A method of electrical property tomography based on physical neural networks

Abstract

The application discloses a physical neural network-based electrical property tomography method, which spatially obtains permittivity and conductivity of all points through four steps.The application does not need to simplify the core equation of MR-EPT to solve, and does not need to use a numerical method to solve by discretizing the derivative of the dielectric property in space, thereby avoiding the discretization error, so that the application has higher precision in obtaining the permittivity and the conductivity.The physical neural network-based electrical property tomography method uses a neural network to construct a mapping relationship between a transmit field and its spatial derivative and electrical properties, so that the precision of the obtained result is higher.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of nuclear magnetic resonance imaging, and particularly relates to an electrical property tomography method based on a physical neural network. BACKGROUND

[0002] With the increase of the main field strength, the frequency of the radio frequency field of the high-field nuclear magnetic resonance imaging system becomes higher, and the wavelength becomes shorter to the same level as the scanned object. This can cause the interaction between the radio frequency field and the scanned object to be enhanced. Thus, new imaging functions can be realized by utilizing the interaction. Dielectric property tomography (MR-EPT) based on high-field magnetic resonance is a new quantitative nuclear magnetic resonance imaging method.

[0003] The dielectric property of biological tissue is the response characteristic of biological tissue as a substance to an electromagnetic field. The dielectric property can not only be used as a tissue contrast to display the structure image of different tissues in the human body, but also be used as a quantitative index for early cancer detection. In addition, the dielectric property can affect the propagation of electrical signals in the brain and be highly related to the type, activity, function, aging, and pathological changes of brain tissue. Therefore, MR-EPT is expected to provide a new research tool for the study of brain cognition and brain diseases. The distribution of the dielectric property in the human body is also the basis for calculating the specific absorption rate (SAR) for evaluating the radio frequency safety of any exposure to radio frequency / microwave. A big safety problem faced by the ultra-high-field magnetic resonance imaging system is the thermal effect of biological tissue in a strong magnetic field. Accurate imaging of the dielectric property of human tissue can help the study of SAR and promote the further development of the ultra-high-field magnetic resonance system.

[0004] MR-EPT generally includes two steps. The first step is the acquisition of the radio frequency field amplitude and phase. The second step is to obtain the core equation of MR-EPT based on Maxwell's equation, then construct a mathematical model of the inverse problem, and use the acquired radio frequency field data as the input of the inverse problem to reconstruct the distribution of the electrical property. The second step is the core of the electrical property imaging algorithm, especially how to reconstruct the core equation of MR-EPT to form a mathematical model, and accurately reconstruct the distribution of the electrical property according to the constructed mathematical model.

[0005] At present, the mainstream algorithm for electrical property reconstruction is to simplify the core equation of MR-EPT by using some assumptions to form a simple solving equation. One assumption is that the electrical property distribution inside the measured object is uniform, and this method is called Helmholtz EPT. Although this assumption can greatly simplify the solving model and quickly estimate the electrical property distribution, using this uniformity assumption has obvious disadvantages, because the electrical properties inside the measured object generally have a complex distribution, not uniform. This will make the uniformity assumption invalid, resulting in serious calculation errors, especially in the edge part of the tissue. At the same time, the Helmholtz equation itself is a high-order nonlinear nonhomogeneous partial differential equation, and the radio frequency field and electrical properties are both in the form of complex numbers in the equation. Therefore, without using assumptions to simplify, it is difficult to directly solve the equation.

[0006] Therefore, in view of the deficiencies of the prior art, it is very necessary to provide a nano-fiber surface porous bone repair scaffold and a preparation method thereof to solve the deficiencies of the prior art. SUMMARY

[0007] The purpose of the present application is to avoid the deficiencies of the prior art and provide an electrical property tomography method based on a physical neural network. The electrical property tomography method based on the physical neural network uses a neural network to construct a mapping relationship from the emission field and its spatial derivative to the electrical property, and the accuracy of the result is higher.

