Implicit neural network magnetic resonance image reconstruction method based on sensitivity matrix constraint

Abstract

The invention provides an implicit neural network magnetic resonance image reconstruction method based on sensitivity matrix constraint. The method comprises the following steps: performing Fourier coding on coordinates of a to-be-reconstructed image as network input; training an implicit neural network magnetic resonance image reconstruction model based on the sensitivity matrix constraint; and performing data combination by using the output of the model and the sampling data to obtain a final image. The deep learning parallel magnetic resonance reconstruction method based on self-supervised learning has the advantages that high-quality image reconstruction can be achieved, and meanwhile dependence on a large amount of full-sampling data is reduced.

Description

Technical Field

[0001] The invention belongs to the technical field of undersampling and reconstruction of magnetic resonance images, and in particular relates to an implicit neural network magnetic resonance image reconstruction method based on sensitivity matrix constraints. Background Art

[0002] Magnetic Resonance Imaging (MRI) is a commonly used examination method in clinical medicine. It has the advantages of no radiation, non-invasive, non-damage, high resolution, and multiple contrasts, and can provide a lot of valuable information for clinical diagnosis.

[0003] However, for magnetic resonance imaging, the time required for magnetic resonance scanning is longer than other imaging methods (such as computed tomography, X-ray imaging, etc.). The time required to obtain a magnetic resonance image is often at least hundreds of milliseconds, and some even take several minutes. Long-term scanning will cause serious discomfort to patients with claustrophobia. In addition, long-term scanning will cause image distortion due to natural reasons such as heartbeat and breathing. Therefore, researchers proposed to shorten the sampling time by reducing the sampling data and obtain high-definition magnetic resonance images by reconstructing the unsampled data. In order to reconstruct a more accurate image, Pruessmann proposed to reconstruct MRI by assuming the existence of coil sensitivity (Pruessmann, Klaas P., et al. "SENSE: sensitivity encoding for fast MRI." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 42.5 (1999): 952-962.), which is the classic parallel MRI imaging.

[0004] With the development of deep learning, Wang et al. first proposed to introduce deep learning into fast magnetic resonance reconstruction (Wang, Shanshan, et al. "Accelerating magnetic resonance imaging via deeplearning." 2016IEEE 13th international symposium on biomedical imaging (ISBI). IEEE, 2016.). However, this reconstruction requires a large amount of fully sampled MRI data for long-term training. Recently, Shen et al. used the implicit neural representation (INR) network to reconstruct MRI by mapping image coordinates to corresponding intensity values ​​(Shen, Liyue, John Pauly, and Lei Xing. "NeRP: implicit neural representation learning with prior embedding for sparsely sampled image reconstruction." IEEE Transactions on Neural Networks and Learning Systems 35.1 (2022): 770-782). In order to better represent the details and structure of the image, Shen et al. used Fourier coding on the input (Tancik, Matthew, et al. "Fourier features let networks learn high frequency functions in low dimensional domains." Advances in neural information processing systems 33 (2022): 7537-7547.); on the network, instead of using the traditional multilayer perceptron (MLP) as the INR network structure, they used the sinusoidal representation network SIREN, which is a multilayer perceptron MLP with a periodic sine function as the activation function. (Sitzmann, Vincent, et al. "Implicit neural representations with periodic activation functions." Advances in neural information processing systems 33 (2020): 7462-7473.).However, NeRP still requires complete pre-scanned images as prior images for training, and is clinically only applicable to situations where long-term scanning is required.

[0005] To this end, Feng et al., based on NeRP, generated coil sensitivity and images through INR, and regularized the images to perform parallel MRI reconstruction (Feng, Ruimin, et al. "IMJENSE: scan-specific implicit representation for joint coil sensitivity and image estimation in parallel MRI." IEEE Transactions on Medical Imaging, 2023.). On this basis, Hemidi et al. expanded the two-dimensional MRI image reconstruction to three dimensions (adding the time dimension). (Al-Haj Hemidi, Ziad, et al. "CineJENSE: Simultaneous Cine MRI Image Reconstruction and Sensitivity Map Estimation Using Neural Representations." International Workshop on Statistical Atlases and Computational Models of the Heart. Cham: Springer Nature Switzerland, 2023.). Summary of the invention

[0006] In view of the actual needs of magnetic resonance image reconstruction and the need to adapt to development and further promote application, the purpose of the present invention is to provide an implicit neural network magnetic resonance image reconstruction method based on sensitivity matrix constraints, which strengthens the network's learning ability for high-frequency information through Fourier encoding and regularization constraints, optimizes the sensitivity matrix, and thus reconstructs high-quality magnetic resonance images. In this scheme, the coordinates of the image to be reconstructed are Fourier encoded as network input; an implicit neural network magnetic resonance image reconstruction model based on sensitivity matrix constraints is established; the implicit neural network magnetic resonance image reconstruction model based on sensitivity matrix constraints is trained; and the output of the model and the sampled data are used to combine data to obtain the final image. This deep learning parallel magnetic resonance reconstruction method through self-supervised learning has the characteristics of being able to achieve high-quality image reconstruction while reducing dependence on a large amount of full sampling data.

