Compressed sensing nuclear magnetic resonance imaging reconstruction method based on ADMM-CNN

Abstract

The invention provides a compressed sensing nuclear magnetic resonance imaging reconstruction method based on an ADMM-CNN, and aims to solve the technical problems that under the limitation of an MRI imaging principle, the speed of an MRI imaging process is low, motion artifacts are generated, resolution is distorted, and MRI imaging cannot provide accurate pathological information. The method comprises the following steps: converting an image reconstruction process into an optimization problem according to a compressed sensing theory, and solving an iterative solution of the optimization problem by using an ADMM algorithm; enabling an iterative solution of the solved optimization problem to correspond to three parts of the CNN network; converting a sub-problem of an auxiliary variable of the optimization problem into a denoising model; solving the de-noising model through an existing trained CNN de-noising algorithm to obtain a solution of a sub-problem of an auxiliary variable of the optimization problem; and constructing a training sample set, and training parameters of the CNN network to obtain a reconstructed image. According to the method, a better recovery effect can be obtained under the condition that no obvious artifact is generated, the visual effect is better, and distortion is smaller.

Description

Technical Field

[0001] The present invention relates to the technical field of magnetic resonance imaging (MRI), and in particular to a compressed sensing MRI reconstruction method based on ADMM-CNN. Background Art

[0002] After years of development, magnetic resonance imaging (MRI) technology has become a vital component of clinical medical imaging. It offers advantages such as low radiation, high soft tissue resolution, and multimodal imaging. However, due to limitations in MRI imaging principles, the imaging process is slow, and motion artifacts and resolution distortion can occur during acquisition, preventing MRI imaging from providing accurate pathological information. Compressed sensing, on the other hand, can accurately reconstruct the original sample using far fewer sample data points than conventional methods without significantly degrading image quality. Compressed sensing magnetic resonance imaging (CS-MRI) sparsely samples sample data in k-space (i.e., Fourier space) using a signal sparsity prior, and then reconstructs the image using compressed sensing theory. Regularization associated with the data prior is a key component of the CS-MRI model for reducing imaging artifacts and improving imaging accuracy, making CS-MRI an effective method.

[0003] Currently, there are multiple methods for reconstructing CS-MRI images. For example, sparse regularization, expressed in the wavelet and total variation domains, is used to achieve fast MRI image reconstruction. Total variation (TV) regularization in the gradient domain is also widely used in MRI imaging. However, while these methods offer rapid and convenient reconstruction, they can also produce staircase artifacts during the reconstruction process. Furthermore, dictionary learning methods are used to improve reconstruction accuracy, and nonlocal methods are used to approximate local patches for reconstruction. Both methods can reconstruct CS-MRI images and better preserve image details. However, in terms of performance, classic CS-MRI methods are fast but yield less accurate reconstruction results. Methods based on nonlocality and dictionary learning typically produce higher-quality MRI images, but at the expense of slower reconstruction speed. The alternating direction method of multipliers (ADMM) has been demonstrated to be an effective variable separation algorithm with guaranteed convergence. However, while the ADMM method can achieve the desired results, it typically requires hundreds of iterations to converge, and determining the update rate and penalty parameters in CS-MRI methods is not easy.

[0004] Nowadays, deep learning methods have become an effective tool for learning mappings in image restoration and have been introduced into CS imaging tasks. The basic idea is to use deep neural networks to learn network parameters from a large amount of training data and directly establish nonlinear reconstruction mappings. Based on this idea, many effective results have been invented. For example, the ADMM-Net algorithm uses the parameters in ADMM as network parameters and optimizes the ADMM algorithm through gradient descent, which can effectively reconstruct and restore CS-MRI images. However, this method still has disadvantages such as long training time. Therefore, based on the above reasons, finding a method with short training time and more accurate reconstruction effect is the focus of the improvement of the present invention.

[0005] The invention patent application number 202310429341.X discloses a self-supervised fast CS-MRI reconstruction method based on a twin neural network, which includes the following steps:

[0006] (1) Pre-training of the CS-MRI reconstruction network: The undersampled k-space measurement values are divided into two mutually disjoint vectors, namely the training component and the loss component. The reconstruction network uses the training component as input and the loss component as a constraint. The output obtained by the network and the loss component are penalized for consistency. The reconstruction network is trained by randomly dividing different training components and loss components in the same proportion multiple times to obtain a pre-trained CS-MRI reconstruction network. At the same time, the parameters of the reconstruction network are retained to obtain a learnable twin network that is exactly the same as it.