[0008] The above-mentioned purpose of the present application is realized by the following technical measures:

[0009] The present application provides an electrical property tomography method based on a physical neural network, comprising the following steps:

[0010] Step (1), obtaining an emission field B from a radio frequency field B;

[0011] Step (2), using a numerical method to solve the first-order derivative and the second-order derivative of each point in the emission field B obtained in step (1) on the x-axis, y-axis and z-axis in space;

[0012] Step (3), constructing a fully connected neural network and combining and transforming the Maxwell equation to obtain a loss function;

[0013] Step (4), using the data set consisting of the transmit field B, the first derivative and the second derivative obtained in step (2) and the loss function obtained in step (3) to train the full connection neural network obtained in step (3), to obtain a trained full connection neural network, the trained full connection neural network takes the transmit field B at any point in space and the first derivative at the point obtained in step (2) as input, and the output of the trained full connection neural network is the permittivity and conductivity at the point, thereby obtaining the permittivity and conductivity at all points in space.

[0014] In the step (1), the radio frequency field B is generated by a radio frequency coil, and B = (B x ,B y ,B z ), wherein B x is the x-axis component of the radio frequency field in space, B y is the y-axis component of the radio frequency field in space, and B z is the z-axis component of the radio frequency field in space, the transmit field B is obtained from the radio frequency field B, and there is

[0015] In the step (3), the method for obtaining the loss function is specifically:

[0016] Step (3.1), combining and transforming the Maxwell equations to obtain formula (1)

[0017]

[0018] wherein ε c = ωε-iσ is the complex permittivity, B is the imaginary unit, ε is the permittivity, σ is the conductivity, ω is the Larmor precession frequency corresponding to the nuclear magnetic resonance device, μ is the magnetic permeability, is the Nabla operator, is the Laplace operator;

[0019] Step (3.2), setting B z to 0 and separating and recombining formula (1) of step (3.1) to obtain the core equation (2) of MR-EPT;

[0020]

[0021] wherein B x is the first derivative of the transmit field B in the x-axis in space, B y is the first derivative of the transmit field B in the y-axis in space, B z is the first derivative of the transmit field B in the z-axis in space, ε c,x is the first derivative of the complex permittivity in the x-axis in space, and ε c,yis the first derivative of the complex permittivity on the y-axis in space, and PDE is the symbol for a partial differential equation.

[0022] Step (3.3): Obtain the loss function according to equation (2).

[0023] Preferably, the loss function described above is represented by equation (3).

[0024]

[0025] in As expressed by equation (4),

[0026]

[0027] in Equation (5) represents that

[0028]

[0029] Where α is the equilibrium and The weights are given and α>0. The corr function is the cross-correlation function for calculating the correlation between the permittivity gradient and the conductivity gradient, and is expressed by equation (6).

[0030]

[0031] in Let X be the mean. Let Y be the mean of Y, N be the number of points to be calculated, and k be the kth point.

[0032] Preferably, step (2) above specifically involves using numerical methods to solve for the first derivative B of each point in the launch field B in the three spatial directions. x First derivative B y First derivative B z Second derivative B xx Second derivative B xy Second derivative B xz Second derivative B yz Second derivative B yy and second derivative B zz ;

[0033] First derivative B x As expressed by equation (7),

[0034]

[0035] First derivative B y Equation (8) represents that,

[0036]

[0037] First derivative B z As expressed by equation (9),

[0038]

[0039] Second derivative B xx Represented by equation (10);

[0040]

[0041] Second derivative B xy As expressed by equation (11),

[0042]

[0043] Second derivative B xz As expressed by equation (12),

[0044]

[0045] Second derivative B yz It can be expressed by equation (13),

[0046]

[0047] Second derivative B yy As expressed by equation (14),

[0048]

[0049] Second derivative B zz Represented by equation (15);

[0050]

[0051] Where (x) k ,y k ,z k Let be the coordinates of the k-th point, and dx, dy, and dz be the discretized grids in the three directions of space, respectively.

[0052] Preferably, the above fully connected neural network consists of two parts: network A and network B.

[0053] Preferably, the above network A constructs f ε The function, wherein network B constructs f σ The function takes as input a transmission field B at any point in space and a first derivative B at that point as input to both network A and network B. x First derivative B y and the first derivative B z The outputs are the permittivity and conductivity at that point, respectively.

