Magnetic resonance fingerprint reconstruction method based on double-domain geometric structure and space priori depth expansion

Abstract

The invention discloses a magnetic resonance fingerprint reconstruction method based on a double-domain geometric structure and spatial prior depth expansion, and relates to the technical field of magnetic resonance fingerprint imaging. The method comprises the following steps: (1) acquiring magnetic resonance fingerprint undersampling k space data; (2) establishing a magnetic resonance fingerprint reconstruction model based on fingerprint domain manifold structure prior and tissue characteristic domain sparse prior; (3) iteratively solving the reconstruction model by using a dominant minimization and near-end gradient algorithm; (4) adopting an effective strategy to obtain a closed-form solution of a near-end mapping sub-problem, and expanding all iteration steps to the deep neural network; (5) training the network by using the existing data to obtain a network model; and (6) reconstructing magnetic resonance fingerprints and various tissue parameter diagrams by using the trained network. According to the method, the structured geometric information of the fingerprint data and the spatial characteristics of the organization parameter diagram are fully utilized by expanding the network, and an excellent reconstruction effect can be presented under the condition of high-power undersampling of the data.

Description

Technical Field

[0001] The present invention relates to the technical field of magnetic resonance fingerprint imaging, and particularly to a magnetic resonance fingerprint reconstruction method based on a dual-domain geometric structure and spatial prior depth unfolding. Background Art

[0002] Magnetic resonance fingerprinting (MRF) [1] is an emerging fast quantitative magnetic resonance imaging technique that can achieve parallel imaging of multiple tissue characteristics in a single scan. This technique uses a pseudo-randomly varying pulse sequence to acquire data, so as to obtain magnetic resonance images with different contrast weightings and the unique evolution signals of different tissues over time (i.e., magnetic resonance fingerprints). At the same time, a fingerprint dictionary of theoretical magnetic resonance signals of all possible tissues under the same pulse excitation conditions is constructed based on the Bloch equation. Then, the collected magnetic resonance fingerprints are pattern-matched with the dictionary entries to achieve simultaneous imaging of multiple tissue characteristics. In order to further shorten the acquisition time, magnetic resonance fingerprints acquire k-space data at a sampling rate far lower than that specified by the Nyquist sampling theorem, resulting in severe aliasing artifacts in the reconstructed images and further degrading the quality of tissue parameter imaging.

[0003] For this reason, researchers have proposed many magnetic resonance fingerprint reconstruction methods to improve the accuracy of tissue parameter estimation. Dictionary-based model-driven methods mainly construct regularization penalty terms by utilizing the structural prior of the magnetic resonance fingerprint data matrix / tensor. These methods have inherent discretization errors, slow inference speed, and low reconstruction accuracy. Existing deep learning-based methods directly estimate tissue parameters from the preliminarily reconstructed fingerprints using neural networks with black-box properties, lacking effective utilization of the physical acquisition process and the prior information of fingerprint data. In recent years, deep unfolding networks [2] have been introduced into magnetic resonance fingerprint reconstruction due to their interpretability similar to model-driven methods and powerful learning ability of neural networks, in order to further improve the imaging quality.

[0004] However, existing methods rarely utilize the structured geometric information of high-dimensional magnetic resonance fingerprint data and the additional spatial prior unfolding research in the tissue characteristic domain. Therefore, it is necessary to design a magnetic resonance fingerprint reconstruction method that fully considers the dual-domain data prior of the fingerprint domain and the tissue characteristic domain and has high processing speed and high reconstruction quality. Summary of the Invention

[0005] The technical problem to be solved by the present invention is:

[0006] Aiming at the limitations of the current magnetic resonance fingerprint reconstruction algorithm, the present invention provides a magnetic resonance fingerprint reconstruction method based on a dual-domain geometric structure and spatial prior depth unfolding to further improve the quality of the reconstructed image and shorten the reconstruction time.

[0007] The technical solution adopted by the present invention to solve the above problems is as follows:

[0008] (1) Obtain undersampled k-space data of magnetic resonance fingerprint.

[0009] (2) Establish a magnetic resonance fingerprint reconstruction model based on the prior of the manifold structure in the fingerprint domain and the sparse prior in the tissue property domain.