[0007] (2) Training of CS-MRI reconstruction network: A twin self-supervised training network is established, and the front and rear series networks are initialized by the pre-trained network of step (1); in the training stage, the gradient stopping method is first used to freeze the parameters of one of the CS-MRI reconstruction networks, hereinafter referred to as the frozen network, and the other learnable twin neural network that is exactly the same as it is, hereinafter referred to as the reconstruction network; the collected undersampled k-space data is input into the frozen network to obtain a pseudo-clear MRI reconstructed image; then the sampling rules are pre-set to resample the pseudo-clear MRI image in k-space using the obtained binary mask to obtain an artificially enhanced view, and the resampled data is input into the reconstruction network to obtain an MRI reconstructed image, and a consistency penalty is imposed on the reconstructed MRI image and the accurately measured k-space value to optimize the reconstruction network. After the reconstruction ability of the reconstruction network becomes stronger, the parameters of the frozen network are replaced, and the reconstruction network is continuously iterated and updated to finally obtain a CS-MRI reconstruction network with good reconstruction effect;

[0008] (3) Testing of the CS-MRI reconstruction network: The CS-MRI reconstruction network trained in step (2) is tested by inputting only undersampled k-space measurement data into the CS-MRI reconstruction network and outputting high-definition and high-quality MRI reconstructed images.

[0009] The above invention can achieve fast, high-definition and high-quality MRI image reconstruction and reduce motion artifacts, but the reconstruction results are not very accurate. Summary of the Invention

[0010] In response to the technical problems that the existing CS-MIR reconstruction method has a slow imaging process due to the limitations of MRI imaging principles, motion artifacts and resolution distortion are generated during the acquisition process, and MRI imaging cannot provide accurate pathological information, the present invention proposes a compressed sensing nuclear magnetic resonance imaging reconstruction method based on ADMM-CNN, which solves a sub-problem in the alternating direction multiplier method (ADMM) using a convolutional neural network. By constructing a loss function and optimizing the penalty parameters in ADMM and the update rate of the Lagrange multiplier using the gradient descent method, a trained ADMM model is obtained, which can effectively reconstruct nuclear magnetic resonance image signals from compressed sensing measurements, with short training time and more accurate reconstruction effect. The present invention combines the CNN prior with the CS-MRI reconstruction model, adopts the CNN structure of residual learning (RL) for processing Gaussian noise and batch normalization (BN) to solve the neighbor mapping problem, and is a new nuclear magnetic resonance imaging reconstruction method.

[0011] In order to achieve the above object, the technical solution of the present invention is implemented as follows: a compressed sensing MRI reconstruction method based on ADMM-CNN, the steps of which are as follows:

[0012] Step 1: According to the compressed sensing theory, the image reconstruction process is transformed into an optimization problem, and the ADMM algorithm is used to solve the iterative solution of the optimization problem;

[0013] Step 2: Construct a CNN network: Correspond the iterative solution of the optimization problem to the three parts of the CNN network; transform the auxiliary variable sub-problem of the optimization problem into a denoising model;

[0014] Step 3: Solve the denoising model using the existing trained CNN denoising algorithm to obtain the solution to the auxiliary variable sub-problem of the optimization problem;

[0015] Step 4: Construct a training sample set, train the parameters of the CNN network, and obtain the reconstructed image.

[0016] Preferably, the image reconstruction process is converted into an optimization problem:

[0017]

[0018] Among them, the natural image x∈R N Represents the original signal, y∈R MRepresents the measurement results in the undersampled k-space, M and N represent the dimensions of the natural image x and the measurement result y respectively, M<<N, the ratio M / N represents the sampling rate, Φ∈R M×N is the projection matrix; D l Represents the transformation matrix for convolution operation, L represents the number of filters, λ l >0 represents the regularization parameter; g(·) is the regularization function, represents the optimal reconstructed image of the optimization problem, Represents the square of the l2 norm.

[0019] Preferably, the method of using the ADMM algorithm to solve the iterative solution of the optimization problem is:

[0020] Introducing an auxiliary variable z in the image domain, the optimization problem is equivalent to:

[0021]

[0022] Get the Lagrange augmented function:

[0023]

[0024] Among them, ρ>0 is the penalty parameter, α is the Lagrange multiplier, and <,> represents the inner product;

[0025] Using scaled Lagrange multipliers The ADMM algorithm solves the Lagrangian augmented function by iteratively optimizing three sub-problems, which are expressed as:

[0026]

[0027] in, Indicates the update rate.

[0028] Preferably, I is used to represent the identity matrix. Then, the following methods are used: 1. Derivative the quadratic term of the biological subproblem of natural image x and solve the equation to obtain an explicit solution; 2. Convert the subproblem of auxiliary variable z into a denoising problem, using a CNN network to replace the traditional proximal operator; 3. Directly derive the multiplier update rule based on the ADMM algorithm. The three subproblems are expressed as the following solutions:

[0029]

[0030] Where superscript n∈{1,2,…,N s} represents the nth iteration, N s represents the final number of iterations, and the superscript T represents the Hermitian transpose operator; is a regularization function g(·), transformation matrix D l and the regularization parameter λ lThe nonlinear operator determined by z (n-1) , β (n-1) They are the iterative results of the auxiliary variable z and the Lagrange multiplier β for the n-1th iteration, respectively. (n) , β (n) 、x (n) are the iterative results of the nth iteration auxiliary variable z, Lagrange multiplier β and natural image x. (n) 、Z (n) 、M (n) They represent the reconstruction layer, auxiliary variable update layer and multiplier update layer in the CNN network respectively.