[0054] Preferably, the activation functions of the input layer of network A, the intermediate layer of network A, the input layer of network B, and the intermediate layer of the input layer of network B are all tanh functions.

[0055] And the tanh function is represented by equation (16),

[0056]

[0057] Preferably, the activation functions of the output layer of network A and the output layer of network B are both sigmoid functions, and the sigmoid function is represented by equation (17).

[0058]

[0059] The tanh function maps the input to the range (-1, 1), while the sigmoid function maps the input to the range (0, 1).

[0060] When the inputs to network A and network B are both the emission field B at any point in space and the first derivative B at that point... x First derivative B y and the first derivative B z When the permittivity at that point is expressed by equation (18), the conductivity at that point is expressed by equation (19);

[0061] ε=(81*O a +1)E c Equation (18);

[0062] σ=3*O b Equation (19);

[0063] Where ε is conductivity, σ is permittivity, and O a For the output of network A, O b This is the output of network B.

[0064] Preferably, the mapping of the fully connected neural network described above is represented by equations (20) and (21).

[0065] ε(x k ,y k ,z k )

[0066] =f ε (B(x k ,y k ,x k ),B x (x k ,y k ,z k ),B y(x k ,y k ,z k ),B z (x k ,y k ,x k );w ε )

[0067] Equation (20); σ(x) k ,y k ,z k )

[0068] =f σ (B(x k ,y k ,z k ),B x (x k ,y k ,z k ),B y (x k ,y k ,z k ),B z (x k ,y k ,z k );w σ )

[0069] Equation (21);

[0070] Where w ε and w σ All are parameters of the neural network, ε(x) k ,y k ,z k ) is a coordinate in space (x k ,y k ,z k The permittivity value on ) σ(x k ,y k ,z k ) is a coordinate in space (x k ,y k ,z k The conductivity value on ).

[0071] Preferably, step (4) above specifically includes:

[0072] Step (4.1): Using launch site B, the dataset consisting of the first and second derivatives obtained in step (2) is input into the fully connected neural network obtained in step (3) to obtain the output value of the fully connected neural network. The output value of the fully connected neural network is then input into the loss function obtained in step (3). Let n = 1 and perform forward and backward operations to obtain the current average value of the dataset. average value Update the fully connected neural network and proceed to step (4.2);

[0073] Step (4.2): Let n = n + 1, then proceed to step (4.3);

[0074] Step (4.3): Input the dataset consisting of the first and second derivatives obtained in step (2) into the updated fully connected neural network using the launch site B, and obtain the output value of the fully connected neural network. Proceed to step (4.4).

[0075] Step (4.4): Input the output value of the fully connected neural network into the loss function obtained in step (3), and then perform forward and backward operations to obtain the average value of the dataset. judge The relationship with the iteration threshold, when If the result is greater than the iteration threshold, proceed to step (4.5); otherwise, proceed to step (4.6).

[0076] Step (4.5): Calculate the average value obtained in step (4.4). Update the fully connected neural network to obtain the updated fully connected neural network and return to step (4.2);

[0077] Step (4.6): Define the current updated fully connected neural network as a trained fully connected neural network.

[0078] Preferably, the numerical method described above is the finite difference method;

[0079] Preferably, the above iteration threshold is 0.001.

[0080] Preferably, both network A and network B consist of five fully connected layers, with 256 neurons in each layer.

[0081] Preferably, the fully connected neural network described above is written in Python 3.6 and built using the TensorFlow deep learning library.

[0082] A physical neural network-based electrical characteristic tomography method includes the following steps: Step (1) Obtain the transmission field B based on the radio frequency field B; Step (2) Solve the first and second derivatives of each point in the transmission field B obtained in step (1) in three spatial directions using numerical methods; Step (3) Construct a fully connected neural network and perform a combination transformation on Maxwell's equations to obtain a loss function; Step (4) Train the fully connected neural network obtained in step (3) using the dataset composed of the transmission field B, the first and second derivatives obtained in step (2), and the loss function obtained in step (3) to obtain a trained fully connected neural network. The trained fully connected neural network takes the transmission field B at any point in space and the first derivative obtained at that point through step (2) as input, and the output of the trained fully connected neural network is the permittivity and conductivity of that point, thereby obtaining the permittivity and conductivity of all points in space. This invention does not require simplification of the core equations of MR-EPT for solving, and it avoids discretization and numerical methods when calculating the spatial derivative of dielectric properties, thus avoiding discretization errors. Therefore, this invention yields higher accuracy in permittivity and conductivity. This physical neural network-based electrical property tomography method uses a neural network to construct a mapping relationship between the emission field and its spatial derivative to electrical properties, resulting in even higher accuracy. Attached Figure Description

[0083] The invention will be further described with reference to the accompanying drawings, but the contents of the drawings do not constitute any limitation on the invention.