[0010] (3) Use the leading minimization and proximal gradient algorithms to iteratively solve the reconstruction model.

[0011] (4) Adopt an effective strategy to obtain a closed-form solution of the proximal mapping sub-problem and expand all iterative steps into a deep neural network.

[0012] (5) Use the existing data to train the network to obtain a network model.

[0013] (6) Use the trained network to reconstruct magnetic resonance fingerprints and various tissue parameter maps.

[0014] Furthermore, in the above step (1), the undersampled k-space data of magnetic resonance fingerprint is obtained by using a magnetic resonance scanner, and the data acquisition process can be expressed as:

[0015]

[0016] Among them, represents the undersampled k-space data collected, N c represents the number of coils used in the acquisition, N s represents the number of samples collected by each coil, and L represents the length of the magnetic resonance fingerprint data; represents the magnetic resonance fingerprint data to be reconstructed, N v represents the number of voxels involved (i.e., the product of the length and width dimensions of the magnetic resonance image); represents the noise introduced during the acquisition process; represents the undersampling operator of k-space related to the coil sensitivity.

[0017] Furthermore, the magnetic resonance fingerprint reconstruction model based on the prior of the manifold structure in the fingerprint domain and the sparse prior in the tissue property domain in the above step (2) can be expressed as:

[0018]

[0019] Among them, represents the geometric structure of the magnetic resonance fingerprint data, i.e., the fingerprint manifold; represents the prior regularization term of the manifold structure in the fingerprint domain; Denote the sparse prior regular term of the tissue feature domain; λ1 and λ2 are adjustable regularization parameters. The prior regular term of the manifold structure in the fingerprint domain and the sparse prior regular term of the tissue feature domain can be specifically expressed as:

[0020]

[0021] Among them, Denote a certain sparsification transformation; Denote the mapping from the fingerprint to the tissue parameters, satisfying Denote l tissue parameter maps; ||·|| p (0 < p ≤ 2) and ||·||1 respectively denote the element-wise L p and L1 norms. According to the central position The magnetic resonance fingerprint data X is divided into N p overlapping spatial blocks with a spatial size of d×d. These image blocks are then arranged into a matrix Each row of this matrix corresponds to a vectorized image block B is the weighted incidence matrix corresponding to the discrete approximation graph of the manifold The node set of this graph ( Denote the number of nodes), the edge set (|ε| denotes the number of edges), and the weight elements in the weighted adjacency matrix W are calculated from the similarity between parameter patches, expressed as follows:

[0022]

[0023] Among them, the hyperparameter σ is used to adjust the similarity weight. The element in the s-th row and t-th column of B corresponds to the edge e s and the node v t , defined as follows:

[0024]

[0025] Furthermore, in the above step (3), the magnetic resonance fingerprint reconstruction model constructed is iteratively solved using the majorization-minimization [3] and proximal gradient algorithm [4]. At the k-th iteration, the following three iterative steps need to be solved:

[0026]

[0027] Among them, and respectively denote and 's adjoint operators; denotes the projection operator on the fingerprint manifold; μ is the step size of gradient descent; W (k-1) is resized to be the same as in dimension ⊙ denotes the Hadamard product.

[0028] Furthermore, in the above step (4), the present invention adopts an effective strategy to obtain the closed-form solution of the proximal mapping sub-problem in step (3) and expands all iterative steps into a deep neural network, specifically as follows:

[0029] For the iterative step shown in Equation (6), it is corrected to be carried out in the following order successively:

[0030]

[0031] where Based on the imaging theory of magnetic resonance fingerprinting, the proximal mapping sub-problem shown in Equation (8) is adjusted, and the adjusted form is as follows:

[0032]

[0033] where θ = α / 2μ, and α is used to correct this approximation relationship. The closed-form solution of this problem is shown as follows:

[0034]

[0035] where soft(x,λ) = sign(x)max(0,|x| - λ) denotes the soft threshold operator with a threshold of λ; denotes the adjoint operator of denotes the mapping from tissue parameters to fingerprints.

