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 invention relates to a direct image reconstruction method based on the capacitance tomography of a round sensor. The method comprises the following operation steps: 1. N electrodes of an ECT sensor, which are arranged on the same cross section, are counterclockwise marked as electrodes i (i is larger than or equal to 1 and is smaller than or equal to N) and a traditional 1-by-1 excitation measuring mode is adopted, i.e. a scanning process comprises N-1 actions; 2. measured capacitance values are firstly pre-treated by linear transform to obtain N*(N-1)/2 independent measured charge change values; 3. the scattering transformation t(s) of discrete electrodes is calculated; and 4. the specific inductive capacity variation of any point is rebuilt. The invention aims at a round measuring region, overcomes the defects that a traditional algorithm needs to calculate a sensitivity matrix and can not independently calculate the rebuilding result of a subregion, directly rebuilds images according to capacitance measuring values and can directly rebuild images in the subregion. The method has high real-time performance, can be popularized to ECT and has important practical values and application prospects.

Description

Direct image reconstruction method based on capacitance tomography of circular sensor

(I) technical field

The invention relates to an image reconstruction method, in particular to a direct image reconstruction method based on capacitance tomography of a circular sensor. Belonging to the technical field of image reconstruction.

(II) background of the invention

Electrical capacitance tomography (electrical capacitance tomography-ECT) is the reconstruction of the distribution of the spatial permittivity of an insulating material from measured capacitance data. ECT has several advantages over other tomographic techniques, such as: low cost, fast response, portability, non-invasiveness, and robustness. The ECT has a wide application prospect in many industrial fields, and the core technology of the ECT is to reconstruct the variation of the internal dielectric constant distribution by adopting an image reconstruction method according to the variation of a group of measured values of the boundary capacitance.

In the past, most image reconstruction algorithms were based on the sensitivity theorem derived from the 1971 article by Geselowitz "application of lead theory of electrocardiography to impedance plethysmography" society of Electrical and electronics Engineers, Proc. biomedical Engineers, BME-1838-41. (Geselowitz D1971 An application of electrochemical lead the invention of electrochemical simulation. IEEE trans.biomed. Eng. BME-1838-41) and Lehr in 1972 "a Vector Derivation method for Impedance volume field calculation", the institute of Electrical and electronics Engineers, Proc. biomedical engineering, BME-19156-7 (Lehr J1972A Vector Derivation Useful in Impedance in electrochemical simulation IEEE trans. biomed. Eng. BME-19156-7) are proposed as a linearization method using perturbation principles. The 2005 article by Soleimani M and Lionheart W R B, "non-linear image reconstruction by capacitance tomography using experimental data," measures science and technology 161987-96. (Soleimani M and Lionheart W R B2005 Nonlinear image retrieval for electrical capacitance characterization using experimental data. Meas. Sci. technol. 161987-96).

Since ECT has a 'soft' field characteristic, i.e. the distribution of spatial sensitivity varies with the variation of the spatial permittivity distribution. Over the last few years, there have been researchers performing iterative image reconstruction using updated sensitivity matrices, but this is very time consuming and the convergence of such iterations has not been demonstrated. See the 2004 article by Fang W, "a nonlinear algorithm for tomographic reconstruction of images", measurement science and technology 152124-32. The article in Li Y and Yang W Q2008, "image reconstruction for complex distributed nonlinear Landweber iteration", measures science and technology 1994014. Smok W et al, "actual data validation of capacitance Tomography Image reconstruction algorithm for updating sensitivity matrix", 2006, fourth International Process Tomography conference (Wash. Chalan.) pages 85-89 (Fang W2004A nonlinear Image reconstruction algorithm for electronic capacitance Tomography Meas. Sci. technol. 152124-32. Li Y and Yang W Q2008 Image reconstruction by nonlinear Image comparison for compatible distribution Meas. Sci. technol. 1994014. Smok W, Mirkowski J, Ozewski T and Szatin R2006 version of Image reconstruction algorithm for electronic sensitivity mapping with actual data Verification of sensitivity matrix and program, in. polarization in. 9. polarization analysis. Polish. simulation for Image reconstruction. program. in. Polish. incorporated. Polish. simulation program.85. the third paper of FIGS.

