An MRI image reconstruction method based on the estimation of structural group tensor singular values

Abstract

The present invention discloses an MRI image reconstruction method based on the estimation of structural group tensor singular values, belonging to the technical field of digital image processing. It is an MRI image reconstruction method that constructs a structural group tensor for a set of similar image patches and uses multi-layer priors to estimate the tensor singular values. First, a third-order structural group tensor of the set of similar image patches of the target image patch is constructed, then the third-order tensor is unfolded into a matrix according to the Tucker mode and the multi-layer prior model is used to estimate the tensor low-rank regularization term, and finally an MRI image reconstruction model based on the estimation of structural group tensor singular values is constructed and solved using the alternating direction iteration method; the present invention represents the set of similar image patches in tensor form, which can fully retain the structural characteristics of the image patches themselves and the correlation between the patches. Unfolding the tensor into a matrix according to the mode to estimate its rank can reduce the computational complexity, and the multi-layer priors are used to efficiently and accurately estimate the tensor singular values. The MRI images reconstructed by the present invention are clearer and have fewer artifacts, so it can be used for the reconstruction of medical images.

Description

Technical Field

[0001] The present invention belongs to the technical field of digital image processing, and particularly relates to an MRI image reconstruction method for constructing a third-order structure group tensor, unfolding the structure group tensor into a matrix according to the Tucker mode, and estimating the tensor singular value by using a multi-layer prior model, which is used for high-quality restoration of medical images. Background Art

[0002] Magnetic resonance imaging (MRI) is a method for drawing an internal image of an object by using the principle of nuclear magnetic resonance and realizing an electromagnetic wave emitted by an externally applied gradient magnetic field, and has been widely used in the fields of medicine, chemical industry, aerospace, etc. Compared with the previous medical imaging technologies, magnetic resonance imaging has higher resolution for soft tissues, no ionizing radiation damage to the human body, and higher imaging flexibility. The reconstructed image is the closest to the actual anatomical structure of the human body. Therefore, it has currently become one of the most important medical imaging technologies. However, the slow scanning rate of magnetic resonance imaging will cause motion artifacts to appear in the reconstructed image due to the physiological movement of the subject, resulting in a decrease in imaging quality and a great interference to the diagnostic information, which has hindered the further development of magnetic resonance imaging technology to a certain extent.

[0003] To shorten the magnetic resonance imaging time, in addition to improving the hardware performance of the system itself to accelerate the data sampling speed, the MRI method based on the compressed sensing theory (CS-MRI) acquires signals in an undersampled manner and simultaneously performs signal acquisition and data compression, thereby greatly reducing the scanning time. The previous CS-MRI reconstruction methods based on two-dimensional structure group matrices effectively suppress noise and achieve good reconstruction effects. However, these methods only utilize the non-local similarity of images and ignore the structural characteristics and internal correlations of the image blocks themselves. Considering that tensors have the advantage of retaining the potential structure of multi-dimensional data, how to accurately and effectively represent image information based on tensors is the key to further improving the image reconstruction quality.

[0004] In addition, effectively constraining the low-rank characteristics presented by non-local similar image blocks and accurately estimating the singular value also affect the final reconstruction quality. Most of the existing methods use the nuclear norm to achieve low-rank constraints and have achieved certain results. However, this empirical low-rank regular term lacks adaptability to different images. The image reconstruction method based on Bayesian estimation can more accurately and efficiently characterize the prior information of different images. Therefore, how to design a suitable regular term by using statistical ideas is the key to effectively constraining the low-rank characteristics. Summary of the Invention

[0005] The object of the present invention is to propose an MRI image reconstruction method based on the estimation of the singular values of the structural group tensor in view of the deficiencies existing in the existing MRI image reconstruction methods. This method fully considers the structural characteristics of similar image patches themselves and the non-local similarity between similar image patches, represents the structural group obtained from the set of similar image patches by using a third-order tensor, unfolds the third-order tensor into three matrices according to the Tucker mode, and adaptively estimates the singular values of the structural group tensor by using a multi-layer prior, so as to improve the estimation efficiency and accuracy. The specific steps are as follows:

[0006] (1) Input an original MRI K-space observation data, perform an initial reconstruction of the input data y by using traditional compressive sensing, and obtain an initial reconstructed image

[0007] (2) Extract the target image patches from the initially reconstructed image Then, within the search area centered on the target image patch calculate and compare the Euclidean distances between all the image patches in this area and the target image patch to obtain the image patches similar to the target image patch and stack them in the form of a two-dimensional matrix to form an observed structural group tensor where denotes the operator for extracting similar image patches from the image to construct the i-th structural group tensor;

[0008] (3) The i-th structural group tensor in the image x to be reconstructed has the following relationship with the observed structural group tensor :

[0009]

[0010] where ε i denotes the noise tensor and is assumed to follow a Gaussian distribution with a mean of 0 and a variance of σ 2 ; then unfold the third-order structural group tensor into matrices according to the Tucker mode-k where k can be 1, 2, and 3, and unfold k (·) denotes the operator for unfolding a third-order tensor into a matrix according to the Tucker mode-k;

[0011] (4) Perform singular value decomposition on the three matrices unfolded in step (3) respectively, where k is 1, 2, and 3, that is:

[0012]

[0013] where P i,k and V i,k ​are respectively the left and right singular vector matrices, where \(i\) represents the serial number of the structure group, \(k\) represents the mode number of the Tucker mode expansion of the tensor, and \([\cdot]^H\) H denotes the conjugate transpose operation of a matrix; \(\sum\) i,k is the singular value matrix of, which can be specifically expressed as:

[0014]

[0015] where \(diag(\cdot)\) represents the vector diagonalization operator, denotes the first to the \(r\)th singular values of, and \(r\) represents the number of singular values of;

[0016] (5) Use the multi-layer prior model to adaptively estimate the singular values of the structure group tensor The specific steps are as follows:

[0017] (5a) Assume that any singular value of follows a Gaussian distribution with zero mean. Therefore, the probability density function of is expressed as:

[0018]

[0019] where denotes the variance of the Gaussian distribution with respect to, and \(exp(\cdot)\) represents the exponential function with the natural constant as the base; further constrain the sparsity, assume that the variance follows an exponential distribution. Therefore, the probability density function of is expressed as:

[0020]

[0021] where denotes the expectation of the exponential distribution with respect to; since is an unknown hyperparameter, assume it follows an uninformative Jeffreys distribution. Therefore, the probability density function of is expressed as:

[0022]

[0023] (5b) Based on step (5a), the joint probability distribution of can be obtained: is:

[0024]

[0025] Therefore, with respect to Pi,k , V i,k The joint maximum a posteriori probability estimate is:

[0026]

[0027]

[0028] where denotes the square of the Frobenius norm of the matrix;

[0029] (5c) Since the sum of reciprocals and logarithms of variables makes the optimization process unstable, three auxiliary variables are introduced The joint maximum a posteriori probability estimate can be rewritten as:

[0030]

[0031]

[0032] where denotes the square of the vector two-norm, and respectively denote the vectors corresponding to the auxiliary variables and [·] T denotes the transpose operation of the matrix; A i,k = diag(α i,k )、Q i,k = diag(q i,k ) and C i,k = diag(c i,k ) respectively denote three diagonal matrices with diagonal elements a i,k 、q i,k and c i,k ; ε is a small positive constant to maintain logarithmic stability, log(q i,k + ε) and log(c i,k + ε) respectively denote and l represents the sequence number of the elements in the vector;

[0033] (6) Based on the joint maximum a posteriori probability estimate, the augmented Lagrangian function corresponding to the CS-MRI reconstruction model based on tensor singular value estimation can be expressed as:

[0034]

[0035] where λ represents the regularization parameter, F u denotes the downsampled Fourier encoding matrix, x represents the image to be reconstructed, y represents the downsampled K-space data, B i,k denotes the matrix

[0034]