[0031] Preferably, each stage of the CNN network consists of three parts, corresponding to each sub-problem in the ADMM solution, namely, the reconstruction layer X (n) , multiplier update layer M (n) and auxiliary variable update layer Z (n) , which corresponds to the sub-problem of natural image x, auxiliary variable z and Lagrange multiplier β in one iteration of ADMM algorithm; the auxiliary variable layer Z (n) Including adding layer A (n) and convolutional layer C (n) Two parts, add layer A (n) Perform a sum operation: a (n) =x (n) -β (n-1) , convolutional layer C (n) Responsible for executing the operations of the CNN denoising algorithm.

[0032] Preferably, the method of converting the auxiliary variable sub-problem of the optimization problem into a denoising model is:

[0033] Let the intermediate variable a (n) =x (n) -β (n-1) , rewrite the subproblem of auxiliary variables into

[0034]

[0035] The above subproblems of auxiliary variables are considered as regularization functions An approximate mapping or denoising function, the intermediate variable a (n) is regarded as a noisy image, z is regarded as the original image to be estimated, and z (n) As the corresponding denoised image, It is a mathematical model that characterizes image attributes.

[0036] Preferably, the method for solving the denoising model is: combining the CNN prior with the CS-MRI reconstruction model, using residual learning for processing Gaussian noise and a batch processing normalized CNN network to solve the neighbor mapping problem; using the ReLU activation function to ensure nonlinear expression capability; setting the kernel size of all convolutional layers to 3×3×64, the step size of each layer to 1, and not placing a pooling layer.

[0037] Preferably, the objective function of the CNN denoising algorithm is to minimize the error between the training residual and the actual residual, and:

[0038]

[0039] Among them, x i and u i are the clean image and noisy image of the i-th training sample, N T is the total number of training samples, x i -u i represents the actual residual between the clean image and the noisy image, g(u i ; Θ) represents the residual error after training through the CNN network, u i is the input of CNN network, Θ is the parameter in CNN network;

[0040] Use an off-the-shelf deep convolutional neural network to learn the input noise image u i And the output denoised image x i The relationship between the two, directly from the noisy image a (n) Generate denoised image: z (n) =CNN g (a (n) )=a (n) -g(a (n) ; Θ(σ)); where CNN g (·) represents the CNN denoising algorithm with regularization function g(·).

[0041] Preferably, the method for constructing the training sample set is: obtaining high-quality MRI images x gt ; For MRI image x gt Perform pseudo radial undersampling in k space to obtain an undersampled image y; compare the undersampled image y with the corresponding MRI image x gt Pair them to form the training set Γ;

[0042] The normalized mean square error is selected as the loss function for training the CNN network. The loss function between the network output and the true value is:

[0043]

[0044] in, represents the network output generated based on the network parameters and the undersampled image y,

[0045] By minimizing the loss function using the gradient descent method based on the L-BFGS optimizer, the required parameters are learned through backpropagation. ρ (n) is the penalty parameter that can be learned in the nth iteration, is the update rate that can be learned in the nth iteration; the penalty parameter ρ is initialized using the empirical value of the traditional ADMM algorithm (n) and update rate Minimize the normalized mean square error loss function through the back propagation algorithm and optimize the penalty parameter ρ (n) and update rate

[0046] Preferably, the parameter Θ in the CNN network is determined by the noise level σ. Different models are trained under different noise levels to obtain different parameter values of the parameter Θ. The noise image a (n) The noise level is determined by the sampling rate of compressed sensing;

[0047] The projection matrix Φ is a random Gaussian matrix that satisfies the orthogonality ΦΦ t =I; transformation matrix D l Using gradient transform or discrete wavelet transform, the regularization function g(·) is l q Sparse regularizer: q∈[0, 1], which promotes sparsity of the solution by imposing a penalty on the model parameters.

