A dense attention physical information neural network-based electrical impedance image reconstruction method

Abstract

The application belongs to the field of electrical impedance tomography, and particularly relates to an electrical impedance image reconstruction method based on a dense attention physical information neural network, and establishes a double-branch EIT image reconstruction model fusing a dense attention mechanism and physical information constraints, which is named as DA-PINN. The DA-PINN is composed of an encoder based on a dense attention network (DA-Net), a double-branch decoder and a physical information neural network (PINN) module. The encoder comprises a dilated convolution block and a plurality of cascaded dense attention modules. The double-branch decoder comprises a feature conversion layer, a potential output branch and a conductivity output branch. The PINN module comprises control equation constraints and boundary condition constraints. The DA-PINN method provided in the application has better reconstruction performance in terms of quantitative indexes and visual visual effects.

Description

Technical Field

[0001] This invention belongs to the field of Electrical Impedance Tomography (EIT), and in particular, it is a method for electrical impedance image reconstruction based on a dense attention physical information neural network. Background Technology

[0002] Electrical Impedance Tomography (EIT) is a non-destructive imaging technique that reconstructs the conductivity distribution within a measured region by applying a safe-amplitude excitation current to the surface of the object under test and measuring the boundary voltage response. Due to its advantages such as no ionizing radiation, low equipment cost, fast response speed, portability, and the ability to perform continuous real-time monitoring, EIT can be widely used in non-destructive medical monitoring, industrial inspection, and geophysical exploration.

[0003] EIT image reconstruction, as an inverse problem, utilizes boundary current excitation and voltage measurement data to invert the conductivity distribution within the measured region. When the position, shape, and electrical parameters of the medium within the region change, the current propagation path and potential distribution change accordingly, resulting in measurable differences in the boundary voltage response. However, the number of effective measurement data available at the boundary is far less than the number of conductivity parameters to be determined; simultaneously, changes in conductivity at any location within the region can affect the overall potential distribution and boundary voltage response, making the mapping between boundary voltage and internal conductivity highly nonlinear, ill-conditioned, and nonlocal. Furthermore, measurement noise, boundary shape deviations, electrode position errors, and numerical discretization errors may be amplified during the reconstruction process, leading to blurred target boundaries, local artifacts, omission of small targets, and quantitative deviations in conductivity.

[0004] Traditional EIT image reconstruction methods typically transform the solution of the internal conductivity distribution into a constrained optimization problem, reducing instability in the inversion process by introducing prior information or regularization terms. Commonly used methods include the Tikhonov regularization method, the Landweber iterative method, the Newton-Raphson method, and the conjugate gradient least squares (CGLS) algorithm. These methods generally require establishing a mathematical model between the boundary voltage and the internal conductivity, and updating the reconstruction results iteratively through multiple iterations based on a given initial conductivity distribution. Their reconstruction performance largely depends on parameters such as the regularization form, regularization coefficient, iteration step size, and termination condition. Since these parameters are usually selected empirically based on the specific reconstruction task, different parameter settings may produce different reconstruction results. When multiple targets exist in the measured area, the conductivity differences between targets are large, or the boundary voltage measurement data is affected by noise, traditional methods are prone to problems such as slow convergence speed, long computation time, blurred target edges, and insufficient recovery of local details, making it difficult to simultaneously meet the requirements of reconstruction accuracy, stability, and real-time performance.

[0005] With the development of deep learning technology, researchers have begun to utilize neural networks to learn the nonlinear correspondence between boundary voltage measurement data and internal conductivity distribution. A trained neural network can directly output conductivity reconstruction results based on boundary voltage data, thereby reducing the repetitive iterative process in traditional methods. Existing methods typically employ convolutional neural networks to extract features from boundary voltage data, achieving certain results in improving reconstruction speed, suppressing image artifacts, and improving target shape representation.

[0006] (1) Existing pure data-driven EIT reconstruction methods typically train the network based on the data error between the predicted conductivity and the actual conductivity, without fully utilizing the steady-state electric field control equations, current excitation conditions, and boundary measurement conditions. Although the reconstruction results obtained can fit the training samples well, the prediction results may not necessarily satisfy the corresponding physical laws. When there are differences between the target location, number of targets, conductivity range, noise level, or measurement conditions and the training samples, the reconstruction results may suffer from decreased stability and insufficient generalization ability.

[0007] (2) Existing physical prior-based EIT reconstruction methods typically use prior information such as the smoothness, sparsity, or edge structure of conductivity distribution to constrain the reconstruction process. The reconstruction effect largely depends on the degree of matching between the preset prior and the actual target distribution. When the number, shape, size, or conductivity contrast of targets changes, inappropriate prior constraints may cause excessive edge smoothing, omission of small targets, and quantitative bias. In addition, the prior type and its weight parameters usually need to be manually adjusted, making it difficult to adapt to imaging conditions such as complex targets, noise variations, and model mismatch.

[0008] (3) Existing feature extraction networks are insufficient in representing the complex relationships between different voltage measurement channels and between boundary measurement data and internal conductivity distribution. Conventional convolutional structures mainly extract features within local neighborhoods. As the number of network layers increases, key features may gradually decay during transmission. Some networks lack adaptive filtering mechanisms for features from different measurement channels and different levels, which can easily lead to feature redundancy or insufficient key responses. When there are multiple targets of different sizes, shapes, or conductivity in the imaging region, the existing methods still need to improve their ability to express weak response information, small targets, and complex edge features.

[0009] Therefore, how to fully extract the effective features from the boundary voltage measurement data, reduce the dependence of the reconstruction process on fixed priors and artificial parameters, and effectively integrate the laws of electric field physics into network training to improve the reconstruction accuracy, physical consistency and generalization ability under complex target conditions are urgent problems to be solved in the existing EIT image reconstruction technology. Summary of the Invention

[0010] The purpose of this invention is to overcome the shortcomings of the prior art and provide a method for electrical impedance image reconstruction based on a dense attention physical information neural network. This method can fully extract multi-scale features and key channel information from boundary voltage measurement data, integrate the laws of electric field physics into network training, reduce reconstruction artifacts and target edge blurring, and improve reconstruction accuracy, physical consistency and generalization ability under complex target conditions.

