Method for reconstructing the distribution of electrical properties of materials using electrical impedance tomography

Abstract

Electrical Impedance Tomography Measurement Method. The invention relates to a method for reconstructing the distribution of electrical properties of at least one material in a body comprising a cylindrical portion containing a fluid, using a neural network and electrical value data of the body previously measured by electrical impedance tomography using electrodes arranged around a periphery of the cylindrical portion of the body, each electrode having been excited by a potential of predefined shape. The neural network comprises: an autoencoder comprising several levels (L1,…,LJ), each having at least one convolution layer and at least one dropout layer, and an attention module comprising attention gates (AG1,…, AGJ-1), each having an associated attention signal (gj). Figure for the abstract: Fig. 2

Description

Title of the invention: Method for reconstructing the distribution of electrical properties of materials using electrical impedance tomography. Technical field

[0001] The present invention relates to the field of electrical impedance tomography.

[0002] The invention relates more particularly to a method for reconstructing the distribution of electrical properties of materials of a body comprising a cylindrical part containing a fluid, using a neural network and data on the electrical properties of the body previously measured by electrical impedance tomography.

[0003] The invention also relates to a computer program product configured to implement this process.

[0004] The main application targeted by the present invention relates to the monitoring of fluid flows that may vary abruptly, as may be the case for fluids flowing under high pressure and high temperature.

[0005] One application of particular interest relates to the monitoring of conduits in nuclear installations, but other applications can be envisaged within the scope of the invention. Previous technique

[0006] Electrical Impedance Tomography (EIT) is a non-invasive, non-destructive technique that allows for real-time, continuous visualization of the interior of an object by measuring the electrical properties (potential and electric current) of the object at its surface. This robust approach is particularly well-suited for performing non-intrusive measurements in high-pressure and / or high-temperature environments.

[0007] Compared to MRI and CT, TIE offers a fast image rate of several kHz, with lower spatial resolution due to the non-linearity of the inverse problem.

[0008] More specifically, TIE consists of injecting electrical currents or potentials by means of a set of non-intrusive electrodes arranged on the surface of the monitored object and then measuring the electrical potentials or currents on the surface of the object.

[0009] The electrodes may only be in contact with the outer surface of the object. However, if the surface of the object is made of metal, the electrodes must pass through the wall and be in contact with the fluid.

[0010] The excitation model applied across the electrode pairs, along with the voltage measurements, results in a current-voltage mapping. The impedance map inside the object is reconstructed by solving the associated inverse problem, in order to recover the internal distribution of conductivity and permittivity, in accordance with Ohm's law.

[0011] Initially, the algorithms used to solve the nonlinear inverse problem of the TIE were primarily iterative mathematical theories, such as Tikhonov regularization (TR), linear backprojection (LBP), the Gauss-Newton method (GNM), and Newton's one-step error method (NOSER). These methods use the first-order linear approximation to conjugate the nonlinear mapping, which leads to low resolution and high sensitivity to noise during measurements, not to mention that the computation time increases exponentially with the data.

[0012] Over the past thirty years, methods using artificial neural networks have enabled major advances in the field of image generation, due to their ability to map complex nonlinear relationships with good convergence and low errors. This translates into higher spatial resolution and shorter execution time, not only for the inverse problem, but also for image denoising, super-resolution, and segmentation.

[0013] French patent application FR 3 121 234 describes an image reconstruction method using the NOSER algorithm and implementing frequency multiplexing in which the excitation signals are applied simultaneously to all electrodes. To allow signal discrimination, each electrode is excited by a trigonometric signal. Exciting all electrodes simultaneously avoids the redundancy found in data obtained by sequentially exciting paired electrodes. However, the spatial resolution obtained is sometimes insufficient to detect certain objects.

[0014] Article [1] proposes the AD ALINE network, based on an adaptive linear element. The work described in article [2] applied backpropagation networks, and that of article [3] studied Bayesian multilayer perceptrons. These studies were initiated on linear formation reconstruction operators, measuring the difference between voltage measurements and avoiding non-linearity between voltage and conductivity.