[0007] The technical solution specifically adopted by the present invention to solve the technical problem is:

[0008] An implicit neural network magnetic resonance image reconstruction method based on sensitivity matrix constraints: Fourier encoding the coordinates of the image to be reconstructed as network input; training an implicit neural network magnetic resonance image reconstruction model based on sensitivity matrix constraints; and combining the model output and sampled data to obtain the final image.

[0009] Furthermore, the specific process of performing Fourier encoding on the coordinates of the image to be reconstructed is:

[0010] γ(θ)=[cos(2πBθ),sin(2πBθ)] T

[0011] Among them, θ represents the normalized coordinate set of the image, c ij ∈θ represents the pixel coordinates of the i-th row and j-th column in the normalized image coordinates to be reconstructed, c ij =(x i ,y j ),x i ∈[0,1)y j ∈[0,1); the matrix B represents the coefficients of the Fourier characteristic transform and is a matrix of size (E,N θ ); E represents the output size of Fourier encoding; N θ Represents the dimension size of the coordinates; each item in the matrix B is derived from a Gaussian distribution N(0,σ 2 ) for sampling.

[0012] Furthermore, the implicit neural network magnetic resonance image reconstruction model based on sensitivity matrix constraints is expressed as:

[0013]

[0014] The model consists of two implicit neural networks, namely the image implicit neural network and the sensitivity matrix implicit neural network, which generate X α and S β ; The input of the model is the Fourier-encoded coordinate γ(θ);

[0015] The image implicit neural network is used to generate X α , where X α represents a single-channel magnetic resonance image reconstructed by the implicit neural network, and α represents the network parameter to be optimized in the image implicit neural network;

[0016] The sensitivity matrix implicit neural network is used to generate S β , where S β represents the multi-channel coil sensitivity matrix generated by the implicit neural network, S β =[S β1 , ..., Sβn , ..., S βN ],S βn represents the coil sensitivity of the nth channel, β represents the network parameter to be optimized in the sensitivity matrix implicit neural network;

[0017] Y represents multi-channel k-space data with under-sampling and zero-filling operation at unsampled locations, Y = [Y1, ..., Y n , ..., Y N ],Y n represents the k-space data of the nth channel; F represents the operator for performing a two-dimensional fast Fourier transform on the multi-channel image; represents an operator that performs undersampling and zero-filling operations at unsampled locations; R represents the operation of X α and S β Each channel of the image is smoothed and regularized; F(g, θ) is a k-space filter whose weights gradually decrease from the center to the outside, where g is a parameter that controls the rate at which the filter weights decrease; θ is the coordinate of the image after normalization; ⊙ represents the Hadamard product; λ1 and λ2 are regularization weight parameters; The square of the sum of the 2-norms of the quantities.

[0018] Furthermore, the training of the implicit neural network magnetic resonance image reconstruction model based on sensitivity matrix constraints is specifically performed by minimizing the total loss function L total To estimate the optimal values ​​of parameters α, β in the implicit neural network The total loss function is defined as:

[0019]

[0020] The loss function is minimized using the gradient descent algorithm and the standard back-propagation algorithm.

[0021] The above total loss function is divided into the error correction loss part L DC , using the smoothing loss part L calculated by R TV And the filtering loss L after adding the filter F(g,θ) HDF :

[0022]

[0023]

[0024] Therefore, the total loss function can also be expressed as:

[0025] L total =L DC +λ1*L TV +λ2*L HDF .

[0026] Furthermore, the output of the model and the sampled data are combined to obtain a final image representation as follows:

[0027]

[0028] in and Represents the sensitivity matrix and single-channel image of the nth channel output when the model is optimal; Represents the final multi-channel combined image; Represents the operator for performing two-dimensional fast inverse Fourier transform on a multi-channel image; Representation and The operator for complementary undersampling. Sampling location Perform zero-fill operation. Sampling location Perform zero-fill operation.