[0011] The technical solution adopted by the present invention to solve the above-mentioned technical problems is as follows:

[0012] This invention provides a method for electrical impedance tomography (EIT) image reconstruction based on a dense attention physical information neural network. A dual-branch EIT image reconstruction model, named DA-PINN, is established, integrating a dense attention mechanism with physical information constraints. DA-PINN consists of a dense attention network-based encoder, a dual-branch decoder, and a physical information neural network module. The encoder includes dilated convolutional blocks and multiple cascaded dense attention modules. The dual-branch decoder includes a feature transformation layer, a potential output branch, and a conductivity output branch. The physical information neural network module includes control equation constraints and boundary condition constraints.

[0013] After feature extraction by the encoder, the boundary voltage measurement sequence is synchronously output by the dual-branch decoder, showing the node potential distribution and node conductivity distribution within the imaging area. Based on the node potential distribution and node conductivity distribution, a partial differential equation residual loss is constructed, and an electrode boundary measurement consistency loss is constructed based on the node potential distribution. The partial differential equation residual loss, electrode boundary measurement consistency loss, and conductivity data loss together form the total network loss function.

[0014] Furthermore, in the control equation constraints, the control equations for the potential field and conductivity field within the imaging region are spatially discretized using a point electrode model and the triangular finite element method. A global finite element stiffness matrix is ​​assembled based on the predicted node conductivity, and an electrode current load vector is constructed based on the current excitation mode. The discrete residual vector of the control equation is calculated using the predicted node potential and predicted node conductivity, thereby constructing the residual loss of the partial differential equation.

[0015] Furthermore, in the boundary condition constraints, the finite element mesh nodes corresponding to the outflow electrodes of each current excitation mode are set as reference nodes, and the predicted potential value of the reference node is used as a reference potential to correct the predicted potential of other nodes. The predicted voltage is calculated based on the predicted potential of each measurement electrode to the corresponding node after correction, and the predicted voltage is compared with the measured voltage to construct the electrode boundary measurement consistency loss.

[0016] Furthermore, the total network loss function is a weighted combination of the conductivity data loss, the partial differential equation residual loss, and the electrode boundary measurement consistency loss, wherein the weight coefficients of the partial differential equation residual loss and the electrode boundary measurement consistency loss are determined by a Bayesian optimization algorithm.

[0017] Furthermore, the dilated convolutional block includes an initial convolutional layer and a merging operation unit. The initial convolutional layer receives the boundary voltage measurement sequence after dimension reconstruction. The merging operation unit contains multiple parallel dilated convolutional branches, each with a different dilation rate. The output of each dilated convolutional branch is connected to a channel splicing layer, and the output of the channel splicing layer is connected to a fusion convolutional layer.

[0018] Furthermore, there are multiple dense attention modules, which are cascaded in sequence; each dense attention module includes a dense block and an attention transition layer. The dense block contains multiple bottleneck layers connected in series, and the bottleneck layers are densely connected to achieve feature reuse; the attention transition layer is used to generate channel attention weights and to recalibrate the output features of the dense block using the channel attention weights.

[0019] Furthermore, the bottleneck layer includes a first convolutional layer, a second convolutional layer, a batch normalization layer, and an activation function layer. The newly generated features of each bottleneck layer are concatenated with its input features along the channel dimension and used as the input of the subsequent bottleneck layer.

[0020] Furthermore, the feature conversion layer includes a fully connected layer, a batch normalization layer, and an activation function layer; the potential output branch includes a first fully connected layer and a second fully connected layer; the conductivity output branch includes a first fully connected layer and a second fully connected layer.

[0021] Furthermore, in the multiple cascaded dense attention modules, the outputs of each dense attention module except the last one are pooled and input into the next dense attention module. The output of the last dense attention module is then global average pooled and flattened to form the encoder's global feature vector.

[0022] Furthermore, the feature transformation layer converts the encoder's global feature vector into shared features, and the potential output branch and the conductivity output branch output the node potential distribution and node conductivity distribution based on the shared features, respectively.

[0023] The advantages and positive effects of this invention are:

[0024] 1. This invention sets up four parallel dilated convolution branches with different dilation rates, which can extract boundary voltage features of different sensing ranges while keeping the feature space size unchanged. This is beneficial to enhancing the network's ability to express local responses and information related to different measurement channels.

[0025] 2. This invention achieves continuous transmission and reuse of features at different levels through dense connections, reducing the attenuation of key features during transmission in deeper layers of the network; and performs adaptive recalibration of different feature channels through an attention transition layer, enhancing the response of important features and suppressing redundant information.

[0026] 3. This invention uses potential output branch and conductivity output branch to simultaneously obtain the node potential distribution and node conductivity distribution, providing interrelated prediction parameters for the constraints of the governing equations and the consistency constraints of electrode boundary measurements, which is beneficial to improving the physical consistency of the reconstruction results.

[0027] 4. This invention uses the triangular finite element method to spatially discretize the control equations, assembles the global finite element stiffness matrix based on the predicted node conductivity, and constructs the discrete residuals of the control equations. This can integrate the electric field control relationship within the imaging region into the network training, reducing the insufficient physical constraints caused by relying solely on conductivity label errors.

[0028] 5. This invention calculates the predicted voltage based on the predicted node potential and uses the difference between the predicted voltage and the measured voltage to construct the electrode boundary measurement consistency loss, so that the network output is simultaneously constrained by the internal control equations and the boundary measurement data.

[0029] 6. This invention uses conductivity data loss, partial differential equation residual loss, and electrode boundary measurement consistency loss together for network training, and determines the loss weight coefficients through Bayesian optimization algorithm, which helps to reduce the process of manually adjusting loss weights and improves reconstruction stability and generalization ability under different target distributions and measurement conditions. Attached Figure Description

[0030] Figure 1 This is a schematic diagram of the point electrode EIT model of the 16-electrode adjacent excitation and adjacent measurement of the present invention;

[0031] Figure 2 This is a flowchart illustrating the overall process of the electrical impedance image reconstruction method of the present invention.

[0032] Figure 3 This is a schematic diagram of the overall structure of the DA-PINN model of the present invention;

[0033] Figure 4 This is a schematic diagram of the structure of the DA-PINN encoder of the present invention;

[0034] Figure 5 This is a schematic diagram of the DA-PINN dual-branch decoder of the present invention;

[0035] Figure 6 This is a flowchart illustrating the physical information constraints and network loss construction process of this invention.