[0035] where λ represents the regularization parameter, F u denotes the downsampled Fourier encoding matrix, x represents the image to be reconstructed, y represents the downsampled K-space data, B i,k denotes the matrix denotes the matrix The corresponding Lagrange multiplier, β represents the penalty parameter, i represents the serial number of the structure group, and k represents the mode number of the Tucker mode expansion of the tensor; the reconstruction model is solved according to the alternating direction iteration method, and taking x, α i,k ,q i,k ,c i,k ,P i,k ,V i,k the seven variables as the optimization objects to minimize the augmented Lagrangian function:

[0036] (6a) The sub-problem with respect to P i,k and V i,k can be expressed as:

[0037]

[0038] Performing singular value decomposition on yields:

[0039]

[0040] where U i,k and G i,k are the left singular vector matrix and the right singular vector matrix of respectively, and Γ i,k is the singular value matrix of ; the optimal solution of the sub-problem with respect to P i,k and V i,k is:

[0041] P i,k = U i,k

[0042] V i,k = G i,k

[0043] (6b) The sub-problem with respect to α i,k can be expressed as:

[0044]

[0045] Substituting the optimal solutions of P i,k and V i,k into it, this sub-problem can be rewritten as:

[0046]

[0047] Furthermore, this sub-problem can be decomposed into a scalar optimization problem with respect to :

[0048]

[0049] where is Γ i,k the l-th diagonal element of; Using the first-order optimization condition, a closed-form solution to the scalar optimization problem regarding can be obtained;

[0050] (6c) The sub-problem regarding q i,k can be expressed as:

[0051]

[0052] Substituting the optimal solutions of P i,k and V i,k , this sub-problem can be rewritten as:

[0053]

[0054] Similarly, this sub-problem can be decomposed into a scalar optimization problem regarding for solution;

[0055] (6d) The sub-problem regarding c i,k can be expressed as:

[0056]

[0057] The solution method for this sub-problem is the same as that for the sub-problem of q i,k ;

[0058] (6e) The sub-problem regarding can be expressed as:

[0059]

[0060] This sub-problem is a least squares problem, and the closed-form solution of can be directly obtained by solving the corresponding normal equations;

[0061] (6f) The sub-problem regarding x can be expressed as:

[0062]

[0063] This sub-problem is a least squares problem and can be solved using the conjugate gradient method;

[0064] (7) Repeat step (6) while updating the Lagrange multiplier B i,k , the penalty parameter β, and the variance σ 2 . After a certain number of iterations, the finally reconstructed MRI image can be obtained.

[0065] The innovation of the present invention is to directly stack similar image patches in the form of a two-dimensional matrix to construct a third-order structured group tensor; expand the third-order structured group tensor into three matrices according to the Tucker mode and use a multi-layer prior model to estimate the tensor singular values, improving the estimation efficiency and accuracy; use the alternating direction iteration method to solve the entire reconstruction model and apply this method to the reconstruction of magnetic resonance images (MRI).

[0066] The beneficial effects of the present invention are as follows: using tensors to represent a set of similar image patches fully preserves the structural characteristics of the image patches themselves and the mutual correlation between the image patches; converting the rank of a high-order tensor into the sum of the ranks of multiple matrices expanded according to the Tucker mode reduces the computational complexity; using a multi-layer prior model to estimate the tensor singular values improves the estimation accuracy. Therefore, the finally estimated image not only has a good overall visual effect but also retains a large amount of details inside the image, making the entire estimation result closer to the true value, and the running time is also within an acceptable range.