[0015] To overcome the non-linearity of the TIE, the method in article [4] uses the Radial-Basis-Functions neural network, and that in article [5] explored a neural network dense with simulated EIDORS data. Recent studies have focused more on complex architectures, as described in article [6], where a finely tuned autoencoder method is applied to EIT pulmonary monitoring, or in article [7], which proposes a multilayer autoencoder. The researchers in article [8] investigated the use of energy-based antecedents with a physics-informed neural network to improve model learning, while those in article [9] used hybrid-fusion learning for high-resolution reconstruction.

[0016] WO 2022 / 77866 describes an electrical impedance imaging method using an original set of voltage data measured on an area to be tested. An initial conductivity distribution sequence of the area forms a corresponding training data set, to which noise is added. A variational autoencoder is trained with this training data. Depending on the input voltage data set, an encoder of the variational autoencoder is trained to obtain an input voltage data characteristic, and a matching relationship is established between the input voltage data set and a corresponding conductivity distribution sequence using a decoder of the trained variational autoencoder.

[0017] With the continued progress of deep learning in TIE, better results are being explored compared to traditional algorithms.

[0018] In a known way, the inverse problem is solved without taking into account the interest of giving indications to artificial intelligence (AI) on the points to prioritize in order to improve the spatial resolution of the result.

[0019] Improving image reconstruction is nevertheless essential. Identifying the best match between the injected current or potential and the conductivity distribution can lead to low spatial resolution due to the ill-posed nature of the problem.

[0020] There is therefore a need to propose a method for processing measurements by TIE which remedies the disadvantages of the prior art, in particular to improve the spatial resolution of the images.

[0021] The object of the invention is to meet at least partially this need. Description of the invention

[0022] To this end, the invention relates, according to one of its aspects, to a method for reconstructing the distribution of electrical properties of a body comprising a cylindrical part within which a fluid flows, using a neural network and electrical value data of the body previously measured by electrical impedance tomography using electrodes arranged around a periphery of the cylindrical part of the body, each electrode having been excited by a potential of predefined shape,

[0023] the neural network comprising: - an autoencoder comprising several levels Lb... ,Lj, each having at least one convolution layer and at least one dropout layer, and - an attention module comprising attention gates AGi,..., AGj.i, each having an associated attention signal gj,

[0024] the process comprising the following steps:

[0025] i) the electrical value data are processed successively by each level Li,...,Lj of convolution and dropout layers of the autoencoder of the neural network so as to encode the data to obtain encoded data xb ..., Xj for each layer level LB...,Lj,

[0026] ii) In order to calculate the coefficient aj of an attention gate AGj, the latter is configured to add the encoded data xb..., x; to the attention signal gj of the corresponding attention gate AGj, a function at least of the concatenation of the output data xj+i of the higher-rank attention gate AGj+i and the decoded data from the higher-rank level Lj+2 to that of the higher-rank attention gate AGj+i, the result of this addition being transformed by at least one mathematical function so as to obtain the coefficient

[0027] iii) the output data Xj from a layer level Lj of the autoencoder are multiplied by the coefficient Oj of the corresponding attention gate AGj to obtain output data Xj from the first attention gate AGj,

[0028] iv) the previous step is repeated up to a first attention gate AGi in order to obtain the output data Xj from the first attention gate AGh

[0029] v) said output data Xj from the first attention gate AGi are concatenated with the decoded data from the second level L2 in order to obtain a data matrix representative of an image of the distribution of electrical properties of at least one material of the body.

[0030] Thanks to the invention, it is possible to reconstruct the distribution of the electrical properties of materials, in particular conductivity and permittivity, with better spatial resolution.

[0031] The method according to the invention uses an end-to-end neural network, which may be called an Autoencoder Improved Attention-Net (AIA-Net). The first segment is an autoencoder used as a remodeling feature extraction architecture for the voltage input, and the second segment is an enhanced attention module that solves the inverse problem and provides the distribution image of the electrical properties of the materials under test.