[0036] Figure 7 This is a schematic diagram comparing the reconstruction results of different reconstruction methods under multiple conductivity and multiple object datasets in this invention. Detailed Implementation

[0037] The present invention will be further described in detail below through specific embodiments. The following embodiments are merely descriptive and not limiting, and should not be used to limit the scope of protection of the present invention.

[0038] The overall inventive concept of this invention is as follows:

[0039] 1. Adopting a reconstruction approach that combines data-driven and physical constraints, the boundary voltage measurement sequence obtained by the multi-electrode EIT system is used as the network input. Multi-scale features and key channel information are extracted through a dense attention encoder, and the node potential distribution and node conductivity distribution in the imaging area are predicted simultaneously using a dual-branch decoder.

[0040] 2. Parallel dilated convolutional branches with different dilation rates are set in the encoder, and multiple dense attention modules are cascaded to enable the continuous transfer and reuse of shallow boundary voltage response features and deep global features. At the same time, channel attention weights are used to enhance important features and suppress redundant features.

[0041] 3. By utilizing the shared feature relationship between the potential output branch and the conductivity output branch, without using the node potential inside the imaging region as a known supervision label, the node potential and node conductivity are obtained synchronously based solely on the boundary voltage measurement data, and physical information constraints are constructed accordingly.

[0042] 4. The steady-state electric field control equations are discretized using a point electrode model and the triangular finite element method. The global finite element stiffness matrix is ​​assembled based on the predicted node conductivity. The discrete residuals of the control equations and the predicted voltages are calculated based on the predicted node potentials. The conductivity data loss, the partial differential equation residual loss, and the electrode boundary measurement consistency loss are all used for DA-PINN training.

[0043] This invention takes a two-dimensional circular imaging region as an example. For the specific principles of the 16-electrode adjacent excitation and adjacent measurement point electrode EIT model, please refer to [reference needed]. Figure 1 Sixteen electrode positions are uniformly arranged on the boundary of a two-dimensional circular imaging region with a diameter of 10 cm, each electrode position corresponding to a boundary finite element mesh node. This embodiment employs adjacent current excitation and adjacent voltage measurement methods. The incentive model is directed towards the first Current is injected into the nodes corresponding to each electrode, and a current of equal amplitude flows out from the nodes corresponding to the next adjacent electrode. The current load on the remaining finite element nodes is zero. Thirteen sets of effective adjacent voltage differences are obtained for each excitation mode, and a total of 208 boundary voltage measurements are obtained for 16 excitation modes.

[0044] The overall workflow of electrical impedance image reconstruction methods can be found in [reference needed]. Figure 2 The main components include boundary voltage data acquisition, finite element mesh and point electrode model establishment, DA-PINN model construction, composite loss calculation, model training, and conductivity distribution reconstruction of the object under test.

[0045] In the point electrode model, the measured voltage is determined by the potential difference between the corresponding measuring electrode nodes. This model does not introduce electrode contact impedance and electrode coverage length; the electrode potential is directly represented by the potential of the corresponding finite element node. Based on the node current load vectors corresponding to each excitation mode and the stiffness matrix of the triangular element, the global finite element stiffness matrix is ​​assembled to form the finite element discrete equations, thereby constructing the finite element physical constraints of DA-PINN.

[0046] Under steady-state conditions, the first The electric potential field and conductivity field corresponding to the current excitation mode satisfy the following governing equations:

[0047]

[0048] in, For the imaging area, The spatial coordinates within the imaging region, For the first The electric potential field corresponding to the current excitation mode. The conductivity field within the imaging region, For gradient operators, It is a divergence operator.

[0049] The imaging region was then discretized using a triangular finite element mesh. The mesh contains 882 finite element nodes, with the 16 point electrodes corresponding to 16 boundary nodes. For the... The batch of samples, the first The average conductivity of each triangular unit cell is calculated using the following formula:

[0050]

[0051] in, For the first The batch of samples in the first... Average conductivity of the triangular unit cell, For the first The first batch of samples corresponding to the first batch of samples The first triangular unit The predicted conductivity values ​​for each node, This represents the total number of triangular units.

[0052] After the mesh geometry information is determined, the first The element representation of the unit conductivity template stiffness matrix for each triangular element is:

[0053]

[0054] in, For the first Stiffness matrix of unit conductivity template for triangular elements. The stiffness matrix of the template with unit conductivity is the first... Line 1 Column matrix elements, For the first The area of ​​each triangular unit and The first The first triangular unit The first derivative coefficients of the linear shape functions with respect to the first and second spatial coordinates. and The first The first triangular unit The first derivative coefficients of the linear shape functions with respect to the first and second spatial coordinates.

[0055] Multiplying the average conductivity of each element by the stiffness matrix of the unit conductivity template yields the element stiffness matrix for each triangular element. Then, using the degree-of-freedom mapping between local and global nodes, all element stiffness matrices are accumulated to the corresponding global location to form the global finite element stiffness matrix. The above processes are represented as follows:

[0056]

[0057]

[0058] in, For the first The first batch of samples corresponding to the first batch of samples The element stiffness matrix of a triangular element. For the first Extraction matrix from global node degrees of freedom to local node degrees of freedom of each triangular element. To extract the transpose of the matrix, For the first The global finite element stiffness matrix corresponding to each batch of samples.

[0059] After establishing the global finite element stiffness matrix, the point electrode current load vectors for each current excitation mode are constructed according to adjacent excitation methods. The loads of the nodes corresponding to the injected electrode, the nodes corresponding to the outflowing electrode, and the remaining nodes are set as follows:

[0060]

[0061] in, For the first The current load values ​​of the finite element mesh nodes corresponding to the injected electrodes under various current excitation modes. For the first The current load values ​​of the finite element mesh nodes corresponding to the outflowing electrode under the current excitation mode. For the first The current load values ​​of the remaining finite element mesh nodes under the current excitation mode. Inject current amplitude into the electrodes; electrode model includes There are several adjacent current excitation modes, each of which includes The measurement electrode pairs and the input boundary measurement voltage sequence include... Each measured voltage value.

[0062] During training, the current amplitude can be normalized to In practical applications, when using other safe excitation amplitudes, it is only necessary to use consistent scale normalization for the measured voltage and current loads. The unit conductivity template stiffness matrix, triangular element node index, local-global degree of freedom mapping, and current load vectors for each excitation mode can all be pre-calculated before network training.