[0084] Figure 1 This is a flowchart of the electrical characteristic tomography method based on physical neural networks of the present invention.

[0085] Figure 2 This is a diagram of a fully connected neural network structure.

[0086] Figure 3 This is the result of reconstructing the capacitance image on the Duke human brain tumor simulation model using the present invention and the Helmholtz EPT method.

[0087] Figure 4 The results of this invention and the Helmholtz EPT method on the reconstructed conductivity image of the Duke human brain tumor simulation model are presented. Detailed Implementation

[0088] The technical solution of the present invention will be further described in conjunction with the following embodiments.

[0089] Example 1.

[0090] An electrical property tomography method based on physical neural networks, such as Figure 1 As shown, it includes the following steps:

[0091] Step (1): Obtain the transmission field B based on the radio frequency field B;

[0092] Step (2): Use numerical methods to solve for the first and second derivatives of each point in the launch field B obtained in step (1) on the x-axis, y-axis and z-axis in space;

[0093] Step (3): Construct a fully connected neural network and perform a combination transformation on Maxwell's equations to obtain the loss function;

[0094] Step (4): Using the dataset consisting of the emission field B, the first derivative and the second derivative obtained in step (2), and the loss function obtained in step (3), train the fully connected neural network obtained in step (3) to obtain the trained fully connected neural network. The trained fully connected neural network takes the emission field B at any point in space and the first derivative obtained at that point through step (2) as input. The output of the trained fully connected neural network is the permittivity and conductivity at that point, thereby obtaining the permittivity and conductivity of all points in space.

[0095] In step (1), the radio frequency field B is generated by the radio frequency coil, and B = (B x B y B z ), where B x B represents the x-axis component of the radio frequency field in space. y B represents the y-axis component of the radio frequency field in space. z Let B be the z-axis component of the radio frequency field in space. Based on the radio frequency field B, the transmitted field B is obtained, and thus exists...

[0096] Step (2) of this invention specifically involves using numerical methods to solve for the first derivative B at each point in the launch field B. x First derivative B y First derivative B z Second derivative B xx Second derivative B xy Second derivative B xz Second derivative B yz Second derivative B yy and second derivative B zz ;

[0097] First derivative B x As expressed by equation (7),

[0098]

[0099] First derivative B y Equation (8) represents that,

[0100]

[0101] First derivative B z As expressed by equation (9),

[0102]

[0103] Second derivative B xx Represented by equation (10);

[0104]

[0105] Second derivative B xy As expressed by equation (11),

[0106]

[0107] Second derivative B xz As expressed by equation (12),

[0108]

[0109] Second derivative B yz It can be expressed by equation (13),

[0110]

[0111] Second derivative B yy As expressed by equation (14),

[0112]

[0113] Second derivative B zz Represented by equation (15);

[0114]

[0115] Where (x) k ,y k ,z k Let be the coordinates of the k-th point, and dx, dy, and dz be the discretized grids in the three directions of space, respectively.

[0116] In step (3) of this invention, the method for obtaining the loss function specifically includes the following steps:

[0117] Step (3.1): Perform a combined transformation on Maxwell's equations to obtain equation (1).

