Magnetic resonance spectroscopy reconstruction method based on structured low-rank matrix of local signal blocks

Abstract

The application provides a magnetic resonance spectrum reconstruction method based on a local signal block structured low-rank matrix, and relates to magnetic resonance spectrum, and comprises the following steps: 1) establishing a magnetic resonance spectrum reconstruction model based on a local signal block structured matrix; 2) establishing a reconstruction model of an improved reconstruction model without singular value decomposition; 3) establishing a solution algorithm of the improved reconstruction model without singular value decomposition; and 4) performing two-dimensional Fourier transform on the obtained reconstructed time domain signal to obtain a final magnetic resonance spectrum. According to the technical scheme, a small-scale structured matrix is constructed by using a local signal block to replace a large-scale structured matrix, so that the reconstruction calculation speed is accelerated, and the required internal memory for reconstruction is reduced.

Description

Technical Field

[0001] This invention relates to the field of magnetic resonance spectroscopy undersampling and spectrum reconstruction technology, and in particular to a magnetic resonance spectroscopy reconstruction method based on a structured low-rank matrix of local signal blocks. Background Technology

[0002] Magnetic resonance spectroscopy has become a widely used analytical technique in clinical medicine and biotechnology research. However, acquiring high-dimensional magnetic resonance spectra typically requires lengthy data acquisition times, with typical three-dimensional magnetic resonance spectroscopy acquisition taking several days. Undersampling and reconstruction techniques can effectively reduce acquisition time and have been applied in commercial spectrometer systems. Undersampling and reconstruction techniques sample data at a rate lower than the Nyquist sampling rate and then reconstruct the unsampled data using reconstruction methods. To achieve accurate reconstruction, researchers have proposed using the self-sparseness of the spectrum for reconstruction (Xiaobo Qu, Xue Cao, Di Guo, Zhong Chen, "Compressed sensing for sparse magnetic resonance spectrocopy," International Society for Magnetic Resonance in Medicine 18th Scientific Meeting. Stockholm, Sweden, pp. 3371, 2010.; Xiaobo Qu, Di Guo, Xue Cao, Shuhui Cai, Zhong Chen, "Reconstruction of self-sparse 2D NMR spectra from undersampled data in indirect dimension," Sensors, vol. 11, no. 9, pp. 8888-8909, 2011.). However, numerous experiments have found that methods based on spectral sparsity cannot reliably reconstruct broad-peak data. Furthermore, at lower sampling rates, methods based on spectral sparsity may introduce artifacts and distortions.

[0003] The magnetic resonance spectroscopy reconstruction method based on low-rank structured matrices can effectively overcome the above-mentioned shortcomings. This method utilizes the low-rank characteristics of the structured matrix of the time-domain free-induction decay signal for reconstruction (Xiaobo Qu, Maxim Mayzel, Jian-Feng Cai, Zhong Chen, Vladislav Orekhov. "Accelerated NMR spectroscopy with low-rank reconstruction," Angewandte Chemie International Edition, vol. 54, no. 3, pp. 852-854, 2015.; Hengfa Lu, Xinlin Zhang, Tianyu Qiu, Jian Yang, Jiaxi Ying, DiGuo, Zhong Chen, Xiaobo Qu. "Low rank enhanced matrix recovery of hybrid time and frequency data in fast magnetic resonance spectroscopy," IEEE Transactions on Biomedical Engineering, vol. 65, no. 4, pp. 809-820, 2017.). Compared with methods based on spectral sparsity, the low-rank structured matrix method significantly improves the reconstruction of low-intensity broad spectral peaks and performs better at lower sampling rates.

[0004] However, the aforementioned methods based on low-rank structured matrices require the construction of large-scale structured matrices, resulting in long computation times and high memory consumption during reconstruction. Currently, to address these issues, some studies have proposed using sliding windows to reduce the size of the structured matrix, thereby accelerating reconstruction (Jianfan Wu, Runmin Xu, Yihui Huang, Jiaying Zhan, Zhangren Tu, Xiaobo Qu, Di Guo. "Fast NMR spectroscopy reconstruction with a sliding window based Hankel matrix," Journal of Magnetic Resonance, vol. 342, p. 107283, 2022.). However, this research is only applicable to one-dimensional spectral reconstruction problems. For two-dimensional spectral reconstruction problems, how to construct small-scale structured matrices remains unexplored. Summary of the Invention

[0005] In view of this, the purpose of this invention is to provide a magnetic resonance spectrum reconstruction method based on a structured low-rank matrix of local signal blocks. This method uses a small-scale structured matrix constructed from local signal blocks to replace a large-scale structured matrix, thereby accelerating the reconstruction calculation speed and reducing the memory required for reconstruction.