[0048] Compared with the prior art, the present invention has the following advantages: the iterative form of the ADMM algorithm is expanded and generalized into a deep network structure. Each stage of the network consists of three parts, corresponding to each sub-problem solved by the ADMM algorithm. Each stage of the network includes the reconstruction layer X (n) , multiplier update layer M (n) and auxiliary variable update layer Z (n) , which corresponds to an iterative process of the ADMM algorithm, has strong generalization ability, is easy to integrate and has high flexibility. The present invention mainly addresses the second sub-problem of the multiplier update layer M (n) To improve the problem of solving complex nonlinear operators in the , the auxiliary variable update layer is divided into the addition layer A (n) and convolutional layer C (n) Two parts, adding layers to perform simple sum operations: a (n) =x (n) -β (n-1)The convolution layer is responsible for executing the CNN denoising algorithm, which can simplify the calculation and improve flexibility and scalability. The present invention uses the gradient descent method based on the L-BFGS algorithm to minimize the loss function and learn the required parameters. To achieve the goal of training and optimizing the ADMM algorithm, the L-BFGS algorithm can accelerate convergence, reduce memory consumption, and be easy to implement and tune. To find the optimal noise level, the present invention conducted experiments on different noise levels for sampling rates of 10% to 50%. The average PSNR value has an optimal setting for the σ value of each CS sampling rate. At a sampling rate of 10%, σ = 15 has the best recovery effect, while at a sampling rate of 20% to 50%, σ = 10 has an even better recovery effect. Therefore, the optimal σ parameter can be found for different CS sampling rates. The ADMM-CNN algorithm of the present invention can achieve better recovery results without producing obvious artifacts, with better visual effects and less distortion. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0050] Figure 1 This is a flowchart of the CNN denoising algorithm used in this invention, where Convolution is the convolution layer, BN is the batch normalization layer, and ReLU is the modified linear unit activation function. Here, the network has a total of 17 convolutional layers.

[0051] Figure 2 This is the data flow diagram during the ADMM algorithm iteration process.

[0052] Figure 3 The average PSNR results obtained by using the ADMM-CNN algorithm at different σ values with a sampling rate of 10% to 50%, where (a) is 10%, (b) is 20%, (c) is 30%, (d) is 40%, and (e) is 50%.

[0053] Figure 4 The average PSNR results obtained by using the ADMM-CNN algorithm with different σ values for sampling rates of 10% to 20%, where (a) is NMSE: 0.0459, PSNR: 37.5711, (b) is NMSE: 0.0561, PSNR: 35.8500, and (c) is the original image.

[0054] Figure 5Visual comparison of the reconstructed images and difference images of ADMM-Net and ADMM-CNN of the present invention at a sampling rate of 20%, including: (a) original image, (b) ADMM-Net (35.58dB), (c) ADMM-Net difference image, (d) ADMM-CNN (37.57dB), (e) ADMM-CNN difference image, (f) original image, (g) ADMM-Net (32.89dB), (h) ADMM-Net difference image, (i) ADMM-CNN (34.66dB), and (j) ADMM-CNN difference image. DETAILED DESCRIPTION

[0055] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without creative work are within the scope of protection of the present invention.

[0056] Example 1

[0057] like Figure 1 As shown in FIG, a compressed sensing MRI reconstruction method based on ADMM-CNN is shown, and the steps are as follows:

[0058] Step 1: According to the compressed sensing theory, the image reconstruction process is transformed into an optimization problem, and the ADMM algorithm is used to solve the iterative solution of the optimization problem.

[0059] Use x∈R N To represent the original signal, that is, the natural image, use y∈R M Represents the measurement results in the undersampled k-space, where M and N represent the dimensions of the natural image x and the measurement result y, respectively, M<<N, and the ratio M / N represents the sampling rate or sub-rate. Through compressed sensing theory, the process of reconstructing the image can be regarded as a problem of solving the following optimization equation:

[0060]

[0061] Among them, the projection matrix Φ∈R M×N , for a given sampling rate, Φ is usually generated as a random Gaussian matrix that satisfies orthogonality, that is, ΦΦ T =I. D l Represents the transformation matrix for convolution operation, L represents the number of filters, λ l >0 indicates the regularization parameter. As a special case, the transformation matrix D lGradient transform or discrete wavelet transform (DWT) can be used, and the regularization function g(·) can be defined as l q Sparse regularizer: q∈[0, 1], which promotes sparsity of the solution by imposing a penalty on the model parameters. It represents the original signal x that makes the optimization equation (1) reach the optimal solution, that is, the reconstructed image. Represents the square of the l2 norm.

[0062] For the above optimization problem, the ADMM algorithm can be used to solve it. By introducing an auxiliary variable z in the image domain, formula (1) is equivalent to:

[0063]

[0064] By introducing an auxiliary variable method on the image domain, its Lagrangian augmented function can be obtained:

[0065]

[0066] Where ρ>0 is the penalty parameter, α is the Lagrange multiplier, and <,> represents the inner product. For simplicity, we use the scaled Lagrange multiplier The ADMM algorithm solves the following three sub-problems {x, z, β} by iterative optimization. The three sub-problems can be expressed as:

[0067]

[0068] in, Denotes the update rate. Using I to denote the identity matrix, the explicit solution is obtained by: 1. Differentiating the quadratic term of x and solving the equation; 2. Converting the subproblem of z into a denoising problem, using CNN to replace the traditional proximal operator; 3. Directly deriving the multiplier update rule based on ADMM. The above three subproblems can be expressed as the following solutions:

[0069]

[0070] Where superscript n∈{1,2,…,N s} represents the nth iteration, N s represents the final number of iterations, and the superscript T represents the Hermitian transpose operator. For the second subproblem, is a regularization function g(·), transformation matrix D l and the regularization parameter λ l The nonlinear operator determined by z (n-1) , β (n-1) They are the iterative results of the auxiliary variable z and the Lagrange multiplier β for the n-1th iteration, respectively. (n) , β(n) 、x (n) They are the iterative results of the nth iteration auxiliary variable z, Lagrange multiplier β and natural image x, X (n) 、Z (n) 、M (n) They represent the reconstruction layer, auxiliary variable update layer and multiplier update layer in the CNN network respectively.

[0071] Mapping the ADMM iterative process in formula (5) into a data flow graph yields Figure 2 , in particular, in the first layer (ie n = 1), X (1) =(Φ T Φ+ρ (1) I) -1 (Φ T y). In order to facilitate calculation, the auxiliary variable update layer Z (n) Divided into additional layer A (n) and convolutional layer C (n) Two parts.

[0072] Step 2: Build a CNN network and correspond the iterative solution of the optimization problem to the three parts of the CNN network; transform the auxiliary variable sub-problem of the optimization problem into a denoising model.

[0073] The proposed ADMM and CNN combined algorithm is defined by expanding and generalizing the iterative ADMM algorithm into a deep structure. Each stage of the network consists of three parts, corresponding to each sub-problem in the ADMM solution, namely, the reconstruction layer X (n) , multiplier update layer M (n) and auxiliary variable update layer Z (n) , which corresponds to an iterative process of the ADMM algorithm, such as Figure 2 shown.

[0074] The present invention mainly addresses the second sub-problem of multiplier update layer M (n) To improve the complex nonlinear operator solution problem, let the intermediate variable a (n) =x (n) -β (n-1) , then the second sub-problem of the optimization problem (4) can be rewritten as

[0075]

[0076] The sub-problems of the above optimization problem can be viewed as regularization functions An approximate mapping or denoising function of . The intermediate variable a (n) can be regarded as a noisy image, z as the original image to be estimated, and z (n) As the corresponding denoised image, It is a mathematical model that characterizes image attributes.

[0077] Step 3: Solve the denoising model in step 2 using the existing trained CNN denoising algorithm to obtain the solution to the auxiliary variable sub-problem of the optimization problem.

[0078] Inspired by the success of deep convolutional neural networks (CNN) in image restoration, the CNN prior is combined with the CS-MRI reconstruction model, and the CNN structure of residual learning (RL) and batch normalization (BN) for processing Gaussian noise is adopted to solve the above-mentioned neighbor mapping problem; the ReLU activation function is used to ensure nonlinear expression ability; the input and output are added through jump connections to avoid gradient disappearance; a BN layer is added after each convolution layer and before the ReLU activation function; the kernel size of all convolution layers is set to 3×3×64, and the step size of each layer is set to 1. In order to maintain the spatial resolution of the image, no pooling layer is placed. It should be noted that the CNN denoising algorithm used in the present invention outputs the noise information of the image, that is, the residual between the original clean image and the noisy image. Therefore, the objective function of training the CNN denoising algorithm of the present invention is to minimize the error between the training residual and the actual residual, and its formula is as follows:

[0079]

[0080] Among them, x i and u i are the clean image and noisy image of the i-th training sample, N T is the total number of training samples, x i -u i represents the actual residual between the clean image and the noisy image, g(u i ; Θ) represents the residual error after training after passing through the CNN feedforward network, u i is the network input, and Θ is the parameter in the network.

[0081] Specifically, compared with the explicit image mathematical model, the existing deep convolutional neural network is used to learn the relationship between the input noisy image and the output denoised image, so that the following calculation can be performed directly from the noisy image a (n) Generate denoised image z (n) :

[0082] z (n) =CNN g (a (n) )=a (n) -g(a (n) ;Θ(σ)) (8)

[0083] Among them, CNN g(·) represents the CNN denoising algorithm with the regularization function g(·), Θ is a parameter in the network, which is determined by the noise level σ.

[0084] The CNN denoising algorithm (7) in the above deep network structure has been trained. Formula (7) calculates the error of each parameter through the back propagation algorithm and then solves it using the gradient descent method. Although the entire calculation process requires a lot of computer computing power and time, it is completely independent of the ADMM-CNN algorithm. This means that once it is trained, the only calculation for each image processing process is only calculated through the feedforward network, that is, Formula (8). Its parameter Θ is determined by the noise level σ. Under different noise levels, different models are trained separately, and different parameter values of Θ can be obtained by Formula (7). Therefore, in the proposed algorithm, it is necessary to evaluate the noise image a (n) The noise level of the noise image a is used to select the best CNN denoising model. (n) The noise level is basically determined by the sub-rate of compressed sensing. The larger the sub-rate, the smaller the noise level of the image.

