A multi-index feature extraction method for ECT image reconstruction

Abstract

A multi-exponential feature extraction ECT image reconstruction method first reconstructs the image grayscale matrix and sensitivity matrix through zero vector expansion to construct a sparse observation equation. The grayscale vector obtained by the Tikhonov regularization algorithm is then modeled as a finite innovation rate signal. This grayscale vector is filtered using an exponential regeneration sampling kernel and uniformly sampled to extract feature information. After obtaining measurement values from the sampled samples, the sparse observation matrix is combined with the FRI observation matrix, and the sparse observation vector is combined with the FRI observation vector. The row vectors of the integrated observation matrix and integrated observation vector are randomly reconstructed to obtain the integrated observation equation. The capacitance tomography image reconstruction problem is converted into an L0 norm minimization problem. The L0 norm minimization problem is solved to obtain an estimate of the sparse vector, and the grayscale vector estimate is calculated. Finally, the original image is reconstructed. The present invention achieves good image quality and high reconstruction accuracy.

Description

Technical Field

[0001] The present invention relates to the technical field of image processing, and in particular to an ECT image reconstruction method based on multi-index feature extraction. Background Art

[0002] Electrical capacitance tomography (ECT) is a process imaging technology that uses capacitance sensors as measuring electrodes. With the development of industry, there is an increasing demand for visual detection of flow parameters of two-phase flow within pipeline cross-sections. In recent years, ECT has been widely used in industrial process imaging and has become a research hotspot in process tomography. ECT, based on capacitance sensors, offers many advantages, including low equipment cost, non-invasiveness, simple structure, and safety, making it an effective solution for two-phase flow measurement.

[0003] The basic idea behind electrical capacitance tomography is that, in reality, the fluid in a pipeline is not a single substance, but is usually a mixture of two or more substances, known as two-phase flow and multiphase flow. Each fluid has a different dielectric constant. The dielectric constant of two-phase flow changes as the fluid distribution changes. A series of capacitance sensors are set up around the pipeline to obtain measurement values. The data acquisition system uploads the capacitance measurement values obtained by the capacitance sensor to the imaging computer, and converts the dielectric constant distribution of the internal cross-section of the pipeline into an image through an image reconstruction algorithm. The typical ECT system structure is shown in Figure 1, where 100 is the computer, 200 is the detection coil, 300 is the shielding layer, and 400 is the excitation coil.

[0004] The sensitivity matrix between the capacitive sensors ij can be described by formula (1):

[0005]

[0006] Among them, C i,j (k) represents the dielectric constant of the kth pixel on the two-phase flow pipe cross section, which is ε h , the dielectric constant of the remaining pixels is ε l The capacitance measurement value between the capacitance sensors ij when k = 1, 2, ..., m. μ(k) is the correction factor. They represent that all pixels on the two-phase flow pipe cross section are filled with the dielectric constant ε h and ε l When , the capacitance measurement value between capacitance sensors ij.

[0007] The dielectric constant distribution ε(x,y) of the two-phase fluid in the cross section of the measured pipeline and the capacitance vector C between different measuring electrodes obtained from the capacitance sensor i The relationship between can be expressed as:

[0008] C i =∫∫ D S i (x,y)ε(x,y)dxdy (2)

[0009] Where D represents the pipeline area to be inspected, (x, y)∈D means that the values of the pixel points are all on the cross section of the pipeline to be inspected; ε(x, y) is the dielectric constant distribution function, C i (i=1,2...m) is the m sets of capacitance measurement values obtained by capacitance sensors, m is the number of pairs of capacitance sensors, and m is an integer. i (x,y) is the sensitivity distribution function, corresponding to C i .

[0010] It can be seen from formula (2) that C i The relationship between ε(x,y) and ε(x,y) is not linear. Only when the dielectric constants are not much different can they be approximated as a linear relationship. First, discretize Equation (2), then linearize it, and finally normalize it to obtain the matrix form mathematical model of ECT:

[0011] C=Sg (3)

[0012] Where C is the normalized capacitance measurement, an m×1 capacitance vector. g is an n×1 normalized grayscale value vector corresponding to the permittivity distribution within the pipe cross section. m is an integer, representing the number of pairwise combinations between the measuring plates, and n is an integer, representing the number of pixels in the cross section. S is an m×n normalized sensitivity matrix. ECT image reconstruction involves solving for the grayscale vector g given the measured capacitance vector C, thereby deriving the permittivity distribution within the pipe. The results are typically presented as an image, hence the term image reconstruction.

