A magnetic resonance symmetric spectrum reconstruction method based on non-uniform sampling templates

Abstract

The application discloses a magnetic resonance symmetric spectrum reconstruction method based on a non-uniform sampling template, which comprises the following steps: constructing a sampling template; constructing a magnetic resonance spectrum reconstruction model with sparse constraints; converting the magnetic resonance spectrum reconstruction model with sparse constraints into a magnetic resonance symmetric spectrum reconstruction model with strict symmetric constraints and common constraints of sparse constraints; and reconstructing a spectrum diagram in two steps by using a truncated Newton interior point method. The application combines the sampling template and the reconstruction method, can reconstruct a magnetic resonance symmetric spectrum with high cross-peak quality, fully utilizes the symmetry prior of the magnetic resonance symmetric spectrum in the sampling template and the reconstruction method, is simple to operate, has strong pertinence, and has excellent effect, and finally realizes high-quality reconstruction of the magnetic resonance symmetric spectrum by non-uniform sampling, especially high-quality reconstruction of the cross-peak in the magnetic resonance symmetric spectrum.

Description

Technical Field

[0001] This invention relates to non-uniform sampling of magnetic resonance spectra, and particularly to a method for reconstructing symmetric magnetic resonance spectra based on non-uniform sampling templates. Background Technology

[0002] Magnetic resonance spectroscopy is a common non-invasive analytical technique in fields such as chemistry, biology, and materials science. Symmetrical magnetic resonance spectra refer to a special type of spectrum characterized by a symmetrical structure, consisting of diagonal and cross-peaks. As an important branch of multidimensional magnetic resonance spectroscopy, symmetric magnetic resonance spectra, with their symmetrical structure, play a crucial role in the structural analysis and dynamic analysis of biological macromolecules such as proteins. However, multidimensional magnetic resonance experiments require significant data acquisition time. While non-uniform sampling can significantly shorten experimental time, it necessitates effective sampling templates for obtaining sampling points and efficient reconstruction methods for reconstructing the spectrum.

[0003] Currently, the L1 norm-based compressed sensing reconstruction method is considered the most effective reconstruction method. However, this method still suffers from insufficient cross-peak accuracy in the reconstruction of magnetic resonance symmetric spectra. This is because the L1 norm sparsity constraint focuses on strong peaks while ignoring the contribution of weak peaks when reconstructing the spectrum. In magnetic resonance symmetric spectra, this results in better reconstruction quality for diagonal peaks but poorer reconstruction quality for cross-peaks. Summary of the Invention

[0004] The main objective of this invention is to overcome the aforementioned deficiencies in the prior art and propose a magnetic resonance symmetric spectrum reconstruction method based on a non-uniform sampling template. This method is highly targeted and can effectively solve the problem of insufficient accuracy of cross peaks in magnetic resonance symmetric spectra, improve the reconstruction quality of magnetic resonance symmetric spectra, especially the reconstruction quality of cross peaks, and has broad application prospects.

[0005] The present invention adopts the following technical solution:

[0006] A magnetic resonance symmetric spectrum reconstruction method based on a non-uniform sampling template includes:

[0007] S103, Construct a magnetic resonance spectrum reconstruction model with sparse constraints, as follows:

[0008]

[0009] Where X represents the magnetic resonance spectrum, Y represents the data obtained from the magnetic resonance experiment, F represents the inverse Fourier transform matrix, Ω represents the sampling template, and P... Ω This represents the sampling operation that obtains sampling points based on the sampling template, where λ represents the regularization parameter used to balance the fidelity term and the regularization term; for magnetic resonance symmetric spectra, x T =x, where the superscript T represents the transpose operation;

[0010] S105 transforms the sparsely constrained magnetic resonance spectrum reconstruction model into a symmetric magnetic resonance spectrum reconstruction model with both strict symmetry constraints and sparse constraints, as follows:

[0011]

[0012] In equation (1), X is replaced by Sx, where x is the vectorized form of the symmetric matrix X after the reduction of variables, and S is the symmetric permutation operation, which converts the vector x into the symmetric matrix X; the relationship between x, X and S is shown in equation (3) below:

[0013]

[0014] Where n represents the number of rows and columns of the magnetic resonance symmetric spectrum matrix X;

