Cardiac magnetic resonance image reconstruction method based on algorithm unfolding

Abstract

The application discloses a kind of based on algorithm development cardiac magnetic resonance image reconstruction method, belong to image processing field, this method includes the following steps: S1: obtaining original cardiac magnetic resonance image;S2: establish based on local convolution dictionary learning cardiac magnetic resonance image reconstruction model;S3: using Nesterov acceleration proximal gradient method to solve based on local convolution dictionary learning cardiac magnetic resonance image reconstruction model;S4: according to step S2 and S3 construct cardiac magnetic resonance image reconstruction neural network model;S5: cardiac magnetic resonance image reconstruction neural network model is trained;S6: according to new cardiac magnetic resonance image obtains corresponding image to be reconstructed.This application can effectively capture the structural information and detail features of cardiac magnetic resonance image, obtain high-quality reconstructed image.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of image processing, and particularly relates to a cardiac magnetic resonance image reconstruction method based on algorithm unfolding. BACKGROUND

[0002] Cardiac magnetic resonance (CMR) imaging is one of the important detection tools in clinical practice. However, the traditional CMR technique usually requires a long sampling process, which is challenging for patients with limited breath-holding capacity. A commonly used technique to solve this problem is the compressed sensing cardiac magnetic resonance (CS-CMR) imaging technique, which uses the compressed sensing (CS) theory to reconstruct CMR images from highly undersampled k-space data. In 2008, Lustig et al. first introduced the CS theory into the magnetic resonance imaging (MRI) reconstruction task, and proposed a non-coherent undersampling scheme for MRI, which significantly reduced the scanning time and improved the resolution of the image. In 2011, Wech et al. combined the CS theory with CMR and proposed an accelerated magnetic resonance image reconstruction method based on CS, which used sparse transform to significantly reduce the scanning time while obtaining high spatial and temporal resolution CMR images. In 2013, Wang et al. proposed a dynamic CMR image reconstruction method based on a convolutional dictionary learning model for compressed sensing dynamic CMR image reconstruction, which used a three-dimensional spatiotemporal dictionary based on image blocks to sparsely represent dynamic sequences, and obtained higher spatial and temporal resolution CMR images while improving the imaging speed. However, the above-mentioned model-based optimization methods often have high computational complexity, and there are difficulties in selecting the optimal algorithm and tuning the parameters. In order to overcome the above defects, Ke et al. proposed an unsupervised deep learning method for multi-coil CMR based on a time-interleaved sampling strategy. However, the CMR image reconstruction method based on deep learning usually has a complex internal structure, which makes it difficult to understand its internal mechanism and lacks interpretability. SUMMARY

[0003] The purpose of the present application is to provide a cardiac magnetic resonance image reconstruction method based on algorithm unfolding, which can effectively capture the structural information and detailed features of the cardiac magnetic resonance image, thereby significantly improving the overall quality of the reconstructed image.

[0004] To achieve the above-mentioned purpose, the present application discloses a cardiac magnetic resonance image reconstruction method based on algorithm unfolding, which comprises the following steps:

[0005] Step S1: obtaining an original cardiac magnetic resonance image;

[0006] The original cardiac magnetic resonance image is an undersampled cardiac magnetic resonance image b acquired from k-space, which is saved in a row vector;

[0007] S2: establishing a cardiac magnetic resonance image reconstruction model based on local convolution dictionary learning;

[0008] The cardiac magnetic resonance image reconstruction model based on local convolution dictionary learning is described as follows:

[0009]

[0010] wherein, represents an undersampled Fourier encoding matrix; represents a row of undersampled CMR images acquired from k-space, is a row of CMR images to be reconstructed corresponding to b, represented by a row vector; represents a convolution dictionary learning model based on local processing, N represents the signal dimension, which is the same as N in MxN, represents a local convolution dictionary, is an atom in the local convolution dictionary, j represents the index of the dictionary atom, is the sparse coefficient of the CMR image to be reconstructed in the local convolution dictionary D L ; is an operator that places D L α i in the i-th position and fills the remaining items with zeros; is a data consistency term error variable;‖α i ‖1 is an l1 norm constraint on the vector α i , representing the sum of the absolute values of all non-zero elements in α i ; represents the square of the l2 norm constraint on the intra-symbol vector, ρ, r1, r2 represent the first, second and third regularization parameters respectively; Ω(D L ) represents a unit norm ball constraint that ensures the length of the dictionary element is 1; in the cardiac magnetic resonance image reconstruction model based on local convolution dictionary learning, x, α i , z and D L are the to-be-optimized variables to be solved;

[0011] S3: solving the cardiac magnetic resonance image reconstruction model based on local convolution dictionary learning using the Nesterov accelerated proximal gradient method, including the following steps:

[0012] S31: decomposing the objective function of the reconstruction model into two parts, one of which is a gradient Lipschitz continuous C1 a lower semi-continuous convex function g(a i ,z) and a lower semi-continuous convex function g(a i ,D L ), specifically:

[0013]

[0014] S32: calculating the extrapolation point of the to-be-optimized variable, updating the extrapolation point of the to-be-optimized variable using the proximal gradient descent method according to the gradient function of the to-be-optimized variable, and using φ k (x)=r3‖x-x k ‖1 as the improved dry friction potential function in the proximal gradient descent method;

[0015] S33: solving the proximal mapping function of the to-be-optimized variable, including:

[0016] S331: solving the composite proximal mapping of updating the sparse coefficient a i ;

[0017] S332: solving the composite proximal mapping of updating the local convolution dictionary D L ;

[0018] S333: solving the composite proximal mapping of updating the error variable z;

[0019] S334: solving the proximal mapping of updating the to-be-reconstructed CMR image x;

[0020] S4: constructing a cardiac magnetic resonance image reconstruction neural network model according to steps S2 and S3;

[0021] S5: training the cardiac magnetic resonance image reconstruction neural network model;

[0022] S6: obtaining the corresponding to-be-reconstructed image according to the new cardiac magnetic resonance image.

[0023] Preferably, Ω(D L ) in the formula 1 represents a unit norm ball constraint for ensuring that the length of the dictionary element is 1:

[0024]

[0025] wherein m represents the total column number of D L , j represents the index of the column number of D L , the set W represents a unit norm ball for limiting the length of the dictionary element, and D L [:,j] represents all elements of the jth column of D L .

[0026] Preferably, the step S32 specifically includes:

[0027] as shown and is the extrapolation point of the to-be-optimized variable at the kth iteration, specifically:

[0028]

[0029] wherein, represents an inertia term coefficient, and a is a viscous damping parameter, and the value of a is a positive number, z k and x k are the to-be-optimized variables a i , D L , z, x and the extrapolation point at the kth iteration, the initial value of a is zero, the initial value of z k is a value of standard normal distribution, and the initial value of x k is obtained by processing k-space undersampling data b using a partial Fourier transform reconstruction operator to obtain an initial to-be-reconstructed CMR image

[0030] The gradient functions of the to-be-optimized variables are as follows:

[0031]

[0032]

[0033] wherein, δ = 1 + β, P j , α j represent a specific point of P i , α i , and ρ represents a first regularization parameter.

[0034] The proximal gradient descent method is used to update z k+1 and x k+1 , and the update formulae are as follows:

[0035]

[0036] wherein, φ k (z) = r3‖z-x k ‖1 is an improved dry friction potential function, r3 represents a fourth regularization parameter, λ represents a step parameter of the proximal gradient descent, and prox() is a proximal mapping function.

[0037] Preferably, the S331 updates the composite proximal mapping of the sparse coefficient a i , which comprises:

[0038] Define an intermediate variable is:

[0039]

[0040] According to the definition of the proximal mapping function, we have

[0041]

[0042] where u denotes a candidate solution to the minimization problem above;

[0043] which is simplified to a one-dimensional L1-norm regularization problem:

[0044]

[0045] where is an element in , is an element in , is a real number; Let According to formula (19), we have T h (w) = -T -h (-w); therefore, we only consider the case of h ≥ 0. Analysis shows that when h ≥ 0, the unique closed-form solution of T h (w) is:

[0046]

[0047] Therefore, the closed-form solution of the composite proximal mapping is:

[0048]

[0049] Preferably, the S332 updates the composite proximal mapping of the local convolution dictionary D L , comprising: defining an intermediate variable

[0050]

[0051]

[0052] According to the definition of the proximal mapping function, we have

[0053] where V denotes a matrix for approaching

[0054] ;

[0055] Solved by using the scaled ADMM algorithm.

[0056] ​​​​​Preferably, the S333 updates the composite proximal mapping of the error variable z, including:

[0057] Defining an intermediate variable z k′ as:

[0058]

[0059] According to the definition of the proximal mapping function, prox λ φ k (z k′ ) is written as:

[0060]

[0061] The proximal mapping function is solved using a soft threshold algorithm to obtain z k+1 :

[0062]

[0063] Where S represents a soft threshold operator.

[0064] Preferably, the S334 updates the proximal mapping of the to-be-reconstructed CMR image x, including: defining an intermediate variable x k′ as:

[0065]

[0066] According to the definition of the proximal mapping function, prox λ φ k (x k′ ) is written as:

[0067]

[0068] Where q is a parameter for approaching x k .

[0069] The proximal mapping function is solved using a soft threshold algorithm to obtain x k+1 :

[0070]

[0071]

[0072] Where S represents a soft threshold operator.