[0013] As can be seen from formula (3), the inverse matrix of S exists only when m=n. In actual ECT systems, the number of capacitance sensors is generally small, resulting in the number of measured values m being far less than the number of pixels n, making S irreversible. The equation must be solved when the measured capacitance value is insufficient. This is an underdetermined problem, and underdetermined problems do not have a unique solution. The key to capacitance tomography is the image reconstruction algorithm. Currently, many relatively mature image reconstruction algorithms have been proposed, such as the linear back projection method (LBP), the Landweber algorithm, and the Tikhonov regularization algorithm. The LBP algorithm has relatively low quality for complex flow patterns. However, it is a simple and fast algorithm. Compared with the LBP algorithm, the Landweber algorithm has better image quality and has been widely used in recent years. However, the core of this algorithm is the steepest descent method. If the step size parameter is inappropriate, it will lead to divergent results. If the step size is too large, the optimal point may not be found or it may fall into a local minimum. If the step size is too small, too many iterations are required, which takes too long. The Tikhonov regularization algorithm can obtain an approximately stable image, but the image edges are not ideal. To date, improving the quality and accuracy of electrical capacitance tomography image reconstruction remains a key technical issue. Summary of the Invention

[0014] In order to overcome the shortcomings of existing technologies, the present invention proposes a multi-exponential feature extraction ECT image reconstruction method for the problem of electrical capacitance tomography. The image grayscale matrix and sensitivity matrix are reorganized through zero vector expansion to construct a sparse observation equation. The grayscale vector of the ECT image is then modeled as a finite innovation rate (FRI) signal, which is uniformly sampled after filtering using exponential regeneration sampling to extract feature information from the input signal. After obtaining measurement values from the sampled samples, the sparse observation matrix is combined with the FRI observation matrix, the sparse observation vector is combined with the FRI observation vector, and the row vectors of the comprehensive observation matrix and the comprehensive observation vector are randomly reorganized to obtain the comprehensive observation equation. The problem of electrical capacitance tomography image reconstruction is converted into an L0 norm minimization problem. In order to obtain an estimate of the sparse vector, the grayscale of the original image is estimated by solving the L0 norm minimization problem.

[0015] The technical solution adopted by the present invention to solve its technical problem is:

[0016] A multi-index feature extraction ECT image reconstruction method comprises the following steps:

[0017] Step 1: Initialize the data, obtain and normalize the data. The problem of ECT image reconstruction is to estimate the dielectric constant distribution in the pipeline by solving the gray vector g through the process of measuring the capacitance vector C;

[0018] Step 2: Zero-padding and reorganization to construct a sparse observation equation. In actual ECT systems, the number of capacitive sensors is generally small, and the number of pixels n is much larger than the number of measured capacitance values m, resulting in a low sampling rate and reduced reconstruction accuracy. Virtual electrodes are added through zero vector expansion to obtain more measured capacitance values and construct a sparse observation equation. The mathematical model of the ECT system is written as:

[0019] λ0=S0g (4)

[0020] Among them, λ0 is the capacitance vector expanded by the zero-filling reorganization method; S0 is the sensitivity matrix expanded by the zero-filling reorganization method;

[0021] Step 3: Modeling the finite innovation rate FRI signal. Since the grayscale vector ranges from [0,1], it is a typical discrete FRI signal. Therefore, the grayscale vector g obtained by the Tikhonov regularization algorithm is v Modeled as a Dirac pulse train signal g(x):

[0022] g v =(S T ·S+α·I) -1 ·S T C (5)

[0023] Among them, g v is a grayscale value vector of size n×1, α is a regularization parameter, I is a unit matrix of size n×n, since S T ·S may be irreversible, so the parameter α is added to solve S T For S-ill-posed problems, the regularization algorithm can effectively solve the ill-posed problem of irreversible sensitivity matrix;

[0024]

