High-fidelity magnetic resonance spectrum reconstruction method

Abstract

The invention discloses a high-fidelity magnetic resonance spectrum reconstruction method, and relates to a magnetic resonance spectrum reconstruction method. Comprising the following steps: 1) acquiring a to-be-reconstructed magnetic resonance spectrum time domain signal; 2) proposing a magnetic resonance spectrum reconstruction model based on symmetric rank-one Hankel matrix decomposition; and 3) solving the model in the step 2) by using a projection gradient algorithm to obtain a magnetic resonance spectrum time domain signal, and performing Fourier transform on the magnetic resonance spectrum time domain signal to obtain a final spectrum signal. According to the method, the high-fidelity reconstruction of the magnetic resonance spectrum is realized based on the symmetry of the Hankel matrix and the rank-one characteristic of the magnetic resonance spectrum signal.

Description

Technical Field

[0001] The present invention relates to a magnetic resonance spectrum reconstruction method, in particular to a high-fidelity magnetic resonance spectrum reconstruction method based on symmetric rank-one Hankel matrix decomposition. Background Art

[0002] Magnetic resonance spectroscopy (MRI) is a molecular detection technique that combines noninvasive, non-invasive, and radiation-free properties, with important applications in biomedicine, chemistry, and materials science (Xiaobo Qu, Maxim Mayzel, Jian-Feng Cai, Zhong Chen, and Vladislav Orekhov, "Accelerated NMR spectroscopy with low-rank reconstruction," Angewandte Chemie International Edition, vol. 54, no. 3, pp. 852-854, 2015). Undersampling the temporal signal of MRI spectroscopy can reduce acquisition time, but the missing signal from undersampling requires reconstruction to eliminate artifacts caused by undersampling.

[0003] Researchers have exploited the low-rank nature of the Hankel matrix of time-domain magnetic resonance spectroscopy signals and proposed a series of low-rank Hankel matrix reconstruction methods. For example, to address the computational overhead of singular value decomposition, a matrix decomposition-accelerated reconstruction method was proposed (Di Guo, Hengfa Lu, and Xiaobo Qu, "A fast low rank Hankel matrix factorization reconstruction method for non-uniformly sampled magnetic resonance spectroscopy," IEEE Access, vol 5, pp. 16033-16039, 2017.); and for high-dimensional cases, a tensor decomposition-based reconstruction method was proposed (Jiaxi Ying, Hengfa Lu, Qingtao Wei, Jian-Feng Cai, DiGuo, Jihui Wu, Zhong Chen, and Xiaobo Qu, "Hankel matrix nuclear norm regularized tensor completion for N-dimensional exponential signals," IEEE Transactions on Signal Processing, vol 65, pp. 3702-3717, 2017.). With the development of artificial intelligence, a low-rank Hankel deep learning method based on matrix decomposition was proposed (Yihui Huang, Jinkui Zhao, Zi Wang, Vladislav Orekhov, Di Guo, and Xiaobo Qu, "Exponential signal reconstruction with deep Hankelmatrix factorization," IEEE Transactions on Neural Networks and Learning Systems, vol. 34, no. 9, pp. 6214–6226, 2023.).

[0004] However, none of the above methods exploit the symmetry and rank-one properties of the Hankel matrix, and may result in large reconstruction errors, long reconstruction time, and high memory usage in challenging scenarios. Summary of the Invention

[0005] The present invention addresses the problem of reconstructing high-fidelity magnetic resonance spectra by proposing a high-fidelity magnetic resonance spectroscopy reconstruction method based on a symmetric rank-one Hankel matrix decomposition. This method, leveraging the symmetry of the Hankel matrix and the rank-one nature of magnetic resonance spectroscopy signals, enables rapid reconstruction of high-fidelity spectra.

[0006] The present invention comprises the following steps:

[0007] 1) obtaining a magnetic resonance spectroscopy time domain signal to be reconstructed;

[0008] 2) A magnetic resonance spectroscopy reconstruction model based on symmetric rank-1 Hankel matrix decomposition is proposed;

[0009] 3) using the projected gradient algorithm to solve the model in step 2) to obtain a magnetic resonance spectrum time domain signal, and performing a Fourier transform on the signal to obtain the final spectrum signal;

[0010] In step 1), the specific steps of obtaining the magnetic resonance spectrum time domain signal to be reconstructed may be:

[0011] Let y represent the time domain signal of the fully sampled one-dimensional magnetic resonance spectrum, which is modeled as a superposition of exponential signals:

[0012]

[0013] Where R is the number of spectral peaks, f r ∈[0,1) and represent the amplitude, normalized frequency and damping factor of the rth spectral peak respectively; is a plural set, is a set of positive real numbers; y(n) represents the nth data point of the magnetic resonance spectroscopy time domain signal y, and Δt is the sampling time interval;

[0014] If the length of signal y is an even number, zeros are added to the end of signal y to make the signal length an odd number. If the signal length is an odd number, no change is made. The processed one-dimensional magnetic resonance spectroscopy time domain signal is expressed as Where N is a natural number.