[0036] The present invention unfolds the above model-based iterative algorithm into a deep neural network. Each iteration step involving the three sub-problems shown in equations (7)-(9) is unfolded into a stage of the deep network. Each stage contains four modules, namely: the Manifold Projection module (MP), the Learned Sparsity module (LS), the p-norm enhanced Manifold Structure regularization module (pMS), and the Information Fusion module (IF). The Manifold Projection module uses a pair of encoder-decoder networks to model the manifold projection step in equation (7), where the encoder learns the mapping from the reconstructed fingerprint to the corresponding tissue parameters, and the decoder re-projects the original reconstructed fingerprint onto the fingerprint manifold via the estimated tissue parameters. The Learned Sparsity module uses two independent convolutional neural networks and a soft threshold operator with channel-wise learnable thresholds to enhance the sparse feature constraints on the parameter map. In addition, the Manifold Projection module is coupled with the Learned Sparsity module to achieve flexible conversion between the fingerprint data domain and the tissue property domain. The p-norm enhanced Manifold Structure regularization module with a learnable norm order p dynamically extracts manifold structure information and further improves the reconstruction quality by using data-adaptive manifold smoothness constraints. The Information Fusion module realizes the effective fusion of data consistency regarding the physical acquisition model and the manifold structure information learned by the p-norm enhanced Manifold Structure regularization module.

[0037] Furthermore, the network constructed by training with a total of 408 sets of magnetic resonance fingerprint data in the above step (5) is used to obtain the optimal network parameters for subsequent fast and accurate reconstruction of magnetic resonance fingerprints and various tissue parameter maps.

[0038] Furthermore, by inputting the acquired undersampled k-space data into the trained network in the above step (6), the finally reconstructed magnetic resonance fingerprint data and the corresponding tissue parameter maps where N t represents the number of stages in the network.

[0039] The present invention has the following beneficial technical effects:

[0040] Technical key points of the present invention: (1) Obtain undersampled k-space data of magnetic resonance fingerprint; (2) Establish a magnetic resonance fingerprint reconstruction model based on the prior of manifold structure in the fingerprint domain and the sparse prior in the tissue property domain; (3) Use the dominant minimization and proximal gradient algorithm to iteratively solve the reconstruction model; (4) Adopt an effective strategy to obtain the closed-form solution of the proximal mapping sub-problem and expand all iterative steps into a deep neural network; (5) Train the network with existing data to obtain a network model; (6) Use the trained network to reconstruct magnetic resonance fingerprint and various tissue parameter maps. By expanding the network, the present invention can make full use of the structured geometric information of fingerprint data and the spatial features of tissue parameter maps, and can present excellent reconstruction effects under the condition of high-fold undersampling of data.

[0041] The present invention is a magnetic resonance fingerprint reconstruction method based on the dual-domain geometric structure and spatial prior under the deep learning theory. The magnetic resonance fingerprint reconstruction method based on the deep unfolding of the dual-domain geometric structure and spatial prior provided by the present invention can quickly reconstruct high-quality magnetic resonance fingerprint signals and various tissue parameter maps from highly undersampled data. The algorithm proposed by the present invention has high computational efficiency and can be conveniently accelerated by using GPU, which is an efficient algorithm. Experimental results show that the present invention is superior to several advanced methods in terms of reconstruction accuracy, and at the same time maintains a processing time comparable to that of existing deep learning-based methods. Description of the Drawings

[0042] Figure 1 It is a flowchart of the method of the present invention.

[0043] Figure 2 It is a schematic structural diagram of the network proposed by the present invention. The top of the image is the overall structure of the network composed of N t stages. Each stage includes a (a) Manifold Projection module (MP), a (b) Learned Sparsity module (LS), a (c) p-norm enhanced Manifold Structure regularization module (pMS), and a (d) Information Fusion module (IF).

[0044] Figure 3 It is a schematic structural diagram of the basic block in the manifold projection module and the channel attention-based soft threshold operator in the learned sparsity module. (a) Structures of basic block I and basic block II, (b) Structure of the channel attention-based soft threshold operator.

[0045] Figure 4This is a comparison chart of the reconstruction results of the method of the present invention and other advanced methods on a set of test data. Detailed implementation manners

[0046] The present invention will be described in detail below in conjunction with the accompanying drawings and examples.