[0063] After completing the boundary data organization and finite element model establishment, according to Figure 3 Construct DA-PINN. The model consists of an encoder based on a dense attention network, a two-branch decoder, and a Physical Information Neural Network (PINN) module. The encoder is responsible for extracting multi-scale features from the boundary voltage measurement sequence and forming shared global features; the two-branch decoder outputs the node potential distribution and node conductivity distribution in parallel based on the shared features; the PINN module uses the prediction results of the two branches to calculate the governing equation constraints and boundary condition constraints.

[0064] The overall structure of the DA-PINN model can be found by referring to Figure 3 The model consists of an encoder based on a dense attention network, a dual-branch decoder, and a PINN module. After the boundary voltage measurement sequence is input into the encoder, the encoder extracts multi-scale features and key channel information, forming global features that are shared by both output branches. The dual-branch decoder outputs the node potential distribution and node conductivity distribution within the imaging region in parallel based on these global features. The PINN module constructs governing equation constraints and boundary condition constraints based on the prediction results of the two branches, and uses the corresponding physical constraint loss for network training.

[0065] For the specific structure of the encoder, please refer to Figure 4 First, the boundary voltage measurement sequences were reconstructed according to the order of 16 excitation modes and 13 sets of measurement voltages for each excitation mode, converting the 208-dimensional boundary voltage sequence of each sample into a sequence of size [size missing]. The input feature tensor. The dimension reconstruction process and the feature extraction process of the initial convolutional layer are respectively represented as:

[0066]

[0067]

[0068] in, For the boundary measurement voltage sequence of the input encoder, The input feature tensor is obtained by reconstructing the dimension of the boundary measurement voltage sequence. The output features of the initial convolutional layer, This refers to the number of samples processed in a batch. The number of measured voltage values ​​for each sample. and These are the height and width of the input feature tensor, respectively. , and These represent the number of channels, height, and width of the output features from the initial convolutional layer, respectively. These are the kernel weights of the initial convolutional layer. and These represent the height and width of the initial convolutional layer kernel, respectively. This represents the convolution operation. This indicates batch normalization operations. This indicates the linear rectification activation operation. This indicates a dimension reconstruction operation.

[0069] Based on 16 excitation modes and 13 sets of measured voltages for each excitation mode, the height of the input feature tensor is determined. and width The numbers are 16 and 13 respectively, representing the number of measured voltage values ​​per sample. The value is 208. The initial convolutional layer sequentially performs convolution, batch normalization, and... Activation yields initial output features. .

[0070] Initial output features The next step is to merge and run the dilated convolution block. This unit consists of four parallel dilated convolution branches with dilation rates of 1, 2, 3, and 4, respectively. Each branch performs dilated convolution and... The activation and convolution stride are both 1, and a padding method is used to keep the spatial size of the output feature map unchanged. The feature extraction process of each dilated convolution branch is represented as follows:

[0071]

[0072] in, For the first The output features of each dilated convolution branch For the first The dilation rate of each dilated convolution branch, For the first The kernel weights of each dilated convolutional branch, For the first The number of output channels of each dilated convolution branch and The first The height and width of the kernel of each dilated convolution branch. For expansion rate The dilated convolution operation.

[0073] After obtaining features from different receptive fields through four dilated convolutional branches, these features are fed into the channel concatenation layer along the channel dimension. The feature tensor output by the channel concatenation layer serves as the input to the fusion convolutional layer; the fusion convolutional layer then sequentially performs convolution, batch normalization, and... Activation is performed, ultimately outputting multi-scale fused features. The corresponding channel concatenation and fusion calculations are expressed as follows:

[0074]

[0075]

[0076] in, The output features of the channel splicing layer The number of channels for the output features of the channel splicing layer. This indicates a concatenation operation performed along the channel dimension. To fuse the output features of the convolutional layers, , and These represent the number of channels, height, and width of the output features from the fused convolutional layer, respectively. To merge the convolutional kernel weights of the convolutional layers, and These represent the height and width of the convolutional kernel in the fused convolutional layer, respectively.

[0077] Multi-scale fused features are then sequentially fed into multiple cascaded dense attention modules. Each dense attention module consists of a dense block and an attention transition layer. The dense block contains... A series of bottleneck layers are connected sequentially, and each bottleneck layer undergoes batch normalization in turn. activation, Convolution, Batch Normalization Activation and Convolution is used to perform non-linear feature transformation. Both convolutional layers have a stride of 1. Convolution employs a padding method that maintains the spatial size of the output feature map. The new features generated by each bottleneck layer are concatenated with the input features of that bottleneck layer along the channel dimension. The concatenated result serves as the input to subsequent bottleneck layers, allowing features extracted from preceding layers to be directly passed to later layers. The new feature generation process and dense connection process of each bottleneck layer are respectively represented as follows:

[0078]

[0079]

[0080] in, For the first Input features of each bottleneck layer For the first New features generated at the bottleneck layer For the first Densely connected output of bottleneck layers The number of bottleneck layers contained in the dense block. and The first The number of channels in the bottleneck layer's input features and the densely connected output. and These represent the height and width of the feature map within the dense block, respectively. The number of growth channels for each bottleneck layer. For the first The convolutional kernel weights of the first convolutional layer in the bottleneck layer This represents the number of output channels of the first convolutional layer. For the first The convolutional kernel weights of the second convolutional layer in the bottleneck layer.

[0081] After the dense blocks complete feature generation and reuse, the attention transition layer adaptively adjusts the importance of each channel. The attention transition layer consists of a global average pooling layer, a dimensionality-reducing convolutional layer, an dimensionality-increasing convolutional layer, and... Activation layer composition. First, global average pooling is used to compress spatial information, and the output of the dimensionality-reduced convolution is then... After activation, input into the up-dimensional convolutional layer, and then through... The function obtains the attention weights for each channel and multiplies these weights channel-by-channel with the dense block output to form recalibrated features. The processing steps are as follows:

[0082]

[0083] in, The output features of dense blocks, The output features of the global average pooling layer. For the output features of the dimensionality reduction convolutional layer, for Attention weights output by the activation layer. The output features of the attention transition layer, The number of channels for the output features of the dense block. The number of channels for the output features of the dimensionality reduction convolutional layer. The dimensionality reduction ratio is a positive integer and It is an integer. For the convolution kernel weights of the dimensionality reduction convolutional layer, For the convolution kernel weights of the up-dimensional convolutional layer, This indicates a global average pooling operation. express Activation function This indicates element-wise multiplication operations broadcast by channel.