[0067] The present invention is mainly verified by means of simulation experiments, and all steps and conclusions are verified to be correct on MATLAB9.5. Brief Description of the Drawings

[0068] Figure 1 is the workflow block diagram of the present invention;

[0069] Figure 2 is the original MRI cardiac image used in the simulation of the present invention;

[0070] Figure 3 is the reconstruction result of the MRI image with a sampling rate of 10% using the FDLCP method;

[0071] Figure 4 is the reconstruction result of the MRI image with a sampling rate of 10% using the PANO method;

[0072] Figure 5 is the reconstruction result of the MRI image with a sampling rate of 10% using the NLR method;

[0073] Figure 6 is the reconstruction result of the MRI image with a sampling rate of 10% using the BGSR method;

[0074] Figure 7 is the reconstruction result of the MRI image with a sampling rate of 10% using the method of the present invention. Detailed Embodiment

[0075] Referring to Figure 1 , the present invention is an MRI image reconstruction method based on the estimation of structured group tensor singular values, and the specific steps are as follows:

[0076] Step 1: Perform initial restoration on the image, construct the structural group tensor, and then expand the tensor in the Tucker mode-k.

[0077] (1a) Use the total variation method to perform initial reconstruction on the input original K-space data y to obtain the initial reconstructed image

[0078] (1b) Extract all target image patches from the initially reconstructed image Then, within the search range centered on the target image patch Calculate and compare the Euclidean distances between all image patches in this region and the target image patch The smaller the Euclidean distance, the higher the similarity degree. Through this method of matching similar image patches, find the first (m - 1) most similar image patches to the target image patch ; (1c) Stack the retained (m - 1) similar image patches and the target image patch

[0079] directly in the form of a two-dimensional matrix to construct the observed structural group tensor where represents the operator for extracting similar image patches from the image to construct the i-th structural group tensor, which can fully exploit the non-local similarity of the image and preserve the structural information of the image patches;

[0080] (1d) Since the non-local similar image patches share highly similar contour structures, obviously, there is also a very high correlation, that is, low-rank property, between the slices of the i-th structural group tensor in the image x to be reconstructed. Therefore, there is a relationship between the observed structural group tensor and the structural group tensor ; and compared with directly calculating the rank of a high-order tensor, the rank of the matrix obtained by expanding the tensor in the Tucker mode is easier to calculate. For this reason, expand the third-order structural group tensor in the Tucker mode-k into a matrix where k can be 1, 2, and 3. Thus, the low-rank constraint problem regarding the structural group tensor is transformed into the problem of estimating the singular values of these three matrices;

[0081] Step 2: Different from the previous methods based on empirical low-rank regularization, in order to more accurately characterize the low-rank property of the third-order structural group tensor theoretically, use Bayesian inference to adaptively estimate the singular values of the third-order structural group tensor ;

[0082] (2a) Perform singular value decomposition on the three matrices respectively:

[0083]

[0084] Intuitively, the more zeros there are in the singular values, the smaller the rank of the matrix. To statistically describe the structural sparsity of the singular value matrix ∑ i,k , according to 's low-rank property, assume that 's any singular value follows a Gaussian distribution with a mean of zero and a variance of :

[0085]

[0086] where represents the variance of the Gaussian distribution with respect to , and exp(·) represents the exponential function with the natural constant as the base; to enhance 's sparsity, further assume that the variance of the Gaussian distribution follows an exponential distribution with an expectation of :

[0087]

[0088] where represents the expectation of the exponential distribution with respect to ; to facilitate the determination of the unknown hyperparameter , assume that it follows an uninformative Jeffreys distribution:

[0089]

[0090] It is not difficult to see that when tends to infinity, tends to zero. Therefore, the prior distribution of this layer can also further promote the low-rank property of the structural group tensor ;

[0091] (2b) After constructing the multi-layer prior model, the joint probability distribution of can be obtained:

[0092]

[0093] Substituting the probability density functions of the prior distributions of each layer into the above formula, we can get:

[0094]

[0095] Therefore, the joint maximum a posteriori probability estimation of P i,k ,V i,k :

[0096]

[0097]

[0098] where represents the square of the matrix Frobenius norm;

[0099] (2c) Since the sum of reciprocals and logarithms of variables makes the optimization process unstable, three auxiliary variables are introduced The joint maximum a posteriori probability estimation can be rewritten as:

[0100]

[0101]