[0006] To achieve the above objectives, the present invention adopts the following technical solution: a magnetic resonance spectrum reconstruction method based on a structured low-rank matrix of local signal blocks, comprising the following steps:

[0007] Step 1) Establish a magnetic resonance spectral reconstruction model based on a structured matrix of local signal blocks;

[0008] Step 2) Establish an improved reconstruction model that does not require singular value decomposition;

[0009] Step 3) Establish a solution algorithm for the improved reconstruction model that does not require singular value decomposition;

[0010] Step 4) The reconstructed time-domain signal is obtained from step 3). The magnetic resonance spectrum can be obtained by performing a two-dimensional Fourier transform on the reconstructed time-domain signal.

[0011] In a preferred embodiment, in step 1), the specific method for establishing the magnetic resonance spectral reconstruction model based on the structured low-rank matrix of local signal blocks is as follows:

[0012]

[0013] Where X represents the two-dimensional time-domain magnetic resonance spectrum signal to be reconstructed; operator This represents extracting multiple local signal blocks from a two-dimensional time-domain signal, where the size of each local signal block is m×n, and the step sizes in the horizontal and vertical directions are S, respectively. m S n ,Right now Among them, X r This represents the r-th local signal block that has been extracted. This represents an operator that converts multiple small signal blocks into a block Hankel matrix and arranges them column-wise, i.e. Operator This indicates that the matrix is ​​arranged as a block Hankel matrix; Y represents the time-domain spectral signal that is undersampled and zero-filled at the unsampled locations. This represents an operator that undersamples and fills the unsampled locations with zeros; ||·|| * The nuclear norm of a matrix, ||·|| F λ refers to the Frobenius norm of the matrix, where λ is the regularization parameter.

[0014] In a preferred embodiment, in step 2), the specific method for establishing an improved reconstruction model without singular value decomposition is as follows: the model in formula (1) is rewritten using matrix decomposition as follows:

[0015]

[0016] Here, P and Q are two decomposition matrices, and the superscript "H" is the complex conjugate transpose of the matrix.

[0017] In a preferred embodiment, in step 3), the specific method for establishing the solution algorithm for the improved reconstruction model without singular value decomposition is as follows: the reconstruction model in formula (2) is solved using the alternating direction multiplier method as follows:

[0018]

[0019] Where D is the Lagrange multiplier, <·,·> is the inner product, and β is a parameter greater than zero; the variables in (3) are iteratively updated according to the following formula:

[0020]

[0021] When the maximum number of iterations K or X is reached, the error between two adjacent iterations is... The iteration ends when the value is less than the set positive threshold μ; the superscript "-1" indicates finding the matrix inverse, the superscript "*" indicates the adjoint operator, and the subscript "k" indicates the solution of the k-th iteration. k P k Q k D k Let X, P, Q, and D represent the values ​​of variables X, P, Q, and D respectively at the k-th iteration, where τ is a parameter greater than zero; in the initialization algorithm, that is, when k=1, P k and Q k The initial matrix is ​​a random matrix, D k The initial value is a matrix consisting entirely of 1s.

[0022] Compared with existing technologies, this invention has the following advantages: This invention reduces the size of conventional structured matrices by constructing structured matrices from local signal blocks, thereby accelerating reconstruction computation and reducing the memory required for reconstruction. First, a spectral reconstruction model based on a structured matrix of local signal blocks is established. Then, an improved reconstruction model without singular value decomposition is established. Next, an iterative algorithm is used to reconstruct the magnetic resonance time-domain signal. Finally, a two-dimensional Fourier transform is performed on the reconstructed time-domain signal to obtain the final spectrum. This invention constructs structured matrices from local signal blocks, reducing the size of structured matrices in conventional methods, thus significantly reducing computation time and memory usage. Attached Figure Description

[0023] Figure 1This is the sampling template in the embodiments of the present invention. Figure 1 In the diagram, white represents sampled points, and black represents unsampled points.