[0084] The encoder contains four dense attention modules cascaded in sequence. The outputs of the first, second, and third dense attention modules are respectively processed by kernels with a size of [missing information]. The first step is average pooling with a stride of 2, and the resulting downsampled features are used as input to the next dense attention module. The output of the fourth dense attention module is not further processed. Instead of average pooling, global average pooling is used to compress the spatial dimension, followed by flattening to form the encoder's global feature vector. The average pooling process and the formation process of the global feature vector in the first three dense attention modules are respectively represented as follows:

[0085]

[0086]

[0087] in, For the first The output features of a dense attention module For the first The output features obtained by average pooling of dense attention modules and The average pooling kernel size and step size of the pooling layer are both , The fourth dense attention module outputs the encoder's global feature vector, which is obtained after global average pooling and flattening. This represents the average pooling operation. This represents the flattening operation; the first dense attention module receives the output features of the fused convolutional layer. The pooling outputs of the first three dense attention modules are input into the next dense attention module, which is the encoder's global feature vector. Input a dual-branch decoder.

[0088] The output of the fourth dense attention module is global average pooled and flattened to form a 1024-dimensional global feature vector, which serves as the input to the dual-branch decoder.

[0089] The structure of the dual-branch decoder can be referenced. Figure 5 This includes a feature transformation layer, a potential output branch, and a conductivity output branch. The global feature vector is first input to the feature transformation layer, and then sequentially passes through a fully connected layer, batch normalization, and... The feature is activated and converted into a 2048-dimensional shared feature. This shared feature is then input into the potential and conductivity output branches, respectively, for parallel prediction of the nodal potential and conductivity distributions within the imaging region. The global feature transformation process is represented as follows:

[0090]

[0091] in, The output features of the feature transformation layer, The feature transformation layer outputs the dimension of the features. This is the weight matrix of the feature transformation layer.

[0092] The potential output branch includes, in sequence, the first fully connected layer, the batch normalization layer, and... The activation layer and the second fully connected layer. The 2048-dimensional shared features input to this branch are first transformed into 1024-dimensional features by the first fully connected layer, and then sequentially processed by batch normalization and... After activation, a second fully connected layer outputs a potential feature vector containing 14112 elements. This potential feature vector is then reconstructed according to 16 excitation modes and 882 finite element nodes to obtain a size of... The nodal potential matrix. The nodal potential mapping process is represented as:

[0093]

[0094]

[0095]

[0096] in, The output characteristics of the first fully connected layer in the potential output branch are shown. The flattened node potential output by the second fully connected layer of the potential output branch. To flatten the nodal potential distribution obtained by dimensional reconstruction, The dimension of the output feature of the first fully connected layer in the potential output branch is [value]. and These are the weight matrices for the first and second fully connected layers of the potential output branch, respectively. This represents the number of current-driven excitation modes. This represents the total number of nodes in the finite element mesh.

[0097] The conductivity output branch includes, in sequence, the first fully connected layer, the batch normalization layer, and... The activation layer and the second fully connected layer. The 2048-dimensional shared features input to this branch are first transformed into 1024-dimensional features by the first fully connected layer, and then successively processed by batch normalization and... After activation, a node conductivity vector of length 882 is output via a second fully connected layer. Each element in this vector corresponds to the conductivity value of a finite element node, thus obtaining the node conductivity distribution within the imaging region. The node conductivity mapping process is expressed as follows:

[0098]

[0099]

[0100] in, The output characteristics of the first fully connected layer in the conductivity output branch are given. The node conductivity distribution of the second fully connected layer output for the conductivity output branch. Let be the dimension of the output feature of the first fully connected layer in the conductivity output branch. and These are the weight matrices for the first and second fully connected layers of the conductivity output branch, respectively.

[0101] During the training phase, only the conductivity output branch is provided with real nodal conductivity labels to supervise the nodal conductivity distribution output by that branch. The potential output branch does not use the nodal potentials within the imaging region as supervision labels; instead, its output nodal potentials are trained using constraints from the governing equations and electrode boundary measurement consistency constraints.

[0102] The construction process for physical information constraints and network loss can be referenced. Figure 6 After obtaining the nodal potential matrix and nodal conductivity vector, the global finite element stiffness matrix is ​​first assembled based on the nodal conductivity vector. For each excitation mode, the global finite element stiffness matrix is ​​multiplied by the corresponding nodal potential vector, and the point electrode current load vector corresponding to that excitation mode is subtracted to obtain the discrete residual vector of the governing equations. The formation process is expressed as follows:

[0103]

[0104] in, For the first The batch of samples in the first... Discrete residual vectors of the control equations under various current excitation modes The first node in the conductivity distribution The predicted node conductivity vector corresponding to each batch of samples. The first in the nodal potential distribution The batch of samples in the first... The predicted node potential vector corresponding to each current excitation mode. This is the global finite element stiffness matrix assembled from the predicted nodal conductivity vectors. For the first The electrode current load vector corresponding to each current excitation mode.

[0105] The L2 norm squared value is calculated for the discrete residual vector of the control equation under each batch sample and each excitation mode, and then averaged over the batch sample dimension and the excitation mode dimension to obtain the partial differential equation residual loss, which has the following form:

[0106]

[0107] in, For the residual loss of partial differential equations, This represents the square of the second norm of a vector.

[0108] Boundary constraints include reference potential correction and electrode boundary measurement consistency constraints. Because the steady-state potential solution has additive constant uncertainty—meaning that simultaneously increasing the potential of each finite element mesh node by an arbitrary constant will not change the electric field or the voltage difference between electrodes—for each current excitation mode, the finite element mesh node corresponding to the outflowing electrode is selected as the reference node, and the predicted potential of this reference node is used as the reference potential. For the same batch of processed samples and the same current excitation mode, the predicted potential of the reference node is subtracted from the predicted potential of each finite element mesh node to obtain the node predicted potential after reference potential correction. The reference potential correction process is expressed as follows:

[0109]

[0110] in, For the first The batch of samples in the first... The first current excitation mode Predicted potential values ​​of each finite element mesh node after reference potential correction. For the first The batch of samples in the first... The first current excitation mode Predicted potential values ​​for each finite element mesh node. For the first The batch of samples in the first... The predicted potential value of the reference node corresponding to the outflow electrode under the current excitation mode.

