Magnetic resonance spectrum reconstruction method based on structured matrix null space
By constructing a structured matrix null space model and utilizing singular value decomposition and the conjugate gradient method, the spectral aliasing problem in magnetic resonance spectral undersampling reconstruction was solved, achieving fast and high-fidelity signal recovery.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-12-10
- Publication Date
- 2026-03-10
AI Technical Summary
Existing technologies suffer from spectral aliasing and artifacts in magnetic resonance spectral undersampling reconstruction, making it difficult to achieve efficient and accurate signal recovery.
The reconstruction method based on the null space of the structured matrix obtains calibration data from non-uniformly sampled signals, constructs a structured matrix, calculates the null space using singular value decomposition, and solves the reconstructed signal using the conjugate gradient method to build a model that includes data consistency and null space constraints.
It achieves rapid and high-fidelity reconstruction of magnetic resonance spectra, improves computational efficiency and robustness, reduces the number of iterations, and enhances reconstruction quality.
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Figure CN121633951A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The application belongs to the field of magnetic resonance spectroscopy under-sampling and reconstruction, and particularly relates to a magnetic resonance spectroscopy reconstruction method based on structured matrix null space. BACKGROUND
[0002] Magnetic resonance spectroscopy is a non-invasive analysis technology based on magnetic resonance phenomenon, which can be used for atomic resolution structure analysis and has wide application in the fields of physics, chemistry and biological science, and has important value in tumor diagnosis, metabolic disease analysis and nervous system research.
[0003] In nuclear magnetic resonance spectroscopy experiment, free induction decay (FID) is the original form of signal acquisition. In order to reduce the sampling time, non-uniform sampling is often used in practice to obtain part of the data according to a specific sparse sampling mode. However, too few sampling points do not meet the Nyquist sampling theorem, which will cause serious aliasing and artifacts in the spectrum, and the under-sampled FID signal must be recovered through an efficient reconstruction algorithm.
[0004] Traditional methods usually construct constraint conditions by exploiting the sparsity and low-rank of FID signals, and obtain the reconstruction results through multiple iterations. Donoho et al. proposed compressed sensing algorithm, which utilized the sparsity of signals in the transform domain to achieve reconstruction under low sampling rate by minimizing the L1 norm (Donoho D L. Compressed sensing[J]. IEEE Transactions on Information Theory, 2006, 52(4): 1289-1306.). Bostock et al. proposed an improved compressed sensing algorithm, which improved the sampling flexibility and reconstruction resolution by randomly collecting orthogonal component data, and was more advantageous in high-dimensional experiments (Bostock M J, Holland D J, Nietlispach D. Improving resolution in multidimensional NMR using random quadrature detection with compressed sensing reconstruction[J]. Journal of Biomolecular NMR, 2017, 68:67-77.). Qu et al. proposed to convert FID signals into structured low-rank Hankel matrix, which utilized the physical property that the matrix rank is equal to the number of spectral peaks, and achieved high-fidelity spectral reconstruction through low-rank approximation (Qu X, Mayzel M, Cai J F, et al. Accelerated NMR Spectroscopy with Low‐Rank Reconstruction[J]. Angewandte Chemie International Edition, 2015, 54(3): 852-854.). Further, Wu et al. proposed to use sliding window to construct structured low-rank Hankel matrix, which effectively reduced the matrix size and accelerated the reconstruction speed (Wu J, Xu R, Huang Y, et al. Fast NMR spectroscopy reconstruction with a sliding window based Hankel matrix[J]. Journal of Magnetic Resonance, 2022, 342: 107283.).
