A hybrid method for electrical impedance tomography image reconstruction

Abstract

The application discloses a kind of electrical impedance tomography image reconstruction hybrid method, comprising the following steps: according to the difference b of the measured value of boundary voltage obtained from experiment meas And sensitivity matrix A;Propose objective function;Propose the equivalent transformation of objective function;Introduce auxiliary variable, in image reconstruction, by minimizing objective function to solve optimal conductivity variation value, propose optimization model;In order to obtain the solution closer to real conductivity, convert optimization model;Using alternating minimization algorithm to solve auxiliary variable and optimal conductivity variation value;Combined with Tikhonov method, solve initial conductivity variation value;Respectively solve auxiliary variable and optimal conductivity variation value;According to the optimal conductivity variation value solved, the reconstruction of image is completed.The application shows superior performance in improving image reconstruction quality, significantly suppresses the reconstruction image artifact, and the boundary of reconstruction image is clear, and the size of the inclusion reconstructed is accurate.

Description

Technical Field

[0001] This invention belongs to the field of electrical tomography technology, specifically relating to a hybrid method for image reconstruction in electrical impedance tomography. Background Technology

[0002] Electrical tomography (ET), which emerged in the late 1980s, is a process tomography technique based on the sensing mechanism of electrical properties (conductivity / dielectric constant / complex admittance / magnetic permeability). It derives the distribution information of the medium within the measured region through boundary measurements, and then images the distribution information of electrical properties. ET is commonly used in industrial inspection and also has broad application prospects in biomedical imaging. The main ET techniques include Electrical Resistance Tomography (ERT), Electrical Impedance Tomography (EIT), Electrical Magnetic Tomography (EMT), and Electrical Capacitance Tomography (ECT). As an important branch of ET, Electrical Impedance Tomography has received widespread attention due to its non-invasive, radiation-free, and low-cost advantages. However, image reconstruction in Electrical Impedance Tomography presents a serious ill-posed problem. Currently, regularization methods are commonly used to solve ill-posed problems. Regularization methods improve the stability of the solution by adding a penalty function to the objective function. Due to its simple structure, stable solution, and fast reconstruction speed, the Tikhonov method is considered the most classic regularization method. The Tikhonov method applies damping to higher-order eigenvectors, resulting in favorable conditions for the solution. However, because the Tikhonov method uses the L2 norm as a regularization term, the reconstructed image becomes overly smooth, losing sharp edge information. This leads to numerous artifacts, blurred boundaries, and inaccurate object size reconstruction, significantly affecting the spatial resolution of the image reconstruction. To overcome this problem, many methods have been proposed to improve the performance of the Tikhonov method. For example, BYSun et al. proposed an improved Tikhonov regularization method for lung cancer electrical impedance tomography (EIT) monitoring. This method incorporates prior information about the conductivity distribution of patient lung tissue into the EIT imaging process. Compared to the Tikhonov method, this method improves the quality of the reconstructed image, but a large number of artifacts still exist in the reconstructed image. Furthermore, the location and size of inclusions cannot be accurately reconstructed. L. Hu et al. proposed a method combining Tikhonov regularization and Krylov subspaces to address the ill-posed problem in electrical impedance tomography. However, artifacts appear in the background of the reconstructed image, and the boundaries of intrusions in the reconstructed image are blurred. WRFan et al. proposed a method combining Tikhonov regularized maximum entropy with normalized sensitivity mapping to solve the inverse problem of electrical impedance tomography. However, the reconstructed images using this method are not ideal in terms of size and location.Therefore, it is still necessary to develop a new method to improve the reconstruction performance of the Tikhonov method. Summary of the Invention

[0003] The technical problem addressed by this invention is to provide a hybrid method for electrical impedance tomography (EIT) image reconstruction, aiming to improve the performance of the Tikhonov method in EIT image reconstruction. This method employs an alternating minimization algorithm to solve the optimization problem and introduces a fast Fourier transform to accelerate the solution speed. The proposed method results in reconstructed images with fewer artifacts, clearer boundaries, and more accurate reconstructed inclusion sizes. Compared to the Tikhonov method, this invention demonstrates superior performance in improving image reconstruction quality, significantly suppressing reconstructed image artifacts, and producing clearer boundaries and more accurate reconstructed inclusion sizes.