[0118]

[0119] Where ε c =ωε-iσ is the complex permittivity, i is the imaginary unit, ε is the permittivity, σ is the conductivity, ω is the Larmor precession frequency corresponding to the nuclear magnetic resonance equipment, and μ is the permeability. For the Nabla operator, For the Laplace operator;

[0120] Step (3.2), let B z Set the value to 0 and separate and recombine equation (1) from step (3.1) to obtain the core equation (2) of MR-EPT;

[0121]

[0122] Among them B x Let B be the first derivative of the launch site B along the x-axis in space. y Let B be the first derivative of the y-axis in space within launch site B. z Let ε be the first derivative of the launch field B along the z-axis in space. c,x Let ε be the first derivative of the complex permittivity on the x-axis in space. c,y is the first derivative of the complex permittivity on the y-axis in space, and PDE is the symbol for a partial differential equation.

[0123] Step (3.3): Obtain the loss function according to equation (2).

[0124] The loss function is represented by equation (3).

[0125]

[0126] in As expressed by equation (4),

[0127]

[0128] in Equation (5) represents that

[0129]

[0130] Where α is the equilibrium and The weights are given and α>0. The corr function is the cross-correlation function for calculating the correlation between the permittivity gradient and the conductivity gradient, and is expressed by equation (6).

[0131]

[0132] in Let X be the mean. Let Y be the mean of Y, N be the number of points to be calculated, and k be the kth point.

[0133] It should be noted that, because when using birdcage coils, then B z The amplitude is much smaller than the components on the x-axis and y-axis (B). x and By Therefore, it can be ignored.

[0134] It should also be noted that the loss function of the present invention consists of two parts. The first part consists of the core equation of EPT, the purpose of which is to make the dielectric constant of the model output approximate the equation (i.e., equation (4)).

[0135] The loss function involves calculating the first-order derivatives of conductivity and permittivity in space, which, according to the chain rule, can be expressed as follows:

[0136]

[0137] Where f = wf ε -if σ ,but,

[0138]

[0139] because Modulo operation on complex numbers is required, but neural network operations are real number operations. Therefore, in practical applications, the real part of the PDE is calculated separately. I and the imaginary part PDE R Then substitute In China, due to Therefore, the complex operations of the original loss function can be converted into real number operations. This includes the derivatives of the dielectric constant with respect to B and the gradient of B. It can be easily obtained using the automatic differentiation method of neural networks.

[0140] The second part of the loss function references the image matching algorithm, based on the fact that conductivity and permittivity distributions in the human body have similar structural features, thus the gradient information of conductivity and permittivity are highly correlated. Therefore, this invention calculates the cross-correlation value between the gradients of permittivity and conductivity, and uses this value as the network's loss function, so that the network's output conductivity and permittivity have high structural similarity.

[0141] The fully connected neural network of this invention consists of two parts: network A and network B. Network A constructs f ε The function, wherein network B constructs f σ The function takes as input a transmission field B at any point in space and a first derivative B at that point as input to both network A and network B. x First derivative B y and the first derivative B z The outputs are the permittivity and conductivity at that point, respectively.

[0142] The activation functions of the input layer of network A, the intermediate layer of network A, the input layer of network B, and the intermediate layer of the input layer of network B are all tanh functions, and the tanh function is represented by equation (16).

[0143]

[0144] The activation functions of the output layer of network A and the output layer of network B are both sigmoid functions, and the sigmoid function is represented by equation (17).

[0145]

[0146] The tanh function maps the input to the range (-1, 1), while the sigmoid function maps the input to the range (0, 1).

[0147] When the inputs to network A and network B are both the emission field B at any point in space and the first derivative B at that point... x First derivative B y and the first derivative B z When the permittivity at that point is expressed by equation (18), the conductivity at that point is expressed by equation (19);

[0148] ε=(81*O a +1)E c Equation (18);

[0149] σ=3*O b Equation (19);

[0150] Where ε is the permittivity, σ is the conductivity, and O a For the output of network A, O b This is the output of network B.

[0151] It should be noted that since the range of conductivity and capacitance of human tissue has been experimentally measured, the capacitance of human tissue in the range of 1.5T-7T is Ec-82Ec, where Ec is the vacuum capacitance and the conductivity range is 0-3S / m, in this invention, the outputs of a simple linear transformation network A and network B are mapped to these ranges to obtain the final capacitance obtained by equation (18) and the conductivity obtained by equation (19).

[0152] The mapping of the fully connected neural network is represented by equations (20) and (21).