[0005] On the other hand, the model based on null space constraint has been widely used in the reconstruction of Magnetic Resonance Imaging (MRI). Shin et al. proposed a simultaneous auto-calibration k-space estimation method, which reorganized the multi-channel k-space data into a single structured matrix and achieved automatic calibration and reconstruction by low-rank completion (Shin P J, Larson P E Z, Ohliger M A, et al. Calibrationless parallel imaging reconstruction based on structured low-rank matrix completion[J]. Magnetic Resonance in Medicine, 2014, 72(4): 959–970). Haldar proposed the LORAKS framework for MRI reconstruction, which modeled the low-rank of local k-space neighborhood and combined with regularization and low-rank promotion algorithm to improve the reconstruction quality (Haldar JP. Low-rank modeling of local k-space neighborhoods (LORAKS) for constrained MRI[J]. IEEE Transactions on Medical Imaging, 2013, 33(3): 668–681.). Further, Haldar proposed AC-LORAK, which used the auto-calibration signal to estimate the null space basis of LORAKS matrix, effectively accelerated the calculation speed and improved the reconstruction quality (Haldar J P. Autocalibrated LORAKS for fast constrained MRI reconstruction[C], IEEE 12th International Symposium on Biomedical Imaging (ISBI). IEEE, 2015: 910–913.). In high-dimensional complex data, Qian et al. proposed to decompose the two-dimensional matrix into multiple one-dimensional sub-matrices, and divide them into strong subspace and uncertain subspace, which is conducive to improve the reconstruction speed and protect the details of the reconstructed image (Qian C, Han M, Zhu L, et al. Fast and ultra-high shot diffusion MRI image reconstruction with self-adaptive Hankel subspace[J]. Medical Image Analysis, 2025, 102: 103546.).Current LORAKS and SAKE are only widely used in the field of MRI reconstruction, but the core is the zero space constraint of structured matrix, and this idea is also applicable to nuclear magnetic resonance spectroscopy. Tu et al. proposed a fast self-learning subspace method to realize high-quality reconstruction of magnetic resonance spectroscopy (Tu Z, Liu H, Zhan J, et al. A fast self-learning subspace reconstruction method for non-uniformly sampled nuclear magnetic resonance spectroscopy[J]. Applied Sciences, 2020, 10(11):3939.). This is a subspace-based method in magnetic resonance spectroscopy, but this method aims to improve the reconstruction quality with subspace, rather than realizing computational acceleration in the algorithm structure. SUMMARY
[0006] The purpose of the present application is to provide a structured matrix zero space based magnetic resonance spectroscopy reconstruction method, which utilizes the low rank of time domain nuclear magnetic resonance spectroscopy signal and the linear consistency in the zero space to realize fast and high-fidelity reconstruction of magnetic resonance spectroscopy.
[0007] To achieve the above purpose, the technical scheme of the present application is: a structured matrix zero space based magnetic resonance spectroscopy reconstruction method, comprising:
[0008] Step 1), obtaining calibration data from the full sampling region of non-uniformly sampled time domain nuclear magnetic resonance spectroscopy signal;
[0009] Step 2), constructing a structured matrix by sliding window rearrangement on the calibration data;
[0010] Step 3), calculating the zero space based on the singular value decomposition of the structured matrix;
[0011] Step 4), constructing a magnetic resonance spectroscopy reconstruction model containing data consistency term and zero space constraint term;
[0012] Step 5), solving the to-be-reconstructed signal by using conjugate gradient method.
[0013] Further, in step 1), the method for obtaining calibration data from the full sampling region of the sampled time domain nuclear magnetic resonance spectroscopy signal is: for one-dimensional signal, the calibration data is ; for two-dimensional signal, the calibration data is ; for three-dimensional signal, the calibration data is ; and higher dimensional signal is similar; wherein represents the low frequency part of the full sampling signal or the pre-reconstruction result using other fast reconstruction method, These represent the size of the data extracted in each dimension.
[0014] Furthermore, for two-dimensional signals, the calibration data is as follows: The collected data is rapidly pre-reconstructed using compressed sensing algorithms. That is, from the pre-reconstruction signal A 100x100 area is selected as the calibration data. .
[0015] Furthermore, in step 2), the method for constructing a structured matrix by rearranging the calibration data using a sliding window is as follows:
[0016]
[0017] in This refers to the calibration data obtained in step 1). This represents an operator that transforms a signal into a structured low-rank matrix; a one-dimensional magnetic resonance spectroscopy signal is processed through... Constructing a Hankel matrix, the two-dimensional magnetic resonance spectral signal is transmitted through... By constructing a block Hankel matrix, higher-dimensional magnetic resonance signals can be transmitted through... Construct a multidimensional Hankel matrix; Indicates through operator Will The resulting structured matrix.