[0032] Since the excitation potential of the electrodes has a known predefined shape, it is possible to reconstruct the electrical conductivity and / or the electrical permittivity. Thanks to the attention gates of the invention, which have knowledge of the correspondence between the measured electrical values ​​and the electrical properties of the materials, the spatial resolution of the output image is very satisfactory. Auto-encoder

[0033] Dropout layers are used, in a known manner, to activate or deactivate neurons. This technique reduces overfitting during model training. Certain neurons can be temporarily deactivated in the network, along with all corresponding incoming and outgoing connections. The choice of neurons to be deactivated is advantageously random.

[0034] In a preferred embodiment, maximum value sampling is implemented from one layer level Li,...,Lj to the next in order to compress the electrical value data. This process is known as "max pooling." The max-pooling process is performed by passing a window, or filter, of predetermined dimensions over the entire input. The window moves across the image and selects the maximum value from each portion of the image, called the pooling window or region. The window slides across the input, and for each window position, the largest value is selected and placed in the output matrix.

[0035] Jump connections are advantageously made between the layer levels Li,...,Lj to obtain the encoded data Xi,..., x; at the input of the attention gates. This technique, as is known, allows convolutional neural networks to bypass certain layers and connect directly to deeper or shallower layers.

[0036] Preferably, in step ii), the encoded data (xb..., Xj) are, according to their spatial dimension, convolved or transposed to adapt to the spatial dimension of the attention signal considered. Attention module

[0037] The attention gates of the attention module in the decoding section are advantageously used as filters for cross-information between the encoding section, by transporting features through the jump connections, and the decoding section. This cross-information can be improved by adding to the upper layers the features extracted from each layer in the decoding section.

[0038] Known electrical impedance tomography methods use artificial intelligence by focusing on the non-linear correspondence between voltage and conductivity, without giving the neural network any instructions on where to look.

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[0052] The attention mechanism was introduced in article

[10] , where it is formulated as follows for so-called soft attention: Preferably, the coefficient aj of the corresponding attention gate AGj is calculated using the following equations: th] + w Ig LJ + kJ) Where . q1 is the attention quotient used to determine the attention coefficients Oj, °i and ct2 are respectively rectified linear unit activation functions, denoted ReLU, and sigmoid, wJ, W] and ipT are linear transformations intended to reduce the consumption of parameters and computing resources, combined with the bias terms 1¾. bxy, and bg.. to form a set of parameters 0att. The attention module makes it possible to highlight regions important for feature extraction when running convolution layers, and to remove irrelevant regions from the image. Preferably, the encoded data addition operation is performed on all input data with each attention signal gj, primarily to increase aligned weights and decrease non-aligned weights, thereby reducing spatial information and highlighting important features. The features extracted from the input are advantageously selected thanks to the information contained in the attention signals gj. The input features xb..., Xj are advantageously scaled with respect to the attention coefficients Oj, in order to identify the significant areas of the image, after passing through a trilinear interpolation wJ, wJ and tpT. In a preferred embodiment, the rp transformation produces a weight matrix which is given to the sigmoid activation function in order to limit the values ​​of the attention coefficients (ai,l' ^id) e [0, 1]. A resampler can be used to expand the dimensions of the weight matrix. At the output of each attention gate, a concatenation operation can be performed between the attention signal gj and the output data x, given that they are of different dimensions.