[0153] ε(x k ,y k ,z k )

[0154] =f ε (B(x k ,y k ,z k ),B x (x k ,y k ,z k ),B y (x k ,y k ,z k ),B z (x k ,y k ,z k );w ε )

[0155] Equation (20); σ(x) k ,y k ,z k )

[0156] =f σ (B(x k ,y k ,z k ),B x (z k ,y k ,z k ),B y (x k ,y k ,z k ),B z (x k ,y k ,z k );w σ )

[0157] Equation (21);

[0158] Where w ε and w σ All are parameters of the neural network, ε(x) k ,y k ,z k ) is a coordinate in space (x k ,y k ,z k The permittivity value on ) σ(x k ,y k ,z k ) is a coordinate in space (x k ,y k ,z k The conductivity value on ).

[0159] Step (4) of the present invention is specifically as follows:

[0160] Step (4.1): Using launch site B, the dataset consisting of the first and second derivatives obtained in step (2) is input into the fully connected neural network obtained in step (3) to obtain the output value of the fully connected neural network. The output value of the fully connected neural network is then input into the loss function obtained in step (3). Let n = 1 and perform forward and backward operations to obtain the current average value of the dataset. average value Update the fully connected neural network and proceed to step (4.2);

[0161] Step (4.2): Let n = n + 1, then proceed to step (4.3);

[0162] Step (4.3): Input the dataset consisting of the first and second derivatives obtained in step (2) into the updated fully connected neural network using the launch site B, and obtain the output value of the fully connected neural network. Proceed to step (4.4).

[0163] Step (4.4): Input the output value of the fully connected neural network into the loss function obtained in step (3), and then perform forward and backward operations to obtain the average value of the dataset. judge The relationship with the iteration threshold, when If the result is greater than the iteration threshold, proceed to step (4.5); otherwise, proceed to step (4.6).

[0164] Step (4.5): Calculate the average value obtained in step (4.4). Update the fully connected neural network to obtain the updated fully connected neural network and return to step (4.2);

[0165] Step (4.6): Define the current updated fully connected neural network as a trained fully connected neural network.

[0166] The numerical method used is the finite difference method. The iteration threshold is 0.001. Both network A and network B consist of five fully connected layers, with 256 neurons in each layer.

[0167] The fully connected neural network of this invention is written in Python 3.6 and built using the TensorFlow deep learning library.

[0168] Example 2.

[0169] The purpose of this example is to specifically illustrate the implementation steps of an electrical characteristic tomography method based on a physical neural network, as described in Example 1, using the Duke human brain tumor simulation model. The steps include:

[0170] Step (1): Based on the radio frequency field B, obtain the transmission field B, and at the same time obtain the reference permittivity and conductivity image data;

[0171] Step (2): Use numerical methods to solve for the first and second derivatives of each point in the launch field B obtained in step (1) on the x-axis, y-axis and z-axis in space;

[0172] Step (3): Construct a fully connected neural network and perform a combination transformation on Maxwell's equations to obtain the loss function. In this embodiment, the loss function is specifically selected as α = 4e-7.

[0173] Step (4): Using the dataset composed of the first and second derivatives obtained in step (2) and the loss function obtained in step (3), train the fully connected neural network obtained in step (3) to obtain the trained fully connected neural network. In this embodiment, the optimizer used for training the neural network is Adam, with a learning rate of 1e-6; the neural network weights are initialized using a normal distribution with a standard deviation of 1e-4; the weights of each layer of the neural network are standardized to accelerate the convergence of the loss function; 13864 data points are input simultaneously for each training iteration. When the loss function no longer decreases (specifically...) (At time), the fully connected neural network training is completed to obtain the trained fully connected neural network. The trained fully connected neural network of the present invention takes the emission field B at any point in space and the first derivative at that point obtained through step (2) as input, and the output of the trained fully connected neural network is the permittivity and conductivity at that point, thereby obtaining the permittivity and conductivity of all points in space.

[0174] In this embodiment, the permittivity and conductivity images in step (4) are compared with the EPT method (Helmholtz EPT) based on the uniformity assumption, both visually and quantitatively. Figure 3 The results of the two methods in permittivity reconstruction are shown. Figure 4 The figures above show the results of two methods for reconstructing conductivity. As can be seen from the two figures above, the reconstructed permittivity and conductivity obtained by this invention are highly consistent with the simulated permittivity and conductivity, and are visually significantly superior to methods based on the uniformity assumption.