[0025] Where x=1,2...,N is the pixel position, L is the number of non-zero grayscale values of g(x), a l ∈[0,1] is the pixel x l Gray value, x l ∈{1,2,....,N} is the pixel position with non-zero gray value;

[0026] Step 4: Finite Innovation Rate (FRI) sampling. First, use the exponential regeneration sampling kernel Filter the Dirac pulse sequence signal g(x) and then uniformly sample it to obtain K sample values y k (k=1,2,...,K);

[0027] After uniform sampling, we get K samples y k :

[0028]

[0029] Where y(x) = g(x)*h(x) is the result of filtering the FRI signal g(x), <·,·> represents the inner product, y k is the sample value uniformly sampled after filtering, T is the sampling interval, and K is the number of sample values;

[0030] Step 5: Get the measured values from the FRI sampling sample and construct the FRI observation equation. Use the spline coefficient C m,k The sampling value y of FRI k After weighted summation, M complex exponentials can be calculated K measurements can be obtained from K samples y k Calculation yields:

[0031]

[0032] Formula (8) is written in matrix form:

[0033]

[0034] Since the pixel position x l belongs to the set {1,2,...,N}, a l ∈[0,1] is the pixel x l Gray value; so formula (8) is written as:

[0035]

[0036] U=Ag (11)

[0037] Where U=[τ0,τ1,...,τ M-1 ] T ∈R M×1 is the FRI measurement vector, g=[g1,g2,...,g N ] T is a grayscale vector of size N×1 in the ECT system, and A is composed of the set Construct an M×N FRI measurement matrix;

[0038] Step 6: Construct the comprehensive observation equation. The steps are as follows:

[0039] Step 6.1: Combine the sparse observation matrix with the FRI observation matrix. Since both formulas (4) and (10) are multidimensional observations of g, the two observation equations can be combined to obtain a new comprehensive observation equation. Combining formulas (4) and (11) yields the new comprehensive observation equation:

[0040]

[0041] Step 6.2: Randomly reorganize the row vectors of the integrated observation vector and the integrated observation matrix using the same rule. Therefore, the mathematical model of the ECT system is rewritten as:

[0042] λ new =S new g (13)

[0043] Among them, λ new is the comprehensive observation vector; S new is the comprehensive observation matrix;

[0044] Step 7: Find the sparse solution of the comprehensive observation equation. The steps are as follows:

[0045] Step 7.1: Sparse Transformation. Only sparse signals can meet the requirements of compressed sensing. For ECT systems, the sparsity of signals often cannot meet the requirements of compressed sensing. To ensure that the input signal is suitable for compressed sensing, an orthogonal transform is performed on the input signal to convert it into a sparse signal. The orthogonal transform is expressed by the following formula:

[0046] g=ψs (14)

[0047] Among them, the matrix ψ of size N×N is the sparse basis, and the original signal g is projected onto the sparse basis to obtain a sparse vector s of size N×1;

[0048] Step 7.2: Find the sparse solution. The capacitance tomography image reconstruction problem is transformed into an L0 norm optimization problem. Since the signal s is sparse, the sparse solution is obtained. The most direct method is to solve the L0 norm optimization problem using the OMP algorithm:

[0049]

[0050] Among them, the L0 norm ||s||0 represents the number of non-zero coefficients in the vector s, which is solved by the orthogonal matching pursuit algorithm OMP, and the matrix ψ of size N×N is the sparse basis;

[0051] Step 8: Image reconstruction. The original image signal is estimated as:

[0052]

[0053] Furthermore, in step 4, the exponential regeneration sampling kernel is used After filtering the Dirac pulse sequence signal g(x), uniform sampling is performed to obtain the sample y k (k=1,2,...,K);

[0054] The formula for the M-1 order exponential reproducing sampling kernel is as follows:

[0055]

[0056] Among them, M-1=3 is the order of the regeneration sampling kernel, and the M-1 order regeneration sampling kernel can regenerate M exponential C m,k It is derived from the following formula:

[0057]

[0058] in yes The dual function of is a quasi-orthogonal function.