[0053] Multiplication by elements between the input features and the attention coefficients advantageously leads to the output data (Xj ii2, x^, ^4). Preliminary measures

[0054] In a preferred embodiment, the electrical impedance tomographic measurement of the body, comprising a cylindrical part containing a fluid, in order to obtain the previously measured electrical values ​​of the body, comprises the following steps:

[0055] i. arrangement of a number ne of electrodes around a periphery of the cylindrical part of the body,

[0056] ii. simultaneous excitation of each of the neelectrodes, each electrode being excited by a potential Vnexc having the form:

[0057] „ r - / « \ i ( t ) = ÆJ^COS ( 2 / rfmt ) [ ô^cos ( m6„ ) + ô^sin ( ) j

[0058] where A is a signal amplitude, 0n is the angular position of the electrode En, fm = m * f0 is an oscillation frequency, f0 is a fundamental frequency chosen such that fm is less than the Nyquist frequency of the system for all m,

[0059] iii. measurement of the electrical properties of the body using electrodes,

[0060] iv. processing of data from measurement step iii, comprising the sub-steps following: a) For each electrode En, calculate the data points Mn defined by:

[0061] [Math.6] mm =

[0062] where R is the value of the resistance used for the measurement of Vnmeas with Vnmeas = RL across the resistor, P is the number of points in a discrete sequence of measurement of the current In, p is the discrete time, k is a Fourier coefficient between 1 and (ne - 1) and [3P = (2irp / P). Use of the process

[0063] The invention also relates to the use of the method just described for the reconstruction of the distribution of electrical properties of a nuclear installation conduit, in particular a two-phase flow passing through the conduit. computer program product

[0064] The invention also relates, in another of its aspects, to a computer program product comprising a medium and, recorded on this medium, instructions readable by a processor so that, when executed, they allow the reconstruction process according to the invention to be implemented. Device

[0065] The invention also relates to a device for implementing the method according to the invention, comprising: • an acquisition system comprising at least one programmable logic network, an analog signal generation module and an analog signal measurement module, with a plurality of electrodes being connected to the acquisition system, • a computer configured to control the acquisition system, • a neural network integrated into or connected to the computer, the neural network comprising: - an autoencoder comprising several levels Lb... ,Lj, each having at least one convolution layer and at least one dropout layer, and - an attention module comprising attention gates AGi,..., AGj.i, each having an associated attention signal gj.

[0066] The characteristics stated above for the method apply to the use, the computer program product and the device, and vice versa. Brief description of the drawings

[0067] [Fig-1] Fig. 1 represents a diagram illustrating an example of implementation steps work of the process according to the invention,

[0068] [Fig.2] Fig.2 illustrates an example of architecture for implementing the method according to the invention.

[0069] [Fig.3a] [[Fig.3b]) Figures 3a and 3b show in detail an example of attention gates used in the invention,

[0070] [Fig. 4] Figure 4 illustrates comparative results between the process according to the invention and the prior art, and

[0071] [Fig.5] Fig.5 represents a curve illustrating the mean squared error obtained when using the method according to the invention. Detailed description of an example

[0072] A diagram representing the steps of an example of implementation of the process according to the invention is illustrated in [Fig.1].

[0073] In a first step, experimental measurements by electrical impedance tomography are performed on a body comprising a cylindrical portion through which a fluid flows, using electrodes arranged around a periphery of the cylindrical portion of the body, each electrode having been excited by a predefined shape potential. The electrodes are arranged non-intrusively on a circular periphery of the body, as shown in [Fig. 2]. In this example, the electrodes are angularly and regularly distributed around the periphery of the body.

[0074] The electrodes are connected to a printed circuit board, which is itself connected to a data acquisition system. A screen displays the data and the images produced from this data. The data acquisition system contains the Linux operating system (HOTE) which controls a programmable FPGA logic array, also contained within the data acquisition system.

[0075] The acquisition system makes it possible to generate the analog excitation signals and to measure the analog measurement signals from the electrodes.

[0076] Electrical value data Mn(k) are thus obtained.

[0077] In a second step, this data is processed by the autoencoder of the neural network according to the invention in order to reshape it to obtain encoded data xi, ..., Xj for each layer level Lb, ..., Lj of the autoencoder, as shown in [Fig. 2]. Each level comprises at least one convolution layer and at least one dropout layer. The data is processed layer by layer and concatenated together.

[0078] As seen in [Fig.2], a maximum value sampling, called "max-pooling", is implemented from one layer level Li,...,Lj to another in order to compress the electrical value data.