[0073] Preferably, the S4 constructs a cardiac magnetic resonance image reconstruction neural network model according to steps S2 and S3, specifically:

[0074] The end-to-end iterative cardiac magnetic resonance image reconstruction neural network model is constructed through an algorithm expansion technique, and is realized by an iteration module, which is sequentially composed of a calculation kth iteration extrapolation point submodule for realizing the functions of formulas (5)-(16), a gradient calculation submodule, a gradient descent submodule, and a proximal solution submodule, a convolution dictionary module is shared in each iteration, and the convolution dictionary is updated in the neural network back propagation process.

[0075] Preferably, the S5 trains the cardiac magnetic resonance image reconstruction neural network model; specifically:

[0076] An undersampled cardiac magnetic resonance image b is obtained, and the undersampled cardiac magnetic resonance image b is reconstructed by using a partial Fourier transform reconstruction operator The undersampled cardiac magnetic resonance image b is processed to obtain an initial to-be-reconstructed CMR image of the undersampled cardiac magnetic resonance image b And inputting the initial to-be-reconstructed CMR image and the undersampled cardiac magnetic resonance image b into the cardiac magnetic resonance image reconstruction neural network model;

[0077] In the training stage, the convolution dictionary and the regularization parameter are optimized by using training data {x 0 ,x *} through a back propagation algorithm, x * is a known full-sampled image;

[0078] Finally, a loss function is used to supervise the network training process, and the cardiac magnetic resonance image reconstruction is completed.

[0079] Wherein, x * represents a true value label in the training process; Θ represents a parameter set to be learned in the network training, including a convolution dictionary, a coefficient of viscous damping, a regularization parameter of the model, and a coefficient of dry friction function; represents a network output, i.e., a reconstructed image in the Kth iteration; Num represents the number of training images, i represents the index of the image order, and N s represents the size of each CMR image.

[0080] Compared with the prior art, the present application has the following beneficial effects:

[0081] The convolution dictionary learning model based on local processing is used as a sparse regularization constraint of the to-be-reconstructed CMR image, and the structural information and detail features of the CMR image are accurately captured; the Nesterov accelerated proximal gradient method is used to solve the CMR image reconstruction model, and the model solving steps are converted into an end-to-end iterative neural network through an algorithm expansion technique, so that the reconstruction speed and accuracy of the CMR image are significantly improved on the premise of ensuring that the network is completely interpretable. BRIEF DESCRIPTION OF DRAWINGS

[0082] Figure 1 The overall flowchart of the cardiac magnetic resonance image reconstruction method based on the algorithm developed by the present application;

[0083] Figure 2 The architecture diagram of the cardiac magnetic resonance image reconstruction neural network model of the present application;

[0084] Figure 3 The architecture diagram of the iteration module in the present application;

[0085] Figure 4A A raw cardiac magnetic resonance image in the present application;

[0086] Figure 4B The error image of the raw cardiac magnetic resonance image in the present application;

[0087] Figure 5A The CMR image to be reconstructed obtained in the present application;

[0088] Figure 5B The error image of the CMR image to be reconstructed in the present application. DETAILED DESCRIPTION

[0089] In order to make the technical personnel in the art better understand the technical solutions in the present specification, the technical solutions in the present specification will be clearly and completely described below in conjunction with the drawings in the embodiments of the present specification. Obviously, the described embodiments are only a part of the embodiments of the present specification, not all the embodiments. Based on the embodiments in the present specification, all other embodiments obtained by those of ordinary skill in the art without creative labor should belong to the scope of protection of the present specification.

[0090] In the following description, specific details are set forth in order to provide a thorough understanding of the embodiments of the present application. However, persons of ordinary skill in the art will readily recognize that the application can be practiced without these specific details. In other instances, well-known methods have not been described in detail in order not to obscure the present application.

[0091] Many modifications and variations of the specific implementation of the present specification can be made without departing from the scope or spirit of the present application, which will be apparent to those skilled in the art. Other implementations derived from the present specification will be apparent to those skilled in the art. The present specification and examples are only exemplary.

[0092] As used herein, "comprise", "include", "have", "contain", and the like, are open-ended terms, i.e., meaning "including but not limited to".

[0093] The application will be further described in detail below with reference to the accompanying drawings and embodiments.

[0094] As shown in the figure, a cardiac magnetic resonance image reconstruction method based on algorithm unfolding includes: Figure 1

[0095] Step S1: Obtain an original cardiac magnetic resonance image.

[0096] The original cardiac magnetic resonance image is an undersampled cardiac magnetic resonance image obtained from k-space, and the undersampled cardiac magnetic resonance image b is saved in a row vector.