[0015] S107 uses the truncated Newton interior point method to reconstruct the spectrum in two steps, as detailed below:

[0016] 1) Transform equation (2) into a convex quadratic equivalent model with linear inequality constraints, as shown in equation (4):

[0017]

[0018] in, y is the vectorized form of Y; u i x represents i The boundary;

[0019] 2) Replace the linear inequality constraint in equation (4) with a logarithmic barrier function, and transform equation (4) into equation (5):

[0020]

[0021] Where, Φ(x,u)=-∑ i log(u i +x i )-∑ i log(u i -x i ), used to control the proportion of the barrier function in the model; let u means u i The vector formed;

[0022] 3) Initialize parameters t, x, and u;

[0023] 4) Repeat the following sub-steps:

[0024] ① The pre-processed conjugate gradient method is used to calculate the Newton linear equation system HΔp=-g, and the solution of the equation system is used as the optimization direction. In the direction of descent, In expression (5), F t (p) The Hessian matrix of the current iteration value p, In expression (5), F t (p) is the gradient vector at the current iteration value p;

[0025] ② The descent step size s = β is calculated using the backtracking straight line method. k , where k is equal to the smallest positive integer that satisfies equation (6):

[0026] F t (p+β k △p)≤F t (p)+αβ k g T △p (6)

[0027] Where 0 < α < 0.5 and 0 < β < 1 are the parameters of the algorithm;

[0028] ③ Update the iteration value based on the descent direction and descent step size:

[0029] p=p+sΔp (7)

[0030] ④ Calculate the duality gap θ:

[0031]

[0032] in, It is the dual function of equation (2), and the variable v in the dual function is the dual feasible point, which is obtained through equation (9):

[0033] v=2σ(Ρ Ω FSx-y) (9)

[0034] Where, σ=min{λ / |2((Ρ Ω FS) T (Ρ Ω FS)x) i -2y i |}, where the subscript i represents the i-th element;

[0035] ⑤ Determine whether θ / G(v) is less than the preset precision value. If it is, exit the loop; otherwise, execute the next loop step.

[0036] ⑥ Update t and jump back to step ① of the loop. The update rule is:

[0037]

[0038] Where μ = 2, σ min =0.5, where m is the length of vector y;

[0039] ⑦ Output the optimal solution x*;

[0040] 5) Obtain the optimal solution x by truncating Newton's interior point method. * Then, only the diagonal peaks of the spectrum are retained, using x d This indicates that the cross-peak time-domain signal Y c Obtained through equation (11):

[0041] Y c = Y-P Ω F Sx d (11)

[0042] 6) Obtain the cross peak x of the spectrum through the optimized formula (12). c :

[0043]

[0044] 7) Obtain the vectorized form x of the final reconstructed spectrum through equation (13). f :

[0045] x f =x d +x c (13)

[0046] 8) x f The magnetic resonance symmetry spectrum is rearranged to the corresponding size.

[0047] Preferably, before constructing the magnetic resonance spectrum reconstruction model with sparse constraints, the method further includes:

[0048] S101, Construct the sampling template; details are as follows:

[0049] S1011, constructing an ordered "symmetric pair";

[0050] All data points are placed in a two-dimensional matrix, which is divided into upper and lower parts along the diagonal. Two data points that are symmetrical about the diagonal or one data point on the diagonal form a "symmetrical pair", which is represented by the same number. The "symmetrical pairs" are numbered sequentially.

[0051] S1012, using a one-dimensional Poisson gap sampling template with "symmetric pair" selected;

[0052] The “symmetric pairs” in the data point matrix are arranged into a one-dimensional vector according to the numbering order. A portion of the “symmetric pairs” are selected by ratio using a one-dimensional Poisson gap, where the check mark and cross mark indicate that the “symmetric pair” is selected and not selected, respectively.

[0053] S1013, If the selected “symmetric pair” contains two data points, then one of the data points is randomly selected for sampling with equal probability;

[0054] The selected "symmetric pair" is represented by a checkmark. The "symmetric pair" located on the diagonal contains only one data point, while the "symmetric pair" located at other positions contains two data points. Since the "symmetric pair" contains two data points, one data point in the selected "symmetric pair" is selected for sampling in a form of equal probability random selection. The vector form of the "symmetric pair" is rearranged into a two-dimensional matrix form to obtain the magnetic resonance symmetric spectrum sampling template.