[0079] In a third step, the attention module according to the invention is used to solve the inverse problem. This attention module comprises attention gates AGi,..., AGj.i, each having an associated attention signal gj.

[0080] As shown in Figures 3a and 3b, in order to calculate the coefficient aj of an attention gate AGj, the latter is configured to add the encoded data Xi,..., Xj to the attention signal gj of the corresponding attention gate AGj, a function at least of the concatenation of the output data xj+1 of the higher-rank attention gate AGj+i and the decoded data from the higher-rank level Lj+2 to that of the higher-rank attention gate AGj+i. The result of this addition is transformed by at least one mathematical function so as to obtain the coefficient a,.

[0081] Preferably, and in the example considered, the coefficient Oj of the corresponding attention gate AGj is calculated using the following equations:

[0082] qU=iWM+) + Wggÿ+k) ) + N

[0083]

[0084] where qlatt is the attention quotient used to determine the attention coefficients aj, CTi and a2 are respectively rectified linear unit activation functions, denoted ReLU, and sigmoid, wJ, Wg and T1 are linear transformations intended to reduce the consumption of parameters and computing resources, combined with the bias terms b^, bXiJ, and bg;, to form a set of parameters 0att.

[0085] The rp transformation produces a weight matrix which is given to the sigmoid activation function in order to limit the values ​​of the attention coefficients, a resampler is used to extend the dimensions of the weight matrix.

[0086] In the illustrated example, as seen in [Fig.2], jump connections are made between the layer levels Li,...,Lj to obtain the encoded data xb ..., Xj at the input of the attention gates.

[0087] During the step of calculating the attention coefficients, the encoded data xb..., Xj are, according to their spatial dimension, convolved or transposed to adapt to the spatial dimension of the attention signal considered, as shown in figures 3a and 3b.

[0088] For example, in the attention gate AGb, the spatial dimension of Xi is twice that of the attention signal gb. To be able to add them, it is therefore necessary to divide the dimension of xi by 2 (2D convolution with stride = 2). As for the other input signals Xj, their spatial dimension is less than that of the attention signal gb, except for x2, which has the same dimension. It is therefore necessary to perform an inverse convolution with different steps (stride = 1 for x2, 2 for x3, and 4 for x4) in order to add the signals with gb.

[0089] The same principle continues for AG2, AG3, AG4.

[0090] In this figure, F, H, W, D, Fint, are respectively the features, height, width, inlet channel or depth, in the case of 3D data, and intermediate features.

[0091] The output data x3 from a layer level Lj of the autoencoder are multiplied by the coefficient Oj of the corresponding attention gate AGj, calculated as shown, in order to obtain output data Xj from the first attention gate AGj, as illustrated in Figures 3a and 3b. This step is repeated up to a first attention gate AGi in order to obtain the output data Xj from the first attention gate AGb

[0092] As shown in Figure 2, the output data Xj from the first attention gate (AGi) are concatenated with the decoded data from the second level (L2) in order to obtain, in the last step of [Fig. 1], a data matrix representing an image of the distribution of electrical properties of at least one material of the body. In this example, the image is a 32 x 32 matrix.

[0093] Figure 4 presents some of the results obtained through the implementation of the process according to the invention and compared to other known architectures: 2D-CNN, Variational-Auto-Encoder (VAE), Generative DNN with U-Net, Dense ResNet, Conv ResNet, U-Net3++, Attention-Net.

[0094] The images in [Fig. 4] correspond to tests carried out on a body having a cylindrical part which may include, from left to right:

[0095] - two rods with a circular cross-section (1 cm and 2 cm in diameter),

[0096] - two sticks with a circular cross-section (1 cm in diameter),

[0097] - a stick with a circular cross-section (2 cm in diameter),

[0098] - a stick with a rectangular cross-section (2.5 x 1 cm surface area), and

[0099] - two sticks with a circular cross-section (1 cm and 2 cm in diameter).

[0100] It should be noted that, for the sake of brevity, only the results of certain tests have been illustrated, but the conclusions are the same for all tests carried out.