[0175] Peak signal-to-noise ratio (PSNR) and structural similarity were selected as the quantification metrics to evaluate the results of this invention and the Helmholtz EPT method. Higher PNR and structural similarity values ​​indicate a closer reconstructed image to the reference image. Table 1 shows the quantification metrics obtained by the two methods.

[0176] Table 1. Quantitative Indicators of Helmholtz EPT and Results Obtained by This Invention

[0177]

[0178] As can be seen from Table 1, the present invention also shows that the peak signal-to-noise ratio and structural similarity values ​​are significantly higher than those of the Helmholtz EPT method. Therefore, the present invention is significantly better than the method based on the uniformity assumption in terms of reconstruction accuracy.

[0179] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit the scope of protection of the present invention. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the essence and scope of the technical solutions of the present invention.

Claims

1. A method for electrical characteristic tomography based on physical neural networks, characterized in that, Includes the following steps: Step (1): Obtain the transmission field B based on the radio frequency field B; Step (2): Use numerical methods to solve for the first and second derivatives of each point in the launch field B obtained in step (1) on the x-axis, y-axis and z-axis in space; Step (3): Construct a fully connected neural network and perform a combination transformation on Maxwell's equations to obtain the loss function; Step (4): Use the dataset consisting of the emission field B, the first derivative and the second derivative obtained in step (2), and the loss function obtained in step (3) to train the fully connected neural network obtained in step (3) to obtain the trained fully connected neural network. The trained fully connected neural network takes the emission field B at any point in space and the first derivative obtained at that point through step (2) as input. The output of the trained fully connected neural network is the permittivity and conductivity at that point, thereby obtaining the permittivity and conductivity of all points in space. In step (1), the radio frequency field B is generated by the radio frequency coil, and B = (B x B y B z ), where B x B represents the component of the radio frequency field along the x-axis in space. y B represents the component of the radio frequency field on the y-axis in space. z Let B be the component of the radio frequency field on the z-axis in space. The transmitted field B is obtained from the radio frequency field B, and thus exists. In step (3), the method for obtaining the loss function is as follows: Step (3.1): Perform a combined transformation on Maxwell's equations to obtain equation (1). Where ε c =ωε-iσ is the complex permittivity, B is the imaginary unit, ε is the permittivity, σ is the conductivity, ω is the Larmor precession frequency corresponding to the nuclear magnetic resonance equipment, and μ is the permeability. For the Nabla operator, For the Laplace operator; Step (3.2), let B z Set the value to 0 and separate and recombine equation (1) from step (3.1) to obtain the core equation (2) of MR-EPT; Among them B z Let B be the first derivative of the launch site B along the x-axis in space. y Let B be the first derivative of the y-axis in space within launch site B. z Let ε be the first derivative of the launch field B along the z-axis in space. c,x Let ε be the first derivative of the complex permittivity on the x-axis in space. c,y is the first derivative of the complex permittivity on the y-axis in space, and PDE is the symbol for a partial differential equation. Step (3.3): Obtain the loss function according to equation (2); The loss function is represented by equation (3). in As expressed by equation (4), in Equation (5) represents that Where α is the equilibrium and The weights are given and α>

0. The corr function is the cross-correlation function for calculating the correlation between the permittivity gradient and the conductivity gradient, and is expressed by equation (6). in Let X be the mean. Let Y be the mean of Y, N be the number of points to be calculated, and k be the kth point.

2. The electrical characteristic tomography method based on physical neural networks according to claim 1, characterized in that: Step (2) specifically involves using numerical methods to solve for the first derivative B at each point in the launch site B. x First derivative B y First derivative B z Second derivative B zz Second derivative B zy Second derivative B xz Second derivative B yz Second derivative B yy and second derivative B zz ; First derivative B x As expressed by equation (7), First derivative B y Equation (8) represents that, First derivative B z As expressed by equation (9), Second derivative B xx Represented by equation (10); Second derivative B xy As expressed by equation (11), Second derivative B xz As expressed by equation (12), Second derivative B yz It can be expressed by equation (13), Second derivative B yy As expressed by equation (14), Second derivative B zz Represented by equation (15); Where (x) k ,y k ,z k Let be the coordinates of the k-th point, and dx, dy, and dz be the discretized grids in the three directions of space, respectively.