[0015] In step 2), the magnetic resonance spectroscopy reconstruction model based on symmetric rank-one Hankel matrix decomposition is as follows:

[0016]

[0017] in, is a sampling operator on the sampling set Ω, defined as: if n∈Ω, then otherwise is a weighted sampling operator on the sampling set Ω, satisfying ⊙ represents the Hadamard product; ||·|| F represents the Frobenius norm of the matrix; To estimate the number of spectral peaks, is the identity operator, is the decomposition vector of the rth spectral peak;

[0018] Lifted Hankel operator Defined as: for a vector So is the lifted Hankel matrix corresponding to vector a, satisfying:

[0019]

[0020] Among them, a n is the nth element in vector a;

[0021] Lifted Hankel adjoint operator Defined as: For the matrix So is the lifted Hankel adjoint vector corresponding to the matrix A;

[0022]

[0023] Among them, a ij is the element in the i-th row and j-th column of matrix A.

[0024] In step 3), the projected gradient algorithm is used to solve the model in step 2) to obtain a magnetic resonance spectrum time domain signal, which is then subjected to Fourier transform to obtain the final spectrum signal. The specific steps may be:

[0025] For simplicity of expression, the matrix is decomposed into Defined as:

[0026]

[0027] The feasible region C is defined as:

[0028]

[0029] Where 2N-1 is the signal length, To estimate the number of spectral peaks, ||Z|| 2,∞ =max(||Z (1,:) ||2,||Z (2,:) ||2,…,||Z (N,:) ||2),σ is The maximum singular value of , ||·||2 represents the two-norm of the vector;

[0030] The penalty factor is introduced to transform model (2) into an unconstrained optimization problem for solution; the loss function f(Z) is defined as:

[0031]

[0032] Where λ is the regularization parameter;

[0033] Use the projected gradient algorithm to update the variable Z. In the kth iteration, Update through formula (8);

[0034]

[0035] where η is the step size, gradient Calculated by formula (9):

[0036]

[0037] in express conjugation of;

[0038] make pass Map the updated result into its feasible domain; is the projection operator on the feasible domain C, defined as:

[0039]

[0040] Let the solution of step k+1 be in Set the iteration stopping criteria to reach the maximum number of iterations W or the solution x of two consecutive iterations k and x k+1 The difference in the relative two-norm of ||x k+1 -x k ||2 / ||x k ||2 is less than the threshold η, where η is a number greater than 0; after the iteration is terminated, the output of the last iteration is As the final reconstructed complete time domain signal, x last The reconstructed spectrum can be obtained by performing Fourier transform.

[0041] Compared with the prior art, the present invention has the following outstanding technical effects and advantages:

[0042] 1. The reconstruction accuracy of the present invention is high. Compared with the cutting-edge SHGD algorithm, the correlation coefficient between the reconstructed spectrum and the spectral peak of the full sampling wave is as high as 0.9995, far exceeding the 0.9936 of the SHGD algorithm. It can accurately capture subtle spectral peak features such as small peaks, greatly improving the fidelity of spectral reconstruction and providing a more reliable data foundation for subsequent spectral-based analysis and diagnosis.

[0043] 2. The present invention deeply exploits the symmetry of the Hankel matrix and the rank-one characteristic of the magnetic resonance spectroscopy signal. Compared with traditional methods, when processing undersampled data, it solves the model through an optimized projected gradient algorithm, reduces unnecessary calculation steps, greatly reduces the computational complexity, significantly shortens the reconstruction time, and effectively improves the efficiency of magnetic resonance spectroscopy reconstruction. It is more suitable for scenarios with high time requirements such as clinical rapid testing.

[0044] 3. The present invention avoids the large amount of memory consumption caused by high-dimensional data processing and complex calculations by reasonably defining the feasible domain and optimizing the model structure. While ensuring high-fidelity reconstruction effects, it is more friendly to hardware resources, reduces operating costs, has good scalability, and can run stably on devices with different configurations.