[0101] We can see that prior to U-Net3++, known architectures were not sufficiently optimized for reconstructing conductivity, mainly due to the leakage gradient problem, since there are no skipped connections. The exception would be the DNN+U-Net architecture in which U-Net includes these skipped connections, but feature extraction is initiated with a deep neural network, which is not optimized for image data compared to a convolutional neural network. For U-Net3++, looking at the second image from left to right, the network lacked spatial feature identification for accurate reconstruction. This was improved with the classical Attention-Net, but this neural network had problems with the conductivity of the water in the background, as can be seen in Figure 5 of [Fig. 4].The network used in the invention shows very good reconstruction results with better identification of spatial edges.

[0102] Table 1 shows that the measurements give better results with the invention. Even though R2 has a high value for the proposed architecture, it can lead to misinterpretations. The RMSE and SER values ​​are quite low, which is a sign of good regression, and are within the range of the MSE loss function, as illustrated in [Fig. 5].

[0103] [Tables] Metrics R2 RMSE SER Architectures (%) (qS / cm) (qS / cm) 2D-CNN 68.3 0.3001 0.3240 VAE 70.9 0.2769 0.2831 DNN + U-Net 76.3 0.2547 0.2342 Dense ResNet 80.4 0.2010 0.1912 Conv ResNet 85.9 0.1832 0.1447 U-Net3++ 91.7 0.1521 0.1272 Attention-Net 92.3 0.1308 0.1187 AIA-Net 98.5 0.1012 0.1009

[0104] Figure 5 shows that the mean squared error loss function is very small, indicating a good match between the voltage input data and the conductivity output data. The validation loss exhibits small variations, indicating that the implemented neural network model is stable. Furthermore, the absence of overfitting or underfitting is explained by the close values ​​between the two losses.

[0105] The invention is not limited to the examples just described; in particular, features of the illustrated examples can be combined in unillustrated variants.

[0106] Other variations and improvements may be envisaged without departing from the scope of the invention. In particular, the method according to the invention may be implemented using a neural network different from that described, including additional layers.

[0107] The invention can be used in the nuclear field, particularly as a method for preventing the presence of bubbles in the primary circuit and cavitation in the pumps of the various circuits. The invention can be applied to the medical field, particularly for lung monitoring and the detection of anomalies, notably by implementing simultaneous excitation.

[0108] The method according to the invention can find application in the agri-food sector, in particular for counting fruits and vegetables, in the petroleum sector, in particular for detecting fouling in production wells, or in the pharmaceutical sector, for example for detecting foreign bodies during the processing of medicines. List of cited references

[0109] R. Guardo & al., “A Neural Network Approach To Image Reconstruction In Electrical Impedance Tomography”, IEEE, 1991

[0110] A. Nejatali & al., "An itérative algorithm for electrical impédance imaging using neural networks", IEEE, 1998

[0111] J. Lampinen & al., "Application of Bayesian neural network in electrical impédance tomography", IEEE, 1999

[0112] Chao Wang & al., "RBF neural network image reconstruction for electrical impédance tomography", IEEE, 2004

[0113] Xiuyan Li & al., "An image reconstruction framework based on deep neural network for electrical impédance tomography", IEEE, 2017

[0114] Jin K. S. & al., "A Learning-Based Method for Solving Ill-Posed Nonlinear Inverse Problems: ASimulation Study of Lung EIT", SIAM J. Imaging sciences, 2019

[0115] Xiaoyan C. & al., "Deep Autoencoder Imaging Method for Electrical Impédance Tomography", IEEE, 2021

[0116] Akarsh P. & al., "Improved Training of Physics-Informed Neural Networks Using Energy-Based Priors: a Study on Electrical Impédance Tomography", ICLR, 2023

[0117] Hao Y. & al., "High-resolution conductivity reconstruction by electrical impédance tomography using structure-aware hybrid-fusion learning", Sciencedirect, 2024