[0085] In order to facilitate subsequent calculations, the auxiliary variable update layer is divided into two parts, namely, adding layer A (n) and convolutional layer C (n) , where the addition layer performs a simple summation operation: a (n) =x (n) -β (n-1) , while the convolution layer is responsible for executing the CNN denoising algorithm, as shown in formula (8). The convolution layer is an auxiliary variable update layer Z (n) core operations.

[0086] It should be noted that the trained CNN network is directly used to solve the sub-problem, so no additional training is required, which saves a lot of training time and computer computing power. At the same time, when solving formula (5) to obtain the ADMM network, there is no need to update the multiplier layer M. (n) This also saves a lot of training time.

[0087] Step 4: Construct a training sample set, train the parameters of the CNN network, and obtain the reconstructed image.

[0088] Use the fully sampled reconstructed MRI image in k-space as the ground truth x gt , the undersampled image y in k-space is used as input, and then a training set Γ is constructed, which contains randomly selected undersampled data and MRI images of the corresponding true data. The normalized mean square error (NMSE) is selected as the loss function for training the network. Given a training data pair, first obtain a high-quality MRI image xgt ; Secondly, MRI images x gt Perform pseudo radial undersampling to obtain an undersampled image y; then compare the undersampled image y with the corresponding MRI image x gt Paired to form the training set Γ. The loss between the network output and the true value is defined as:

[0089]

[0090] in, Represents the network output generated based on the network parameters and the undersampled image y, obtained according to formula (1). In the present invention, the required parameters are learned by minimizing the loss function using the gradient descent method based on the L-BFGS algorithm. ρ (n) is the penalty parameter that can be learned in the nth stage (iteration), is the update rate that can be learned in the nth stage. The penalty parameter ρ is initialized using the empirical value of the traditional ADMM algorithm. (n) and update rate The normalized mean square error (NMSE) loss function is minimized by the back propagation algorithm, and the penalty parameter ρ is optimized. (n) and update rate Thereby achieving the purpose of training and optimizing the ADMM algorithm.

[0091] The proposed ADMM-CNN algorithm was extensively evaluated experimentally, where it was trained and tested using brain and chest MR image sets. The image resolution was 256×256. For each dataset, 100 images were randomly selected for training and 50 images were selected for testing. The training was performed for sampling rates of 10%, 20%, 30%, 40%, and 50%. The specific process is as follows: 1. Pseudo-radial sampling was applied to the fully sampled images to generate undersampled data with different sampling rates. 2. The parameter ρ was initialized based on the model or randomly. (n) and 3. Using the L-BFGS optimizer, parameters were updated via backpropagation to minimize the NMSE loss. The commonly used pseudo-radial sampling pattern in k-space was chosen to accelerate scanning, improve image quality, reduce motion artifacts, and mitigate the effects of inhomogeneous fields. Image reconstruction accuracy was evaluated using the NMSE and peak signal-to-noise ratio (PSNR) averaged over the sampled images.

[0092] From the above discussion, it can be seen that the noise level σ is the only parameter in the entire deep network that needs to be manually set. For the convenience of processing, we only use the CNN denoising network pre-trained for the noise level range of {10, 15, 20, 25, 30} according to formula (7). In order to find the optimal noise level, experiments were conducted on different noise levels with a sampling rate of 10% to 50%. The average PSNR value is as follows: Figure 3 As shown. It can be clearly seen from Figure 3 As can be seen from the figure, there is an optimal setting for the value of σ for each CS sampling rate. At a 10% sampling rate, σ = 15 achieves the best recovery effect, while at a sampling rate of 20% to 50%, σ = 10 achieves even better recovery. Therefore, the optimal σ parameter can be found for different CS sampling rates.

[0093] In the experimental part, the ADMM-CNN algorithm of the present invention is compared with the ADMM-Net algorithm based on the ADMM algorithm, which is specifically used to restore CS-MRI. Using the same training set and test set of medical brain images as the ADMM-Net algorithm, the program is run to obtain CS-MRI restoration results with sampling rates of 20%, 30%, 40% and 50% and compared respectively. Figure 4 、 Figure 5 and Table 1.