[0029] And, an implicit neural network magnetic resonance image reconstruction system based on sensitivity matrix constraints, including: a Fourier encoding module, used to Fourier encode the coordinates of the reconstructed image as network input; a training module, used to train the implicit neural network magnetic resonance image reconstruction model based on sensitivity matrix constraints; a final image generation module, used to use the output of the model and the sampling data to combine data to obtain the final image.

[0030] Furthermore, the specific process of performing Fourier encoding on the coordinates of the image to be reconstructed is:

[0031] γ(θ)=[cos(2πBθ),sin(2πBθ)] T

[0032] Among them, θ represents the normalized coordinate set of the image, c ij ∈θ represents the pixel coordinates of the i-th row and j-th column in the normalized image coordinates to be reconstructed, c ij =(x i ,y j ),x i ∈[0,1)y j ∈[0,1); the matrix B represents the coefficients of the Fourier characteristic transform and is a matrix of size (E,N θ ); E represents the output size of Fourier encoding; N θ Represents the dimension size of the coordinates; each item in the matrix B is derived from a Gaussian distribution N(0,σ 2 ) for sampling.

[0033] Furthermore, the implicit neural network magnetic resonance image reconstruction model based on sensitivity matrix constraints is expressed as:

[0034]

[0035] The model consists of two implicit neural networks, namely the image implicit neural network and the sensitivity matrix implicit neural network, which generate X α and S β ; The input of the model is the Fourier-encoded coordinate γ(θ);

[0036] The image implicit neural network is used to generate X α , where X α represents a single-channel magnetic resonance image reconstructed by the implicit neural network, and α represents the network parameter to be optimized in the image implicit neural network;

[0037] The sensitivity matrix implicit neural network is used to generate S β , where S β represents the multi-channel coil sensitivity matrix generated by the implicit neural network, S β =[S β1 , ..., S βn , ..., S βN ],S βn represents the coil sensitivity of the nth channel, β represents the network parameter to be optimized in the sensitivity matrix implicit neural network;

[0038] Y represents multi-channel k-space data with under-sampling and zero-filling operation at unsampled locations, Y = [Y1, ..., Y n , ..., Y N ],Y n represents the k-space data of the nth channel; F represents the operator for performing a two-dimensional fast Fourier transform on the multi-channel image; represents an operator that performs undersampling and zero-filling operations at unsampled locations; R represents the operation of X α and S β Each channel of the image is smoothed and regularized; F(g, θ) is a k-space filter whose weights gradually decrease from the center to the outside, where g is a parameter that controls the rate at which the filter weights decrease; θ is the coordinate of the image after normalization; ⊙ represents the Hadamard product; λ1 and λ2 are regularization weight parameters; The square of the sum of the 2-norms of the quantities.

[0039] An electronic device comprises a memory, a processor and a computer program stored in the memory and executable on the processor, wherein when the processor executes the program, the steps of the implicit neural network magnetic resonance image reconstruction method based on sensitivity matrix constraints as described above are implemented.

[0040] A non-transitory computer-readable storage medium stores a computer program, which, when executed by a processor, implements the steps of the implicit neural network magnetic resonance image reconstruction method based on sensitivity matrix constraints as described above.

[0041] Compared with the prior art, the present invention and its preferred embodiment have the characteristics of being able to achieve high-quality image reconstruction while reducing the dependence on a large amount of full sampling data through the deep learning parallel magnetic resonance reconstruction method of self-supervised learning. Through Fourier coding and regularization constraints, the network's learning ability for high-frequency information is enhanced, the sensitivity matrix is ​​optimized, and high-quality magnetic resonance images are reconstructed. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] The present invention is further described in detail below with reference to the accompanying drawings and specific embodiments:

[0043] Figure 1 Schematic diagram of the image model reconstruction process in an embodiment of the present invention.

[0044] Figure 2 The experimental results of MRI brain image reconstruction by different methods in the embodiments of the present invention are shown in Figure 1. (a) is the original image; (b) is the result image reconstructed by the method of the present invention; (c) is the sampling template M used, with a sampling rate of 25.94%; (d) is the complementary template used in step 4). DETAILED DESCRIPTION

[0045] In order to make the features and advantages of this patent more obvious and easy to understand, the following embodiments are specifically described in detail as follows:

[0046] It should be noted that the following detailed descriptions are illustrative and are intended to provide further explanation of the present application. Unless otherwise specified, all technical and scientific terms used in this specification have the same meanings as those commonly understood by those skilled in the art to which the present application belongs.