[0045] 4. The present invention has strong universality and can effectively reconstruct both one-dimensional and two-dimensional magnetic resonance spectroscopy data, providing strong technical support for the widespread application of magnetic resonance spectroscopy technology in multiple fields such as biomedicine, chemistry and materials. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 The two-dimensional magnetic resonance full sampling spectrum and the reconstructed spectrum with 20% sampling data volume are shown in Figure 1. (a) is the full sampling spectrum, (b) is the reconstructed spectrum using the cutting-edge algorithm (SHGD algorithm), and (c) is the reconstructed spectrum of the present invention. DETAILED DESCRIPTION

[0047] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the following embodiments will be further described in conjunction with the accompanying drawings. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0048] In this embodiment, the size of the acquired two-dimensional magnetic resonance spectrum data to be reconstructed is 116×255, and the one-dimensional time domain signals of the magnetic resonance spectrum are reconstructed one by one along the indirect dimension.

[0049] 1) obtaining a magnetic resonance spectroscopy time domain signal to be reconstructed;

[0050] Let y represent the time domain signal of the fully sampled one-dimensional magnetic resonance spectrum, which is modeled as a superposition of exponential signals:

[0051]

[0052] Where R is the number of spectral peaks, f r ∈[0,1) and represent the amplitude, normalized frequency and damping factor of the rth spectral peak respectively. is a plural set, is a set of positive real numbers. y(n) represents the nth data point of the magnetic resonance spectroscopy time domain signal y, and Δt is the sampling time interval.

[0053] If the length of signal y is even, zeros are added to the end of signal y to make the signal length odd. If the signal length is odd, no change is made. In this example, the processed one-dimensional magnetic resonance spectroscopy time domain signal is represented as

[0054] 2) A magnetic resonance spectroscopy reconstruction model based on symmetric rank-1 Hankel matrix decomposition is proposed;

[0055] The magnetic resonance spectroscopy reconstruction model based on symmetric rank-one Hankel matrix decomposition is constructed as follows:

[0056]

[0057] in, is a sampling operator on the sampling set Ω, defined as: if n∈Ω, then otherwise is a weighted sampling operator on the sampling set Ω, satisfying ⊙ represents the Hadamard product. ||·|| F represents the Frobenius norm of the matrix. To estimate the number of spectral peaks, is the identity operator, is the decomposition vector of the rth spectral peak.

[0058] In this embodiment, the lifting Hankel operator Defined as: for a vector So is the lifted Hankel matrix corresponding to vector a, satisfying:

[0059]

[0060] Among them, a n is the nth element in vector a.

[0061] Lifted Hankel adjoint operator Defined as: For the matrix So is the lifted Hankel adjoint vector corresponding to the matrix A,

[0062]

[0063] Among them, a ij is the element in the i-th row and j-th column of matrix A.

[0064] 3) Using the projected gradient algorithm to solve the model in step 2) to obtain the magnetic resonance spectrum time domain signal, and performing Fourier transform on it to obtain the final spectrum signal.

[0065] For simplicity of expression, the matrix is decomposed into Defined as:

[0066]

[0067] The feasible region C is defined as:

[0068]

[0069] where ||Z|| 2,∞ =max(||Z (1,:) ||2,||Z (2,:) ||2,…,||Z (N,:) ||2), σ is The maximum singular value of .

[0070] The penalty factor is introduced to transform model (2) into an unconstrained optimization problem for solution. The loss function f(Z) is defined as:

[0071]

[0072] Where λ=1 is the regularization parameter.

[0073] Use the projected gradient algorithm to update the variable Z. In the kth iteration, Update through formula (8);

[0074]

[0075] Where η = 0.05 is the step size, r = 1, 2, ..., 10, and the gradient Calculated by formula (9):

[0076]

[0077] in express The conjugation of .

[0078] make pass Map the updated result to its feasible domain. is the projection operator on the feasible region C, which is defined in this embodiment as:

[0079]

[0080] The iteration stopping criterion is set to the maximum number of iterations 1000 or the iteration tolerance error η = 10 -4 . Where η=||x k+1 -x k ||2 / ||x k ||2, After the iteration is terminated, the output of the last iteration is As the final reconstructed complete time domain signal, x last The reconstructed spectrum can be obtained by performing Fourier transform.

[0081] The reconstruction fidelity is determined by the peak correlation coefficient R between the reconstructed spectrum and the full sampling spectrum. 2 To evaluate, R 2 The larger the value, the higher the reconstruction fidelity. Figure 1 As shown, Figure 1 (a) in the figure is the fully sampled spectrum. Figure 1 (b) is the spectral reconstruction result of the cutting-edge SHGD algorithm (literature: Jinsheng Li, Wei Cui, and Xu Zhang, "Projected gradient descent forspectral compressed sensing via symmetric Hankel factorization," IEEE Transactions on Signal Processing, vol 72, pp. 1590-1606, 2024.), and the spectral peak correlation coefficient is 0.9936. Figure 1 (c) is the reconstructed spectrum of the present invention, and the peak correlation coefficient is 0.9995. Figure 1 For the reconstructed small peaks marked by the middle arrows, the method of the present invention is superior to the control method and can achieve high-fidelity spectrum reconstruction.