[0055] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0056] This invention improves upon the insufficient accuracy of cross-peaks from two aspects: sampling template and reconstruction method. Regarding the sampling template, a sampling template suitable for symmetric magnetic resonance spectra is proposed, taking into account the symmetry characteristics of symmetric magnetic resonance spectra. This template effectively avoids redundant information acquisition, improves information acquisition efficiency, and thus reconstructs higher-quality cross-peaks. Regarding reconstruction, a reconstruction model with both sparsity and symmetry constraints is proposed, making the reconstructed spectrum more consistent with the actual characteristics of symmetric magnetic resonance spectra. Furthermore, the model solution employs a two-step reconstruction approach, addressing diagonal peaks and cross-peaks separately, mitigating the side effects of L1 norm sparsity constraints and further improving the quality of cross-peaks in the reconstructed spectrum. By combining the sampling template and reconstruction method, high-quality symmetric magnetic resonance spectra can be reconstructed. This invention is simple to operate, highly targeted, and effectively solves the problem of insufficient accuracy of cross-peaks in symmetric magnetic resonance spectra, improving the reconstruction quality of symmetric magnetic resonance spectra, especially the reconstruction quality of cross-peaks, and has broad application prospects. Attached Figure Description

[0057] Figure 1 This is a flowchart of a magnetic resonance symmetry spectrum reconstruction method based on a non-uniform sampling template according to an embodiment of the present invention;

[0058] Figure 2 This is a schematic diagram of the magnetic resonance symmetric spectrum sampling template implementation method according to an embodiment of the present invention;

[0059] Figure 3 The Monte Carlo test results of cross-peak RLNE values ​​reconstructed from simulated magnetic resonance symmetric spectrum data using four sampling templates through the SCREEN method and the method of the present invention are shown in the embodiment of the present invention; where (a) represents the SCREEN method and (b) represents the method of the present invention.

[0060] Figure 4 This is a comparison chart of the full sampling spectrum and the reconstructed spectrum with 5% non-uniform sampling in an embodiment of the present invention. Detailed Implementation

[0061] The present invention will be further illustrated below with reference to specific embodiments. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. Furthermore, it should be understood that after reading the teachings of this invention, those skilled in the art can make various alterations or modifications to the invention, and these equivalent forms also fall within the scope defined by the appended claims.

[0062] The following examples will reconstruct α-synuclein. 15 N- 15 N-projection spectrum.

[0063] A magnetic resonance symmetric spectrum reconstruction method based on a non-uniform sampling template includes:

[0064] S101, Construct a sampling template.

[0065] (1) Constructing ordered "symmetric pairs"

[0066] All data points are placed in a 620×620 two-dimensional matrix, which is divided into upper and lower parts along the diagonal. Two data points symmetrical about the diagonal are called a "symmetric pair" and are represented by the same number. Figure 2 (a) shows the sequential numbering of the “symmetrical pairs”.

[0067] (2) Use a one-dimensional Poisson gap sampling template and select "symmetric pair".

[0068] The "symmetric pairs" in the 620×620 data point matrix are arranged according to... Figure 2 (b) The numbering is arranged in order into a one-dimensional vector of size 192510×1. One-dimensional Poisson gaps are used to select 19220 pairs of “symmetric pairs”, where the checkmark and cross symbols indicate that the “symmetric pair” is selected and not selected, respectively.

[0069] (3) If the selected “symmetric pair” contains two data points, then one of the data points is randomly selected with equal probability for sampling.

[0070] The "symmetric pairs" selected for sampling are represented by a checkmark. Symmetric pairs located on the diagonal contain only one data point, while those in other positions contain two data points. Since each symmetric pair contains two data points, one of the selected symmetric pairs is chosen as the actual sampling point through a random selection process with equal probability. "Above" and "below" represent the data points sampled from the upper and lower halves of the data point matrix, respectively. Finally, the vector-like symmetric pairs are rearranged... Figure 2 (c) shows a two-dimensional matrix form, which yields a sampling template of size 620×620.