[0047] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit the exemplary embodiments according to the present application. As used herein, unless the context clearly indicates otherwise, the singular form is also intended to include the plural form. In addition, it should be understood that when the terms "comprise" and / or "include" are used in this specification, it indicates the presence of features, steps, operations, devices, components and / or combinations thereof.

[0048] The following example will further illustrate the present invention in conjunction with the accompanying drawings. This example will reconstruct a 2D MRI image with 14 channels downloaded from the fastMRI Initiative database of New York University, and the data size is 320*320*14. The sampling template is composed of Figure 2(c) shows an ACS with 24 uniform sampling templates with a sampling rate of 25.94%. The specific steps of this embodiment are as follows:

[0049] 1) Establish Fourier coding:

[0050] γ(θ)=[cos(2πBθ),sin(2πBθ)] T (1)

[0051] Among them, θ represents the normalized coordinate set of the image, c ij ∈θ represents the pixel coordinates of the i-th row and j-th column in the normalized image coordinates, c ij =(x i ,y j ), x i ∈[0,1)y j ∈[0, 1). In this embodiment, the single-channel image data is 320*320, and the normalized coordinates are to divide the x-axis and y-axis into 320 points respectively. B represents the coefficient of Fourier feature transform, which is a value of (E, N θ ) is a real matrix; E represents the output size of Fourier encoding, N θ Represents the dimension size of the coordinates. Each item in the matrix B is derived from a Gaussian distribution N(0, σ 2 ) is sampled. In this embodiment, E=256, N θ =2.

[0052] 2) Establish an implicit neural network magnetic resonance image reconstruction model based on sensitivity matrix constraints:

[0053]

[0054] The model consists of two implicit neural networks, namely the image implicit neural network and the sensitivity matrix implicit neural network, which generate X α and S β The image implicit neural network is used to generate X α , where X α represents a single-channel magnetic resonance image reconstructed by the implicit neural network, and α represents the network parameter to be optimized in the image implicit neural network. The sensitivity matrix implicit neural network is used to generate S β , where S β represents the multi-channel coil sensitivity matrix generated by the implicit neural network, that is, S β =[S β1 , ..., S βn , ..., S βN ],S βnrepresents the coil sensitivity of the nth channel, β represents the network parameter to be optimized in the sensitivity matrix implicit neural network; Y represents the multi-channel k-space data with under-sampling and zero-filling operation at the unsampled position, that is, Y = [Y1, ..., Y n , ..., Y N ],Y n represents the k-space data of the nth channel; F represents the operator for performing two-dimensional fast Fourier transform on the multi-channel image; represents an operator that performs undersampling and zero-filling operations at unsampled locations; R represents the operation of X α and S β Each channel of the image is smoothed and regularized; F(g, θ) is a k-space filter whose weights gradually decrease from the center to the outside, where g is a parameter that controls the rate at which the filter weights decrease; θ is the coordinate of the image after normalization; ⊙ represents the Hadamard product; λ1 and λ2 are regularization weight parameters; The square of the sum of the 2-norms of the directivity. In this embodiment, N = 14, g = 1, λ1 = 2e -4 ,λ2=1e -4 .

[0055] 3) Train the implicit neural network magnetic resonance image reconstruction model based on sensitivity matrix constraints by minimizing the total loss function L total To estimate the optimal values ​​of parameters α, β in the implicit neural network The total loss function is defined as:

[0056]

[0057] The total loss function can be divided into three parts: error correction loss part (L DC ), using the smoothing loss part (L TV ) and the filtering loss after adding the filter F(g,θ) (L HDF ):

[0058]

[0059] Therefore, the total loss function can also be expressed as:

[0060] L total =L DC +λ1*L TV +λ2*L HDF (7)

[0061] The minimization of the loss function is achieved by the gradient descent algorithm and the standard back propagation algorithm. The model consists of two implicit neural networks. The input of the model is the Fourier encoded coordinates γ(θ).

[0062] 4), the output of the model and the sampled data are combined to obtain the final image:

[0063]

[0064] in and Represents the sensitivity matrix and single-channel image of the nth channel output when the model is optimal; Represents the final multi-channel combined image; Represents the operator for performing two-dimensional fast inverse Fourier transform on a multi-channel image; Representation and The operator for complementary undersampling (i.e. Sampling location Perform zero-fill operation. Sampling location Perform zero-fill operation).