In 1980, Calderon proposed a new linearization method to solve the two-dimensional inverse problem. The Calderon's algorithm implemented by Allers and Santosa solves the conductivity in electrical resistance tomography by reducing the inverse problem to a moment problem, developing the conductivity into a form of Zernike polynomials. See article A. Allers and F. Santosa for "Stability and resolution analysis of a linearization problem in electrical resistance tomography", Inverse problem 7, 515-. Bikowski and Mueller apply the Calderon method to two-dimensional ERT to reconstruct the conductivity distribution. See in detail j.bikowski, and j.l.mueller article "two-dimensional EIT image reconstruction with Calderon method", Inverse problem and image, 2, 43-61(2008) (j.bikowski, and j.l.mueller, "2D EIT recovery using Calderon' method," Inverse problem and image 2, 43-61 (2008)).

However, the algorithm of Calderon cannot be directly applied to ECT because the ECT sensor has a shielding layer, and meanwhile, the resistance tomography in the literature adopts an excitation measurement strategy of current excitation and voltage measurement, while the ECT adopts an excitation measurement mode of voltage excitation and current measurement. Because the method belongs to a direct method and has high real-time performance, the method is popularized to ECT, namely image reconstruction can be directly carried out on partial regions, and the method has important application value.

Disclosure of the invention

1. The purpose is as follows: the invention aims to provide a direct image reconstruction method based on the capacitance tomography of a circular sensor, which overcomes the defects of the prior art and can realize rapid image reconstruction.

2. The technical scheme is as follows: the invention relates to a direct image reconstruction method based on capacitance tomography of a circular sensor, which comprises the following specific operation steps:

the method comprises the following steps: for an ECT sensor with N electrodes on the same cross 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 mode, and a traditional 1-by-1 excitation measurement mode is adopted, namely, N-1 actions are included in one measurement process. Step 1, applying alternating voltage with the amplitude of V to an electrode 1, grounding the rest N-1 electrodes or keeping the same potential with the ground, and measuring to obtain N-1 capacitance values of the electrode 1 and electrodes 2 to N respectively; step 2, applying alternating voltage with the amplitude V to the electrode 2, grounding the rest N-1 electrodes or keeping the same potential with the ground, and measuring to obtain 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 to obtain N-3 capacitance values of the electrode 3 and the electrodes 4 to N respectively; in the same way, in the step N-1, the electrode N-1 is applied with an alternating voltage with the amplitude of V, and the other N-1 electrodes are grounded or kept at the same voltage with the groundAnd measuring to obtain 1 capacitance value of the electrode N-1 and the electrode N. The total measurement results in N x (N-1)/2 independent measurement capacitance change values. Such as Δ C_i，jIs the capacitance variation between electrode pairs i-j (i ≠ j, j is greater than or equal to 1 and less than or equal to N).

Step two: firstly, preprocessing the measured capacitance value through linear transformation to obtain N (N-1)/2 independent measured charge change values. Such as Δ q_j ^kIs the amount of change in charge on the jth electrode at the kth measurement.

Wherein,

<math><mrow><mi>Δ</mi><msub><mi>C</mi><mrow><mi>i</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><munderover><mi>Σ</mi><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>j</mi><mo>&NotEqual;</mo><mi>i</mi></mrow><mi>N</mi></munderover><mi>Δ</mi><msub><mi>C</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow></math>

is the ith electrodeSelf capacitance change amount, Δ C_i，jIs the amount of capacitance change between electrode pairs i-j (i ≠ j). V_j ^kThe 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,

indicating the amount of change in charge generated at the jth electrode during the kth measurement.

a_{n} (s) = \frac{{(is)}^{n}}{n!},

<math><mrow><msub><mi>a</mi><mi>m</mi></msub><mrow><mo>(</mo><mover><mi>s</mi><mo>&OverBar;</mo></mover><mo>)</mo></mrow><mo>=</mo><mfrac><msup><mrow><mo>(</mo><mi>i</mi><mover><mi>s</mi><mo>&OverBar;</mo></mover><mo>)</mo></mrow><mi>m</mi></msup><mrow><mi>m</mi><mo>!</mo></mrow></mfrac><mo>,</mo></mrow></math>

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

I = \sqrt{- 1},

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><msup><mrow><mn>2</mn><mi>π</mi></mrow><mn>2</mn></msup></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><msub><mrow><mn>2</mn><mi>s</mi></mrow><mn>1</mn></msub><mi>x</mi><mo>+</mo><msub><mrow><mn>2</mn><mi>s</mi></mrow><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). The theoretical derivation of the calculations in the above image reconstruction method is:

assuming a potential in the sensitive field within the region omega

The equation satisfied is:

wherein z is a complex number representing position (x, y), ε (z) andrespectively, the dielectric constant and the potential distribution.