[0111] After completing the reference potential correction, according to the predetermined measurement electrode sequence, the difference between the predicted potential values ​​of the finite element mesh nodes corresponding to the two electrodes in each measurement electrode pair after reference potential correction is calculated to obtain the corresponding predicted voltage. The calculation process of the predicted voltage is expressed as follows:

[0112]

[0113] in, For the first The batch of samples in the first... The first current excitation mode The predicted voltage corresponding to the set of measurement electrode pairs and The first one is respectively the second one. The predicted potential value is obtained after correction of the reference potential of the finite element mesh nodes corresponding to the two electrodes in the set of measuring electrodes. This represents the number of measurement electrode pairs corresponding to each current excitation mode.

[0114] The electrode boundary measurement consistency loss is obtained by averaging the squared differences between the predicted voltage and the corresponding measured voltage of each measurement electrode pair under each current excitation mode for each batch of samples. Its form is as follows:

[0115]

[0116] in, To measure the consistency loss at the electrode boundary, For the first The batch of samples in the first... The first current excitation mode The measured voltage corresponding to the set of measuring electrode pairs.

[0117] The squared L2 norm of the difference vector between the predicted node conductivity vector and the corresponding true node conductivity vector for each batch of samples is calculated, and then averaged over the batch sample dimension and the finite element mesh node dimension to obtain the conductivity data loss, which takes the form:

[0118]

[0119] in, For conductivity data loss, For the first The predicted node conductivity vector corresponding to each batch of samples. For the first The actual node conductivity vector corresponding to each batch of samples.

[0120] By weighting and combining the conductivity data loss, the partial differential equation residual loss, and the electrode boundary measurement consistency loss, the total network loss function is obtained, and its expression is:

[0121]

[0122] in, Let be the total network loss function. and These are the weighting coefficients for the partial differential equation residual loss and the electrode boundary measurement consistency loss, respectively.

[0123] Weighting coefficient and The search scope is A Bayesian optimization algorithm was employed to search for the two weight coefficients, aiming to minimize the total network loss on the validation set. After 40 Bayesian optimization iterations, the optimal weights were determined. , .

[0124] Training, validation, and testing data were all generated from finite element forward problem simulations. Each sample contains 882 nodal conductivity values, 208 object-field boundary voltage measurements, and 208 object-field and space-field voltage difference values. The object-field and space-field voltage difference sequence serves as the encoder input, the object-field boundary voltage measurements are used to calculate the electrode boundary measurement consistency loss, and the nodal conductivity values ​​serve as supervision labels for the conductivity output branch.

[0125] The conductivity of the background medium in the circular imaging region was set to 20 S / m. Circular objects with different conductivity and numbers were placed within this background medium to construct a multi-conductivity, multi-object dataset. This dataset contains 10,000 samples, with the conductivity of each object selected from 2 S / m, 8 S / m, 34 S / m, and 40 S / m. Two thousand samples were generated for each sample containing 1, 2, 3, 4, and 5 objects. All objects were circular, and their positions and sizes were randomly set while ensuring they were located within the imaging region and met preset spacing conditions.

[0126] To improve the network's adaptability to measurement noise, 20% of the samples were randomly selected from the samples corresponding to the number of objects to be imaged in the multi-conductivity, multi-object dataset. Gaussian white noise with a signal-to-noise ratio of 30 dB was added to the boundary voltage data of the selected samples. Subsequently, the samples corresponding to the number of objects to be imaged were divided into training, validation, and test sets at proportions of 70%, 15%, and 15%, respectively. The training set was used to update the network parameters, the validation set was used to select loss weights and other hyperparameters, and the test set was used to evaluate the model's reconstruction performance on samples not included in the training.

[0127] DA-PINN is built on the PyTorch framework and trained on an NVIDIA RTX 4090 graphics processor. The batch size is set to 128, and the initial learning rate is set to... The training epochs were set to 300. The Adam optimizer, based on adaptive moment estimation, was used to update the network parameters. The exponential decay rates of the first and second moments were set to 0.9 and 0.999, respectively, and the stability constant to prevent division by zero was set to... During training, a stepped learning rate decay strategy is employed, adjusting the learning rate to 50% of the previous stage every 10 training cycles; simultaneously, L2 regularization is used, with the weight decay coefficient set to... This reduces model complexity and suppresses overfitting.

[0128] In each mini-batch training process, the sequence of voltage difference between the object field and the air field is first input into DA-PINN, and the predicted node potential and predicted node conductivity are obtained synchronously through one forward propagation. Then, the conductivity data loss is calculated based on the predicted node conductivity and the actual node conductivity. The global finite element stiffness matrix is ​​assembled based on the predicted node conductivity, and the residual loss of the partial differential equation is calculated in combination with the predicted node potential. After the predicted node potential is corrected by the reference potential, the predicted voltage is calculated, and the electrode boundary measurement consistency loss is further calculated. Finally, the three losses are weighted and combined to obtain the total network loss, and backpropagation and network parameter updates are performed based on the total network loss.

[0129] After network training is complete, the network parameters are fixed and DA-PINN is switched to inference mode. Following the same current excitation sequence, measurement sequence, and normalization parameters as in the training phase, the boundary voltage data of the object under test is processed to form a 208-dimensional sequence of object-field and space-field voltage differences. This sequence is then input into the trained DA-PINN model to obtain the conductivity values ​​of 882 finite element nodes. Based on the finite element mesh node coordinates, the node conductivity distribution can be directly displayed as a triangular mesh diagram, or it can be interpolated to a regular pixel mesh to form a two-dimensional EIT reconstructed image. The potential output branch simultaneously outputs the node potential distribution under 16 current excitation modes for physical consistency verification of the reconstruction results.

[0130] To verify the reconstruction performance of the DA-PINN proposed in this invention, classical numerical solution algorithms and the EIT image reconstruction model based on deep learning were selected as comparison methods.

