{{(i \overset{&OverBar;}{the s})}^{m}}{m!},$ s represents the conjugate function of the complex number s=s ₁ +Is ₂ , $I = \sqrt{- 1},$ Both s ₁ and s ₂ are real numbers.

步骤四：进行任一点介电常数变化量的重建Step 4: Reconstruct the variation of permittivity at any point

$δϵ δϵ ((x x,, y the y)) \approx \approx \frac{11}{{22 π π}^{22}} {&Integral; &Integral; &Integral; &Integral;}_{{R R}^{22}} \frac{t t (({s the s}_{11} + + I I {s the s}_{22}))}{{s the s}_{11}^{22} + + {s the s}_{22}^{22}} {e e}^{I I ((- - {22 s the s}_{11} x x + + {22 s the s}_{22} y the y))} {ds ds}_{11} {ds ds}_{22} - - - - - - ((44))$

其中，δε(x，y)为圆形区域由直角坐标系坐标(x，y)对应的位置上的介电常数变化值。上述图像重建方法中计算的理论推导是：Wherein, δε(x, y) is the change value of the dielectric constant at the position corresponding to the coordinate (x, y) of the rectangular coordinate system in the circular area. The theoretical derivation of the calculations in the above image reconstruction method is:

假设在区域Ω内，敏感场中电势

满足的方程为：Assume that within the region Ω, the potential in the sensitive field

The satisfied equation is:

其中z是一个代表位置(x，y)的复数，ε(z)和分别表示介电常数和电势的分布。where z is a complex number representing the position (x, y), ε(z) and are the distributions of permittivity and potential, respectively.

根据散度定理，有According to the divergence theorem, we have

$= = 00$

其中，v(z)是L₂勒贝格(Lebesgue)空间中任意的连续函数。ds代表在边界上的单位弧长。where v(z) is any continuous function in L ₂ Lebesgue (Lebesgue) space. ds stands for at border The unit arc length on .

当区域Ω包含介电常数分布为ε(z)时，从边界电势到边界电流密度的映射可表示为：When the region Ω contains a dielectric constant distribution ε(z), the mapping from the boundary potential to the boundary current density can be expressed as:

特别地，当介电常数为一常数时，Λ₁可表示为：In particular, when the dielectric constant is a constant, _Λ1 can be expressed as:

对于一个包含扰动的介电常数ε＝1+δε，且扰动仅发生在Ω范围内时，For a permittivity ε=1+δε including disturbance, and the disturbance only occurs in the range of Ω,

同时at the same time

且满足边界条件and satisfy the boundary conditions

如果假设在整个敏感场中满足If the assumption is satisfied in the entire sensitive field

用(9)减去(10)式可得：Subtract (10) from (9) to get:

即：Right now:

根据以上的分析，令

v (z) = e^{i \overset{&OverBar;}{sz}} .

这里s＝s₁+is₂是一个复数，s₁和s₂是实数。According to the above analysis, let

v (z) = e^{i \overset{&OverBar;}{sz}} .

Here s=s ₁ +is ₂ is a complex number, and s ₁ and s ₂ are real numbers.

将方程(14)的左边表示为t(s)，即：Denote the left side of equation (14) as t(s), namely:

$t t ((s the s)) = = {&Integral; &Integral;}_{&PartialD; &PartialD; Ω Ω} {e e}^{i i \overset{&OverBar; &OverBar;}{sz sz}} (({Λ Λ}_{11 + + δϵ δϵ} - - {Λ Λ}_{11})) (({e e}^{isz isz})) dz dz - - - - - - ((1515))$

整理成关于复数s＝s₁+is₂的直角坐标系表示，有Organized into a Cartesian coordinate system representation about the complex number s=s ₁ +is ₂ , there is