According to the principle of divergence, there are

= 0

Wherein v (z) is L₂Arbitrary continuous functions in the Lebesgue (Lebesgue) space. ds represents at the boundaryUpper unit arc length.

When the region Ω contains a dielectric constant distribution of ∈ (z), the mapping from the boundary potential to the boundary current density can be expressed as:

in particular, when the dielectric constant is a constant, Λ₁Can be expressed as:

for a dielectric constant e, which contains perturbations, of 1+ δ e, and perturbations occur only in the range of Ω,

at the same time

And satisfy the boundary conditions

If it is assumed that this is satisfied in the whole sensitive field

Subtracting (10) from (9) yields:

namely:

based on the above analysis, let

<math><mrow><mi>v</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>e</mi><mrow><mi>i</mi><mover><mi>sz</mi><mo>&OverBar;</mo></mover></mrow></msup><mo>.</mo></mrow></math>

Where s is s₁+is₂Is a complex number, s₁And s₂Is a real number.

The left side of equation (14) is denoted as t(s), i.e.:

<math><mrow><mi>t</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>&Integral;</mo><mrow><mo>&PartialD;</mo><mi>Ω</mi></mrow></msub><msup><mi>e</mi><mrow><mi>i</mi><mover><mi>sz</mi><mo>&OverBar;</mo></mover></mrow></msup><mrow><mo>(</mo><msub><mi>Λ</mi><mrow><mn>1</mn><mo>+</mo><mi>δϵ</mi></mrow></msub><mo>-</mo><msub><mi>Λ</mi><mn>1</mn></msub><mo>)</mo></mrow><mrow><mo>(</mo><msup><mi>e</mi><mi>isz</mi></msup><mo>)</mo></mrow><mi>dz</mi><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>15</mn><mo>)</mo></mrow></mrow></math>

arranged as a plurality s ═ s₁+is₂Is expressed by a rectangular coordinate system of

<math><mrow><mi>t</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>≈</mo><mo>-</mo><mn>2</mn><mrow><mo>(</mo><msubsup><mi>s</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>s</mi><mn>2</mn><mn>2</mn></msubsup><mo>)</mo></mrow><msub><mo>&Integral;</mo><mi>Ω</mi></msub><mi>δϵ</mi><msup><mi>e</mi><mrow><mo>-</mo><mi>i</mi><mrow><mo>(</mo><msub><mrow><mo>-</mo><mn>2</mn><mi>s</mi></mrow><mn>1</mn></msub><mo>,</mo><msub><mrow><mn>2</mn><mi>s</mi></mrow><mn>2</mn></msub><mo>)</mo></mrow><mo>·</mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></msup><mi>dxdy</mi><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>16</mn><mo>)</mo></mrow></mrow></math>

Through Fourier inverse transformation, the variation of the dielectric constant distribution can be obtained:

<math><mrow><mo>≈</mo><mfrac><mn>1</mn><msup><mrow><mn>2</mn><mi>π</mi></mrow><mn>2</mn></msup></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><msub><mi>is</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><msub><mrow><mo>-</mo><mn>2</mn><mi>s</mi></mrow><mn>1</mn></msub><mo>,</mo><msub><mrow><mn>2</mn><mi>s</mi></mrow><mn>2</mn></msub><mo>)</mo></mrow><mo>·</mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><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>17</mn><mo>)</mo></mrow></mrow></math>

it can be demonstrated that as δ ε (x, y) approaches zero, the error through δ ε (x, y) reconstruction also approaches zero.

3. The advantages and the effects are as follows: aiming at the circular measuring area, the method overcomes the limitations that the traditional algorithm needs to calculate a sensitivity matrix and can not independently calculate the reconstruction result of a partial area, directly realizes image reconstruction according to the capacitance measured value, and can only directly carry out image reconstruction on the partial area.

(IV) description of the drawings

FIG. 1 is a schematic diagram of a circular sensor used in the practice of the present invention

FIG. 2 is a schematic diagram of a U-shaped simulation model

FIG. 3 is a schematic diagram of a reconstruction result of a U-shaped simulation model

The figure is well illustrated as follows:

1 metal tube layer 2 insulating material layer 3 electrode

(V) detailed description of the preferred embodiments

In the process of image reconstruction by the method of the invention, the applied sensor is shown in figure 1 and mainly comprises three layers of structures, wherein the outer layer is a metal tube layer 1 for fixing and shielding the structure, the middle structure layer is an insulating material layer 2, the inner structure layer 3 is N electrodes attached to the insulating material layer 2 for realizing synchronous measurement of a real part and an imaginary part of electrical impedance, the electrodes are uniformly distributed on the same circumference and are mutually insulated between adjacent electrodes.

