A method for dynamic magnetic resonance imaging using implicit neural representation of low-rank tensors

Abstract

The application discloses a dynamic magnetic resonance imaging method for implicitly representing a low-rank tensor by using a neural network. The method comprises the following steps: learning a continuous feature representation of a three-dimensional discrete coordinate point by using an implicit neural network for a magnetic resonance image; dividing the continuous feature into a plurality of overlapping and non-overlapping basic cubes by using a non-local method of the continuous representation, and performing similarity clustering on the basic cubes to construct a continuous group representation in a unified multi-dimensional space; decomposing the continuous group representation by using a coupled low-rank function, and then training a reconstruction model; and obtaining a reconstructed image by using the trained reconstruction model for actual k-space data. The application improves the quality and efficiency of cardiac movie image reconstruction.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of magnetic resonance imaging, and more particularly to a dynamic magnetic resonance imaging method for implicitly representing a low-rank tensor. BACKGROUND

[0002] Magnetic resonance cardiac dynamic movie imaging has important clinical significance for the diagnosis and treatment of cardiovascular diseases. By non-invasively observing the heart movement, the heart chamber volume, ejection fraction and myocardial mass can be accurately quantified, thereby providing important diagnostic basis for doctors, but this way takes a long scanning time.

[0003] In recent years, methods based on deep learning have been successfully applied to cardiac movie dynamic MRI reconstruction, but most deep learning-based reconstruction methods rely on supervised learning training that requires a large amount of fully sampled data. In cardiac MRI dynamic imaging, it is a great challenge to obtain complete k-space data sampling. Due to the long scanning time, it is easy to cause fluctuations in respiration or heart rate, thereby causing the image quality to decrease. Unsupervised / self-supervised learning methods do not require a large amount of labeled data and can directly learn the prior information of the image from the undersampled data to achieve high-quality image reconstruction. For example, a self-supervised learning method adopts a physical guidance-based self-supervised learning strategy to divide the available undersampled data into training and validation subsets, and only relies on the existing undersampled data for optimization during the training process. These methods can achieve high acceleration rates while maintaining high reconstruction quality. However, the iterative nature of this type of method can easily lead to time-consuming calculations of the algorithm. In unsupervised learning methods, researchers use an untrained deep neural network to guide the network to reconstruct dynamic MR images from accelerated data by combining a regularization term of a physical model. This method can effectively reconstruct dynamic MR images at different acceleration factors and does not require a large amount of training data. However, the setting of the initial conditions and network parameters of this type of method has a significant impact on the final results, and if not properly set, it can lead to convergence to a suboptimal solution or the generation of artifacts. Researchers have introduced an implicit neural (INR) network into the MRI image reconstruction task, reformulated the image reconstruction problem as an optimization problem of a continuous function, reduced the dependence on a large amount of external training data, and combined different prior regularizations to recover images from sparsely sampled images. Although the INR-based reconstruction method can have good reconstruction performance in low-dimensional reconstruction tasks, it often requires high computational resources and slow training speed in high-dimensional reconstruction tasks (such as magnetic resonance cardiac dynamic movie image reconstruction, where the model needs to learn representation in both spatial and temporal domains), and the model is prone to overfitting.

[0004] In summary, the existing fast magnetic resonance cardiac dynamic movie reconstruction methods based on unsupervised or self-supervised learning still have limitations in high acceleration ratio, especially in high undersampling ratio, the reconstruction quality will be significantly reduced. SUMMARY

[0005] The purpose of the present application is to overcome the defects of the prior art, and provide a dynamic magnetic resonance imaging method for implicitly representing low-rank tensors by neural networks. The method comprises the following steps:

[0006] For magnetic resonance images, a continuous feature representation of three-dimensional discrete coordinate points is learned using an implicit neural network;

[0007] The continuous features are divided into a plurality of overlapping and non-overlapping basic cubes by a non-local method of continuous representation, and the basic cubes are similarity clustered to construct a continuous group representation in a unified multi-dimensional space;

[0008] The continuous group representation is decomposed by coupling a low-rank function, and a reconstruction model is trained and obtained;

[0009] For actual k-space data, a reconstructed image is obtained using the trained reconstruction model.