[0004] To solve the above-mentioned technical problems, the present invention adopts the following technical solution: a hybrid method for image reconstruction of electrical impedance tomography, characterized by the following specific steps:

[0005] Step S1: Obtain the difference b between the measured values ​​of the boundary voltage obtained from the experiment. meas With sensitivity matrix A, the experimental system employed a 16-electrode resistivity tomography (RTT) system. The experiment used an excitation and measurement mode of adjacent current excitation and adjacent voltage measurement. Under cyclic excitation and measurement, 208 sets of boundary voltage measurement data were obtained. The measured boundary voltage without inclusions was b1, and the measured boundary voltage with inclusions was b2. The difference b between the measured boundary voltages was calculated. meas That is, b meas =b2-b1;

[0006] The sensitivity matrix A is calculated based on the measured value b1 of the boundary voltage excluding inclusions, combined with sensitivity theory. The formula for calculating the sensitivity matrix A is as follows:

[0007]

[0008] In the formula, A d,m Let u(I) be the element in the d-th row and m-th column of the sensitivity matrix A. d ) and u(I m ) for the d-th electrode group and the m-th electrode group respectively under an excitation current of I d I m The electric potential distribution in the time field, Represent u(I) d ) and u(I m The gradient operator;

[0009] Step S2: The objective function of the proposed method is:

[0010]

[0011] in

[0012]

[0013] In the formula, σ represents the change in optimal conductivity. e Let ||·||² represent the standard deviation of the Gaussian random variable, and ||·||² represent the L2 norm. This is the fidelity term of the objective function. Here, P is the penalty term of the objective function, and P is the ill-conditioned matrix. The initial change in conductivity can be calculated using the Tikhonov method, A. T Let λ denote the transpose of the sensitivity matrix A, where λ is the regularization parameter and I is the identity matrix.

[0014] Step S3: Equivalent transformation of the objective function of the proposed method:

[0015]

[0016] In the formula, P + Let P be the pseudo-inverse of the ill-conditioned matrix P. T Let P be the transpose of the ill-conditioned matrix. It is a seminorm;

[0017] Step S4: Introduce auxiliary variables In image reconstruction, the optimal conductivity change is solved by minimizing the objective function, and its optimization model is expressed as:

[0018]

[0019] Step S5: Use respectively and Replace the optimization model in step S4 and

[0020] The optimization model in step S4 is transformed into:

[0021]

[0022] In the formula, β is the hyperparameter 1;

[0023] Step S6: Use the alternating minimization algorithm to minimize the auxiliary variables. and the change in optimal conductivity Solution:

[0024]

[0025] Optimal conductivity change value The solution is abbreviated as D(·;σ e +β) denotes the noise reduction operator;

[0026]

[0027] To accelerate the auxiliary variables To improve the solution speed, the Fast Fourier Transform is used, with auxiliary variables... The solution in the Fourier domain is expressed as:

[0028]

[0029] in

[0030]

[0031] In the formula, F -1 {·} denotes the inverse fast Fourier transform operator, F{·} denotes the fast Fourier transform operator, and α is the hyperparameter 2;

[0032] The selection of β must satisfy:

[0033]

[0034] The choice of α must satisfy:

[0035]

[0036] The left side of the above equation uses the auxiliary variable η L This indicates that the right side uses the auxiliary variable η. R express;

[0037] Step S7: Solve for the initial change in conductivity. The difference b between the measured values ​​of the boundary voltage obtained in step S1 meas Using the sensitivity matrix A and the Tikhonov method, the initial conductivity change value is solved.

[0038] Step S8: Solve for the auxiliary variables separately and the change in optimal conductivity The iterative process is as follows:

[0039] (1) Input P, σ e , Noise reduction operator D(·; σ e +β), set the iteration count K, maximum iteration count K max ;

[0040] (2) Initialize β, α, the increment of α Δα, and the threshold γ, and set K = 0;

[0041] (3) K = K + 1;

[0042] (4) Update the optimal conductivity change value

[0043] (5) Update auxiliary variables

[0044] (6) Update auxiliary variables

[0045] (7) Update auxiliary variables

[0046] (8) Determine K and η L / η R Does K > 1 and η satisfy the condition? L / η R <γ; If yes, then α=α+△α, K=0, return (3); If no, proceed to the next step;

[0047] (9) Determine whether K satisfies K≥K max If yes, stop iterating; otherwise, return (3).

[0048] Step S9: Based on the solved optimal conductivity change value Complete the image reconstruction.