The invention relates to a direct image reconstruction method based on capacitance tomography of a circular sensor, which comprises the following specific operation steps:

the method comprises the following steps: for an ECT sensor with N electrodes on the same cross 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 mode, and a traditional 1-by-1 excitation measurement mode is adopted, namely, N-1 actions are included in one measurement process. Step 1, applying alternating voltage with the amplitude of V to an electrode 1, grounding the rest N-1 electrodes or keeping the same potential with the ground, and measuring to obtain N-1 capacitance values of the electrode 1 and electrodes 2 to N respectively; step 2, applying alternating voltage with the amplitude V to the electrode 2, grounding the rest N-1 electrodes or keeping the same potential with the ground, and measuring to obtain 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 to obtain 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 of V is applied to the electrode N-1, the rest N-1 electrodes are grounded or keep the same potential with the ground, and the electrode N-1 and the electrode N are obtained through measurement1 capacitance value. The total measurement results in N x (N-1)/2 independent measurement capacitance change values. Such as Δ C_i，jIs the capacitance variation between electrode pairs i-j (i ≠ j, j is greater than or equal to 1 and less than or equal to N).

Wherein,

is the change of self capacitance, Δ C, on the ith electrode_i，jIs the amount of capacitance change between electrode pairs i-j (i ≠ j). V_j ^kThe 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>i</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,

a_{n} (s) = \frac{{(is)}^{n}}{n!},

I = \sqrt{- 1},

s₁and s₂Are all real numbers.

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

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).

Numerical simulations were used to evaluate this new method. Consider a model of the dielectric constant distribution of a U-shaped region, shown in figure 2. The dielectric constant of the white areas is equal to 1 (representing air) and the black areas is equal to 3 (representing oil).

In polar coordinates expressed by the parameters r and theta, equation (4) can be written as

<math><mrow><mo>≈</mo><mfrac><mn>1</mn><msup><mrow><mn>2</mn><mi>π</mi></mrow><mn>2</mn></msup></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><msub><mi>is</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><msub><mrow><mo>-</mo><mn>2</mn><mi>s</mi></mrow><mn>1</mn></msub><mo>,</mo><msub><mrow><mn>2</mn><mi>s</mi></mrow><mn>2</mn></msub><mo>)</mo></mrow><mo>·</mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><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>5</mn><mo>)</mo></mrow></mrow></math>

<math><mrow><mo>=</mo><mfrac><mn>1</mn><msup><mrow><mn>2</mn><mi>π</mi></mrow><mn>2</mn></msup></mfrac><msubsup><mo>&Integral;</mo><mn>0</mn><msub><mi>R</mi><mn>0</mn></msub></msubsup><msubsup><mo>&Integral;</mo><mrow><mo>-</mo><mi>π</mi></mrow><mi>π</mi></msubsup><mfrac><mrow><mi>t</mi><mrow><mo>(</mo><msup><mi>re</mi><mi>iθ</mi></msup><mo>)</mo></mrow></mrow><mi>r</mi></mfrac><msup><mi>e</mi><mrow><mi>i</mi><mrow><mo>(</mo><mo>-</mo><mn>2</mn><mi>r</mi><mi>cos</mi><mi>θ</mi><mo>,</mo><mn>2</mn><mi>r</mi><mi>sin</mi><mi>θ</mi><mo>)</mo></mrow><mo>·</mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></msup><mi>dθdr</mi></mrow></math>

Wherein R is₀Is the area radius for numerical integration, optionally 5 times the pipe radius. The product of the Gaussian-Louvre equations yields δ ε (x, y). The reconstructed image of the model in fig. 2 is shown in fig. 3.

Comparing fig. 2 and fig. 3, it can be seen that the new image reconstruction method of the present invention has a good image reconstruction result in this case. In implementing the algorithm set forth above, the primary computational task is to compute the double numerical integral. The gaussian-like let integral can be used, and the algorithm has good real-time performance because the weight equation and the position of the gaussian point can be predetermined.

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

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>