3. The electrical characteristic tomography method based on physical neural networks according to claim 2, characterized in that: The fully connected neural network consists of two parts: network A and network B. Network A constructed f ε The function, wherein network B constructs f σ The function, wherein the inputs to both network A and network B are the emission field B at any point in space and the first derivative B at that point. x First derivative B y and the first derivative B z The outputs are the permittivity and conductivity at that point, respectively.

4. The electrical characteristic tomography method based on physical neural networks according to claim 3, characterized in that: The activation functions of the input layer of network A, the intermediate layer of network A, the input layer of network B, and the intermediate layer of the input layer of network B are all tanh functions, and the tanh function is represented by equation (16). The activation functions of the output layers of network A and network B are both sigmoid functions, and the sigmoid function is represented by equation (17). The tanh function maps the input to the range (-1, 1), and the sigmoid function maps the input to the range (0, 1). When the inputs to network A and network B are both the emission field B at any point in space and the first derivative B at that point... x First derivative B y and the first derivative B z When the permittivity at that point is expressed by equation (18), the conductivity at that point is expressed by equation (19); ε=(81*O a +1)E c Formula (18); σ=3*O b Formula (19); Where ε is the permittivity, σ is the conductivity, and O a For the output of network A, O b This is the output of network B.

5. The electrical characteristic tomography method based on physical neural networks according to claim 4, characterized in that: The mapping of the fully connected neural network is represented by equations (20) and (21). ε( x k ,y k ,z k ) = f ε (B(x k , y k , z k ), B x (x k , y k , z k ), B y (x k , y k , z k ), B z (x k , y k , z k ); w ε ) Equation (20); σ(x k , y k , z k ) = f σ (B(x k , y k , z k ), B x (x k , y k , z k ), B y (x k , y k , z k ), B z (x k , y k , z k ); w σ ) Equation (21); Where w ε and w σ These are all parameters of the neural network, ε(x) k ,y k ,z k ) is a spatial coordinate (x k ,y k ,z k The permittivity value on ) σ(x k ,y k ,z k ) is a spatial coordinate (x k ,y k ,z k The conductivity value on ).

6. The electrical characteristic tomography method based on physical neural networks according to claim 5, characterized in that, Step (4) specifically involves: Step (4.1): Using launch site B, the dataset consisting of the first and second derivatives obtained in step (2) is input into the fully connected neural network obtained in step (3) to obtain the output value of the fully connected neural network. The output value of the fully connected neural network is then input into the loss function obtained in step (3). Let n = 1 and perform forward and backward operations to obtain the current average value of the dataset. average value Update the fully connected neural network and proceed to step (4.2); Step (4.2): Let n = n + 1, then proceed to step (4.3); Step (4.3): Input the dataset consisting of the first and second derivatives obtained in step (2) into the updated fully connected neural network using the launch site B, and obtain the output value of the fully connected neural network. Proceed to step (4.4). Step (4.4): Input the output value of the fully connected neural network into the loss function obtained in step (3), and then perform forward and backward operations to obtain the average value of the dataset. judge The relationship with the iteration threshold, when If the result is greater than the iteration threshold, proceed to step (4.5); otherwise, proceed to step (4.6). Step (4.5): Calculate the average value obtained in step (4.4). Update the fully connected neural network to obtain the updated fully connected neural network and return to step (4.2); Step (4.6): Define the current updated fully connected neural network as a trained fully connected neural network.

7. The electrical characteristic tomography method based on physical neural networks according to claim 6, characterized in that: The numerical method is the finite difference method; The iteration threshold is 0.001; Both network A and network B consist of five fully connected layers, with 256 neurons in each layer.

8. The electrical characteristic tomography method based on physical neural networks according to claim 7, characterized in that: The fully connected neural network was written in Python 3.6 and built using the TensorFlow deep learning library.

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