[0097] Step S2: Establish a cardiac magnetic resonance image reconstruction model based on local convolution dictionary learning.

[0098] The convolution dictionary learning model based on local processing is introduced into cardiac magnetic resonance image reconstruction, and the local features of the image are captured by learning the sparse representation of each local area, so as to construct a cardiac magnetic resonance image reconstruction model based on local convolution dictionary learning.

[0099] Specifically, the cardiac magnetic resonance image reconstruction model based on local convolution dictionary learning is described as follows:

[0100]

[0101] Wherein, indicates an undersampled Fourier encoding matrix; indicates an undersampled cardiac magnetic resonance image, is a to-be-reconstructed CMR image corresponding to b, also in a row vector format; M indicates the degree number of the undersampled cardiac magnetic resonance image; N indicates the dimension number of the to-be-reconstructed CMR image; indicates a convolution dictionary learning model based on local processing, indicates a local convolution dictionary, and m indicates the dimension number of the local convolution dictionary, is an atom in the local convolution dictionary, j indicates the index of the dictionary atom, and n indicates the dimension number of an atom in the local convolution dictionary; is the sparse coefficient of the to-be-reconstructed CMR image in the local convolution dictionary D L ; is an operator that places D K α i in the i-th position and fills the remaining items with zero; is a data consistent item error variable; indicates a complex number set, indicates an MxN complex matrix, indicates an Nx1 complex matrix; indicates an Mx1 complex matrix; ​represents a set of real numbers, represents an n x m real matrix, represents an n x 1 real matrix; represents an m x 1 real matrix; represents an N x n real matrix; represents an N x 1 real matrix; i ‖1 is the l1 norm constraint on the vector a i , represents the sum of the absolute values of all non-zero elements in a i ; represents the square of the l2 norm constraint on the sign vector, p, r1, r2 represent the first, second, and third regularization parameters, respectively; Ω(D L ) is a regularization constraint imposed on D L , represents a unit norm ball constraint that ensures the length of the dictionary elements is 1:

[0102]

[0103] where m represents the total number of columns of D L , j represents the index of the column number of D L , the set W represents a unit norm ball used to limit the length of the dictionary elements, D L [:,j] represents all elements of the jth column of D L .

[0104] In the cardiac magnetic resonance image reconstruction model based on local convolution dictionary learning, x, a i , z, and D L are the optimization variables to be solved.

[0105] Step S3: The Nesterov accelerated proximal gradient method is used to solve the cardiac magnetic resonance image reconstruction model based on local convolution dictionary learning, i.e., formula (1).

[0106] Specifically, in order to accelerate the convergence speed of the objective function in the cardiac magnetic resonance image reconstruction model based on local convolution dictionary learning, the Nesterov accelerated proximal gradient method is used to solve the reconstruction model, and the steps include:

[0107] First, the objective function of the reconstruction model is decomposed into two parts: a gradient Lipschitz continuous C 1 function and a suitable lower semi-continuous convex function. Second, the extrapolation point of the optimization variable and the gradient function of the optimization variable are calculated. Finally, the proximal mapping function of the optimization variable is solved. The whole process includes the following steps:

[0108] S31: Decompose the objective function of the reconstruction model into two parts, defined as f(x, a i , z) and g(a i , D L ), specifically:

[0109]

[0110] S32: Considering the kth iteration, calculate the extrapolation points z k and x k according to formulas (5)-(8) and

[0111]

[0112] wherein, represents an inertia term coefficient, a is a viscous damping parameter, and its value is a positive number, z k and x k are the to-be-optimized variables a i , D L , z, x, and the extrapolation points at the kth iteration, the initial value of is zero, the initial value of is a value of standard normal distribution, the initial value of z k is zero, and the initial value of x k is obtained by processing the undersampled cardiac magnetic resonance image b using the partial Fourier transform reconstruction operator to obtain the initial to-be-reconstructed CMR image

[0113] and are the extrapolation points of the to-be-optimized variables at the kth iteration, and the gradient functions of the to-be-optimized variables are as follows:

[0114]

[0115] wherein, δ=1+β, P j , a j represent a specific point of P i , a i , and ρ represents a first regularization parameter.

[0116] The proximal gradient descent method is used to update z k+1 and x k+1 , and the update formulas are as follows:

[0117]

[0118] where φ k (x) = r3‖x - x k ‖1 is an improved dry friction potential function, r3 represents a fourth regularization parameter, λ represents a step parameter of proximal gradient descent, and prox() is a proximal mapping function.