[0071] Specifically, Figure 2 In the diagram, (a) data points are divided into upper and lower halves, where "symmetric pairs" are represented by the same numerical designation. (b) Symmetric pairs are arranged in vector form, with each "symmetric pair" represented by a colored square. Checkmark (cross) symbols mark selected (unselected) "symmetric pairs." Above (below) indicates that the sampling points used for sampling come from the upper (lower) part of the data point matrix. (c) Magnetic resonance symmetry spectrum sampling template.

[0072] S103, Construct a magnetic resonance spectrum reconstruction model with sparse constraints, as follows:

[0073]

[0074] in, Let X represent a complex matrix of size 620×620, X represent the magnetic resonance spectrum of size 620×620 to be reconstructed, Y represent the dataset containing 19220 data points obtained from a 5% non-uniform sampling magnetic resonance experiment, F represent the inverse Fourier transform operation, Ω represent the sampling template, and P represent the complex matrix of size 620×620. Ω This represents a non-uniform sampling operation that obtains sampling points based on a sampling template. λ represents a regularization parameter used to balance fidelity and the regularization term, set to 0.1. For symmetric magnetic resonance spectra, X... T =X, where the superscript T represents the transpose operation, so the original 384,400 variables in X can be further reduced to 192,510 independent variables, and the model can be transformed.

[0075] S105 transforms the sparsely constrained magnetic resonance spectrum reconstruction model into a symmetric magnetic resonance spectrum reconstruction model with both strict symmetry constraints and sparse constraints, as follows:

[0076]

[0077] in, Let X represent a set of complex vectors of length 192510. In equation (1), X is replaced by Sx, where x is the vectorized form of the symmetric matrix X after the variable reduction, with a length of 192510. S is a symmetric permutation operation that can convert a complex vector x of length 192510 into a symmetric matrix X. Equation (3) illustrates the relationship between x, X, and S:

[0078]

[0079] Where n = 620 represents the number of rows and columns of the magnetic resonance symmetric spectral matrix X.

[0080] S107, using a two-step reconstruction of the spectrum by truncating Newton's interior points.

[0081] 1) Transform equation (2) into equation (4), a convex quadratic equivalent model with linear inequality constraints:

[0082]

[0083] in, n = 620, y is the vectorized form of Y, where y is the vectorized form of Y; u i x represents i The boundary, since the 1-norm of x in equation (2) is such that each x i The absolute value of is as small as possible, so it can be transformed into the constraint form of the inequality (4), which facilitates the introduction of the barrier function in equation (5).

[0084] 2) Replace the linear inequality constraint in equation (4) with a logarithmic barrier function, and transform equation (4) into equation (5):

[0085]

[0086] Where, Φ(x,u)=-∑ i log(u i +x i )-∑ i log(u i -x i ), used to control the proportion of the barrier function in the model; u represents u i The vector formed by this. For convenience, let... For subsequent explanation.

[0087] 3) Initialize parameters t, x, and u.

[0088] 4) Repeat the following sub-steps:

[0089] ① The pre-processed conjugate gradient method is used to calculate the Newton linear equation system HΔp=-g, and the solution of the equation system is used as the optimization direction. This is the descent direction (often also called the optimization direction). In expression (5), F t (p) The Hessian matrix of the current iteration value p, In expression (5), F t (p) is the gradient vector at the current iteration value p.

[0090] ② The descent step size s = β is calculated using the backtracking straight line method. k , where k is equal to the smallest positive integer that satisfies equation (6):

[0091] F t (p+β k △p)≤F t (p)+αβ k g T △p (6)

[0092] Where 0 < α < 0.5 and 0 < β < 1 are the parameters of the algorithm;

[0093] ③ Update the iteration value based on the descent direction and descent step size:

[0094] p=p+sΔp (7)

[0095] ④ Calculate the duality gap θ:

[0096]

[0097] in It is the dual function of equation (2), and the variable v in the dual function is the dual feasible point, which is obtained through equation (9):

[0098] v=2σ(Ρ Ω FSx-y) (9)where σ=min{λ / |2((P Ω FS) T (Ρ Ω FS)x) i -2y i |}, where the subscript i represents the i-th element.

[0099] ⑤ Determine whether θ / G(v) is less than the preset precision value ε = 10 -5 If the condition is met, the loop will exit; otherwise, the next loop step will be executed.