[0010] Compared with the prior art, the present application has the advantages that a dynamic magnetic resonance image reconstruction model based on implicit neural representation and low-rank tensor decomposition is proposed, which combines a non-local feature decomposition technology based on continuous representation, and can directly recover high-quality images from high acceleration ratio k-space data. Moreover, the present application trains the model in an unsupervised manner, without the need for full-sampling training data, and can recover high-quality cardiac movie images from high-undersampling data, significantly shortening the image reconstruction time.

[0011] Other features and advantages of the present application will become apparent from the following detailed description of exemplary embodiments thereof, taken in conjunction with the accompanying drawings. BRIEF DESCRIPTION OF DRAWINGS

[0012] The accompanying drawings incorporated in and forming a part of the specification, illustrate embodiments of the present application and, together with the description, serve to explain the principles of the application.

[0013] Figure 1 is a flowchart of a dynamic magnetic resonance imaging method for implicitly representing low-rank tensors by neural networks according to an embodiment of the present application;

[0014] Figure 2 is a schematic diagram of a non-local decomposition operation of continuous representation according to an embodiment of the present application;

[0015] Figure 3is a general framework diagram of a method for magnetic resonance cardiac dynamic movie imaging for implicitly neural representation of low-rank tensor according to one embodiment of the present application;

[0016] Figure 4 is a comparison diagram of image reconstruction image and prior art according to one embodiment of the present application;

[0017] In the drawings, loss-loss; zero filling-zero filling; similar cubes-similar cubes; projected results-projected results; training-training; inference-inference. DETAILED DESCRIPTION

[0018] Various exemplary embodiments of the present application will now be described in detail with reference to the accompanying drawings. Note that the relative arrangement, numerical expressions, and numerical values of components and steps set forth in these embodiments are not limiting to the scope of the present application unless otherwise specifically stated.

[0019] The following description of at least one exemplary embodiment is merely illustrative in nature and is in no way limiting to the scope of the application and its applications or uses.

[0020] Techniques, methods, and apparatus known to those of ordinary skill in the relevant art can not be discussed in detail herein, but should be considered as part of the specification, where appropriate.

[0021] In all of the examples shown and discussed herein, any specific values should be interpreted as illustrative only and not as a limitation. Thus, other examples of exemplary embodiments can have different values.

[0022] Note that like reference numerals and letters in the various drawings herein represent similar items unless otherwise specifically stated, and thus, once an item is defined in one drawing, it need not be further discussed in subsequent drawings.

[0023] In general, the dynamic magnetic resonance imaging method for implicitly neural representation of low-rank tensor provided by the present application first learns a continuous feature representation of a three-dimensional discrete coordinate point using an implicit neural network; then, the features are divided into a plurality of overlapping and non-overlapping basic cubes by a non-local method of continuous representation, and similarity clustering is performed thereon to construct a continuous group representation in a unified multi-dimensional space; finally, a high-efficiency representation of continuous groups is realized by coupling low-rank function decomposition. This approach takes into account both the similarity of features between groups and the uniqueness of features within groups, and can achieve fast and high-quality cardiac movie image reconstruction.

[0024] Specifically, referring to Figure 1 As shown, the dynamic magnetic resonance imaging method for implicitly neural representation of low-rank tensor provided includes the following steps:

[0025] Step S110, an implicit neural network for implicit representation of MR images is constructed to learn continuous feature representation of three-dimensional discrete coordinate points.

[0026] Assuming Y and X are multi-coil undersampled k-space data with C coils and the corresponding image to be reconstructed, respectively, the reconstruction model for reconstructing X from undersampled data is:

[0027]

[0028] where Y j represents the true undersampled k-space data of the jth coil, A j represents the encoding operator on the jth coil, A is the encoding operator, which can be represented as A=MFC, M is the data undersampling operator, F is the Fourier operator, and C is the multi-channel coil sensitivity matrix, is a regularization operator. represents the regularization prior of the image to be reconstructed, and λ is the regularization coefficient.