$t t ((s the s)) \approx \approx - - 22 (({s the s}_{11}^{22} + + {s the s}_{22}^{22})) {&Integral; &Integral;}_{Ω Ω} δϵ δϵ {e e}^{- - i i (({- - 22 s the s}_{11},, {22 s the s}_{22})) \cdot &Center Dot; ((x x,, y the y))} dxdy dxdy - - - - - - ((1616))$

通过Fourier逆变换，可以得到介电常数分布的变化量：Through the Fourier inverse transformation, the variation of the permittivity distribution can be obtained:

$δϵ δϵ ((x x,, y the y)) = = δϵ δϵ ((z z))$

$\approx \approx \frac{11}{{22 π π}^{22}} {&Integral; &Integral; &Integral; &Integral;}_{{R R}^{22}} \frac{t t (({s the s}_{11} + + {is is}_{22}))}{{s the s}_{11}^{22} + + {s the s}_{22}^{22}} {e e}^{i i (({- - 22 s the s}_{11},, {22 s the s}_{22})) \cdot &Center Dot; ((x x,, y the y))} {ds ds}_{11} {ds ds}_{22} - - - - - - ((1717))$

可以证明当δε(x，y)接近零时，通过δε(x，y)重建的误差也接近于零。It can be shown that when δε(x, y) approaches zero, the error reconstructed by δε(x, y) also approaches zero.

3、优点及功效：针对圆形测量区域，克服传统算法须计算灵敏度矩阵且无法独立计算部分区域重建结果的局限性，根据电容测量值直接实现图像重建，且可以只对部分区域直接进行图像重建。3. Advantages and efficacy: For circular measurement areas, it overcomes the limitations of traditional algorithms that need to calculate the sensitivity matrix and cannot independently calculate the reconstruction results of some areas, and directly realizes image reconstruction according to the capacitance measurement value, and can only directly perform image reconstruction on some areas .

(四)附图说明(4) Description of drawings

图1.本发明中实施采用的圆形传感器示意图Fig. 1. The schematic diagram of the circular sensor implemented in the present invention

图2.U型仿真模型示意图Figure 2. Schematic diagram of U-shaped simulation model

图3.U型仿真模型的重建结果示意图Figure 3. Schematic diagram of the reconstruction results of the U-shaped simulation model

图中符好说明如下：The symbols in the figure are explained as follows:

1金属管层 2绝缘物质层 3电极1 metal tube layer 2 insulating material layer 3 electrodes

(五)具体实施方式(5) Specific implementation methods

在本发明的方法进行图像重建的过程中，所应用的传感器如附图1所示，它主要由三层结构组成，其外层为起结构固定和屏蔽作用的金属管层1，中间结构层为绝缘物质层2，内部结构层3为附着在绝缘物质层2上实现电阻抗实部和虚部的同步测量的N个电极，所述电极均匀分布在同一圆周上，在相邻电极间相互绝缘。In the process of image reconstruction in the method of the present invention, the applied sensor is as shown in accompanying drawing 1, and it mainly is made up of three-layer structure, and its outer layer is the metal tube layer 1 that plays structural fixation and shielding effect, and middle structural layer It is an insulating material layer 2, and the internal structure layer 3 is N electrodes attached to the insulating material layer 2 to realize the synchronous measurement of the real part and the imaginary part of the electrical impedance. insulation.

本发明是一种基于圆形传感器的电容层析成像的图像重建直接方法，包括下列具体操作步骤：The present invention is a direct image reconstruction method based on electrical capacitance tomography of a circular sensor, comprising the following specific operation steps:

$t t ((s the s)) = = \frac{Δθ Δθ}{A A} {Σ Σ}_{m m = = 11}^{\frac{N N}{22}} {a a}_{m m} ((\overset{&OverBar; &OverBar;}{s the s})) {Σ Σ}_{n no = = 11}^{\frac{N N}{22}} {a a}_{n no} ((s the s)) {Σ Σ}_{j j = = 11}^{N N} [[{e e}^{- - im im {θ θ}_{i i}} ((Δ Δ {q q}_{j j}^{n no} + + IΔ IΔ {q q}_{j j}^{n no + + N N / / 22}))]] - - - - - - ((33))$

其中，δε(x，y)为圆形区域由直角坐标系坐标(x，y)对应的位置上的介电常数变化值。Wherein, δε(x, y) is the change value of the dielectric constant at the position corresponding to the coordinate (x, y) of the rectangular coordinate system in the circular area.