[0119] In the present application, an improved dry friction potential function is proposed. The advantage of dry friction is that it makes the velocity of the dynamic system gradually zero, and the system is stationary, and the corresponding optimization problem reaches an optimal solution. In previous work, dry friction is a nonlinear function of velocity, and now it is considered to act on the weighted sum of the velocity vector and the gradient of the function f. As an advantageous property, the sequence thus generated converges to the critical point of f (global minimum when f is convex), while in the past we could only obtain an approximate critical point of f. In addition, after a finite number of steps, the algorithm changes its nature and transitions from an inertial algorithm to a steepest descent method. The potential function of dry friction is different from the existing potential function, and in order to meet the properties of dry friction, the L1 norm is selected as the potential function in the present embodiment.

[0120] S33: The composite proximal mapping of the to-be-optimized variable is solved as follows.

[0121] S331: The composite proximal mapping of the sparse coefficient α i in the update formula (13) is solved, including:

[0122] The intermediate variable is defined as:

[0123]

[0124] According to the definition of the proximal mapping function, the can be written as:

[0125]

[0126] wherein u represents a candidate solution of the minimization problem in the above formula.

[0127] It is simplified to a one-dimensional L1 norm regularization problem:

[0128]

[0129] wherein is an element in , is an element in , is a real number.

[0130] Let According to formula (19), T h (w) = -T -h(-w). Therefore, only the case of h≥0 is considered. It is shown that when h≥0, T h The closed-form solution of the proximal mapping of (w) is

[0131]

[0132] Therefore, the closed-form solution of the composite proximal mapping is

[0133]

[0134] S332: Solve the local convolution dictionary D L in the update formula (14) of the composite proximal mapping, which includes

[0135] Define the intermediate variable as

[0136]

[0137] According to the definition of the proximal mapping function, write as

[0138]

[0139] where V represents a matrix used to approximate .

[0140] Solve the above formula using the scaled ADMM algorithm (Alternating Direction Method of Multipliers). Since V in (23) exists in three places, two auxiliary variables S1 and S2 need to be introduced, and the following constrained minimization problem is obtained:

[0141]

[0142] Write the Lagrangian function corresponding to the constrained minimization problem:

[0143]

[0144] where U1, U2 are dual variables to ensure the constraints, ρ1, ρ2 are Lagrange multipliers, and Ω() is a function defined in formula (2).

[0145] The minimization of the Lagrangian function can be solved by iteratively solving the following five subproblems:

[0146]

[0147] where t is the iteration number of solving (25), is the re-iteration within the loop k, U1, U2 are easy to solve, and are not discussed in detail here. The specific iteration formulas of variables V, S1 and S2 are given below.

[0148] (1) Update of V.

[0149] In the subproblem, the minimization of V is a least squares problem. By taking the derivative of the objective function and setting it to zero, we can obtain the closed-form solution of V as follows:

[0150]

[0151] (2) Update of S1.

[0152] In the subproblem, the minimization of S1 is a proximal mapping problem, which can be solved by:

[0153]

[0154] (3) Update of S2.

[0155] In the subproblem, the minimization of S2 can be updated by solving the proximal mapping of:

[0156]

[0157] By iterating the alternating optimization formula and reaching the maximum number of iterations set, we can obtain the final local convolution dictionary

[0158] S333: Solve the composite proximal mapping of the error variable z in the update formula (15), including:

[0159] Define the intermediate variable z k′ as:

[0160]

[0161] According to the definition of the proximal mapping function, we can write prox λ φ k (z k′ ) as:

[0162]

[0163] Use the soft threshold algorithm to solve the proximal mapping function to obtain z k+1 :

[0164]

[0165] Where S represents the soft threshold operator.

[0166] S334: Solve the proximal mapping of the CMR image x to be reconstructed in the update formula (16), including:

[0167] Define the intermediate variable x k′ as:​

[0168]

[0169] According to the definition of the proximal mapping function, the prox λ φ k (x k′ ) can be written as:

[0170]

[0171] where q is a parameter for approaching x k .

[0172] The proximal mapping function is solved using a soft threshold algorithm, and x k+1 :

[0173]

[0174] where S represents a soft threshold operator.

[0175] Step S4: Constructing a cardiac magnetic resonance image reconstruction neural network model according to steps S2 and S3.

[0176] In order to overcome the problems of complex calculation, difficult tuning of traditional model-based CMR image reconstruction methods, and complex internal structure and lack of explainability of deep learning-based reconstruction methods, algorithm unfolding technology is used to convert the proposed CMR image reconstruction algorithm into an end-to-end iterative neural network, and a cardiac magnetic resonance image reconstruction neural network model is established, which improves the reconstruction efficiency and accuracy while ensuring the explainability of the network. The construction of the cardiac magnetic resonance image reconstruction neural network model is specifically:

[0177] The end-to-end iterative cardiac magnetic resonance image reconstruction neural network model is constructed by algorithm unfolding technology, and the specific cardiac magnetic resonance image reconstruction neural network model architecture diagram is shown in Figure 2 In this network, the same iteration module needs to be used multiple times, and the iteration module is updated according to formulas (5)-(16) each time, as shown in Figure 3 , which is the calculation of the kth iteration extrapolation point module, the gradient calculation module, the gradient descent module, and the proximal solution module in turn. The number of iterations T can be preset according to actual conditions, and is usually set to between three and twenty. The convolution dictionary is shared in each iteration, and the convolution layer network parameters of the network structure are the convolution dictionary, which is updated in the process of neural network back propagation.