[0100] ⑥ Update t and jump back to step ① of the loop. The update rule is:

[0101]

[0102] Where μ = 2, σ min =0.5, where m is the length of vector y.

[0103] ⑦ Output the optimal solution x*.

[0104] 5) Obtain the optimal solution x by truncating Newton's interior point method. * Then, only the diagonal peaks of the spectrum are retained, using x d This indicates that the cross-peak time-domain signal Y c Obtained through equation (11):

[0105] Y c = Y-P Ω F Sx d (11)

[0106] 6) Obtain the cross peak x of the spectrum through the optimized formula (12). c :

[0107]

[0108] 7) Obtain the vectorized form x of the final reconstructed spectrum through equation (13). f :

[0109] x f =x d +x c (13)

[0110] 8) x f The magnetic resonance symmetry spectrum was rearranged to a size of 620×620.

[0111] See Figure 3 The figure shows 100 Monte Carlo tests of the cross-peak RLNE values ​​reconstructed from synthetic symmetric magnetic resonance spectral data using four sampling templates via SCREEN and the method of this invention. The vertical axis represents the Relative L2 Norm Error (RLNE), with smaller values ​​indicating higher reconstruction quality. The horizontal axis represents the test index, sorted from worst to best.

[0112] from Figure 3 It can be seen that in the reconstructed spectra obtained by the SCREEN method and the reconstruction method of the present invention, the spectrum obtained by using the sampling template proposed in the present invention has the lowest RLNE value for the cross peak, indicating that the sampling template proposed in the present invention has a better effect on the reconstruction of cross peaks of magnetic resonance symmetric spectra than other sampling templates.

[0113] See Figure 4 The image shows a full sample of α-synuclein. 15 N- 15 A comparison chart of the N-projection spectrum and the reconstructed spectrum with 5% non-uniform sampling (including the SCREEN reconstructed spectrum and the reconstructed spectrum of this invention), using the sampling template proposed in this invention. The first row shows the full-sampled spectrum and the reconstructed spectrum; the second row shows an enlarged view of the area within the dashed box in the first row; and the third row shows a correlation analysis of spectral peak intensities.

[0114] contrast Figure 4 As can be seen from the first two rows of spectra and their magnified views, the spectra obtained by the reconstruction method of this invention show better performance in terms of cross-peak reconstruction quality compared to the spectra obtained by the SCREE method. Compared to the full-sample spectrum, the cross-peaks reconstructed by SCREEN suffer from severe peak shrinkage or even omission, while the reconstruction method of this invention can reconstruct the cross-peaks much better. From Figure 3The correlation analysis between the reconstructed peak intensity and the full-sampled peak intensity in the third row shows that the cross-peak intensity of the SCREEN reconstruction is much lower than that of the full-sampled spectrum, with some peak intensities even approaching zero. In contrast, the cross-peak intensity reconstructed by the method of this invention is close to that of the full-sampled spectrum. Therefore, it can be concluded that the reconstruction method of this invention can further improve the quality of spectral reconstruction compared to existing methods.

[0115] The above are merely specific embodiments of the present invention, but the design concept of the present invention is not limited thereto. Any non-substantial modifications made to the present invention using this concept shall be considered as infringing upon the protection scope of the present invention.