[0102] where represents the square of the vector two-norm, and respectively represent the vectors corresponding to the auxiliary variables and respectively; [·] T represents the transpose operation of the matrix; A i,k = diag(α i,k ), Q i,k = diag(q i,k ) and C i,k = diag(c i,k ) respectively represent three diagonal matrices with diagonal elements a i,k , q i,k and c i,k respectively; ε is a small constant greater than zero to maintain logarithmic stability, log(q i,k +ε) and log(c i,k +ε) respectively represent and l represents the element number in the vector; it can be seen that

[0103]

[0104] is the estimation of the singular value of the structure group tensor through the multi-layer prior model; it should be noted that in equation (23), the singular value matrix Σ i,k is expressed as the product of three diagonal matrices C i,k , Q i,k and A i,k , that is, Σ i , k = C i,k Q i,k A i,k , and C i,k and Q i,kSparse constraints are imposed on their respective diagonal elements by logarithmic penalty terms, which also verifies the low-rank property of the structure group tensor ;

[0105] Step 3: Establish a CS-MRI reconstruction model based on the singular value estimation of the structure group tensor.

[0106] (3a) Based on the joint maximum a posteriori probability estimation, the augmented Lagrangian function corresponding to the CS-MRI reconstruction model based on tensor singular value estimation can be expressed as:

[0107]

[0108] where λ represents the regularization parameter, F u represents the downsampled Fourier encoding matrix, x represents the image to be reconstructed, y represents the downsampled K-space data, B i,k represents the Lagrange multiplier corresponding to the matrix , β represents the penalty parameter, i represents the serial number of the structure group, and k represents the mode number of the tensor Tucker mode expansion; the reconstruction model is solved according to the alternating direction iteration method, and the augmented Lagrangian function is minimized with the seven variables of x, α i,k , q i,k , c i,k , P i,k , V i,k as the optimization objects:

[0109] 3a1) The sub-problem regarding P i,k and V i,k can be expressed as:

[0110]

[0111] Performing singular value decomposition on , we can obtain:

[0112]

[0113] where U i,k and G i,k are the left singular vector matrix and the right singular vector matrix of respectively, and Γ i,k is the singular value matrix of ; then according to the von Neumann trace inequality, we have

[0114]

[0115]

[0116]

[0117] The condition for the von Neumann trace inequality to achieve equality is exactly P i,k and V i,k The optimal solution of the sub - problem is:

[0118] P i,k = U i,k

[0119] V i,k = G i,k

[0120] 3a2) The sub - problem regarding α i,k can be expressed as:

[0121]

[0122] Combining the optimal solutions of P i,k and V i,k and Also, because P i,k and V i,k are unitary matrices, according to the unitary invariance of the Frobenius norm, this sub - problem can be transformed into:

[0123]

[0124] Since each coefficient in each diagonal matrix is uncorrelated, this sub - problem can be further decomposed into a scalar optimization problem regarding :

[0125]

[0126] where is the l - th diagonal element of Γ i,k ; Using the first - order optimization condition, the closed - form solution of the scalar optimization problem regarding can be obtained, that is:

[0127]

[0128] 3a3) The sub - problem regarding q i,k can be expressed as:

[0129]

[0130] Following the same simplification steps as in 3a2), this sub - problem can be rewritten as:

[0131]

[0132] Similarly, this sub - problem can be decomposed into a scalar optimization problem regarding for solution:

[0133]

[0134] where the auxiliary variable is non - negative, and the scalar - form optimization problem can be simplified to the following form:

[0135]

[0136] where it can be seen that \(f(q)\) is a typical univariate quadratic optimization problem with parameters, and the optimal solution can be obtained by solving the first - order optimization condition and its non - negative local minimum value is:

[0137]

[0138] By comparing the magnitudes of \(f(0)\) and \(f(q * ) to obtain the global optimal solution of the sub - problem of \(q i,k , the solutions for all cases are given as follows:

[0139]

[0140] 3a4) The sub - problem regarding \(c i,k can be expressed as:

[0141]

[0142] The solution method of this sub - problem is the same as that of \(q i,k , where

[0143] 3a5) The sub - problem regarding can be expressed as:

[0144]