[0029] In the INR method based on unsupervised learning, the input feature is usually a spatial coordinate vector of N=3 (for cardiac cine data). In order to model the internal continuity of the cardiac cine image, in one embodiment, its intensity is represented as a continuous function of three-dimensional spatial coordinates, and is parameterized using a multilayer perceptron (MLP), for example, X(θ) is learned by the continuous feature learned by the parameterized continuous function, and a total variation (TV) term is used as a regularization term, and the INR-based reconstruction model can be rewritten as:

[0030]

[0031] where R TV represents the total variation (TV) regularization term, and the parameter θ represents the structural characteristics of the internal cardiac cine MR image, and at the same time, the TV term is introduced to suppress noise and preserve the edge details of the image, thereby realizing reconstruction.

[0032] Step S120, the features are divided into a plurality of overlapping and non-overlapping basic cubes by a non-local method of continuous representation, and similarity clustering is performed thereon to construct a continuous group representation in a unified multi-dimensional space.

[0033] Step S120 is used to realize the non-local solution based on continuous representation. Specifically, first, a set of observed coordinates For the minimum and maximum values of the observed coordinates of the dth dimension (d=1,…,N), ad 、b d Therefore, the entire N-dimensional continuous domain space is represented as:

[0034]

[0035] In one embodiment, a Continuous Representation-based NonLocal (CRNL) self-similarity measurement function h(·) can be used to compactly and efficiently represent the image to be reconstructed on the N-dimensional continuous domain space, and an implicit neural network is used to approximate it. For example, by the following optimization equation:

[0036]

[0037] where h -1 (·) represents the inverse operation of the CRNL method, ψ represents the regularization function, and v represents the observed initial coordinates.

[0038] The continuous domain can be divided into multiple non-overlapping basic units according to the dimension. If δ d represents the number of partition intervals corresponding to the dimension, and p represents the hyperparameter that determines the size of the basic unit, then it can be combined into T overlapping and L non-overlapping cubes, and their volume V and number T / L are:

[0039] V = pδ1×pδ2×pδ3×…×(b d -a d ) (5)

[0040] T = (n1-p+1)(n2-p+1)(n3-p+1) (6)

[0041]

[0042] where δ1, δ2, and δ3 represent the interval size of dividing the continuous space in each dimension, and n1, n2, and n3 represent the number of sub-regions after dividing the continuous space in each dimension.

[0043] Figure 2 is a schematic diagram of the non-local solution operation of the continuous representation, where n1 = n2 = 5, p = 2, D t represents the basic continuous cube, represents the key continuous cube, represents the two similar continuous cubes extracted from the lth key cube . In this way, the set of overlapping and non-overlapping cubes constructed is and Next, we construct the global autocorrelation among consecutive cubes. Specifically, for each non-overlapping consecutive cube... The desired result is in an overlapping set of cubes. Find the S most similar consecutive cubes among two similar cubes. and D t The similarity is determined by the function value f on the grid coordinates of both. θ The Euclidean distance is calculated and expressed as:

[0044]

[0045] Among them, v key and v is and D t The coordinates corresponding to each set. Select S sets according to the aforementioned distance index. Middle and set The most similar cube. Then, each And S most similar cubes D t Stack them into a single continuous group; in this way, L consecutive groups can be obtained. To ensure that the observation coordinates of each consecutive group lie in the same cubic space, for example, each original observation coordinate can be adjusted. (i.e., similar cubes) The values ​​of the first three dimensions of the observed coordinates (in the set of observation coordinates) are taken and concatenated with the index s of the cube to construct a new coordinate vector w. v,s , is represented as:

[0046]

[0047] Among them, the definition and They are respectively Minimum coordinate values ​​in the 1st, 2nd, and 3rd dimensions.

[0048] Finally, the l-th All new coordinates w v,s Collecting these coordinates yields a new set of observation coordinates:

[0049]

[0050] And define a new multivariable function:

[0051]

[0052] Step S130: The continuous group is characterized by coupled low-rank function decomposition, thereby obtaining the reconstruction model.

[0053] For example, utilize low-rank tensor function decomposition to efficiently fit the multivariate function of each consecutive group Specifically, introduce shared factor functions Share N+1 factor functions (MLP networks) across all groups, while each group retains an independent core tensor C l Finally, the reconstruction model of magnetic resonance cardiac dynamic movie imaging based on implicit neural representation low-rank tensor can be obtained:

[0054]

[0055] Where, × d represents the tensor product operation of the d-th mode, s l (v) represents the low-rank tensor function of the l-th consecutive group, N represents the dimension number of the consecutive space, C l represents the core matrix of the l-th consecutive group.