[0018] Furthermore, in step 3), the null space is calculated using the singular value decomposition of the structured matrix. The method is as follows:
[0019]
[0020]
[0021] in Representing the structured matrix respectively The left singular vector matrix, the singular value diagonal matrix, and the right singular vector matrix obtained by performing singular value decomposition. This indicates the number of the largest singular values that need to be retained. It is the null space composed of the last few columns of the right singular vector matrix.
[0022] Furthermore, in step 4), the method for constructing the magnetic resonance spectral reconstruction model, which includes data consistency terms and null space constraint terms, is as follows:
[0023]
[0024] in This represents the time-domain signal to be reconstructed. This represents the collected data that was undersampled and filled with zeros at the unsampled locations. This represents an undersampled mask matrix where the value is 1 at the sampled position and 0 at the unsampled position. This represents an operator that transforms a signal into a structured low-rank matrix. The null space calculated in step 3), Indicates the weighting coefficient. The square of the 2-norm of a vector. Representing vectors - The square of the norm.
[0025] Furthermore, in step 5), the method for solving the signal to be reconstructed using the conjugate gradient method is as follows: For Find the gradient and set it to 0:
[0026]
[0027] in This represents the time-domain signal to be reconstructed. This represents the collected data that was undersampled and filled with zeros at the unsampled locations. This represents an operator that transforms a signal into a structured low-rank matrix. This represents an operator that restores a structured matrix to its original form. The null space calculated in step 3), Denotes the conjugate transpose of the null space. Represents the undersampling mask matrix. The weighting coefficients are represented by the conjugate gradient method, which is used to solve for the signal to be reconstructed.
[0028] The present invention also provides a magnetic resonance spectrum reconstruction system based on a structured matrix null space, including a memory, a processor, and computer program instructions stored in the memory and executable by the processor. When the processor executes the computer program instructions, it can implement the steps of the method described above.
[0029] The present invention also provides an electronic device, characterized in that it includes a memory, a processor, and computer program instructions stored in the memory and executable by the processor, wherein when the processor executes the computer program instructions, it can implement the steps of the method described above.
[0030] The present invention also provides a computer-readable storage medium having stored thereon computer program instructions that can be executed by a processor, wherein when the processor executes the computer program instructions, it can implement the steps of the method described above.
[0031] Compared with existing technologies, the present invention has the following advantages: The magnetic resonance spectrum reconstruction method based on structured matrix null space proposed in this invention utilizes the linear dependence of signals to construct null space constraints, which has high robustness. Moreover, compared with the reconstruction method of Hankel matrix, this method does not require iterative and repeated matrix decomposition, thus achieving higher computational efficiency and is suitable for fast and accurate reconstruction of magnetic resonance spectrum. Attached Figure Description
[0032] Figure 1 This is a schematic diagram of the method flow of the present invention.
[0033] Figure 2 This is the magnetic resonance spectrum used for reference in the embodiments.
[0034] Figure 3 This is the sampling template used in the embodiment.
[0035] Figure 4 The spectrum was reconstructed using a low-rank Hankel matrix method based on matrix factorization.
[0036] Figure 5 This is the spectrum reconstructed by the method of this invention.
[0037] Table 1 shows the reconstruction time and relative L2-norm error of the singular value decomposition of the Hankel matrix in this invention. Detailed Implementation
[0038] The technical solution of the present invention will now be described in detail with reference to the accompanying drawings.
[0039] like Figure 1 As shown, this invention provides a magnetic resonance spectrum reconstruction method based on a structured matrix null space, comprising:
[0040] Step 1) Obtain calibration data from the full sampling region of the non-uniformly sampled time-domain nuclear magnetic resonance spectroscopy signal;
[0041] Step 2) Rearrange the calibration data using a sliding window to construct a structured matrix;
[0042] Step 3) Calculate the null space based on the singular value decomposition of the structured matrix;
[0043] Step 4) Construct a magnetic resonance spectral reconstruction model that includes data consistency terms and null space constraint terms;
[0044] Step 5) Use the conjugate gradient method to solve for the signal to be reconstructed.
[0045] The following are specific implementation examples of the present invention.