[0047] The flowchart of the method of the present invention is shown in Figure 1 , and the specific implementation steps are as follows:

[0048] (1) Obtain undersampled k-space data of magnetic resonance fingerprint.

[0049] (2) Establish a magnetic resonance fingerprint reconstruction model based on the prior of the manifold structure in the fingerprint domain and the sparse prior in the tissue characteristic domain.

[0050] (3) Use the leading minimization and proximal gradient algorithms to iteratively solve the reconstruction model.

[0051] (4) Adopt an effective strategy to obtain a closed-form solution of the proximal mapping sub-problem, and expand all iterative steps into a deep neural network.

[0052] (5) Use the existing data to train the network to obtain a network model.

[0053] (6) Use the trained network to reconstruct magnetic resonance fingerprints and various tissue parameter maps.

[0054] The above step (1) is carried out in the following manner:

[0055] Use a magnetic resonance scanner to obtain undersampled k-space data of magnetic resonance fingerprint. The data acquisition process can be expressed as:

[0056]

[0057] Among them, represents the undersampled k-space data collected, N c represents the number of coils used in the acquisition, N s represents the number of samples collected by each coil, and L represents the length of the magnetic resonance fingerprint data; represents the magnetic resonance fingerprint data to be reconstructed, N v represents the number of voxels involved (i.e., the product of the length and width dimensions of the magnetic resonance image); represents the noise introduced during the acquisition process; represents the k-space undersampling operator related to the coil sensitivity.

[0058] The above step (2) is carried out in the following manner:

[0059] The magnetic resonance fingerprint reconstruction model based on the prior of the manifold structure in the fingerprint domain and the sparse prior in the tissue characteristic domain can be expressed as:

[0060]

[0061] Among them, represents the geometric structure of the magnetic resonance fingerprint data, that is, the fingerprint manifold; represents the prior regularization term of the fingerprint domain manifold structure; represents the sparse prior regularization term of the tissue property domain; λ1 and λ2 are adjustable regularization parameters. The prior regularization term of the fingerprint domain manifold structure and the sparse prior regularization term of the tissue property domain can be specifically expressed as:

[0062]

[0063] Among them, represents a certain sparsifying transformation; represents the mapping from the fingerprint to the tissue parameters, satisfying represents l tissue parameter maps; ||·|| p (0 < p ≤ 2) and ||·||1 respectively represent the element-wise L p and L1 norms. According to the central position the magnetic resonance fingerprint data X is divided into N p overlapping spatial blocks with a spatial size of d×d. These image blocks are then arranged into a matrix Each row of this matrix corresponds to a vectorized image block B is the weighted incidence matrix corresponding to the discrete approximation graph of the manifold of which the node set ( represents the number of nodes), the edge set (|ε| represents the number of edges), and the weight elements in the weighted adjacency matrix W are calculated from the similarity between the parameter blocks, expressed as follows:

[0064]

[0065] Among them, the hyperparameter σ is used to adjust the similarity weight. The element in the s-th row and t-th column of B corresponds to the edge e s and the node v t , defined as follows:

[0066]

[0067] The above step (3) is carried out in the following manner:

[0068] The magnetic resonance fingerprint reconstruction model constructed in step (2) can be solved iteratively using the majorization-minimization and proximal gradient algorithms. At the k-th iteration, the following three iterative steps need to be solved:

[0069]

[0070] Among them, and respectively represent and the adjoint operators of; represents the projection operator on the fingerprint manifold; μ is the step size of gradient descent; W (k-1) is resized to be the same dimension as ⊙ represents the Hadamard product.

[0071] The above step (4) is carried out in the following manner:

[0072] For the iterative step shown in equation (6), it is corrected to be carried out in the following order successively:

[0073]

[0074] Among them, Based on the imaging theory of magnetic resonance fingerprint, the proximal mapping sub-problem shown in equation (8) is adjusted, and the adjusted form is as follows:

[0075]

[0076] Among them, θ = α / 2μ, and α is used to correct this approximation relationship. The closed-form solution of this problem is shown as follows:

[0077]

[0078] Among them, soft(x,λ) = sign(x)max(0,|x|-λ) represents the soft threshold operator with a threshold of λ; represents the adjoint operator of, represents the mapping from tissue parameters to fingerprints.