[0178] When the cardiac magnetic resonance image reconstruction neural network model is used, first, the number of iterations is determined, and then the to-be-reconstructed image x and the undersampled image b are input into the iteration module, second, the optimization variables are updated according to formulas (5)-(16), wherein the inputs of formulas (5)-(8) are the variable values of the last time and the variable values of the time two times ago, and the output is the value of the extrapolation point, the inputs of formulas (9)-(12) are the extrapolation point, the undersampled image, the data consistency error variable and the convolution dictionary, and the output is the gradient value of the variable, the inputs of formulas (13)-(16) are the extrapolation point and the gradient value, and the output is the proximal function value with dry friction, the iteration module is repeated T times, and the to-be-reconstructed image x is obtained.

[0179] Step S5: training the cardiac magnetic resonance image reconstruction neural network model.

[0180] The original cardiac magnetic resonance image and the initial to-be-reconstructed CMR image are input into the cardiac magnetic resonance image reconstruction neural network model to obtain the to-be-reconstructed CMR image, and the cardiac magnetic resonance image reconstruction neural network model is adjusted by using the fully sampled image, and specifically:

[0181] An original cardiac magnetic resonance image is acquired, the original cardiac magnetic resonance image is an undersampled cardiac magnetic resonance image b acquired from k-space, the undersampled cardiac magnetic resonance image b is processed by using a partial Fourier transform reconstruction operator to obtain an initial to-be-reconstructed CMR image of the undersampled cardiac magnetic resonance image b and both are input into the cardiac magnetic resonance image reconstruction neural network model as network inputs.

[0182] In the training phase, the convolution dictionary and the regularization parameter are constantly optimized on a large amount of training data {x 0 ,x *} by using a back propagation algorithm, x * is a known fully sampled image.

[0183] Finally, the network training process is supervised by using a loss function to ensure that the network output can accurately approximate the true value, and the cardiac magnetic resonance image reconstruction is completed.

[0184] In the above content, x * represents a true value label in the training process; Θ represents a parameter set to be learned in the network training, including a convolution dictionary, a coefficient of viscous damping, a regularization parameter of the model and a coefficient of dry friction function; represents a network output, that is, a reconstructed image of the Kth iteration; Num represents the number of training images, i represents the index of the image order, and N s represents the size of each CMR image.

[0185] Step S6: obtaining a corresponding image to be reconstructed according to the new cardiac magnetic resonance image.

[0186] The new cardiac magnetic resonance image is also an undersampled cardiac magnetic resonance image obtained from k-space, and an initial reconstructed CMR image x of the new undersampled cardiac magnetic resonance image b is first obtained 0 , and then the undersampled cardiac magnetic resonance image b and the initial reconstructed CMR image x 0 are input into the trained cardiac magnetic resonance image reconstruction neural network model in S5 to obtain the image to be reconstructed x.

[0187] For example, a CMR image dataset composed of 216 CMR images, each CMR image being N x ×N y = 320 × 320 in size is used. The dataset is divided into a training set, a validation set and a test set in a ratio of 144 / 36 / 36, and the ratio of the first type of image and the second type of image in the dataset is 4:1. The experiment uses 36 spokes to retrospectively simulate the initial k-space data containing 12 coils, obtains undersampled images from fully sampled images, and additionally adds Gaussian noise with a standard deviation of σ = 0.02.

[0188] Figure 5A and 5B The CMR images and error images obtained by the cardiac magnetic resonance image reconstruction method based on algorithm expansion in the application can be compared with Figure 4A and 4B It can be seen that the application can effectively reconstruct the structural information and detailed features of the CMR image, significantly improve the clarity and accuracy of the image, and provide more comprehensive and reliable information support for clinical diagnosis.