[0049] Compared with existing technologies, this invention has the following advantages and beneficial effects: This invention employs an alternating minimization algorithm to solve the optimization problem and introduces a fast Fourier transform to accelerate the solution speed. This invention provides qualitative and quantitative analysis of the performance of this hybrid method for electrical impedance tomography image reconstruction. The results show that the hybrid method proposed in this invention produces higher image quality, clearer backgrounds, and the size of inclusions in the reconstructed image is closer to the real image. The hybrid method proposed in this invention improves the performance of the Tikhonov method for reconstruction. Attached Figure Description

[0050] Figure 1 This is a flowchart of the image reconstruction process using the hybrid method of the present invention;

[0051] Figure 2 This forms the basic structure of the EIT system;

[0052] Figure 3 The images show the results of image reconstruction under noise-free conditions using the Tikhonov method and hybrid methods for the six selected classic models.

[0053] Figure 4 A comparison table of fuzzy radius (BR) and structural similarity (SSIM) calculated by the Tikhonov method and hybrid methods under noise-free conditions; Figure 4In the middle (a), the BR values ​​of the reconstructed image under noise-free conditions are shown. Figure 4 (b) shows the SSIM value of the reconstructed image under noise-free conditions;

[0054] Figure 5 The images show the reconstruction results of six classical models using the Tikhonov method and a hybrid method under a noise condition with a signal-to-noise ratio of 20dB.

[0055] Figure 6 A comparison table of fuzzy radius (BR) and structural similarity (SSIM) calculated for six classic models when reconstructed using the Tikhonov method and a hybrid method under a noise condition with a signal-to-noise ratio of 20 dB; Figure 6 In the middle (a), the BR value of the reconstructed image is given under noise conditions with a signal-to-noise ratio of 20 dB. Figure 6 (b) SSIM value of the reconstructed image under noise conditions with a signal-to-noise ratio of 20dB. Detailed Implementation

[0056] The electrical impedance tomography image reconstruction and mixing method of the present invention will be described in detail with reference to the accompanying drawings.

[0057] The present invention provides a hybrid method for image reconstruction using electrical impedance tomography, addressing the problems of numerous artifacts, unclear boundaries, and inaccurate inclusion sizes that occur when reconstructing images using the Tikhonov method. This method improves the performance of the Tikhonov method to enhance the quality of the reconstructed images.

[0058] like Figure 1 The diagram shown is a flowchart of the mixing method of the present invention.

[0059] like Figure 2 The diagram shows the basic structure of an EIT system. The basic structure of an EIT system mainly consists of a sensing electrode array, a data acquisition and control unit, and an image reconstruction unit. The electrode array is formed by a finite number of electrodes uniformly attached to the boundary of the measured area. An excitation current is injected into two adjacent electrodes, and then the boundary voltage is measured from the remaining adjacent electrodes.

[0060] like Figure 3As shown, this invention selected six different models to reconstruct the conductivity distribution under noise-free conditions. The first column in the figure represents the reconstructed real model, while the second and third columns represent the conductivity distribution images reconstructed using the Tikhonov method and the hybrid method, respectively. The results show that the images reconstructed using the Tikhonov method are unsatisfactory. The size of inclusions in the reconstructed images is often much larger than the actual size of the inclusions. The boundaries of the inclusions are also blurred. Simultaneously, some artifacts are observed in the background. In contrast, the hybrid method significantly improves the quality of image reconstruction. The hybrid method reconstructs inclusions with more accurate sizes, clearer boundaries, and no artifacts in the background. Therefore, the image reconstruction results demonstrate that the hybrid method outperforms the Tikhonov method under noise-free conditions.

[0061] Because the Tikhonov method uses the L2 norm as a regularization term, the reconstructed image becomes overly smooth, losing sharp edge information. This results in numerous artifacts, blurred boundaries, and inaccurate object size in the reconstructed image, significantly impacting the spatial resolution of the image reconstruction. To overcome this problem, this invention proposes a hybrid method for electrical impedance tomography image reconstruction. To solve for the optimal conductivity variation value... The initial conductivity change value g was further optimized using a proposed hybrid method for image reconstruction from electrical impedance tomography. The specific implementation steps are as follows:

[0062] Step S1: Obtain the difference b between the measured values ​​of the boundary voltage obtained from the experiment. meas With sensitivity matrix A, the experimental system employed a 16-electrode resistivity tomography (RTT) system. The experiment used an excitation and measurement mode of adjacent current excitation and adjacent voltage measurement. Under cyclic excitation and measurement, 208 sets of boundary voltage measurement data were obtained. The measured boundary voltage without inclusions was b1, and the measured boundary voltage with inclusions was b2. The difference b between the measured boundary voltages was calculated. meas That is, b meas =b2-b1;

[0063] The sensitivity matrix A is calculated based on the measured value b1 of the boundary voltage excluding inclusions, combined with sensitivity theory. The formula for calculating the sensitivity matrix A is as follows:

[0064]