[0079] The present invention expands the above model-based iterative algorithm into a deep neural network, attached Figure 2 ​Shows the schematic diagram of the network structure. Each iteration step involving the three sub-problems shown in Eqs. (7)-(9) is expanded into a stage of the deep network. Each stage contains four modules, namely: the Manifold Projection module (MP), the Learned Sparsity module (LS), the p-norm enhanced Manifold Structure regularization module (pMS), and the Information Fusion module (IF). The Manifold Projection module uses a pair of encoder-decoder networks to model the manifold projection step in Eq. (7), where the encoder learns the mapping from the reconstructed fingerprint to the corresponding tissue parameters, and the decoder re-projects the original reconstructed fingerprint onto the fingerprint manifold via the estimated tissue parameters. The detailed structure can be found in Appendix Figure 2 (a) and Appendix Figure 3 (a). The Learned Sparsity module uses two independent convolutional neural networks and a soft threshold operator with channel-wise learnable thresholds to enhance the sparse feature constraints on the parameter map. The detailed structure can be found in Appendix Figure 2 (b) and Appendix Figure 3 (b). In addition, the Manifold Projection module is coupled with the Learned Sparsity module to achieve flexible conversion between the fingerprint data domain and the tissue property domain. The p-norm enhanced Manifold Structure regularization module with a learnable norm order p dynamically extracts manifold structure information and further improves the reconstruction quality using data-adaptive manifold smoothness constraints. The detailed structure can be found in Appendix Figure 2 (c). The Information Fusion module realizes the effective fusion of data consistency regarding the physical acquisition model and the manifold structure information learned by the p-norm enhanced Manifold Structure regularization module. For details, see Appendix Figure 2 (d).

[0080] The above step (5) is carried out as follows:

[0081] The constructed network is trained using a total of 408 sets of magnetic resonance fingerprint data to obtain the optimal network parameters for subsequent rapid and accurate reconstruction of magnetic resonance fingerprints and various tissue parameter maps.

[0082] The above step (6) is carried out as follows:

[0083] The acquired undersampled k-space data is input into the trained network to obtain the finally reconstructed magnetic resonance fingerprint data and the corresponding tissue parameter maps where N t represents the number of stages in the network.

[0084] To quantitatively evaluate the performance of the method proposed in the present invention, the Peak Signal-to-Noise Ratio (PSNR) and the Normalized Mean Square Error (NMSE) were used in the experiment to measure the quality of the reconstructed magnetic resonance fingerprint and the accuracy of the estimated multi-tissue parameter maps, respectively. The PSNR is defined as follows:

[0085]

[0086] where N = N v ×L represents the total number of pixels in the magnetic resonance fingerprint data, and X R represents the true magnetic resonance fingerprint data, and MAX is its maximum value. The NMSE is defined as follows:

[0087]

[0088] where Μ R represents the true tissue parameter map.

[0089] Figure 4 Fig. shows the comparison of the reconstruction results of the method proposed in the present invention and other advanced methods on a set of test data. Each column from left to right shows the true parameter map and the parameter maps reconstructed by FLOR[5], SCQ[6], TLR-BM[7], and the method proposed in the present invention. The first row and the third row show the spin-lattice relaxation time (T1) and spin-spin relaxation time (T2) parameter maps reconstructed by each method, respectively. The second row and the fourth row show the corresponding Normalized Mean Square Error (NMSE) maps, respectively. The numbers in parentheses below the second row and the fourth row represent the average values of the corresponding NMSE maps. It can be seen from the figure that the method proposed in the present invention has the smallest reconstruction error.

[0090] As shown in Table 1, Table 1 shows the quantitative comparison results of the method of the present invention and other advanced methods on the test data set.

[0091] Table 1 Quantitative comparison results of different methods on the test data set

[0092]

[0093] Table 1 shows the quantitative comparison results of the method proposed in the present invention and other advanced methods on the test data set, including the peak signal-to-noise ratio of the reconstructed magnetic resonance fingerprint data, the normalized mean square error of the reconstructed parameter map, and the average processing time. The quantitative comparison results show that the present invention is superior to other advanced methods in terms of reconstruction accuracy, while maintaining a processing time comparable to that of existing deep learning-based methods.