[0094] As shown in Table 1, where Init-Net15 represents an untrained 15-layer ADMM-CNN algorithm, it is clear that the proposed algorithm achieves better recovery results at all sampling rates compared to the ADMM-Net and CREAM algorithms. The proposed ADMM-CNN algorithm reduces the NMSE by approximately 0.0033 to 0.0081 compared to the ADMM-Net algorithm, while achieving a PSNR gain of 0.72 to 1.01 dB. The ADMM-Net algorithm comes from the literature [Y.Yang, J.Sun, H.Li, Z.Xu. Deep ADMM-Net for compressive sensing MRI [C]. Advances in Neural Information Processing Systems, 2016.10–18.], and the CREAM algorithm comes from the literature [C.Zhao, J.Zhang, R.Wang, et al. CREAM: CNN-REgularized ADMM framework for compressive-sensed image reconstruction [J]. IEEE Access, 2018, 6: 76838-76853.]. ADMM-CNN 13 , ADMM-CNN14 , ADMM-CNN 15 They represent the ADMM-CNN algorithms with 13 layers, 14 layers, and 15 layers respectively.

[0095] from Figure 4 It can be seen that compared with the ADMM-Net method, the ADMM-CNN algorithm proposed in this invention can achieve better restoration effect without producing obvious artifacts. Figure 5 As shown in the figure, compared with ADMM-Net, the ADMM-CNN algorithm of the present invention has better visual effects and less distortion.

[0096] Table 1 Average NMSE and average PSNR (dB) of ADMM-Net, CREAM and ADMM-CNN for CS-MRI reconstruction

[0097]

[0098] This paper introduces deep learning methods into CS imaging tasks. The basic idea is to use deep neural networks to learn network parameters from a large amount of training data. First, generate full-sampled and under-sampled data pairs to construct a training set; second, based on the model or randomly initialize the parameter ρ (n) and The reconstructed image is then generated using the ADMM-CNN model. Finally, the CNN model is trained independently, selecting optimal parameters based on the sampling rate. A nonlinear reconstruction mapping is directly established by combining the linear subproblem of the ADMM with the nonlinear subproblem of the CNN to form a multi-stage deep network. Residual connections and batch normalization are used to enhance the deep network's nonlinear expressiveness and training stability. The L-BFGS algorithm is used to minimize the NMSE loss, dynamically adjust the ADMM parameters, and learn the optimal nonlinear mapping.

[0099] Example 2

[0100] An electronic device comprises a memory and a processor, wherein the memory is used to store a program that supports the processor to execute the compressed sensing magnetic resonance imaging reconstruction method based on ADMM-CNN described in Example 1, and the processor is configured to execute the program stored in the memory.

[0101] Example 3

[0102] A computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, executes the steps of the ADMM-CNN-based compressed sensing magnetic resonance imaging reconstruction method of Example 1.

[0103] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A compressed sensing magnetic resonance imaging reconstruction method based on ADMM-CNN, characterized in that: The steps are as follows: Step 1: According to the compressed sensing theory, the image reconstruction process is transformed into an optimization problem, and the ADMM algorithm is used to solve the iterative solution of the optimization problem; Step 2: Construct a CNN network: Correspond the iterative solution of the optimization problem to the three parts of the CNN network; transform the auxiliary variable sub-problem of the optimization problem into a denoising model; Step 3: Solve the denoising model using the existing trained CNN denoising algorithm to obtain the solution to the auxiliary variable sub-problem of the optimization problem; Step 4: Construct a training sample set, train the parameters of the CNN network, and obtain the reconstructed image.

2. The compressed sensing MRI reconstruction method based on ADMM-CNN according to claim 1, characterized in that The image reconstruction process is converted into an optimization problem: Among them, the natural image x∈R N Represents the original signal, y∈R M Represents the measurement results in the undersampled k-space, M and N represent the dimensions of the natural image x and the measurement result y respectively, M<<N, the ratio M / N represents the sampling rate, Φ∈R M×N is the projection matrix; D l Represents the transformation matrix for convolution operation, L represents the number of filters, λ l >0 represents the regularization parameter; g(·) is the regularization function, represents the optimal reconstructed image of the optimization problem, Represents the square of the l2 norm.

3. The compressed sensing MRI reconstruction method based on ADMM-CNN according to claim 2, characterized in that: The method for solving the iterative solution of the optimization problem using the ADMM algorithm is: Introducing an auxiliary variable z in the image domain, the optimization problem is equivalent to: Get the Lagrange augmented function: Among them, ρ>0 is the penalty parameter, α is the Lagrange multiplier, and <,> represents the inner product; Using scaled Lagrange multipliers The ADMM algorithm solves the Lagrangian augmented function by iteratively optimizing three sub-problems, which are expressed as: in, Indicates the update rate.