[0059] Furthermore, in step 5, the measurement value is calculated from the FRI sampling sample, and the FRI observation equation is constructed. The relationship between the order of the exponential regeneration sampling kernel and the FRI signal is M-1=2L-1, where L is the number of Dirac pulses. The M-1 order exponential regeneration sampling kernel can regenerate M complex exponentials. Among them, α m =a0+jmλ, where a0 is a freely set complex exponential and λ is a freely set real number.

[0060] Furthermore, in step 7.2, in order to ensure that the original signal of the ECT system can be reconstructed, it is necessary to perform an orthogonal transformation on the original signal so that the input signal becomes sparse, and the sparse basis is a discrete cosine transform basis.

[0061] The present invention proposes an ECT image reconstruction method based on multi-exponential feature extraction. First, the image grayscale matrix and sensitivity matrix are reorganized by zero vector expansion to construct a sparse observation equation. By using this method to increase the number of measured capacitance values, the sampling rate of the signal can be increased, and the accuracy of the reconstructed signal can be improved. Secondly, the grayscale vector obtained by the Tikhonov regularization algorithm is modeled as a finite innovation rate (FRI) signal, which is filtered and uniformly sampled using an exponential regeneration sampling core to extract the characteristic information in the input signal. After the measurement value is calculated from the sampled samples, the sparse observation matrix is combined with the FRI observation matrix, the sparse observation vector is combined with the FRI observation vector, and the row vectors of the comprehensive observation matrix and the comprehensive observation vector are randomly reorganized to obtain the comprehensive observation equation. The image reconstruction problem is converted into an L0 norm minimization problem, and the L0 norm minimization problem is solved to obtain an estimate of the sparse vector. Finally, the grayscale vector of the original image is reconstructed.

[0062] The beneficial effects of the present invention are mainly manifested in: improving the sampling rate of the signal and improving the accuracy of the reconstructed signal. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] Figure 1 This is the ECT system structure diagram.

[0064] Figure 2 This is a principle block diagram of the ECT image reconstruction method based on multi-index feature extraction.

[0065] Figure 3 This is the classic FRI sampling structure diagram.

[0066] Figure 4 These are the original images of two flow types, namely core flow and half-pipe flow.

[0067] Figure 5 This is the image effect diagram reconstructed by the LBP algorithm.

[0068] Figure 6 This is the image effect diagram reconstructed by the Landweber algorithm.

[0069] Figure 7 This is the image effect diagram reconstructed by Tikhonov algorithm.

[0070] Figure 8 This is the image effect reconstructed by the method of the present invention. DETAILED DESCRIPTION

[0071] The present invention will be further described below with reference to the accompanying drawings.

[0072] Reference Figures 2 to 8 , an ECT image reconstruction method based on multi-index feature extraction, comprising the following steps:

[0073] Step 1: Initialize the data, obtain and normalize the data. The problem of ECT image reconstruction is to estimate the dielectric constant distribution in the pipeline by solving the gray vector g through the process of measuring the capacitance vector C;

[0074] Step 2: Zero-padding and reorganization to construct a sparse observation equation. In actual ECT systems, the number of capacitive sensors is generally small, and the number of pixels n is much larger than the number of measured capacitance values m, resulting in a low sampling rate and reduced reconstruction accuracy. Virtual electrodes are added through zero vector expansion to obtain more measured capacitance values and construct a sparse observation equation. The mathematical model of the ECT system is written as:

[0075] λ0=S0g (4)

[0076] Among them, λ0 is the capacitance vector expanded by the zero-filling reorganization method; S0 is the sensitivity matrix expanded by the zero-filling reorganization method;

[0077] Step 3: Modeling the finite innovation rate FRI signal. Since the grayscale vector ranges from [0,1], it is a typical discrete FRI signal. Therefore, the grayscale vector g obtained by the Tikhonov regularization algorithm is v Modeled as a Dirac pulse train signal g(x):

[0078] g v =(S T ·S+α·I) -1 ·S T C (5)

[0079] Among them, g v is a grayscale value vector of size n×1, α is a regularization parameter, I is a unit matrix of size n×n, since S T ·S may be irreversible, so the parameter α is added to solve S T For S-ill-posed problems, the regularization algorithm can effectively solve the ill-posed problem of irreversible sensitivity matrix;

[0080]