[0056] Step S140, training the reconstruction model by using the set loss function.

[0057] In the training process, the continuous representation result is projected back to k-space through the encoding operator (under-sampling matrix, Fourier operator and coil sensitivity matrix combination), and the model parameters are optimized through the data consistency loss and total variation (TV) regularization The final loss function is composed of data consistency and image regularization terms.

[0058] Figure 3 is the overall framework diagram of the magnetic resonance cardiac dynamic movie imaging method based on implicit neural representation low-rank tensor, wherein Figure 3 (a) is a basic cubic partition schematic, Figure 3 (b) is a grouping schematic, Figure 3 (c) is a coupled low-rank tensor decomposition schematic, represents the encoding operator, in the training stage In the inference stage

[0059] Step S150, for the actually acquired k-space data, the trained reconstruction model is used to obtain the reconstruction result.

[0060] In the model inference or application stage, in order to further improve the reconstruction quality, an additional data consistency constraint step is introduced in the k-space domain. For example Figure 3As shown in the inference part of the above equation, the measured k-space undersampled data is used to replace the predicted k-space undersampled data to ensure data consistency. Subsequently, the updated combined k-space data is applied with a 2D inverse Fourier transform to obtain the images of each coil. These coil images are then merged by an adaptive coil combination method to finally generate a high-quality reconstructed image.

[0061] To further verify the effect of the present application, the reconstruction results of the present application (labeled as Ours) and the prior art at high acceleration rates of 12, 16, and 21 are compared. The prior art includes:

[0062] ConvDecode, an unsupervised algorithm combined with a physical model (K.P. Slavkova, et al., “An untrained deep learning method for reconstructing dynamic MR images from accelerated model-based data,” Magnetic Resonance in Medicine, vol. 89, no. 4, pp. 1617-1633, 2023.);

[0063] Modl, a supervised algorithm (H.K. Aggarwal, M.P. Mani, and M. Jacob, “MoDL: Model-based deep learning architecture for inverse problems,” IEEE Transactions on Medical Imaging, vol. 38, no. 2, pp. 394-405, 2018.);

[0064] DCNet, a supervised algorithm (Huang, W., Ke, Z., Cui, Z.X., Cheng, J., Qiu, Z., Jia, S., & Liang, D., “Deep low-rank plus sparse network for dynamic MR imaging,” Med. Image Anal., vol. 73, p. 102190, 2021.);

[0065] Supervised algorithm LSNet (Cheng, J., Chen, H., Ke, Z., Huang, W., Qiu, Z., & Liang, D., “Learning data consistency and its application to dynamic MR imaging,” IEEE Trans. Med. Imag., vol. 40, no. 11, pp. 3140-3153, 2021.). The reconstruction results at high acceleration rates of A = 12, 16, 21 are compared, and the experimental results are shown in Figure 4

[0066] Figure 4 is a schematic diagram of the effect comparison of the present application and the prior art. The first frame and the Y-T reverse dynamic cardiac cine image reconstructed at an undersampling factor of R = 12, 16, 21 are compared. It can be seen from Figure 4 that compared with the prior art, the image error of the present application is the smallest, and the reconstructed cardiac structure is the clearest.

[0067] In summary, compared with the prior art, the present application has the following advantages:

[0068] 1) The present application compactly models complex image patterns in the image recovery task through INR-based low-rank tensor function decomposition, and fully excavates the low-rank characteristics of each dimension of high-dimensional images, thereby efficiently recovering clear images;

[0069] 2) The present application develops an implicit neural representation of low-rank tensors based on unsupervised learning for magnetic resonance cardiac dynamic cine imaging method, which uses a non-local method based on continuous representation to compactly represent the cardiac cine data to be reconstructed, and combines a coupled low-rank function decomposition method to efficiently fit the represented observed data, further improving the quality of image reconstruction.