[0065] Figure 1 This is the image model reconstruction process in the embodiment. Figure 2 For the reconstruction result comparison of the embodiment, Figure 2 It can be seen that the present invention can reconstruct magnetic resonance images with smaller errors.

[0066] Those skilled in the art will appreciate that the embodiments of the present application may be provided as methods, systems, or computer program products. Therefore, the present application may adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment in combination with software and hardware. Moreover, the present application may adopt the form of a computer program product implemented in one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) that include computer-usable program code.

[0067] The present application is described with reference to the flowcharts and / or block diagrams of the methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each process and / or box in the flowchart and / or block diagram, as well as the combination of the processes and / or boxes in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to generate a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0068] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.

[0069] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 The steps for the functions specified in one or more boxes.

[0070] It should be noted that, unless otherwise defined, the technical terms or scientific terms used in the present invention should be understood by people with ordinary skills in the field to which the present invention belongs. The "first", "second" and similar words used in the present invention do not indicate any order, quantity or importance, but are only used to distinguish different components. "Include" or "comprise" and similar words mean that the elements or objects appearing before the word include the elements or objects listed after the word and their equivalents, without excluding other elements or objects. "Connect" or "connected" and similar words are not limited to physical or mechanical connections, but may include electrical connections, whether direct or indirect. "Up", "down", "left", "right" and the like are only used to indicate relative positional relationships. When the absolute position of the described object changes, the relative positional relationship may also change accordingly.

[0071] The above is only a preferred embodiment of the present invention, and does not limit the present invention in other forms. Any technician familiar with the profession may use the above disclosed technical content to change or modify it into an equivalent embodiment with equivalent changes. However, any simple modification, equivalent change and modification made to the above embodiment according to the technical essence of the present invention without departing from the technical solution of the present invention still belongs to the protection scope of the technical solution of the present invention.

[0072] This patent is not limited to the above-mentioned optimal implementation mode. Anyone can derive other various forms of implicit neural network magnetic resonance image reconstruction methods based on sensitivity matrix constraints under the inspiration of this patent. All equal changes and modifications made according to the scope of the patent application of the present invention should be covered by this patent.

Claims

1. An implicit neural network magnetic resonance image reconstruction method based on sensitivity matrix constraints, characterized in that: Fourier encoding of the image coordinates to be reconstructed is used as network input; An implicit neural network magnetic resonance image reconstruction model based on sensitivity matrix constraints is trained; the output of the model and the sampled data are combined to obtain the final image.

2. The method for magnetic resonance image reconstruction based on an implicit neural network with sensitivity matrix constraints according to claim 1, characterized in that: The specific process of Fourier encoding the coordinates of the image to be reconstructed is: γ(θ)=[cos(2πBθ),sin(2πBθ)] T Among them, θ represents the normalized coordinate set of the image, c ij ∈θ represents the pixel coordinates of the i-th row and j-th column in the normalized image coordinates to be reconstructed, c ij =(x i ,y j ),x i ∈[0,1)y j ∈[0,1); the matrix B represents the coefficients of the Fourier characteristic transform and is a matrix of size (E,N θ ); E represents the output size of Fourier encoding; N θ Represents the dimension size of the coordinates; each item in the matrix B is derived from a Gaussian distribution N(0,σ 2 ) for sampling.

3. The method for magnetic resonance image reconstruction based on implicit neural network with sensitivity matrix constraint according to claim 1, characterized in that: The implicit neural network magnetic resonance image reconstruction model based on sensitivity matrix constraints is expressed as: The model consists of two implicit neural networks, namely the image implicit neural network and the sensitivity matrix implicit neural network, which generate X α and S β ; The input of the model is the Fourier-encoded coordinate γ(θ); The image implicit neural network is used to generate X α , where X α represents a single-channel magnetic resonance image reconstructed by the implicit neural network, and α represents the network parameter to be optimized in the image implicit neural network; The sensitivity matrix implicit neural network is used to generate S β , where S β represents the multi-channel coil sensitivity matrix generated by the implicit neural network, S β =[S β1 ,...,S βn ,...,S βN ],S βn represents the coil sensitivity of the nth channel, β represents the network parameter to be optimized in the sensitivity matrix implicit neural network; Y represents the multi-channel k-space data with under-sampling and zero-filling operation in the unsampled areas, Y = [Y1, ..., Y n ,...,Y N ],Y n represents the k-space data of the nth channel; F represents the operator for performing a two-dimensional fast Fourier transform on the multi-channel image; represents an operator that performs undersampling and zero-filling operations at unsampled locations; R represents the operation of X α and S β Each channel of is smoothed and regularized; F(g,θ) is a k-space filter whose weights gradually decrease from the center to the outside, where g is a parameter that controls the rate at which the filter weights decrease; θ is the normalized coordinate of the image; ⊙ represents the Hadamard product; λ1 and λ2 are regularization weight parameters; The square of the sum of the 2-norms of the quantities.