[0065] In the formula, A d,m Let u(I) be the element in the d-th row and m-th column of the sensitivity matrix A. d ) and u(I m ) for the d-th electrode group and the m-th electrode group respectively under an excitation current of I d I m The electric potential distribution in the time field, Represent u(I) d ) and u(I m The gradient operator;

[0066] Step S2: The objective function of the proposed method is:

[0067]

[0068] in

[0069]

[0070] In the formula, σ represents the change in optimal conductivity. e Let ||·||² represent the standard deviation of the Gaussian random variable, and ||·||² represent the L2 norm. This is the fidelity term of the objective function. Here, P is the penalty term of the objective function, and P is the ill-conditioned matrix. The initial change in conductivity can be calculated using the Tikhonov method, A. T Let λ denote the transpose of the sensitivity matrix A, where λ is the regularization parameter and I is the identity matrix.

[0071] Step S3: Equivalent transformation of the objective function of the proposed method:

[0072]

[0073] In the formula, P + Let P be the pseudo-inverse of the ill-conditioned matrix P. T Let P be the transpose of the ill-conditioned matrix. It is a seminorm;

[0074] Step S4: Introduce auxiliary variables In image reconstruction, the optimal conductivity change is solved by minimizing the objective function, and its optimization model is expressed as:

[0075]

[0076] Step S5: Use respectively and Replace the optimization model in step S4 and The optimization model in step S4 is transformed into:

[0077]

[0078] In the formula, β is the hyperparameter 1;

[0079] Step S6: Use the alternating minimization algorithm to minimize the auxiliary variables. and the change in optimal conductivity Solution:

[0080]

[0081] Optimal conductivity change value The solution is abbreviated as D(·;σ e +β) denotes the noise reduction operator;

[0082]

[0083] To accelerate the auxiliary variables To improve the solution speed, the Fast Fourier Transform is used, with auxiliary variables... The solution in the Fourier domain is expressed as:

[0084]

[0085] in

[0086]

[0087] In the formula, F -1 {·} denotes the inverse fast Fourier transform operator, F{·} denotes the fast Fourier transform operator, and α is the hyperparameter 2;

[0088] The selection of β must satisfy:

[0089]

[0090] The choice of α must satisfy:

[0091]

[0092] The left side of the above equation uses the auxiliary variable η L This indicates that the right side uses the auxiliary variable η. R express;

[0093] Step S7: Solve for the initial change in conductivity. The difference b between the measured values ​​of the boundary voltage obtained in step S1 meas Using the sensitivity matrix A and the Tikhonov method, the initial conductivity change value is solved.

[0094] Step S8: Solve for the auxiliary variables separately and the change in optimal conductivity The iterative process is as follows:

[0095] (1) Input P, σ e , Noise reduction operator D(·; σ e +β), set the iteration count K, maximum iteration count Kmax ;

[0096] (2) Initialize β, α, the increment of α Δα, and the threshold γ, and set K = 0;

[0097] (3) K = K + 1;

[0098] (4) Update the optimal conductivity change value

[0099] (5) Update auxiliary variables

[0100] (6) Update auxiliary variables

[0101] (7) Update auxiliary variables

[0102] (8) Determine K and η L / η R Does K > 1 and η satisfy the condition? L / η R <γ; If yes, then α=α+△α, K=0, return (3); If no, proceed to the next step;

[0103] (9) Determine whether K satisfies K≥K max If yes, stop iterating; otherwise, return (3).

[0104] Step S9: Based on the solved optimal conductivity change value Complete the image reconstruction.

[0105] like Figure 4 The table shows a comparison of the fuzzy radius (BR) and structural similarity (SSIM) calculated using the Tikhonov method and a hybrid method under noise-free conditions. (Fuzzy radius) A R A0 represents the area of ​​the region of interest (ROI), while A0 represents the area of ​​the entire detection region. The ROI represents a region with an electrical conductivity greater than one-quarter of the maximum reconstructable electrical conductivity. A lower BR value indicates better image reconstruction. Structural similarity. Where T represents the actual conductivity distribution, R represents the reconstructed conductivity distribution, and μ T and μ R These are the means of T and R, respectively. and , respectively, represent the variances of T and R. The closer the SSIM value is to 1, the higher the structural similarity between the reconstructed image and the real image. It can be observed that the BR value of the hybrid method is significantly lower than that of the Tikhonov method. The SSIM value based on the hybrid method is closer to 1, which also indicates that the hybrid method has an advantage over the Tikhonov method.