[0145] According to where \(fold k (·)\) represents the inverse operation of \(unfold k (·)\), aggregating the inverse process of unfolding the matrix in Tucker mode into a tensor, this sub - problem can be transformed into:

[0146]

[0147] The normal equation corresponding to this least - squares problem is:

[0148]

[0149] The closed - form solution of can be directly obtained by matrix inversion, that is:

[0150]

[0151] 3a6) The sub - problem regarding x can be expressed as:

[0152]

[0153] This sub - problem is a least - squares problem, and the conjugate gradient method can be used to obtain the optimal solution:

[0154]

[0155] (3b) It can be seen from the objective expression in step 3a5) that the reciprocal of the variance 1 / σ 2 controls the data fidelity term. Therefore, 1 / σ 2 and the penalty parameter β play the same role; repeat step (3a), while updating the Lagrange multiplier B i,k , the penalty parameter β and the variance σ 2 , that is

[0156]

[0157] β = ρ1β

[0158]

[0159] where ρ1 and ρ2 are constants greater than 1, exponentially amplifying the penalty parameter β and the reciprocal of the variance 1 / σ 2 respectively; after a certain number of iterations, the finally reconstructed MRI image can be obtained.

[0160] The effect of the present invention can be further illustrated by the following simulation experiments:

[0161] I. Experimental conditions and content

[0162] Experimental conditions: The two - dimensional random sampling model is used in the experiment; the experimental images adopt real MRI images, as Figure 2 shown; the peak signal - to - noise ratio PSNR and structural similarity SSIM are used as evaluation indexes of the experimental results to objectively evaluate the reconstruction results. The higher the PSNR value and SSIM value, the better the reconstruction result and the closer it is to the real image.

[0163] Experimental content: Under the above experimental conditions, the reconstruction results are compared between the present invention method and the representative FDLCP method, PANO method, NLR method and BGSR method in the field of MRI image reconstruction.

[0164] Experiment 1: Use the method of the present invention and the FDLCP method, PANO method, NLR method and BGSR method respectively for Figure 2The sampled image is reconstructed. The FDLCP method is a method for estimating and learning an orthogonal dictionary from classified image patches using a geometric method, and its reconstruction result is Figure 3 ; The PANO method is a typical reconstruction method that performs three-dimensional wavelet transform on the structure group and constrains the sparse coefficients with the l1 norm, and its reconstruction result is Figure 4 ; The NLR method utilizes the low-rank property of the structure group and uses logdet(·) as the non-convex constraint term of the structure group, and its reconstruction result is Figure 5 , The BGSR method is a method for estimating sparse coefficients using statistical ideas and learning an orthogonal dictionary from classified image patches, and its reconstruction result is Figure 6 . In the experiment, the method of the present invention sets the image patch size The number of image patches m in the structure group is 32, and the regularization factor is set to λ = 10 for all methods 6 , The maximum number of iterations T = 100, and the iteration termination threshold η = 5×10 -8 ; The final reconstruction result is Figure 7 .

[0165] From Figure 3 , Figure 4 , Figure 5 , Figure 6 and Figure 7 By observing the reconstruction results and the enlarged local area diagrams of each method, it can be seen that by comparing the FDLCP method, the PANO method, the NLR method, and the BGSR method with the method of the present invention, the method of the present invention is superior to other comparison methods in terms of the details of the reconstruction result.

[0166] Table 1 PSNR indicators of different reconstruction methods

[0167] Image FDLCP Method PANO Method NLR Method BGSR Method Method of the Present Invention MRI Image 26.43 26.89 28.00 26.46 29.74

[0168] Table 1 shows the PSNR indicators of the reconstruction results of each method. The higher the PSNR value, the better the reconstruction effect; it can be seen that the PSNR value of the method of the present invention is significantly higher than that of other methods, and this result is consistent with the reconstruction effect diagram.