[0046] This embodiment will reconstruct a simulated two-dimensional magnetic resonance spectrum signal with random Gaussian noise of intensity 0.001 added. The data size is 256*256. The original spectrum without noise is as follows: Figure 2 As shown. The sampling template is made of Figure 3 As shown, this template uses a Poisson interval sampling of 25%, and the reconstruction result of this method is as follows. Figure 5 As shown. The specific reconstruction steps are as follows:
[0047] Step 1), the method for obtaining magnetic resonance spectral calibration data is as follows: For a two-dimensional signal, its calibration data is... In this embodiment, the collected data is rapidly pre-reconstructed using a compressed sensing algorithm. That is, from the pre-reconstruction signal A 100x100 area is selected as the calibration data. .
[0048] Step 2), the method for constructing a structured matrix by rearranging the calibration data using a sliding window is as follows:
[0049]
[0050] in This represents an operator that transforms a signal into a structured low-rank matrix. In this embodiment, a one-dimensional magnetic resonance spectroscopy signal is transformed through... Construct a Hankel matrix. Indicates through operator calibration data The resulting structured matrix. In this embodiment, a 12x12 sliding window is used to... Convert to structured matrix .
[0051] Step 3), the calculation of the null space by the singular value decomposition of the structured matrix. The method is as follows:
[0052]
[0053]
[0054] in They represent respectively by The left singular vector matrix, the singular value diagonal matrix, and the right singular vector matrix obtained by performing singular value decomposition. This indicates the number of the largest singular values that need to be retained. It is the null space composed of the last few columns of the right singular vector matrix. In this embodiment, the right singular vector matrix... To balance computational speed and reconstruction quality, we chose... That is, to retain The last 70 singular vectors form the basis of the null space, and the final null space is obtained. .
[0055] Step 4), the method for constructing the magnetic resonance spectral reconstruction model including data consistency terms and null space constraint terms is as follows:
[0056]
[0057] in This represents the time-domain magnetic resonance spectrum signal to be reconstructed. This represents the collected data that was undersampled and filled with zeros at the unsampled locations. This represents an undersampled mask matrix where the value is 1 at the sampled position and 0 at the unsampled position. Indicates the weighting coefficient. The square of the 2-norm of a vector. Representing vectors -The square of the norm. In this embodiment, , , .
[0058] Step 5), the method of solving the signal to be reconstructed using the conjugate gradient method is as follows: For Find the gradient and set it to 0:
[0059]
[0060] in This represents an operator that restores a structured low-rank matrix to its original form. Let represent the conjugate transpose matrix of the null space. Finally, the conjugate gradient method is used to solve for the signal to be reconstructed.
[0061] like Figure 5 As shown, the spectral reconstruction time of the method of the present invention is... Figure 4 Table 1 compares the time consumption and relative L2 norm error of the spectrum reconstruction method based on matrix factorization of low-rank Hankel matrix (Guo D, Lu H, Qu X. A fast low rank Hankel matrix factorization reconstruction method for non-uniformly sampled magnetic resonancespectroscopy[J]. IEEE Access, 2017, 5: 16033–16039.). It can be seen that the method of the present invention can achieve the purpose of fast spectrum reconstruction.
[0062] surface
[0063]
[0064] The present invention also provides a magnetic resonance spectrum reconstruction system based on a structured matrix null space, including a memory, a processor, and computer program instructions stored in the memory and executable by the processor. When the processor executes the computer program instructions, it can implement the steps of the method described above.
[0065] The present invention also provides an electronic device, characterized in that it includes a memory, a processor, and computer program instructions stored in the memory and executable by the processor, wherein when the processor executes the computer program instructions, it can implement the steps of the method described above.
[0066] The present invention also provides a computer-readable storage medium having stored thereon computer program instructions that can be executed by a processor, wherein when the processor executes the computer program instructions, it can implement the steps of the method described above.
[0067] The above are preferred embodiments of the present invention. Any changes made to the technical solution of the present invention that do not exceed the scope of the technical solution of the present invention shall fall within the protection scope of the present invention.
Claims
1. A method for magnetic resonance spectroscopy reconstruction based on structured matrix null space, characterized in that, The method comprises the following steps: Step 1), obtaining calibration data from a full sampling region of a non-uniformly sampled time-domain nuclear magnetic resonance spectrum signal; Step 2), constructing a structured matrix by rearranging through a sliding window on the calibration data; Step 3), calculating a null space based on singular value decomposition of the structured matrix; Step 4), constructing a magnetic resonance spectrum reconstruction model comprising a data consistency term and a null space constraint term; Step 5), solving the to-be-reconstructed signal by using a conjugate gradient method.