4. The compressed sensing MRI reconstruction method based on ADMM-CNN according to claim 3, characterized in that: Using I to represent the identity matrix, the explicit solution is obtained by:

1. Differentiating the quadratic term of the natural image x biological subproblem and solving the equation; 2. Convert the subproblem of the auxiliary variable z into a denoising problem, using a CNN network to replace the traditional proximal operator; 3. Directly derive the multiplier update rule based on the ADMM algorithm, and the three subproblems are expressed as the following solutions: Where superscript n∈{1,2,…,N s } represents the nth iteration, N s represents the final number of iterations, and the superscript T represents the Hermitian transpose operator; is a regularization function g(·), transformation matrix D l and the regularization parameter λ l The nonlinear operator determined by z (n-1) , β (n-1) They are the iterative results of the auxiliary variable z and the Lagrange multiplier β for the n-1th iteration, respectively. (n) , β (n) 、x (n) are the iterative results of the nth iteration auxiliary variable z, Lagrange multiplier β and natural image x. (n) , Z (n) 、M (n) They represent the reconstruction layer, auxiliary variable update layer and multiplier update layer in the CNN network respectively.

5. The compressed sensing MRI reconstruction method based on ADMM-CNN according to claim 4, characterized in that: Each stage of the CNN network consists of three parts, corresponding to each sub-problem in the ADMM solution, namely the reconstruction layer X (n) , multiplier update layer M (n) and auxiliary variable update layer Z (n) , which corresponds to the sub-problem of natural image x, auxiliary variable z and Lagrange multiplier β in one iteration of ADMM algorithm; the auxiliary variable layer Z (n) Including adding layer A (n) and convolutional layer C (n) Two parts, add layer A (n) Perform a sum operation: a (n) =x (n) -β (n-1) , convolutional layer C (n) Responsible for executing the operations of the CNN denoising algorithm.

6. The compressed sensing MRI reconstruction method based on ADMM-CNN according to any one of claims 3 to 5, characterized in that: The method of converting the auxiliary variable sub-problem of the optimization problem into a denoising model is: Let the intermediate variable a (n) =x (n) -β (n-1) , rewrite the subproblem of auxiliary variables into The above subproblems of auxiliary variables are considered as regularization functions An approximate mapping or denoising function, the intermediate variable a (n) is regarded as a noisy image, z is regarded as the original image to be estimated, and z (n) As the corresponding denoised image, It is a mathematical model that characterizes image attributes.

7. The compressed sensing MRI reconstruction method based on ADMM-CNN according to claim 6, characterized in that: The method for solving the denoising model is as follows: combining the CNN prior with the CS-MRI reconstruction model, using residual learning for processing Gaussian noise and a batch normalized CNN network to solve the neighbor mapping problem; using the ReLU activation function to ensure nonlinear expression capabilities; setting the kernel size of all convolutional layers to 3′3′64, the step size of each layer to 1, and not placing a pooling layer.

8. The compressed sensing MRI reconstruction method based on ADMM-CNN according to claim 7, characterized in that: The objective function of the CNN denoising algorithm is to minimize the error between the training residual and the actual residual, and: Among them, x i and u i are the clean image and noisy image of the i-th training sample, N T is the total number of training samples, x i -u i represents the actual residual between the clean image and the noisy image, g(u i ; Θ) represents the residual error after training through the CNN network, u i is the input of CNN network, Θ is the parameter in CNN network; Use an off-the-shelf deep convolutional neural network to learn the input noise image u i And the output denoised image x i The relationship between the two, directly from the noisy image a (n) Generate denoised image: z (n) =CNN g (a (n) )=a (n) -g(a (n) ; Θ(σ)); where CNN g (·) represents the CNN denoising algorithm with regularization function g(·).

9. The compressed sensing MRI reconstruction method based on ADMM-CNN according to claim 7 or 8, characterized in that: The method for constructing the training sample set is: obtaining high-quality MRI images x gt ; For MRI image x gt Perform pseudo radial undersampling in k space to obtain an undersampled image y; compare the undersampled image y with the corresponding MRI image x gt Pair them to form the training set Γ; The normalized mean square error is selected as the loss function for training the CNN network. The loss function between the network output and the true value is: in, represents the network output generated based on the network parameters and the undersampled image y, By minimizing the loss function using the gradient descent method based on the L-BFGS optimizer, the required parameters are learned through backpropagation. ρ (n) is the penalty parameter that can be learned in the nth iteration, is the update rate that can be learned in the nth iteration; the penalty parameter ρ is initialized using the empirical value of the traditional ADMM algorithm (n) and update rate Minimize the normalized mean square error loss function through the back propagation algorithm and optimize the penalty parameter ρ (n) and update rate 10. The compressed sensing MRI reconstruction method based on ADMM-CNN according to claim 9, characterized in that: The parameter Θ in the CNN network is determined by the noise level σ. Different models are trained under different noise levels to obtain different parameter values of parameter Θ. The noise image a (n) The noise level is determined by the sampling rate of compressed sensing; The projection matrix Φ is a random Gaussian matrix that satisfies the orthogonality ΦΦ T =I; transformation matrix D l Using gradient transform or discrete wavelet transform, the regularization function g(·) is l q Sparse regularizer: q∈[0, 1], which promotes sparsity of the solution by imposing a penalty on the model parameters.

Images