[0081] Where x=1,2...,N is the pixel position, L is the number of non-zero grayscale values of g(x), a l ∈[0,1] is the pixel x l Gray value, x l ∈{1,2,....,N} is the pixel position with non-zero gray value;

[0082] Step 4: Finite Innovation Rate (FRI) sampling. First, use the exponential regeneration sampling kernel Filter the Dirac pulse sequence signal g(x) and then uniformly sample it to obtain K sample values y k (k=1,2,...,K);

[0083] After uniform sampling, we get K samples y k :

[0084]

[0085] Where y(x) = g(x)*h(x) is the result of filtering the FRI signal g(x), <·,·> represents the inner product, y k is the sample value uniformly sampled after filtering, T is the sampling interval, and K is the number of sample values;

[0086] Step 5: Get the measured values from the FRI sampling sample and construct the FRI observation equation. Use the spline coefficient C m,k The sampling value y of FRI k After weighted summation, M complex exponentials can be calculated K measurements can be obtained from K samples y k Calculation yields:

[0087]

[0088] Formula (8) is written in matrix form:

[0089]

[0090] Since the pixel position x l belongs to the set {1,2,...,N}, a l ∈[0,1] is the pixel x l Gray value; so formula (8) is written as:

[0091]

[0092] U=Ag (11)

[0093] Where U=[τ0,τ1,...,τ M-1 ] T ∈R M×1 is the FRI measurement vector, g=[g1,g2,...,g N ] T is a grayscale vector of size N×1 in the ECT system, and A is composed of the set Construct an M×N FRI measurement matrix;

[0094] Step 6: Construct the comprehensive observation equation. The steps are as follows:

[0095] Step 6.1: Combine the sparse observation matrix with the FRI observation matrix. Since both formulas (4) and (10) are multidimensional observations of g, the two observation equations can be combined to obtain a new comprehensive observation equation. Combining formulas (4) and (11) yields the new comprehensive observation equation:

[0096]

[0097] Step 6.2: Randomly reorganize the row vectors of the integrated observation vector and the integrated observation matrix using the same rule. Therefore, the mathematical model of the ECT system is rewritten as:

[0098] λ new =S new g (13)

[0099] Among them, λ new is the comprehensive observation vector; S new is the comprehensive observation matrix;

[0100] Step 7: Find the sparse solution of the comprehensive observation equation. The steps are as follows:

[0101] Step 7.1: Sparse Transformation. Only sparse signals can meet the requirements of compressed sensing. For ECT systems, the sparsity of signals often cannot meet the requirements of compressed sensing. To ensure that the input signal is suitable for compressed sensing, an orthogonal transform is performed on the input signal to convert it into a sparse signal. The orthogonal transform is expressed by the following formula:

[0102] g=ψs (14)

[0103] Among them, the matrix ψ of size N×N is the sparse basis, and the original signal g is projected onto the sparse basis to obtain a sparse vector s of size N×1;

[0104] Step 7.2: Find the sparse solution. The capacitance tomography image reconstruction problem is transformed into an L0 norm optimization problem. Since the signal s is sparse, the sparse solution is obtained. The most direct method is to solve the L0 norm optimization problem using the OMP algorithm:

[0105]

[0106] Among them, the L0 norm ||s||0 represents the number of non-zero coefficients in the vector s, which is solved by the orthogonal matching pursuit algorithm OMP, and the matrix ψ of size N×N is the sparse basis;

[0107] Step 8: Image reconstruction. The original image signal is estimated as:

[0108]

[0109] Furthermore, in step 4, the exponential regeneration sampling kernel is used After filtering the Dirac pulse sequence signal g(x), uniform sampling is performed to obtain the sample y k (k=1,2,...,K);

[0110] The formula for the M-1 order exponential reproducing sampling kernel is as follows:

[0111]

[0112] Among them, M-1=3 is the order of the regeneration sampling kernel, and the M-1 order regeneration sampling kernel can regenerate M exponential C m,k It is derived from the following formula:

[0113]

[0114] in yes The dual function of is a quasi-orthogonal function.