[0189] In summary, the present application provides a cardiac magnetic resonance image reconstruction method based on algorithm unfolding. Firstly, in the environment where sparse coding model is still widely used to realize image reconstruction task, this paper first proposes to introduce the convolution dictionary learning model based on local processing and slice representation into the CMR image reconstruction task, to capture the local features of the image by learning the sparse representation of each local area, thereby constructing a local convolution dictionary learning based cardiac magnetic resonance image reconstruction model, which can effectively capture the detail information of CMR image. Secondly, the optimization algorithm of the model is based on the dynamics system with dry friction damping, and the proximal gradient descent method based on dry friction damping is innovatively proposed, and further using Nesterov acceleration method, a Nesterov accelerated cardiac magnetic resonance image reconstruction algorithm is proposed. Different from the previous Nesterov accelerated proximal algorithm, the addition of dry friction can greatly reduce the oscillation, and the algorithm can converge to a lower objective function value at a faster speed. Finally, through the algorithm unfolding technology, the newly proposed cardiac magnetic resonance image reconstruction algorithm is converted into an end-to-end iterative neural network, which can improve the efficiency and accuracy of CMR image reconstruction while ensuring the interpretability of the network.

[0190] It can be understood that the present application is described through some embodiments, and those skilled in the art know that various changes or equivalent replacements can be made to the features and embodiments without departing from the spirit and scope of the present application. In addition, under the guidance of the present application, the features and embodiments can be modified to adapt to specific conditions and materials without departing from the spirit and scope of the present application. Therefore, the present application is not limited by the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of the present application are within the scope of the present application.

Claims

1. A cardiac magnetic resonance image reconstruction method based on algorithmic expansion, characterized in that, The method includes the following steps: Step S1: Obtain the raw cardiac magnetic resonance images; The original cardiac magnetic resonance image is an undersampled cardiac magnetic resonance image b obtained from k-space, which is stored as a row vector; S2: Establish a cardiac magnetic resonance image reconstruction model based on local convolutional dictionary learning; The cardiac magnetic resonance image reconstruction model based on local convolutional dictionary learning is described as follows: in, This represents the undersampled Fourier encoding matrix; This represents a row of undersampled CMR images obtained from k-space. is a row of CMR images to be reconstructed corresponding to b, represented by a row vector; This represents a convolutional dictionary learning model based on local processing, where N represents the signal dimension, the same as N in M×N. Represents a dictionary of local convolutions. is an atom in the local convolution dictionary, and j represents the index of the atom in the dictionary. The CMR image to be reconstructed is in the local convolution dictionary D L The sparsity coefficient is below; It is D L α i An operator that places an element at the i-th position and fills the remaining elements with zeros; For the data consistency term error variable; ||α i ||1 is for vector α i The l1 norm constraint represents α i The sum of the absolute values ​​of all non-zero elements in the array; This represents squaring the l2 norm constraint on the in-symmetric vector, where ρ, r1, and r2 represent the first, second, and third regularization parameters, respectively; Ω(D L ) represents the unit norm sphere constraint that ensures the length of dictionary elements is 1; in the cardiac magnetic resonance image reconstruction model based on local convolutional dictionary learning, x, α i z and D L The variable to be optimized is the variable that needs to be solved. S3: The Nesterov accelerated proximal gradient method is used to solve the cardiac magnetic resonance image reconstruction model based on local convolutional dictionary learning. The steps include: S31: Decompose the objective function of the reconstructed model into two parts, a gradient Lipschitz continuous C... 1 function f(x,α) i (z) and a lower semi-continuous convex function g(α) i D L Specifically: S32: Calculate the extrapolated points of the variable to be optimized. Based on the gradient function of the variable to be optimized, update the extrapolated points of the variable to be optimized using the proximal gradient descent method, and use φ in the proximal gradient descent method. k (z)=r3‖xx k ||1 is used as an improved dry friction potential function; S33: Solve for the proximal mapping function of the variable to be optimized, including: S331: Solve for the update sparsity coefficient α i The compound proximal mapping; S332: Solving for updating the local convolution dictionary D L The compound proximal mapping; S333: Solve for the composite proximal mapping of the update error variable z; S334: Solve for updating the proximal mapping of the CMR image x to be reconstructed; S4: Construct a neural network model for reconstructing cardiac magnetic resonance images based on steps S2 and S3; S5: Training the neural network model for reconstructing cardiac magnetic resonance images; S6: Obtain the corresponding image to be reconstructed based on the new cardiac magnetic resonance image.

2. The cardiac magnetic resonance image reconstruction method based on algorithm expansion according to claim 1, characterized in that, In equation (1), Ω(D) L () represents the unity norm sphere constraint that ensures the length of each dictionary element is 1: Where m represents D L The total number of columns, j represents the total number of columns in D. L The index of the column number, the set W represents a unit norm sphere used to limit the length of dictionary elements, D L [:,j] represents D L All elements in the j-th column.