[0070] 3) It has been verified that in the task of cardiac cine dynamic magnetic resonance reconstruction, using the present application can significantly improve the reconstruction performance, for example, high-quality images can be reconstructed from more than 20 times undersampled data, further shortening the cardiac magnetic resonance scanning time.

[0071] The present application can be a system, a method and / or a computer program product. The computer program product can include a computer readable storage medium having computer readable program instructions loaded thereon for causing a processor to implement various aspects of the present application.

[0072] ​Computer readable storage media can be tangible storage media which can retain and store instructions for use by an instruction execution device. Computer readable storage media can be, for example, but is not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination of the foregoing. More specific examples (a non-exhaustive list) of computer readable storage media include the following: a portable computer diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), a static random access memory (SRAM), a portable compact disc read-only memory (CD-ROM), a digital versatile disk (DVD), a memory stick, a floppy disk, a mechanically encoded device such as punch-cards or raised structures in a groove having instructions recorded thereon, and any suitable combination of the foregoing. A computer readable storage medium, as used herein, is not to be construed as being transitory signals per se, such as radio waves or other freely propagating electromagnetic waves, electromagnetic waves propagating through a waveguide or other transmission media (e.g., light pulses passing through a fiber-optic cable), or electrical signals transmitted through a wire.

[0073] Computer readable program instructions described herein can be downloaded to respective computing / processing devices from a computer readable storage medium or to an external computer or external storage device via a network, for example, the Internet, a local area network, a wide area network and / or a wireless network. The network can comprise copper transmission cables, optical transmission fibers, wireless transmission, routers, firewalls, switches, gateway computers and / or edge servers. A network adapter card or network interface in each computing / processing device receives computer readable program instructions from the network and forwards the computer readable program instructions for storage in a computer readable storage medium within the respective computing / processing device.

[0074] Computer readable program instructions for carrying out operations of the present application can be assembler instructions, instruction-set-architecture (ISA) instructions, machine instructions, machine dependent instructions, microcode, firmware instructions, state-setting data, or either source code or object code written in any combination of one or more programming languages, including an object oriented programming language such as Smalltalk, C++ or the like and conventional procedural programming languages, such as the "C" programming language or similar programming languages. The computer readable program instructions can execute entirely on the user's computer, partly on the user's computer, as a stand-alone software package, partly on the user's computer and partly on a remote computer or entirely on the remote computer or server. In the latter scenario, the remote computer can be connected to the user's computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or the connection can be made to an external computer (for example, through the Internet using an Internet Service Provider). In some embodiments, electronic circuitry including, for example, programmable logic circuitry, field-programmable gate array (FPGA), or programmable logic array (PLA) can execute the computer readable program instructions by utilizing state information of the computer readable program instructions to personalize the electronic circuitry, in order to perform aspects of the present application.

[0075] The computer readable program instructions can also be loaded onto a computer, other programmable data processing apparatus, or other device to cause a series of operational steps to be performed on the computer, other programmable apparatus or other device to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide processes for implementing the functions / acts specified in the flowchart and / or block diagram block or blocks.

[0076] The computer readable program instructions can also be loaded onto a computer, other programmable data processing apparatus, or other device to cause a series of operational steps to be performed on the computer, other programmable apparatus or other device to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide processes for implementing the functions / acts specified in the flowchart and / or block diagram block or blocks.

[0077] The computer readable program instructions can also be loaded onto a computer, other programmable data processing apparatus, or other device to cause a series of operational steps to be performed on the computer, other programmable data processing apparatus or other device to produce a computer implemented process such that the instructions which execute on the computer, other programmable data processing apparatus, or other device implement the functions / acts specified in the flowchart and / or block diagram block or blocks.

[0078] The computer readable program instructions can also be loaded onto a computer, other programmable data processing apparatus, or other device to cause a series of operational steps to be performed on the computer, other programmable data processing apparatus or other device to produce a computer implemented process such that the instructions which execute on the computer, other programmable data processing apparatus, or other device implement the functions / acts specified in the flowchart and / or block diagram block or blocks.

[0079] Embodiments of the present application have been described above, and the description is intended to be illustrative, and not restrictive, of the various embodiments of the present application. Many modifications and variations of the described embodiments of the present application are possible, given the benefit of the present disclosure, without departing from the scope and spirit of the described embodiments of the present application. The scope of the present application is defined by the appended claims.