[0131] (1) Classical numerical solution algorithms include: Conjugate Gradient Least Squares (CGLS) [A. Bjorck, T. Elfving, and Z. Strakos, "Stability of Conjugate Gradient and Lanczos Methods for Linear Least Squares Problems", SIAM Journal on Matrix Analysis and Applications, vol. 19, no. 3, pp. 720-736, 1998, doi: 10.1137 / S089547989631202X.], and Landweber's iterative algorithm [L. Landweber, "An Iteration Formula for Fredholm Integral Equations of the First Kind", American Journal of Mathematics, vol. 73, no. 3, pp. 615-624, 1951, doi: 10.2307 / 2372313.], Newton-Raphson algorithm [TJ Yorkey, JG Webster, and WJ Tompkins, "Comparing Reconstruction Algorithms for Electrical Impedance Tomography", IEEE Transactions on Biomedical Engineering, vol. BME-34, no.11, pp. 843-852, 1987, doi: 10.1109 / TBME.1987.326032.], Tikhonov regularization algorithm [M. Vauhkonen, D. Vadasz, PA Karjalainen, E. Somersalo, and JP Kaipio, “Tikhonov Regularization and Prior Information in Electrical Impedance Tomography,” IEEE Transactions on Medical Imaging, vol. 17, no. 2, pp. 285-293, 1998, doi: 10.1109 / 42.[700740.]. Each classical numerical solution algorithm uses the same finite element mesh, electrode nodes, excitation measurement protocol, and test samples as DA-PINN, and determines the number of iterations and regularization parameters based on the validation set results or the corresponding algorithm's stopping criteria.

[0132] (2) Deep learning-based EIT image reconstruction models include: LeNet [C. Tan, S. Lv, F. Dong, and M. Takei, "Image Reconstruction Based on Convolutional Neural Network for Electrical Resistance Tomography", IEEE Sensors Journal, vol. 19, no. 1, pp.196-204, 2019, doi: 10.1109 / JSEN.2018.2876411.], 1D-CNN [X. Li, R. Lu, Q. Wang, J. Wang, X. Duan, Y. Sun, X. Li, and Y. Zhou, “One-Dimensional Convolutional Neural Network (1D-CNN) Image Reconstruction for Electrical Impedance Tomography,” Review of Scientific Instruments, vol. 91, no. 12, Art. no.124704, 2020, doi: [10.1063 / 5.0025881.], CNN-RBF [Y. Wu, B. Chen, K. Liu, C. Zhu, H. Pan, J. Jia, H. Wu, and J. Yao, “Shape Reconstruction With MultiphaseConductivity for Electrical Impedance Tomography Using Improved ConvolutionalNeural Network Method,” IEEE Sensors Journal, vol. 21, no. 7, pp. 9277-9287, 2021, doi: 10.1109 / JSEN.2021.3050845.]. All deep learning models used the same training, validation, and test sets. The input was a 208-dimensional boundary voltage difference sequence arranged in the same order. The same data normalization, sample partitioning, and testing evaluation process was used to reduce the impact of differences in data preprocessing and sample partitioning on the comparison results.

[0133] To quantitatively evaluate the difference between the conductivity distribution obtained by each reconstruction method and the true conductivity distribution, the peak signal-to-noise ratio (PSNR) was used. ), structural similarity index ( ), relative error ( ) and correlation coefficient ( The predicted node conductivity and the corresponding true node conductivity obtained by each reconstruction method were used as evaluation metrics. Using the same interpolation method, the predicted node conductivity and the corresponding true node conductivity obtained by each reconstruction method were converted into regular pixel images of the same size, and the pixel values ​​of the two images were normalized to [value missing]. .

[0134] This is used to evaluate the overall difference between the predicted image and the real image. The mean square error (MSE) is obtained by averaging the squared differences in the conductivity values ​​of corresponding pixels in the two images. Its form is:

[0135]

[0136] in, and Represent the nth image in the predicted image and the real image, respectively. The conductivity value of each pixel. Indicates pixel index, This represents the total number of pixels in a regular pixel image.

[0137] Since the pixel values ​​of both the predicted and real images are normalized to According to mean square error Calculate peak signal-to-noise ratio Its form is:

[0138]

[0139] The larger the value, the smaller the overall difference between the predicted image and the real image, and the higher the overall fidelity of the reconstructed image.

[0140] The standard expression for evaluating the similarity between a predicted image and a real image in terms of brightness, contrast, and structure is:

[0141]

[0142] in, and represent the pixel mean of the predicted image and the real image, respectively. and These represent the pixel standard deviations of the predicted image and the real image, respectively. This represents the pixel covariance between the predicted image and the real image. and This represents the stability constant used to prevent the denominator from being zero, and is taken as follows: and . The closer a value is to 1, the higher the consistency between the predicted image and the real image in terms of brightness, contrast, and structure.

[0143] The normalized deviation used to evaluate the predicted conductivity distribution relative to the true conductivity distribution is in the following form:

[0144]

[0145] in, and These represent the pixel conductivity sequences of the predicted and real images, respectively. The smaller the value, the smaller the relative numerical error between the predicted conductivity distribution and the actual conductivity distribution.

[0146] It is used to evaluate the degree of linear correlation between predicted and real images, and its form is:

[0147]

[0148] When the standard deviation of pixels in both the predicted image and the real image is not zero, The range of values ​​is . The closer the value is to 1, the higher the correlation between the predicted image and the real image in terms of conductivity variation trend and spatial distribution.

[0149] The reconstruction methods were evaluated using a test set with multiple conductivity and multiple object datasets. The performance of each reconstruction method on all test samples was calculated. , , and The average value of each evaluation index was taken to compare the overall reconstruction performance of different reconstruction methods.

[0150] The average quantization metrics of each reconstruction method on the test set of multi-conductivity and multi-object datasets are shown in Table 1. The values ​​in the table are expressed in the form of "mean ± standard deviation".

[0151] Table 1. Average quantitative indicators of test results for multi-conductivity and multi-object datasets ( , , and )

[0152]

[0153] The quantitative indicators shown in Table 1 indicate that DA-PINN... , , and DA-PINN achieved optimal results across all four metrics. The result was 28.5702±2.4566 dB, an improvement of 4.6935 dB compared to the second-best 1D-CNN; The value was 0.8845 ± 0.0355, an improvement of 0.0569 compared to 1D-CNN; The value was 0.0495±0.0156, a decrease of 0.0361 compared to 1D-CNN, representing a reduction of approximately 42.17%. The performance was 0.9787 ± 0.0111, an improvement of 0.0440 compared to 1D-CNN. Meanwhile, DA-PINN... , and The standard deviations on the samples were 0.0355, 0.0156 and 0.0111, respectively, all of which were smaller than those of other comparison methods, indicating that the evaluation index fluctuated less across all test samples and had good reconstruction stability.