[0024] Figure 2 This is the fully sampled two-dimensional magnetic resonance spectrum in this embodiment of the invention.

[0025] Figure 3 It is a two-dimensional magnetic resonance spectrum reconstructed using the method of this invention. Detailed Implementation

[0026] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0027] It should be noted that the following detailed descriptions are illustrative and intended to provide further explanation of this application. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application pertains.

[0028] It should be noted that the terminology used herein is for the purpose of describing particular implementations only and is not intended to limit the exemplary implementations according to this application; as used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise; furthermore, it should be understood that when the terms “comprising” and / or “including” are used in this specification, they indicate the presence of features, steps, operations, devices, components and / or combinations thereof.

[0029] The following embodiments will further illustrate the present invention with reference to the accompanying drawings, and provide reconstruction results. This embodiment reconstructs a two-dimensional noise-free magnetic resonance spectrum with a data size of 256×256, containing 20 spectral peaks. Sampling template (e.g.) Figure 1 If 10% of the data is sampled (as shown), then the sampled data points are 6549. The specific steps are as follows:

[0030] 1) Establish a magnetic resonance spectral reconstruction model based on a structured matrix of local signal blocks:

[0031]

[0032] Where X represents the two-dimensional time-domain magnetic resonance spectrum signal to be reconstructed; operator This represents extracting multiple local signal blocks from a two-dimensional time-domain signal, where the size of each local signal block is m×n, and the step sizes in the horizontal and vertical directions are S, respectively. m S n ,Right now Among them, X r This represents the r-th local signal block extracted, where r = 1, ..., 441. In this embodiment, m = n = 16, S m =Sn =12. This represents an operator that converts multiple small signal blocks into a block Hankel matrix and arranges them column-wise, i.e. Operator This indicates that the matrix is ​​arranged as a block Hankel matrix. Y represents the time-domain spectral signal that is undersampled and zero-filled at the unsampled locations. This represents an operator that undersamples and fills the unsampled locations with zeros; ||·|| * The nuclear norm of a matrix, ||·|| F The Frobenius norm of the matrix refers to the regularization parameter λ, which in this embodiment is 10. 6 .

[0033] 2) Establish an improved reconstruction model that does not require singular value decomposition;

[0034] In step 2), the specific method for establishing an improved reconstruction model that does not require singular value decomposition can be as follows: To avoid the time-consuming singular value decomposition, the matrix factorization method (Di Guo, Hengfa Lu, Xiaobo Qu, "Afast low rank Hankel matrix factorization reconstruction method for non-uniformly sampled magnetic resonance spectroscopy," IEEE Access, vol. 5, pp. 16033-16039, 2017.) can be used to rewrite the model in formula (1) as follows:

[0035]

[0036] Here, P and Q are two decomposition matrices, and the superscript "H" is the complex conjugate transpose of the matrix.

[0037] 3) Establish an improved solution algorithm for the reconstruction model that does not require singular value decomposition;

[0038] In step 3), the specific method for establishing the solution algorithm of the improved reconstruction model without singular value decomposition can be as follows: using the alternating direction multiplier method (Zhifang Zhan, Jian-Feng Cai, Di Guo, Yunsong Liu, Zhong Chen, Xiaobo Qu, "Fast multi-class dictionaries learning with geometricaldirections in MRI reconstruction," IEEE Transactions on Biomedical Engineering, vol.63, pp.1850-1861, 2016.) to solve the reconstruction model in formula (2) as follows:

[0039]

[0040] Where D is the Lagrange multiplier, and <·,·> is the inner product. β is a parameter greater than zero; in this embodiment, β = 1. The variables in (3) can be iteratively updated according to the following formula:

[0041]

[0042] When the maximum number of iterations K or X is reached, the error between two adjacent iterations is... The iteration ends when the value is less than the set positive threshold μ. In this embodiment, K = 100 and μ = 10. -5 The superscript "-1" indicates finding the inverse of the matrix, the superscript "*" indicates the adjoint operator, and the subscript "k" indicates the solution of the k-th iteration. X k P k Q k D k Let X, P, Q, and D represent the values ​​of variables X, P, Q, and D respectively at the k-th iteration. τ is a parameter greater than zero; in this embodiment, τ = 1. In the initialization algorithm, that is, when k = 1, P... k and Q k The initial matrix is ​​a random matrix, D k The initial value is a matrix consisting entirely of 1s.