[0082] The above embodiments are only preferred embodiments of the present invention and should not be considered to limit the scope of the present invention. All equivalent changes and improvements made within the scope of the present invention should still fall within the scope of the patent of the present invention.

Claims

1. A high-fidelity magnetic resonance spectroscopy reconstruction method, characterized by the following steps: 1) obtaining a magnetic resonance spectroscopy time domain signal to be reconstructed; 2) Propose a magnetic resonance spectroscopy reconstruction model based on symmetric rank-one Hankel matrix decomposition; 3) Using the projected gradient algorithm to solve the model in step 2) to obtain the magnetic resonance spectrum time domain signal, and performing Fourier transform on it to obtain the final spectrum signal.

2. A magnetic resonance spectroscopy reconstruction method based on symmetric rank-one Hankel matrix decomposition as claimed in claim 1, characterized in that In step 1), the specific steps of obtaining the magnetic resonance spectrum time domain signal to be reconstructed are: Let y represent the time domain signal of the fully sampled one-dimensional magnetic resonance spectrum, which is modeled as a superposition of exponential signals: Where R is the number of spectral peaks, f r ∈[0,1) and represent the amplitude, normalized frequency and damping factor of the rth spectral peak respectively; is a plural set, is a set of positive real numbers; y(n) represents the nth data point of the magnetic resonance spectroscopy time domain signal y, and Δt is the sampling time interval; If the length of signal y is an even number, zero is added to the end of signal y to make the signal length an odd number. If the signal length is an odd number, no change is made. The processed one-dimensional magnetic resonance spectroscopy time domain signal is expressed as Where N is a natural number.

3. A magnetic resonance spectroscopy reconstruction method based on symmetric rank-one Hankel matrix decomposition as claimed in claim 1, characterized in that In step 2), the magnetic resonance spectroscopy reconstruction model based on symmetric rank-one Hankel matrix decomposition is as follows: in, is a sampling operator on the sampling set Ω, defined as: if n∈Ω, then otherwise is a weighted sampling operator on the sampling set Ω, satisfying ⊙ represents the Hadamard product; ||·|| F represents the Frobenius norm of the matrix; To estimate the number of spectral peaks, is the identity operator, is the decomposition vector of the rth spectral peak; Lifted Hankel operator Defined as: for a vector So is the lifted Hankel matrix corresponding to vector a, satisfying: Among them, a n is the nth element in vector a; Lifted Hankel adjoint operator Defined as: For the matrix So is the lifted Hankel adjoint vector corresponding to the matrix A, Among them, a ij is the element in the i-th row and j-th column of matrix A.

4. A magnetic resonance spectroscopy reconstruction method based on symmetric rank-one Hankel matrix decomposition as claimed in claim 1, characterized in that In step 3), the projected gradient algorithm is used to solve the model in step 2) to obtain a magnetic resonance spectrum time domain signal, and the Fourier transform is performed on the signal to obtain the final spectrum signal. The specific steps are: For the sake of simplicity, the matrix is decomposed Defined as: The feasible region C is defined as: Where 2N-1 is the signal length, To estimate the number of spectral peaks, ||Z|| 2,∞ =max(||Z (1,:) ||2,||Z (2,:) ||2,…,||Z (N,:) ||2),σ is The maximum singular value of , ||·||2 represents the two-norm of the vector; The penalty factor is introduced to transform model (2) into an unconstrained optimization problem for solution; the loss function f(Z) is defined as: Where λ is the regularization parameter; Use the projected gradient algorithm to update the variable Z. In the kth iteration, Update through formula (8); where σ is the step size, gradient Calculated by formula (9): in express conjugation of; make pass Map the updated result into its feasible domain; is the projection operator on the feasible domain C, defined as: Let the solution of step k+1 be in Set the iteration stopping criteria to reach the maximum number of iterations W or the solution x of two consecutive iterations k and x k+1 The difference in the relative two-norm of ||x k+1 -x k ||2 / ||x k ||2 is less than the threshold η, where η is a number greater than 0; after the iteration is terminated, the output of the last iteration is As the final reconstructed complete time domain signal, x last The reconstructed spectrum is obtained by performing Fourier transform.

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