[0094] The details of the references cited in this invention are as follows:

[0095] [1] D. Ma, V. Gulani, N. Seiberlich, K. Liu, J. L. Sunshine, J. L. Duerk, and M. A. Griswold, “Magnetic resonance fingerprinting,” Nature, vol. 495, no. 7440, pp. 187–192, Mar. 2013.

[0096] [2] V. Monga, Y. Li, and Y. C. Eldar, “Algorithm unrolling: Interpretable, efficient deep learning for signal and image processing,” IEEE Signal Process. Mag., vol. 38, no. 2, pp. 18–44, Mar. 2021.

[0097] [3] Y. Sun, P. Babu, and D. P. Palomar, “Majorization-minimization algorithms in signal processing, communications, and machine learning,” IEEE Trans. Signal Process., vol. 65, no. 3, pp. 794–816, Aug. 2016.

[0098] [4] N. Parikh and S. Boyd, “Proximal algorithms,” Found. Trends Optim., vol. 1, no. 3, pp. 127–239, Jan. 2014.

[0099] [5] G. Mazor, L. Weizman, A. Tal, and Y. C. Eldar, “Low-rank magnetic resonance fingerprinting,” Med. Phys., vol. 45, no. 9, pp. 4066–4084, Sep. 2018.

[0100] [6]Z.Fang,Y.Chen,M.Liu,L.Xiang,Q.Zhang,Q.Wang,W.Lin,and D.Shen,“Deeplearning for fast and spatially constrained tissue quantification from highlyaccelerated data in magnetic resonance fingerprinting,”IEEE Trans.Med.Imag.,vol.38,no.10,pp.2364–2374,Oct.2019.

[0101] [7]P.Li and Y.Hu,“Learned tensor low-CP-rank and bloch responsemanifold priors for non-Cartesian MRF reconstruction,”IEEE Trans.Med.Imag.,vol.42,no.12,pp.3702–3714,Dec.2023.

Claims

1. A magnetic resonance fingerprint reconstruction method based on a dual-domain geometric structure and spatial prior depth unfolding, characterized in that, The method includes the following steps: (1) Obtain undersampled k-space data of magnetic resonance fingerprint; (2) Establish a magnetic resonance fingerprint reconstruction model based on the prior of the manifold structure in the fingerprint domain and the sparse prior in the tissue property domain; (3) Iteratively solve the reconstruction model by using the majorization-minimization and proximal gradient algorithms; (4) Adopt an effective strategy to obtain the closed-form solution of the proximal mapping sub-problem, and expand all iterative steps into a deep neural network; (5) Train the network with existing data to obtain a network model; (6) Use the trained network to reconstruct magnetic resonance fingerprint and various tissue parameter maps.

2. The method according to claim 1, characterized in that In step (1), use a magnetic resonance scanner to obtain undersampled k-space data of magnetic resonance fingerprint, and the data acquisition process can be expressed as: in, represents the acquired k-space undersampled data, N c Indicates the number of coils used during acquisition, N s represents the number of samples collected by each coil, and L represents the length of the magnetic resonance fingerprint data; Represents the magnetic resonance fingerprint data to be reconstructed, N v represents the number of voxels involved, i.e., the product of the length and width of the magnetic resonance image; represents the noise introduced during the acquisition process; represents the k-space undersampling operator related to coil sensitivity.

3. The method according to claim 1, wherein The specific implementation manner of step (2) above is as follows: The magnetic resonance fingerprint reconstruction model based on the prior of the manifold structure in the fingerprint domain and the sparse prior in the tissue property domain can be expressed as: in, The geometric structure representing the magnetic resonance fingerprint data, i.e., the fingerprint manifold; Represents the prior regularization term of the manifold structure of the fingerprint domain; represents the sparse prior regularization term in the tissue characteristic domain; λ1 and λ2 are adjustable regularization parameters; the fingerprint domain manifold structure prior regularization term and the tissue characteristic domain sparse prior regularization term can be specifically expressed as: wherein, represents a certain sparsification transformation; represents a mapping from fingerprints to tissue parameters, satisfying represents l tissue parameter maps; ||·|| p (0 < p ≤ 2) and ||·||1 respectively represent the element-wise L p and L1 norms; according to the central position the magnetic resonance fingerprint data X is divided into N p overlapping spatial blocks with a spatial size of d×d; these image blocks are then arranged into a matrix each row of this matrix corresponds to a vectorized image block B is the weighted adjacency matrix corresponding to the discrete approximation graph of the manifold of the graph, the node set of the graph (| | represents the number of nodes), the edge set (| | represents the number of edges), and the weight elements in the weighted adjacency matrix W are calculated from the similarity between parameter patches, expressed as follows: Among them, the hyperparameter σ is used to adjust the similarity weight; the element in the sth row and tth column of B corresponds to the edge e s and node v t , defined as follows:

4. The method according to claims 1 and 3, characterized in that, The specific implementation manner of step (3) above is as follows: The magnetic resonance fingerprint reconstruction model constructed in step (2) can be iteratively solved by using the majorization-minimization and proximal gradient algorithms. In the k-th iteration, the following three iterative steps need to be solved: Among them, and respectively represent and adjoint operators; represents the projection operator on the fingerprint manifold; μ is the step size of gradient descent; W (k-1) is readjusted to be of the same dimension as ⊙ represents the Hadamard product.

5. The method according to claims 1 and 4, characterized in that The above step (4) is carried out in the following manner: For the iterative step shown in formula (6), it is corrected to be carried out in the following order successively: in, Based on the imaging theory of magnetic resonance fingerprint, the proximal mapping subproblem shown in equation (8) is adjusted and expressed as follows: where θ = α / 2μ, and α is used to correct this approximation relationship; the closed-form solution of this problem is expressed as follows: where soft(x,λ)=sign(x)max(0,|x|-λ) represents the soft threshold operator with a threshold of λ; denotes the adjoint operator of represents the mapping from tissue parameters to fingerprints; The above model-based iterative algorithm is expanded into a deep neural network; each iteration step involving the three sub-problems shown in equations (7)-(9) is expanded into a stage of the deep network; each stage contains four modules, namely: the manifold projection module MP, the learnable sparse module LS, the p-norm enhanced manifold structure regularization module pMS, and the information fusion module IF; the manifold projection module uses a pair of encoder-decoder networks to model the manifold projection step in equation (7), where the encoder learns the mapping from the reconstructed fingerprint to the corresponding tissue parameters, and the decoder completes the re-projection of the original reconstructed fingerprint onto the fingerprint manifold via the estimated tissue parameters; the learnable sparse module uses two independent convolutional neural networks and a soft threshold operator with channel-wise learnable thresholds to enhance the sparse feature constraints on the parameter map; The manifold projection module is coupled with the learnable sparse module to achieve flexible conversion between the fingerprint data domain and the tissue property domain; the p-norm enhanced manifold structure regularization module with a learnable norm order p dynamically extracts manifold structure information, and uses the data-adaptive manifold smoothness constraint to improve the reconstruction quality; the information fusion module is used to achieve effective fusion of the data consistency regarding the physical acquisition model and the manifold structure information learned by the p-norm enhanced manifold structure regularization module.

6. The method according to claim 5, wherein The specific implementation manner of step (5) above is as follows: The constructed network is trained using a total of 408 groups of magnetic resonance fingerprint data to obtain the best network parameters for subsequent fast and accurate reconstruction of magnetic resonance fingerprint and various tissue parameter maps.

7. The method according to claim 2 and 6, characterized in that, The specific implementation manner of step (6) above is as follows: Input the acquired undersampled k-space data into the trained network to obtain the finally reconstructed magnetic resonance fingerprint data and the corresponding tissue parameter maps where N t represents the number of stages in the network.

8. A magnetic resonance fingerprint reconstruction system based on a dual-domain geometric structure and spatial prior depth unfolding, characterized in that: The system has program modules corresponding to the steps of any one of the above claims 1-7, and executes the steps in the above magnetic resonance fingerprint reconstruction method based on dual-domain geometric structure and spatial prior depth unfolding when running.

9. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores a computer program, and the computer program is configured to implement the steps of a magnetic resonance fingerprint reconstruction method based on dual-domain geometric structure and spatial prior depth unfolding as described in any one of claims 1-7 when called by a processor.

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