[0043] 4) The reconstructed time-domain signal X is obtained from step 3), and the magnetic resonance spectrum can be obtained by performing a two-dimensional Fourier transform on X.

[0044] Table 1 shows a comparison between the reconstruction time of the fast reconstruction method based on the block Hankel matrix and the reconstruction time according to the method of the present invention.

[0045] Table 1

[0046] Magnetic resonance spectroscopy reconstruction method Reconstruction time (in seconds) A Fast Magnetic Resonance Spectrum Reconstruction Method Based on Block Hankel Matrix 11.3 Method of the present invention 0.7

[0047] As shown in Table 1, the reconstruction time of the method of the present invention is significantly shortened. The present invention replaces the large-scale structured matrix with a structured matrix of local signal blocks to accelerate the reconstruction calculation speed and reduce the memory required for reconstruction. By constructing a small structured matrix using local signal blocks, the size of the structured matrix in conventional methods is reduced, thus significantly reducing the computation time and computational memory.

[0048] Figure 2 It is a fully sampled magnetic resonance spectrum. Figure 3 This is the magnetic resonance spectrum reconstructed according to the method of this invention. (Through...) Figure 2 and Figure 3 The comparison shows that the present invention can reconstruct undersampled magnetic resonance spectra with a small error, while the reconstruction time is short and the required computational memory is small.

Claims

1. A magnetic resonance spectral reconstruction method based on a structured low-rank matrix of local signal blocks, characterized in that... Includes the following steps: Step 1) Establish a magnetic resonance spectral reconstruction model based on a structured matrix of local signal blocks; Step 2) Establish an improved reconstruction model that does not require singular value decomposition; Step 3) Establish a solution algorithm for the improved reconstruction model that does not require singular value decomposition; Step 4) The reconstructed time-domain signal is obtained from step 3). The magnetic resonance spectrum can be obtained by performing a two-dimensional Fourier transform on the reconstructed time-domain signal. In step 1), the specific method for establishing the magnetic resonance spectral reconstruction model based on the structured low-rank matrix of local signal blocks is as follows: (1) in, Represents the two-dimensional time-domain magnetic resonance spectrum signal to be reconstructed; operator This represents extracting multiple local signal blocks from a two-dimensional time-domain signal, where the size of each extracted local signal block is... The step lengths for intercepting the horizontal and vertical directions are respectively , ,Right now ,in, Indicates the truncated first A local signal block; This represents an operator that converts multiple small signal blocks into a block Hankel matrix and arranges them column-wise, i.e. The operator This indicates arranging the matrix into a block Hankel matrix; This represents a time-domain spectral signal that is undersampled and zero-filled at the unsampled locations. This operator represents undersampling and zero-filling at unsampled locations; The nuclear norm of a matrix. The Frobenius norm of the matrix. It is the regularization parameter.

2. The magnetic resonance spectrum reconstruction method based on a structured low-rank matrix of local signal blocks according to claim 1, characterized in that... In step 2), the specific method for establishing an improved reconstruction model that does not require singular value decomposition is as follows: the model in formula (1) is rewritten using matrix decomposition as follows: (2) in, and Given two decomposition matrices, with superscript " " is the complex conjugate transpose of the matrix.

3. The magnetic resonance spectrum reconstruction method based on a structured low-rank matrix of local signal blocks according to claim 2, characterized in that... In step 3), the specific method for establishing the solution algorithm of the improved reconstruction model without singular value decomposition is as follows: the reconstruction model in formula (2) is solved using the alternating direction multiplier method as follows: (3) in, For Lagrange multipliers, For inner product, For parameters greater than zero; iteratively update the variables in (3) according to the following formula: (4) When the maximum number of iterations is reached or Error between two adjacent iterations Less than the set positive threshold When the iteration ends; superscript " The expression "*" indicates the inverse of a matrix, the superscript "*" indicates the adjoint operator, and the subscript "" indicates the inverse of a matrix. " indicates the first The solution of the next iteration , , , Representing variables respectively , , , In the The matrix at the next iteration It is a parameter greater than zero; in the initialization algorithm, that is... hour, and Initially a random matrix, The initial value is a matrix consisting entirely of 1s.

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