[0169] Table 2 SSIM indicators of different reconstruction methods

[0170] Image FDLCP Method PANO Method NLR Method BGSR Method Method of the Present Invention MRI Image 0.7179 0.7136 0.8181 0.7282 0.8306

[0171] Table 2 shows the SSIM indicators of the reconstruction results of each method. The higher the SSIM value, the closer the reconstruction result is to the real image; it can be seen that the SSIM value of the method of the present invention is the highest, indicating that the reconstruction result of this method is closer to the real image, and this result is consistent with the reconstruction effect diagram.

[0172] The above experiments show that the reconstruction method of the present invention not only has rich detail information and fewer artifacts, but also has high visual effects and objective evaluation indicators. Thus, it can be seen that the present invention is effective for medical image reconstruction.

Claims

1. An MRI image reconstruction method based on the estimation of structural group tensor singular values, comprising the following steps: (1) Input an original MRI K-space observation data, perform an initial reconstruction of the traditional compressive sensing on the input data y, and obtain an initial reconstructed image (2) In the initial reconstructed image Extract the target image block from Then take the target image block Within the search area centered on the image, all image blocks in the area are calculated and compared with the target image block. The Euclidean distance between the two images is used to obtain the target image patch. Similar image blocks are stacked in the form of a two-dimensional matrix to form an observation structure group tensor in represents an operator for extracting similar image patches from an image to construct the i-th structure group tensor; (3) The $i$-th structural group tensor in the image $x$ to be reconstructed and the observed structural group tensor have the following relationship: where ε i represents the noise tensor and is assumed to follow a Gaussian distribution with a mean of 0 and a variance of σ 2 ; then the third-order structure group tensor is unfolded into a matrix in the Tucker mode-k where k can be 1, 2, and 3, and unfold k (·) represents the operator that unfolds a third-order tensor into a matrix in the Tucker mode-k; (4)For the three matrices expanded in step (3) perform singular value decomposition respectively, where k is 1, 2, and 3, that is: where P i,k and V i,k are respectively the left and right singular vector matrices of H , i represents the serial number of the structural group, k represents the mode number of the tensor Tucker mode expansion, [·] i,k denotes the conjugate transpose operation of the matrix; ∑ is the singular value matrix of where diag(·) represents the vector diagonalization operator, denotes the first to the r-th singular values of and r represents the number of singular values of (5) Use a multi-layer prior model to adaptively estimate the singular values of the structure group tensor, and the specific steps are as follows: ​ (5a) Assume any singular value of obeys a Gaussian distribution with zero mean, so the probability density function of is expressed as: where represents the variance of the Gaussian distribution with respect to , exp(·) represents the exponential function with the natural constant as the base; to further constrain sparsity, assume that the variance follows an exponential distribution, so the probability density function of is expressed as: where represents the expectation of the exponential distribution with respect to ; since is an unknown hyperparameter and is assumed to follow the non-informative Jeffreys distribution, thus the probability density function of (5b) Based on step (5a), the joint probability distribution can be obtained: Therefore, regarding the joint maximum a posteriori probability estimate is: where represents the square of the matrix Frobenius norm; (5c) Since the reciprocal and logarithm of variables make the optimization process unstable, three auxiliary variables are introduced The joint maximum a posteriori probability estimation can be rewritten as: Among them represents the square of the vector two-norm, and respectively represent the auxiliary variables and their respective corresponding vectors; [·] T represents the transpose operation of the matrix; A i,k = diag(α i,k ), Q i,k = diag(q i,k ), and C i,k = diag(c i,k ) respectively represent three diagonal matrices with diagonal elements a i,k , q i,k , and c i,k ; ε is a small constant greater than zero that maintains logarithmic stability, log(q i,k + ε) and log(c i,k + ε) represent respectively and l represents the serial number of the element in the vector; (6) Based on the joint maximum a posteriori probability estimation, the augmented Lagrangian function corresponding to the CS-MRI reconstruction model based on tensor singular value estimation can be expressed as: where λ represents the regularization parameter, F u represents the downsampled Fourier encoding matrix, x represents the image to be reconstructed, y represents the downsampled k-space data, B i,k represents the matrix corresponding Lagrange multiplier, β represents the penalty parameter, i represents the serial number of the structure group, and k represents the mode number of the tensor Tucker mode expansion; the reconstruction model is solved according to the alternating direction iteration method, and x, α i,k , q i,k , c i,k , P i,k , V i,k the seven variables are taken as the optimization objects to minimize the augmented Lagrangian function: (6a) Regarding P i,k and V i,k The sub - problem can be expressed as: For Performing singular value decomposition, we can obtain: where U i,k and G i,k are respectively the left singular vector matrix and the right singular vector matrix of i,k and Γ is the singular value matrix of ; regarding the sub-problems of P i,k and V i,k the optimal solutions are: P i,k = U i,k V i,k = G i,k (6b)Regarding α i,k The sub-problem can be expressed as: Substitute P i,k and V i,k into the optimal solution, and the sub-problem can be rewritten as: Furthermore, this sub-problem can be further decomposed into a scalar optimization problem regarding : wherein is the i,k l-th diagonal element of Γ; Using the first-order optimization condition, a closed-form solution to the scalar optimization problem with respect to can be obtained; (6c) Regarding q i,k The sub-problem can be expressed as: Substitute P i,k and V i,k into the optimal solution, and this sub-problem can be rewritten as: Similarly, this sub-problem can be decomposed into a scalar optimization problem regarding for solution. (6d) Regarding c i,k The sub - problem can be expressed as: The solution method for this sub-problem is the same as that for the sub-problem of q i,k ; (6e)Regarding The sub-problem can be expressed as: This sub-problem is a least squares problem, and the corresponding normal equations can be directly solved to obtain the closed-form solution; (6f) The sub-problem with respect to x can be expressed as: This sub-problem is a least squares problem and can be solved using the conjugate gradient method; (7) Repeat step (6) while updating the Lagrange multiplier B i,k , the penalty parameter β, and the variance σ 2 . After a certain number of iterations, the finally reconstructed MRI image can be obtained.