[0115] Furthermore, in step 5, the measurement value is calculated from the FRI sampling sample, and the FRI observation equation is constructed. The relationship between the order of the exponential regeneration sampling kernel and the FRI signal is M-1=2L-1, where L is the number of Dirac pulses. The M-1 order exponential regeneration sampling kernel can regenerate M complex exponentials. Among them, α m =a0+jmλ, where a0 is a freely set complex exponential and λ is a freely set real number.

[0116] Furthermore, in step 7.2, in order to ensure that the original signal of the ECT system can be reconstructed, it is necessary to perform an orthogonal transformation on the original signal so that the input signal becomes sparse, and the sparse basis is a discrete cosine transform basis.

[0117] To verify the performance of the proposed method, simulation experiments were conducted. A three-dimensional simulation model was established using COMSOL Multiphysics software, generating simulation data: an m×1 capacitance vector C and an m×n sensitivity matrix S. Matlab R2019a simulation tools were used to process the data and reconstruct ECT images. The three-dimensional simulation model was established as follows: the pipeline was square, with eight capacitance sensors. The internal cross-section of the two-phase flow pipeline was divided into 20×20 pixels, and the dielectric constants of the two-phase flow were set to 3 and 1, respectively. The exponential regeneration sampling kernel order was set to 3, meaning M-1 = 3.

[0118] The original images of the two flow types are as follows Figure 4 As shown in the figure, they are core flow and half-pipe flow respectively. The image effect reconstructed by LBP algorithm is as follows Figure 5 As shown, the image reconstructed by the Landweber algorithm is as follows Figure 6 As shown, the image reconstructed by Tikhonov algorithm Figure 7 As shown, the image effect reconstructed by the method of the present invention Figure 8 As shown in the figure, the reconstruction results show that the image reconstructed by the method of the present invention is close to the original image, with clear and smooth image edges. The image quality reconstructed by the method of the present invention is significantly better than that of the traditional algorithm.

[0119] In order to evaluate the performance of the image reconstruction method, the error and correlation coefficient are used as performance indicators. The error is defined as the difference between the grayscale vector of the reconstructed image and the grayscale vector of the original image, and the formula is:

[0120]

[0121] Among them, g is the original grayscale vector, is the reconstructed grayscale vector.

[0122] The correlation coefficient is defined as the linear correlation between the reconstructed image grayscale vector and the original image grayscale vector, which can be defined as:

[0123]

[0124] in, is the mean value of vector g, and is a vector The average value of .

[0125] Table 1 shows the reconstructed image errors and correlation coefficients of three traditional algorithms, namely the LBP algorithm, the Landweber algorithm, the Tikhonov regularization algorithm, and the method of the present invention under two flow types (core flow and half-pipe flow).

[0126]

[0127] Table 1

[0128] Table 1 shows that for two different flow types (core flow and half-pipe flow), the image reconstructed by the method of the present invention has much smaller errors than the traditional algorithm, and the correlation coefficient is higher. This means that the image reconstructed by the method of the present invention is most similar to the original image. The method of the present invention produces better images, outperforming the traditional algorithm, with good image quality and high reconstruction accuracy.