Claims

1. A magnetic resonance symmetric spectrum reconstruction method based on a non-uniform sampling template, characterized in that, include: S103, Construct a magnetic resonance spectrum reconstruction model with sparse constraints, as follows: Where X represents the magnetic resonance spectrum, Y represents the data obtained from the magnetic resonance experiment, F represents the inverse Fourier transform matrix, Ω represents the sampling template, and P... Ω This represents the sampling operation that obtains sampling points based on the sampling template, where λ represents the regularization parameter used to balance the fidelity term and the regularization term; for magnetic resonance symmetric spectra, X T =X, where the superscript T represents the transpose operation; S105 transforms the sparsely constrained magnetic resonance spectrum reconstruction model into a symmetric magnetic resonance spectrum reconstruction model with both strict symmetry constraints and sparse constraints, as follows: In equation (1), X is replaced by Sx, where x is the vectorized form of the symmetric matrix X after the reduction of variables, and S is the symmetric permutation operation, which converts the vector x into the symmetric matrix X; the relationship between x, X and S is shown in equation (3) below: Where n represents the number of rows and columns of the magnetic resonance symmetric spectrum matrix X; S107 uses the truncated Newton interior point method to reconstruct the spectrum in two steps, as detailed below: 1) Transform equation (2) into a convex quadratic equivalent model with linear inequality constraints, as shown in equation (4): in, y is the vectorized form of Y; u i x represents i The boundary; 2) Replace the linear inequality constraint in equation (4) with a logarithmic barrier function, and transform equation (4) into equation (5): Where, Φ(x,u)=-∑ i log(u i +x i )-∑ i log(u i -x i ), used to control the proportion of the barrier function in the model; let u means u i The vector formed; 3) Initialize parameters t, x, and u; 4) Repeat the following sub-steps: ① The pre-processed conjugate gradient method is used to calculate the Newton linear equation system HΔp=-g, and the solution of the equation system is used as the optimization direction. In the direction of descent, In expression (5), F t (p) The Hessian matrix of the current iteration value p, In expression (5), F t (p) is the gradient vector at the current iteration value p; ② The descent step size s = β is calculated using the backtracking straight line method. k , where k is equal to the smallest positive integer that satisfies equation (6): F t (p+β k △p)≤F t (p)+αβ k g T △p (6) Where 0 < α < 0.5 and 0 < β < 1 are the parameters of the algorithm; ③ Update the iteration value based on the descent direction and descent step size: p=p+sΔp (7) ④ Calculate the duality gap θ: in, It is the dual function of equation (2), and the variable v in the dual function is the dual feasible point, which is obtained through equation (9): v=2σ(Ρ Ω FSx-y) (9) Where, σ=min{λ / |2((Ρ Ω FS) T (Ρ Ω FS)x) i -2y i |}, where the subscript i represents the i-th element; ⑤ Determine whether θ / G(v) is less than the preset precision value. If it is, exit the loop; otherwise, execute the next loop step. ⑥ Update t and jump back to step ① of the loop. The update rule is: Where μ = 2, σ min =0.5, where m is the length of vector y; ⑦ Output the optimal solution x*; 5) Obtain the optimal solution x by truncating Newton's interior point method. * Then, only the diagonal peaks of the spectrum are retained, using x d This indicates that the cross-peak time-domain signal Y c Obtained through equation (11): Y c < Y-P Ω F Sx d (11) 6) Obtain the cross peak x of the spectrum through the optimized formula (12). c : 7) Obtain the vectorized form x of the final reconstructed spectrum through equation (13). f : x f =x d +x c (13) 8) x f The magnetic resonance symmetry spectrum is rearranged to the corresponding size.

2. The magnetic resonance symmetry spectrum reconstruction method based on a non-uniform sampling template according to claim 1, characterized in that, Before constructing the magnetic resonance spectrum reconstruction model with sparse constraints, the following steps are also included: S101, Construct the sampling template; details are as follows: S1011, constructing ordered "symmetric pairs"; All data points are placed in a two-dimensional matrix, which is divided into upper and lower parts along the diagonal. Two data points that are symmetrical about the diagonal or one data point on the diagonal form a "symmetrical pair", which is represented by the same number. The "symmetrical pairs" are numbered sequentially. S1012, using a one-dimensional Poisson gap sampling template with "symmetric pair" selected; The "symmetric pairs" in the data point matrix are arranged into a one-dimensional vector according to their numbering order. A portion of the "symmetric pairs" are selected proportionally using a one-dimensional Poisson gap. The checkmark and cross symbols indicate that the "symmetric pair" is selected and not selected, respectively. S1013, If the selected "symmetric pair" contains two data points, then one of the data points is randomly selected for sampling with equal probability; The selected "symmetric pair" is represented by a checkmark. The "symmetric pair" located on the diagonal contains only one data point, while the "symmetric pair" located at other positions contains two data points. Since the "symmetric pair" contains two data points, one data point in the selected "symmetric pair" is selected for sampling in a form of equal probability random selection. The vector form of the "symmetric pair" is rearranged into a two-dimensional matrix form to obtain the magnetic resonance symmetric spectrum sampling template.

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