用数值仿真来评估这种新方法。考虑一种U型区域的介电常数分布模型，由附图2示。白色区域的介电常数等于1(代表空气)，黑色区域等于3(代表油)。Numerical simulations are used to evaluate this new method. Consider a dielectric constant distribution model of a U-shaped region, as shown in Figure 2. The dielectric constant of the white area is equal to 1 (representing air), and the black area is equal to 3 (representing oil).

用参数r和θ表示的极坐标中，方程(4)可以写成In polar coordinates expressed with parameters r and θ, equation (4) can be written as

$δϵ δϵ ((x x,, y the y))$

$\approx \approx \frac{11}{{22 π π}^{22}} {&Integral; &Integral; &Integral; &Integral;}_{{R R}^{22}} \frac{t t (({s the s}_{11} + + {is is}_{22}))}{{s the s}_{11}^{22} + + {s the s}_{22}^{22}} {e e}^{i i (({- - 22 s the s}_{11},, {22 s the s}_{22})) \cdot \cdot ((x x,, y the y))} {ds ds}_{11} {ds ds}_{22} - - - - - - ((55))$

$= = \frac{11}{{22 π π}^{22}} {&Integral; &Integral;}_{00}^{{R R}_{00}} {&Integral; &Integral;}_{- - π π}^{π π} \frac{t t (({re re}^{iθ iθ}))}{r r} {e e}^{i i ((- - 22 r r cos cos θ θ,, 22 r r sin sin θ θ)) \cdot \cdot ((x x,, y the y))} dθdr dθdr$

其中，R₀是用于数值积分的区域半径，可选为5倍的管道半径。用高斯-勒让得求积公式得到δε(x，y)。附图2中模型的重建图像如附图3所示。where _R0 is the region radius used for numerical integration, optionally 5 times the pipe radius. Use the Gauss-Legend quadrature formula to get δε(x, y). The reconstructed image of the model in Figure 2 is shown in Figure 3.

比较附图2和附图3，可以看出本发明中的新图像重建方法在这种情况下有很好的图像重建结果。在实施上面提出的算法过程中，主要计算任务是计算二重数值积分。可采用高斯-勒让得积分，由于权重方程和高斯点的位置可以预先确定，因此该算法具有很好的实时性能。Comparing the accompanying drawings 2 and 3, it can be seen that the new image reconstruction method of the present invention has very good image reconstruction results in this case. During the implementation of the algorithm presented above, the main computational task is to calculate the double numerical integral. Gaussian-Legendre integral can be used, because the weight equation and the position of Gaussian points can be determined in advance, so the algorithm has good real-time performance.

Claims

1. A direct image reconstruction method based on the capacitance tomography of a circular sensor is characterized in that: the method comprises the following specific operation steps:

the method comprises the following steps: for a capacitance tomography sensor with N electrodes on the same section, the N electrodes are marked as an electrode i (i is more than or equal to 1 and less than or equal to N) in a counterclockwise way, and a traditional excitation measurement mode is adopted, namely N-1 actions are included in one scanning process; step 1, applying an AC voltage with an amplitude V to the electrode 1, grounding the rest N-1 electrodes or keeping the same potential with the ground, and measuring the electrode 1 and the electrodes 2 to N together1 capacitance value; step 2, applying alternating voltage with the amplitude V to the electrode 2, grounding the other N-1 electrodes or keeping the same potential with the ground, and measuring the N-2 capacitance values of the electrode 2 and the electrodes 3 to N respectively; step 3, applying alternating voltage with the amplitude of V to the electrode 3, grounding the rest N-1 electrodes or keeping the same potential with the ground, and measuring the N-3 capacitance values of the electrode 3 and the electrodes 4 to N respectively; in the same way, in the step N-1, an alternating voltage with the amplitude V is applied to the electrode N-1, the rest N-1 electrodes are grounded or keep the same potential with the ground, 1 capacitance value is obtained by measuring the electrode N-1 and the electrode N, and the capacitance change values of N x (N-1)/2 independent measurement capacitances, such as delta C, are obtained by measuring in total_i，jIs the capacitance variation between electrode pairs i-j (i is not equal to j, j is more than or equal to 1 and less than or equal to N);