Claims

1. A dynamic magnetic resonance imaging method for implicitly neural representation of low-rank tensors, comprising the steps of: learning a continuous feature representation of three-dimensional discrete coordinate points using an implicit neural network for magnetic resonance images; partitioning the continuous features into a plurality of overlapping and non-overlapping elementary cubes by a non-local method of continuous representation and performing a similarity clustering on the elementary cubes to construct a continuous group representation in a unified multi-dimensional space; obtaining a reconstruction model by decomposing the continuous group representation with a coupled low-rank function and training the reconstruction model; obtaining a reconstructed image using the trained reconstruction model for actually acquired k-space data; wherein the continuous group representation in the unified multi-dimensional space is constructed according to the following steps: Given a set of observed coordinates The minimum and maximum values of the observed coordinates for the dth dimension are a d , b d , and the entire N-dimensional continuous domain is represented as wherein N is the number of dimensions; The image to be reconstructed on the N-dimensional continuous domain space is represented based on a non-local self-similarity measure function h(·) of continuous representation, and an implicit neural network is used to approximate The optimization equation is set as: where c denotes the number of multi-channel coils, Y j denotes the true under-sampled k-space data of the jth coil, A j denotes the encoding operator on the jth coil, h -1 (·) denotes the inverse operation of the CRNL method, v denotes the initial coordinates of the observation, ψ[f θ (·)] denotes the regularization prior of f θ (·). continuous domain Divide the cubes into multiple non-overlapping basic cubes according to their dimensions, and then combine them into T overlapping cubes and L non-overlapping cubes. The volume V and number of the cubes are represented as follows: V = pδ1 x pδ2 x pδ3 x... x pδ d T = (n1-p+1)(n2-p+1)(n3-p+1) wherein δ d represents the number of division intervals in the corresponding dimension, p represents a hyperparameter that determines the size of the basic cube, T groups of overlapping and L groups of non-overlapping cubes are combined, δ1, δ2, and δ3 represent the interval size of dividing the continuous space in each dimension, and n1, n2, and n3 represent the number of sub-regions after dividing the continuous space in each dimension; For each non-overlapping contiguous cube In the overlapping cube set find its most similar S contiguous cubes, where two similar cubes and D t The similarity of two cubes is measured by the distance of the function values f θ over their grid coordinates; Select S sets according to the calculated distance The most similar cube in the set The most similar cube in the set And S most similar cubes D t Stack them to form a continuous group, and get L continuous groups Adjust the values of the first three dimensions of the set of observed coordinates in the similar cuboid and concatenate them with the index s of the cuboid to construct a new coordinate vector w v,s , denoted as: The first All the new coordinates w v,s Collecting them gives a new set of observed coordinates: and defining a new multivariate function: wherein the definitions and are The minimum coordinate values in the 1st, 2nd and 3rd dimensions.

2. The method of claim 1, wherein, the distance is a Euclidean distance, expressed as: where v key and v is and D t corresponding coordinates in the set.

3. The method of claim 1, wherein, the implicit neural network is constructed based on a multi-layer perceptron.

4. The method of claim 1, wherein, the reconstruction model is expressed as: where x d denotes the tensor product operation of the d-th mode, s l (v) denotes the low-rank tensor function of the l-th consecutive group, N denotes the dimension number of the continuous space, C l denotes the core matrix of the l-th consecutive group, R TV denotes the total variation regularization term.

5. The method of claim 1, wherein, In the reconstruction model training phase, the continuous representation results are projected back to k-space by an encoding operator A = MFC, and in the reconstruction model application phase where M is a data undersampling operator, F is a Fourier operator, and C is a multi-channel coil sensitivity matrix.

6. The method of claim 1, wherein, a loss function for training the reconstruction model comprises a data consistency loss and an image regularizer loss.

7. A computer readable storage medium having stored thereon a computer program, wherein, The computer program, when executed by a processor, implements the steps of the method according to any one of claims 1 to 6.

8. A computer device comprising a memory and a processor, having stored on the memory a computer program capable of running on the processor, characterized in that, The processor, when executing the computer program, implements the steps of the method according to any one of claims 1 to 6.

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