[0118] Ozan Oktay & al., “AttentionU-Net : Learning Where to look for the Pancréas,” lst Conférence on MIDL, 2018

Claims

1. Demands A method for reconstructing the distribution of electrical properties of a body comprising a cylindrical portion through which a fluid flows, using a neural network and electrical value data of the body previously measured by electrical impedance tomography using electrodes arranged around the periphery of the cylindrical portion of the body, each electrode having been excited by a predefined shape potential, the neural network comprising: - an autoencoder comprising several levels (Lb... ,Lj) each having at least one convolution layer and at least one dropout layer, and - an attention module comprising attention gates (AGi,..., AGj i), each having an associated attention signal (gj), The process involves the following steps: i) the electrical value data are processed successively by each level (Lb.. .,Lj) of convolution and dropout layers of the neural network autoencoder so as to encode the data to obtain encoded data (xb..., Xj) for each level of layers (Lb.. .,Lj), ii) in order to calculate the coefficient (¾) of an attention gate (AGj), the latter is configured to add the encoded data (xb..., Xj) to the attention signal (gj) of the corresponding attention gate (AGj), a function at least of the concatenation of the output data ^j+i) of the higher-rank attention gate (AGj+i) and the decoded data from the higher-rank level (Lj+2) to that of the higher-rank attention gate (AGj+i), the result of this addition being transformed by at least one mathematical function so as to obtain the coefficient (oij), iii) the output data (xj) of a layer level (Lj) of the autoencoder are multiplied by the coefficient (oij) of the corresponding attention gate (AGj) to obtain output data j^j) of the first attention gate (AGj), iv) the previous step is renewed up to a first attention gate (AGi) in order to obtain the output data (xj of the first attention gate (AGi), v) said output data (xj of the first attention gate (AGi) are concatenated with the decoded data from the second level (L2) in order to obtain a data matrix representative of an image of the distribution of electrical properties of at least one material of the body.

2. A method according to claim 1, wherein maximum value sampling is implemented from one layer level (Lb.. .,Lj) to another in order to compress the electrical value data.

3. A method according to any one of the preceding claims, wherein jump connections are made between layer levels (Lb...,Lj) to obtain the encoded data (xb..., Xj) at the input of the attention gates.

4. A method according to any one of the preceding claims, wherein, in step ii), the encoded data (xb..., Xj) are, according to their spatial dimension, convolved or transposed to adapt to the spatial dimension of the attention signal considered.

5. A method according to any one of the preceding claims, wherein the coefficient (aj) of the corresponding attention gate (AGj) is calculated using the following equations: t= *T(°i( + W88y + M )+b» where q1 is the attention quotient used to determine the attention coefficients aj, ai and are respectively rectified linear unit activation functions, denoted ReLU, and sigmoid, WL WÏ and tpT are linear transformations intended to reduce the consumption of parameters and computing resources, combined with the bias terms 1¼, Kü, and bg, to form a set of parameters 0att.

6. A method according to the preceding claim, wherein the transformation (rp) produces a weight matrix which is given to the activation sigmoid function in order to limit the values ​​of attention coefficients, a resampler is used to extend the dimensions of the weight matrix.

7. Use of the method according to any one of the preceding claims for the reconstruction of the distribution of electrical properties of a nuclear installation conduit, in particular a two-phase flow passing through the conduit.

8. Product computer program comprising a medium and, stored on this medium, instructions readable by a processor so that, when executed, they enable the implementation of the reconstruction process according to any one of claims 1 to 6.

9. Device for implementing the method according to any one of claims 1 to 6, comprising: • an acquisition system including at least one programmable logic network, an analog signal generation module and an analog signal measurement module, a plurality of electrodes being in particular connected to the acquisition system, • a computer configured to control the acquisition system, • a neural network integrated into or connected to the computer, the neural network including: - an autoencoder comprising several levels (Lb... ,Lj) each comprising at least one convolution layer and at least one dropout layer, and - an attention module comprising attention gates (AGi,..., AGj.i), each having an associated attention signal (gj).

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