3. The cardiac magnetic resonance image reconstruction method based on algorithm expansion according to claim 2, characterized in that, Step S32 is as follows: As shown and Let the extrapolation point of the variable to be optimized be the point in the k-th iteration, specifically: in, This represents the inertia coefficient, and α is a viscous damping parameter with a positive value. z k and x k The variables to be optimized are α i D L z, x, and the extrapolated point at the k-th iteration, The initial value is zero. The initial value is the value of the standard normal distribution, z. k The initial value of x is zero. k The initial value is obtained by reconstructing the operator using a partial Fourier transform. Process the undersampled data b in k-space to obtain the initial CMR image to be reconstructed. The gradient functions of the variables to be optimized are as follows: Where δ=1+β, P j α j P i α i At a specific point, ρ represents the first regularization parameter; Update using proximal gradient descent. z k+1 and x k+1 The update formulas are as follows: Where, φ k (x)=r3‖xx k ‖1 is an improved dry friction potential function, r3 represents the fourth regularization parameter, λ represents the step size parameter of the near-end gradient descent, and prox() is the near-end mapping function.

4. The cardiac magnetic resonance image reconstruction method based on algorithm expansion according to claim 1, characterized in that, S331 updates the sparsity coefficient α i The compound proximal mapping includes: Define intermediate variables for: According to the definition of the proximal mapping function, Written as: Where u represents a candidate solution to the problem minimizing the above equation; This simplifies to a one-dimensional L1 norm regularization problem: in, yes One of the elements, yes One of the elements, It is a real number; make According to formula (19), T h (w)=-T -h (-w); therefore, only the case of h≥0 is considered; analysis shows that when h≥0, T h The unique closed-form solution to (w) is: Therefore, the closed-form solution of the compound proximal mapping is:

5. The cardiac magnetic resonance image reconstruction method based on algorithm expansion according to claim 1, characterized in that, S332 updates the local convolution dictionary D L The compound proximal mapping includes: Define intermediate variables for: According to the definition of the proximal mapping function, Written as: Where V represents the approach Matrix; The solution is obtained using the scaling ADMM algorithm.

6. The cardiac magnetic resonance image reconstruction method based on algorithm expansion according to claim 1, characterized in that, The S333 update of the composite proximal mapping of the error variable z includes: Define intermediate variable z k′ for: According to the definition of the proximal mapping function, Written as: The proximal mapping function is solved using a soft thresholding algorithm to obtain z. k+1 : Where S represents the soft threshold operator.

7. The cardiac magnetic resonance image reconstruction method based on algorithm expansion according to claim 1, characterized in that, S334 updates the proximal mapping of the CMR image x to be reconstructed, including: Define intermediate variable x k′ for: According to the definition of the proximal mapping function, Written as: Where q is used to approximate x k Parameters; The near-end mapping function is solved using a soft thresholding algorithm to obtain x. k+1 : Where S represents the soft threshold operator.

8. The cardiac magnetic resonance image reconstruction method based on algorithm expansion according to claim 1, characterized in that, Step S4 constructs a cardiac magnetic resonance image reconstruction neural network model based on steps S2 and S3, specifically as follows: An end-to-end iterative neural network model for cardiac magnetic resonance image reconstruction is constructed using algorithmic unfolding techniques. This is achieved by iterating through the same iterative module, which consists of a sub-module for calculating the extrapolation point of the k-th iteration, a sub-module for calculating the gradient, a sub-module for gradient descent, and a sub-module for proximal solution, all of which implement the functions of formulas (5)-(16). The convolution dictionary module is shared in each iteration, and the convolution dictionary is updated during the backpropagation of the neural network.

9. The cardiac magnetic resonance image reconstruction method based on algorithm expansion according to claim 1, characterized in that, S5 trains the neural network model for reconstructing cardiac magnetic resonance images; specifically: Acquire an undersampled cardiac magnetic resonance image b, and reconstruct the undersampled cardiac magnetic resonance image b using a partial Fourier transform operator. Processing is performed to obtain the initial CMR image to be reconstructed from the undersampled cardiac magnetic resonance image b. Both the sampled cardiac magnetic resonance image b and the undersampled cardiac magnetic resonance image b are input into the cardiac magnetic resonance image reconstruction neural network model. During the training phase, the training data is used to train {x} through the backpropagation algorithm. 0 ,x * } Optimize the convolution dictionary and regularization parameters, x * It is a known fully sampled image; Finally, use the loss function The network training process is supervised to complete the reconstruction of cardiac magnetic resonance images; Where, x * Θ represents the ground truth label during the training process; Θ represents the set of parameters that the network needs to learn during training, including the convolution dictionary, the coefficients of viscous damping, the regularization parameters of the model, and the coefficients of the dry friction function. This represents the network output, i.e., the reconstructed image at the Kth iteration; Num represents the number of training images, i represents the index of the image sequence, and N... s This indicates the size of each CMR image.

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