4. The method for magnetic resonance image reconstruction based on implicit neural network with sensitivity matrix constraint according to claim 3, characterized in that: The implicit neural network magnetic resonance image reconstruction model based on sensitivity matrix constraint is trained by minimizing the total loss function L total To estimate the optimal values ​​of parameters α, β in the implicit neural network The total loss function is defined as: The loss function is minimized using the gradient descent algorithm and the standard back-propagation algorithm.

5. The method for magnetic resonance image reconstruction based on implicit neural network with sensitivity matrix constraint according to claim 4, characterized in that: The final image obtained by combining the output of the model and the sampled data is represented as: in and Represents the sensitivity matrix and single-channel image of the nth channel output when the model is optimal; Represents the final multi-channel combined image; Represents the operator for performing two-dimensional fast inverse Fourier transform on a multi-channel image; Represents an operator that performs complementary undersampling with u.

6. An implicit neural network magnetic resonance image reconstruction system based on sensitivity matrix constraints, characterized in that: include: Fourier encoding module, used to perform Fourier encoding on the coordinates of the image to be reconstructed as network input; The training module is used to train the implicit neural network magnetic resonance image reconstruction model based on the sensitivity matrix constraint; the final image generation module is used to combine the output of the model and the sampling data to obtain the final image.

7. The method for magnetic resonance image reconstruction based on implicit neural network with sensitivity matrix constraint according to claim 6, characterized in that: The specific process of Fourier encoding the coordinates of the image to be reconstructed is: γ(θ)=[cos(2πBθ),sin(2πBθ)] T Among them, θ represents the normalized coordinate set of the image, c ij ∈θ represents the pixel coordinates of the i-th row and j-th column in the normalized image coordinates to be reconstructed, c ij =(x i ,y j ),x i ∈[0,1)y j ∈[0,1); the matrix B represents the coefficients of the Fourier characteristic transform and is a matrix of size (E,N θ ); E represents the output size of Fourier encoding; N θ Represents the dimension size of the coordinates; each item in the matrix B is derived from a Gaussian distribution N(0,σ 2 ) for sampling.

8. The method for magnetic resonance image reconstruction based on implicit neural network with sensitivity matrix constraint according to claim 6, characterized in that: The implicit neural network magnetic resonance image reconstruction model based on sensitivity matrix constraints is expressed as: The model consists of two implicit neural networks, namely the image implicit neural network and the sensitivity matrix implicit neural network, which generate X α and S β ; The input of the model is the Fourier-encoded coordinate γ(θ); The image implicit neural network is used to generate X α , where X α represents a single-channel magnetic resonance image reconstructed by the implicit neural network, and α represents the network parameter to be optimized in the image implicit neural network; The sensitivity matrix implicit neural network is used to generate S β , where S β represents the multi-channel coil sensitivity matrix generated by the implicit neural network, S β =[S β1 ,...,S βn ,...,S βN ],S βn represents the coil sensitivity of the nth channel, β represents the network parameter to be optimized in the sensitivity matrix implicit neural network; Y represents the multi-channel k-space data with under-sampling and zero-filling operation in the unsampled areas, Y = [Y1, ..., Y n ,...,Y N ],Y n represents the k-space data of the nth channel; F represents the operator for performing a two-dimensional fast Fourier transform on the multi-channel image; u represents the operator for undersampling and filling zeros at unsampled positions; R represents the operator for X α and S β Each channel of is smoothed and regularized; F(g,θ) is a k-space filter whose weights gradually decrease from the center to the outside, where g is a parameter that controls the rate at which the filter weights decrease; θ is the normalized coordinate of the image; ⊙ represents the Hadamard product; λ1 and λ2 are regularization weight parameters; The square of the sum of the 2-norms of the quantities.

9. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that: When the processor executes the program, the steps of the implicit neural network magnetic resonance image reconstruction method based on sensitivity matrix constraints as described in any one of claims 1 to 5 are implemented.

10. A non-transitory computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the implicit neural network magnetic resonance image reconstruction method based on sensitivity matrix constraints as described in any one of claims 1 to 5.

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