Claims

1. A multi-index feature extraction ECT image reconstruction method, characterized in that: The method comprises the following steps: Step 1: Initialize the data, obtain and normalize the data. The problem of ECT image reconstruction is to estimate the dielectric constant distribution in the pipeline by solving the gray vector g through the process of measuring the capacitance vector C; Step 2: Fill in zeros and reorganize to construct a sparse observation equation. Add virtual electrodes by zero vector expansion to obtain more measured capacitance values. Construct a sparse observation equation. The mathematical model of the ECT system is written as: λ0=S0g (4) Among them, λ0 is the capacitance vector expanded by the zero-filling reorganization method; S0 is the sensitivity matrix expanded by the zero-filling reorganization method; Step 3: Model the finite innovation rate (FRI) signal and transform the grayscale vector g obtained by the Tikhonov regularization algorithm into v Modeled as a Dirac pulse train signal g(x): g v =(S T ·S+α·I) -1 ·S T C (5) Among them, g v is a grayscale value vector of size n×1, α is the regularization parameter; Where x=1,2…,N is the pixel position, L is the number of non-zero grayscale values of g(x), and a l ∈[0,1] is the pixel x l Gray value, x l ∈{1,2,....,N} is the pixel position with non-zero gray value; Step 4: Finite Innovation Rate (FRI) sampling. First, use the exponential regeneration sampling kernel Filter the Dirac pulse sequence signal g(x) and then uniformly sample it to obtain K sample values y k (k=1,2,...,K); After uniform sampling, we get K samples y k : Where y(x) = g(x)*h(x) is the result of filtering the FRI signal g(x), <·,·> represents the inner product, and y k is the sample value uniformly sampled after filtering, T is the sampling interval, and K is the number of sample values; Step 5: Obtain measurement values from the FRI sampling sample, construct the FRI observation equation, and use the spline coefficient C m,k The sampling value y of FRI k After weighted summation, M complex exponentials can be calculated The measurement value can be obtained from K samples y k Calculation yields: Formula (8) is written in matrix form: Since the pixel position x l belongs to the set {1,2,...,N}, a l ∈[0,1] is the pixel x l Gray value; so formula (8) is written as: U=Ag (11) Where U=[τ0,τ1,...,τ M-1 ] T ∈R M×1 is the FRI measurement vector, g=[g1,g2,...,g N ] T is a grayscale vector of size N×1 in the ECT system, and A is composed of the set Construct an M×N FRI measurement matrix; Step 6: Construct the comprehensive observation equation. The steps are as follows: Step 6.1: Combine the sparse observation matrix with the FRI observation matrix. Since both formulas (4) and (10) are multidimensional observations of g, the two observation equations can be combined to obtain a new comprehensive observation equation. Combining formulas (4) and (11) yields the new comprehensive observation equation: Step 6.2: Randomly reorganize the row vectors of the integrated observation vector and the integrated observation matrix using the same rule, so the mathematical model of the ECT system is rewritten as: l new =S new g (13) Among them, λ new is the comprehensive observation vector; S new is the comprehensive observation matrix; Step 7: Find the sparse solution of the comprehensive observation equation. The steps are as follows: Step 7.1: Sparse transformation, convert the input signal into a sparse signal. The orthogonal transformation is expressed by the following formula: g=ψs (14) Among them, the matrix ψ of size N×N is the sparse basis, and the original signal g is projected onto the sparse basis to obtain a sparse vector s of size N×1; Step 7.2: Find the sparse solution. The capacitance tomography image reconstruction problem is transformed into an L0 norm optimization problem. Since the signal s is sparse, the sparse solution is obtained. The most direct method is to solve the L0 norm optimization problem using the OMP algorithm: Among them, the L0 norm ||s||0 represents the number of non-zero coefficients in the vector s, which is solved by the orthogonal matching pursuit algorithm OMP, and the matrix ψ of size N×N is the sparse basis; Step 8: Image reconstruction. The original image signal is estimated as:

2. The ECT image reconstruction method based on multi-index feature extraction according to claim 1, wherein: In step 4, the exponential regeneration sampling kernel is used After filtering the Dirac pulse sequence signal g(x), uniform sampling is performed to obtain the sample y k (k=1,2,...,K); The formula for the M-1 order exponential reproducing sampling kernel is as follows: Among them, M-1=3 is the order of the regeneration sampling kernel, and the M-1 order regeneration sampling kernel can regenerate M exponential C m,k It is derived from the following formula: in yes The dual function of is a quasi-orthogonal function.

3. The ECT image reconstruction method using multi-index feature extraction according to claim 1 or 2, wherein: In step 5, the measurement value is calculated from the FRI sampling sample, and the FRI observation equation is constructed. The relationship between the order of the exponential regeneration sampling kernel and the FRI signal is M-1=2L-1, where L is the number of Dirac pulses. The M-1 order exponential regeneration sampling kernel can regenerate M complex exponentials. Among them, α m =a0+jmλ, where a0 is a freely set complex exponential and λ is a freely set real number.

4. The ECT image reconstruction method using multi-index feature extraction according to claim 1 or 2, wherein: In step 7.2, in order to ensure that the original signal of the ECT system can be reconstructed, it is necessary to perform an orthogonal transformation on the original signal so that the input signal becomes sparse, and the sparse basis is a discrete cosine transform basis.

Images