2. The structured matrix null space based magnetic resonance spectroscopy reconstruction method of claim 1, wherein, In step 1), the method for acquiring calibration data from the full sampling region of the sampled time-domain NMR spectrum signal is as follows: for a one-dimensional signal, the calibration data is ; for a two-dimensional signal, the calibration data is ; for a three-dimensional signal, the calibration data is ; and for higher-dimensional signals, the same applies; wherein represents the low-frequency part of the full-sampling signal or the result of pre-reconstruction using other fast reconstruction methods, respectively represents the data size after truncation in each dimension.
3. The structured matrix null space based magnetic resonance spectroscopy reconstruction method of claim 2, wherein, For two-dimensional signal, its calibration data is The compressed sensing algorithm is used to quickly pre-reconstruct the collected data, and the pre-reconstruction signal is That is, a 100*100 size region is intercepted from the pre-reconstruction signal as calibration data .
4. The structured matrix null space based magnetic resonance spectroscopy reconstruction method of claim 1, wherein, In step 2), the method for constructing the structured matrix by rearranging through the sliding window on the calibration data is as follows: wherein denotes the calibration data obtained in step 1), denotes an operator that transforms the signal into a structured low-rank matrix, one-dimensional magnetic resonance spectroscopy signals are transformed by into a Hankel matrix, two-dimensional magnetic resonance spectroscopy signals are transformed by into a block-Hankel matrix, higher-dimensional magnetic resonance signals are transformed by into a multi-dimensional Hankel matrix; denotes the structured matrix that is transformed by the operator into a structured matrix. 5. The structured matrix null space based magnetic resonance spectroscopy reconstruction method of claim 1, wherein, In step 3), the null space is computed from the structured matrix singular value decomposition The method is as follows: wherein respectively represent the left singular vector matrix, the singular value diagonal matrix, the right singular vector matrix obtained by singular value decomposition of the structured matrix respectively represent the left singular vector matrix, the singular value diagonal matrix, the right singular vector matrix obtained by singular value decomposition of the structured matrix represents the number of maximum singular values to be retained, is the null space composed of the last columns of the right singular vector matrix.
6. The structured matrix null space based magnetic resonance spectroscopy reconstruction method of claim 1, wherein, In step 4), the method for constructing the magnetic resonance spectrum reconstruction model comprising the data consistency term and the null space constraint term is as follows: wherein denotes the time domain signal to be reconstructed, denotes the acquired data which is undersampled and zero-padded at the non-sampled positions, denotes an undersampling mask matrix which takes the value 1 at the sampling positions and 0 at the non-sampled positions, denotes an operator which transforms the signal into a structured low-rank matrix, is the null space computed for step 3), denotes the weight coefficients, denotes the square of the 2-norm of a vector, denotes the square of the -norm of a vector.
7. The structured matrix null space based magnetic resonance spectroscopy reconstruction method of claim 1, wherein, In step 5), the method for solving the signal to be reconstructed is as follows: taking the derivative of the function with respect to the signal to be reconstructed and setting it to 0: the gradient and setting it to 0: wherein denotes the time domain signal to be reconstructed, denotes the acquired data which is undersampled and zero-padded at the non-sampled positions, denotes an operator which transforms the signal into a structured low-rank matrix, denotes an operator which restores the structured matrix to the original matrix, is the null space calculated for step 3), denotes the conjugate transpose matrix of the null space, denotes the undersampling mask matrix, denotes the weight coefficients which are solved using the conjugate gradient method for the signal to be reconstructed.
8. A structured matrix null space based magnetic resonance spectroscopy reconstruction system, characterized by, The computer program instructions are stored in the memory and can be executed by the processor, and when the processor executes the computer program instructions, the steps of the method according to any one of claims 1-7 can be implemented.
9. An electronic device, comprising: The computer program instructions are stored in the memory and can be executed by the processor, and when the processor executes the computer program instructions, the steps of the method according to any one of claims 1-7 can be implemented. 10.A computer readable storage medium, having stored thereon computer program instructions capable of being executed by a processor, and when the processor executes the computer program instructions, the steps of the method according to any one of claims 1-7 can be implemented.