step two: the measured capacitance values are first pre-processed by linear transformation to yield N x (N-1)/2 independently measured charge changes, e.g.

The change amount of the charge on the jth electrode in the kth measurement;

wherein,

is the change of self capacitance, Δ C, on the ith electrode_i，jIs the capacitance variation between electrode pairs i-j (i ≠ j);

the voltage virtually applied to the jth (j is more than or equal to 1 and less than or equal to N) electrode during the kth measurement is satisfied

Step three: calculating the Scattering transform t(s) of discrete electrodes

<math><mrow><mi>t</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mi>Δθ</mi><mi>A</mi></mfrac><munderover><mi>Σ</mi><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mfrac><mi>N</mi><mn>2</mn></mfrac></munderover><msub><mi>a</mi><mi>m</mi></msub><mrow><mo>(</mo><mover><mi>s</mi><mo>&OverBar;</mo></mover><mo>)</mo></mrow><munderover><mi>Σ</mi><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mfrac><mi>N</mi><mn>2</mn></mfrac></munderover><msub><mi>a</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><munderover><mi>Σ</mi><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mo>[</mo><msup><mi>e</mi><mrow><mo>-</mo><mi>im</mi><msub><mi>θ</mi><mi>j</mi></msub></mrow></msup><mrow><mo>(</mo><mi>Δ</mi><msubsup><mi>q</mi><mi>j</mi><mi>n</mi></msubsup><mo>+</mo><mi>IΔ</mi><msubsup><mi>q</mi><mi>j</mi><mrow><mi>n</mi><mo>+</mo><mi>N</mi><mo>/</mo><mn>2</mn></mrow></msubsup><mo>)</mo></mrow><mo>]</mo><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow></math>

Wherein A represents the area of the electrode,representing the amount of change in charge generated at the jth electrode at the nth measurement;

to representA complex number s ═ s₁+Is₂The function of the conjugate of (a) to (b),

s₁and s₂Are all real numbers;

step four: reconstructing the variation of dielectric constant of any point

<math><mrow><mi>δϵ</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>≈</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><msub><mrow><mo>&Integral;</mo><mo>&Integral;</mo></mrow><msup><mi>R</mi><mn>2</mn></msup></msub><mfrac><mrow><mi>t</mi><mrow><mo>(</mo><msub><mi>s</mi><mn>1</mn></msub><mo>+</mo><mi>I</mi><msub><mi>s</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow><mrow><msubsup><mi>s</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>s</mi><mn>2</mn><mn>2</mn></msubsup></mrow></mfrac><msup><mi>e</mi><mrow><mi>I</mi><mrow><mo>(</mo><mo>-</mo><mn>2</mn><msub><mi>s</mi><mn>1</mn></msub><mi>x</mi><mo>+</mo><mn>2</mn><msub><mi>s</mi><mn>2</mn></msub><mi>y</mi><mo>)</mo></mrow></mrow></msup><msub><mi>ds</mi><mn>1</mn></msub><msub><mi>ds</mi><mn>2</mn></msub><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mrow></math>

Where δ ∈ (x, y) is a dielectric constant change value at a position of the circular region corresponding to the rectangular coordinate system coordinate (x, y).