[0106] like Figure 5 As shown, the reconstruction results of six classical models using the Tikhonov method and a hybrid method under a noise condition with a signal-to-noise ratio of 20 dB are presented. It can be seen that when noise is present, the hybrid method proposed in this invention still outperforms the Tikhonov method in terms of image quality. The hybrid method exhibits strong robustness to noise, making it highly promising for practical applications in electrical impedance tomography.

[0107] like Figure 6 As shown in the table, the blur radius (BR) and structural similarity (SSIM) of six classic models were compared when reconstructed using the Tikhonov method and a hybrid method under a noise condition with a signal-to-noise ratio of 20 dB. Both the BR and SSIM values ​​indicate that the hybrid method achieves better image reconstruction quality. Compared to the Tikhonov method, the hybrid method is less sensitive to noise.

[0108] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A hybrid method for image reconstruction from electrical impedance tomography, characterized in that... The specific steps are as follows: Step S1: Obtain the difference b between the measured values ​​of the boundary voltage obtained from the experiment. meas With sensitivity matrix A, the experimental system employed a 16-electrode resistivity tomography (RTT) system. The experiment used an excitation and measurement mode of adjacent current excitation and adjacent voltage measurement. Under cyclic excitation and measurement, 208 sets of boundary voltage measurement data were obtained. The measured boundary voltage without inclusions was b1, and the measured boundary voltage with inclusions was b2. The difference b between the measured boundary voltages was calculated. meas That is, b meas =b2-b1; The sensitivity matrix A is calculated based on the measured value b1 of the boundary voltage excluding inclusions, combined with sensitivity theory. The formula for calculating the sensitivity matrix A is as follows: In the formula, A d,m Let u(I) be the element in the d-th row and m-th column of the sensitivity matrix A. d ) and u(I m ) for the d-th electrode group and the m-th electrode group respectively under an excitation current of I d I m The electric potential distribution in the time field, Represent u(I) d ) and u(I m The gradient operator of ) Step S2: The objective function of the proposed method is: in In the formula, σ represents the change in optimal conductivity. e Let ||·||² represent the standard deviation of the Gaussian random variable, and ||·||² represent the L2 norm. This is the fidelity term of the objective function. Here, P is the penalty term of the objective function, and P is the ill-conditioned matrix. The initial change in conductivity can be calculated using the Tikhonov method, A. T Let λ denote the transpose of the sensitivity matrix A, where λ is the regularization parameter and I is the identity matrix. Step S3: Equivalent transformation of the objective function of the proposed method: In the formula, P + Let P be the pseudo-inverse of the ill-conditioned matrix P. T Let P be the transpose of the ill-conditioned matrix. It is a seminorm; Step S4: Introduce auxiliary variables In image reconstruction, the optimal conductivity change is solved by minimizing the objective function, and its optimization model is expressed as: Step S5: Use respectively and Replace the optimization model in step S4 and The optimization model in step S4 is transformed into: In the formula, β is the hyperparameter 1; Step S6: Use the alternating minimization algorithm to minimize the auxiliary variables. and the change in optimal conductivity Solution: Optimal conductivity change value The solution is abbreviated as D(·;σ e +β) denotes the noise reduction operator; To accelerate the auxiliary variables To improve the solution speed, the Fast Fourier Transform is used, with auxiliary variables... The solution in the Fourier domain is expressed as: in In the formula, F -1 {·} denotes the inverse fast Fourier transform operator, F{·} denotes the fast Fourier transform operator, and α is the hyperparameter 2; The selection of β must satisfy: The choice of α must satisfy: The left side of the above equation uses the auxiliary variable η L This indicates that the right side uses the auxiliary variable η. R express; Step S7: Solve for the initial change in conductivity. The difference b between the measured values ​​of the boundary voltage obtained in step S1 meas Using the sensitivity matrix A and the Tikhonov method, the initial conductivity change value is solved. Step S8: Solve for the auxiliary variables separately and the change in optimal conductivity The iterative process is as follows: (1) Input P, σ e , Noise reduction operator D(·; σ e +β), set the iteration count K, maximum iteration count K max ; (2) Initialize β, α, the increment of α Δα, and the threshold γ, and set K = 0; (3) K = K + 1; (4) Update the optimal conductivity change value (5) Update auxiliary variables (6) Update auxiliary variable η L : (7) Update auxiliary variable η R : (8) Determine K and η L / η R Does K > 1 and η satisfy the condition? L / η R <γ; If yes, then α=α+△α, K=0, return (3); If no, proceed to the next step; (9) Determine whether K satisfies K≥K max If yes, stop iterating; otherwise, return (3). Step S9: Based on the solved optimal conductivity change value Complete the image reconstruction.

Images