[0154] Typical reconstruction results of different reconstruction methods for multi-conductivity, multi-object datasets are as follows: Figure 7 As shown, the reconstruction results are presented sequentially for the true conductivity distribution and the Conjugate Gradient Least Squares (CGLS) algorithm, the Landweber iterative algorithm, the Newton-Raphson algorithm, the Tikhonov regularization algorithm, LeNet, 1D-CNN, CNN-RBF, and DA-PINN. Classical numerical algorithms, when dealing with multiple targets and significant differences in conductivity between targets, are prone to issues such as target boundary diffusion, local artifacts, and insufficient recovery of small targets. Deep learning-based algorithms like LeNet, 1D-CNN, and CNN-RBF improve the recovery of target position and shape, but still exhibit some deviations in separating adjacent targets, preserving local edges, and quantitatively recovering conductivity. In contrast, DA-PINN provides more consistent reconstruction results for target position, number of targets, and target boundaries with the true conductivity distribution, clearly identifies targets with different conductivity levels and small targets, and exhibits fewer reconstruction artifacts in the background region. The above results show that using a dense attention encoder to extract multi-scale features and key channel information from boundary voltage data, and training the network together with control equation constraints and electrode boundary measurement consistency constraints, can help improve target recognition capability, structure preservation capability, and conductivity distribution reconstruction accuracy under multiple conductivity and multiple object conditions.

[0155] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several modifications and improvements can be made without departing from the inventive concept, and these all fall within the protection scope of the present invention.

Claims

1. A method for electrical impedance image reconstruction based on a dense attention-based physical information neural network, characterized in that, A dual-branch EIT image reconstruction model integrating dense attention mechanism and physical information constraints was established and named DA-PINN. The DA-PINN consists of an encoder based on dense attention network, a dual-branch decoder and a physical information neural network module. The encoder includes dilated convolutional blocks and multiple cascaded dense attention modules. The dual-branch decoder includes a feature conversion layer, an electric potential output branch, and an electric conductivity output branch; the physical information neural network module includes control equation constraints and boundary condition constraints. After feature extraction by the encoder, the boundary voltage measurement sequence is synchronously output by the dual-branch decoder, showing the node potential distribution and node conductivity distribution within the imaging area. Based on the node potential distribution and node conductivity distribution, a partial differential equation residual loss is constructed, and an electrode boundary measurement consistency loss is constructed based on the node potential distribution. The partial differential equation residual loss, electrode boundary measurement consistency loss, and conductivity data loss together form the total network loss function.

2. The electrical impedance image reconstruction method based on dense attention physical information neural network according to claim 1, characterized in that, In the control equation constraints, the control equations for the potential field and conductivity field in the imaging region are spatially discretized using a point electrode model and the triangular finite element method. A global finite element stiffness matrix is ​​assembled based on the predicted node conductivity, and an electrode current load vector is constructed based on the current excitation mode. The discrete residual vector of the control equation is calculated using the predicted node potential and predicted node conductivity, and then the residual loss of the partial differential equation is constructed.

3. The electrical impedance image reconstruction method based on dense attention physical information neural network according to claim 1, characterized in that, In the boundary condition constraints, the finite element mesh nodes corresponding to the outflow electrodes of each current excitation mode are set as reference nodes, and the predicted potential value of the reference node is used as a reference potential to correct the predicted potential of other nodes. The predicted voltage is calculated based on the predicted potential of each measurement electrode to the corresponding node after correction, and the predicted voltage is compared with the measured voltage to construct the electrode boundary measurement consistency loss.

4. The electrical impedance image reconstruction method based on a dense attention-based physical information neural network according to claim 1, characterized in that, The total network loss function is a weighted combination of the conductivity data loss, the partial differential equation residual loss, and the electrode boundary measurement consistency loss, wherein the weight coefficients of the partial differential equation residual loss and the electrode boundary measurement consistency loss are determined by a Bayesian optimization algorithm.

5. The electrical impedance image reconstruction method based on a dense attention-based physical information neural network according to claim 1, characterized in that, The dilated convolutional block includes an initial convolutional layer and a merging operation unit. The initial convolutional layer receives the boundary voltage measurement sequence after dimension reconstruction. The merging operation unit contains multiple parallel dilated convolutional branches, each with a different dilation rate. The output of each dilated convolutional branch is connected to a channel splicing layer, and the output of the channel splicing layer is connected to a fusion convolutional layer.

6. The electrical impedance image reconstruction method based on dense attention physical information neural network according to claim 1, characterized in that, The dense attention modules are multiple and cascaded in sequence; each dense attention module includes a dense block and an attention transition layer. The dense block contains multiple bottleneck layers connected in series, and the bottleneck layers are densely connected to achieve feature reuse; the attention transition layer is used to generate channel attention weights and use the channel attention weights to recalibrate the output features of the dense block.

7. The electrical impedance image reconstruction method based on a dense attention-based physical information neural network according to claim 6, characterized in that, The bottleneck layer includes a first convolutional layer, a second convolutional layer, a batch normalization layer, and an activation function layer. The newly generated features of each bottleneck layer are concatenated with its input features along the channel dimension and used as the input of the subsequent bottleneck layer.

8. The electrical impedance image reconstruction method based on a dense attention-based physical information neural network according to claim 1, characterized in that, The feature transformation layer includes a fully connected layer, a batch normalization layer, and an activation function layer; the potential output branch includes a first fully connected layer and a second fully connected layer; the conductivity output branch includes a first fully connected layer and a second fully connected layer.

9. The electrical impedance image reconstruction method based on a dense attention-based physical information neural network according to claim 1, characterized in that, In a series of cascaded dense attention modules, the outputs of each dense attention module except the last one are pooled and then input into the next dense attention module. The output of the last dense attention module is then global average pooled and flattened to form the encoder's global feature vector.

10. The electrical impedance image reconstruction method based on a dense attention-based physical information neural network according to claim 1, characterized in that, The feature transformation layer converts the encoder's global feature vector into shared features, and the potential output branch and the conductivity output branch output the node potential distribution and node conductivity distribution based on the shared features, respectively.

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