2. The MRI image reconstruction method based on the estimation of the singular value of the structure group tensor according to claim 1, wherein, In step (3), the third-order structure group tensor is unfolded into a matrix according to the Tucker mode-k where k can be 1, 2, and 3. Compared with directly calculating the rank of a high-order tensor, the rank of the matrix obtained by unfolding the tensor according to the Tucker mode is easier to calculate. Therefore, regarding the structure group tensor the low-rank constraint problem is transformed into the problem of estimating the singular values of these three matrices.

3. A method for reconstructing an MRI image based on the estimation of the singular value of a structural group tensor according to claim 1, characterized in that It can be seen from the objective expression in step (5c) that: It is the estimation of the singular values of the structure group tensor through a multi-layer prior model; it should be noted that the singular value matrix Σ in the above formula i,k is represented as the product of three diagonal matrices C i,k , Q i,k and A i,k , that is, Σ i,k = C i,k Q i,k A i,k . The diagonal elements of C i,k and Q i,k are each imposed with a sparse constraint by the logarithmic penalty term, which also verifies the low-rank property of the structure group tensor .

4. A method for reconstructing an MRI image based on the estimation of the singular value of a structural group tensor according to claim 1, wherein Step (6a) obtains a closed solution by combining the von Neumann trace inequality: The condition for the von Neumann trace inequality to achieve equality is exactly P i,k and V i,k the optimal solution of the sub-problem, that is: P i,k = U i,k V i,k = G i,k .

5. A method for reconstructing an MRI image based on structural group tensor singular value estimation according to claim 1, characterized in that, Regarding the scalar optimization problem in step (6c), its specific form is as follows: where the auxiliary variable is a non - negative number, and the optimization problem in scalar form can be simplified to the following form: Among them It can be seen that f(q) is a typical univariate quadratic optimization problem with parameters, and the optimal solution can be obtained by solving the first-order optimization condition The optimal solution is obtained, and one of its non-negative local minima is: Obtain \(q\) by comparing the magnitudes of \(f(0)\) and \(f(q * )\) to get the global optimal solution of the sub - problem of \(q i,k . The solutions for all cases are given as follows: .

Images