Fast Magnetic Resonance Imaging Method Based on Deep Cascade Network with Dense Connections at the Same Layer across Subnets

Abstract

A fast magnetic resonance imaging (MRI) method based on a deep cascaded network with cross-subnet same-layer dense connections. This method makes full use of the rich detailed information in the reference modality to assist in reconstructing the missing information in the target modality: First, a deep cascaded network structure is adopted to cascade multiple MRI reconstruction subnets, enabling the reference modality and target modality data input by the user to directly participate in the auxiliary reconstruction of the network multiple times. Second, "same-layer dense connections" are added inside each subnet to strengthen the feature flow between sub-networks and avoid learning redundant features. Finally, a k-space integrated learning module is added at the end of each subnet to more effectively utilize the frequency-domain data of the reference modality and the reconstruction results of each subnet for MRI reconstruction. Compared with existing MRI auxiliary reconstruction methods, the method provided by the present invention achieves a better reconstruction effect and has strong practicability.

Description

Technical Field

[0001] The present invention relates to the field of medical image processing, especially the medical image technology of magnetic resonance imaging, and is mainly used to achieve the accelerated reconstruction of medical magnetic resonance imaging. Background Art

[0002] Magnetic Resonance Imaging (MRI) is widely used in clinical diagnosis and human disease research due to its high imaging quality, rich contrast information and no radiation damage to the human body. However, its high imaging cost and long acquisition time make it unable to be as popular as CT or X-ray. To accelerate the acquisition time of MRI images, researchers have proposed a series of fast MRI methods, such as designing fast scanning sequences based on the imaging physical principle to reduce the time of k-space repeated filling; parallel imaging based on hardware devices, using multiple MRI receiving coils to reduce the time-consuming phase encoding step; using compressed sensing theory for MRI reconstruction, and only a small part of data can be used to obtain a complete reconstruction through computational methods. In recent years, with the continuous maturity of deep learning technology, the fast MRI method based on deep learning has become a new research hotspot because it can achieve a higher acceleration ratio and better image quality.

[0003] To reconstruct undersampled MRI images using a deep learning model, in addition to selecting an appropriate network structure and training the network with large-scale data, the data characteristics of MRI can also be fully combined, and the prior information of the data can be utilized, such as multi-modal data and k-space data. Due to the special acquisition principle of MRI, different modal images can be obtained according to different set parameters. These multi-modal MRI images have similar anatomical structures and texture information, and the undersampled modality can be assisted in reconstruction through information sharing and complementarity among multi-modal data, that is: a fully sampled reference modality is used to assist in reconstructing the undersampled target modality, so as to achieve a higher acceleration ratio and better reconstruction accuracy.

[0004] However, there are still problems such as detail loss and artifact generation in the MRI images reconstructed based on existing MRI-assisted reconstruction methods. One of the main reasons is that the fully sampled reference modality data and the sampled k-space data in the target modality have a low participation rate in the network reconstruction process. This is because most of the existing work is based on the UNet framework, and the encoding and decoding structure of the UNet framework allows the reference modality data and the target modality data to directly participate in the reconstruction of the target modality only at the beginning or end of the network, which will weaken the auxiliary role of the reference modality and the utilization rate of the sampled k-space data in the target modality. The deep cascaded network has improved the participation rate of the reference modality and the target modality to a certain extent: the network is composed of multiple sub-networks in cascade, and each sub-network can achieve the reconstruction of the target modality once, and the overall network presents a structure of "reconstruction + fine-tuning". This structure can introduce the reference modality data and the sampled k-space data in the target modality into each sub-network respectively, greatly improving the participation rate of the original data of the two modalities in the reconstruction process. However, the intermediate reconstruction structure of the deep cascaded network hinders the information flow between sub-networks: the features of the lower-level sub-networks cannot flow to the higher-level sub-networks, and this short-term memory of the shallow features will further lead to feature redundancy in the higher-level sub-networks: the higher-level sub-networks may repeat the features that the lower-level sub-networks have already learned. Summary of the Invention

[0005] In order to overcome the problems existing in the existing deep cascaded network and make full use of the reference modality data and the sampled data of the target modality for auxiliary reconstruction, the present invention provides a fast magnetic resonance imaging method based on a deep cascaded network with cross-subnetwork same-layer dense connections. In order to improve the learning efficiency of the deep cascaded network, the present invention introduces "same-layer dense connections" between sub-networks and adds a k-space integrated learning module at the end of each sub-network to more effectively utilize the frequency domain data of the reference modality and the target modality.

[0006] The technical solution provided by the present invention is as follows:

[0007] A fast magnetic resonance imaging method based on a deep cascaded network with cross-subnetwork same-layer dense connections, comprising the following steps:

[0008] Step 1: Input the u-fold undersampled k-space MRI data of the target modality and the fully sampled k-space MRI data of the reference modality Here, represents the k-space domain,

[0009] Step 2: Map y u and y' to the image domain : Here, F -1 is the inverse Fourier transform matrix, M u∈{0,1} M×N represents a mask matrix for u-fold undersampling of the fully sampled k-space MRI data y of the target modality;

[0010] Step 3 reconstructs the target MRI image by the deep cascaded network model f Rec (·; Θ)

[0011] Furthermore, in the said Step 3, the deep cascaded network f Rec (·; Θ) is composed of T subnets {f 1 (·; Θ 1 ), f 2 (·; Θ 2 ), …, f T (·; Θ T )}, where Θ = {Θ 1 , Θ 2 , …, Θ T} is the parameter set of f Rec , and Θ t is the parameter set of the subnet f t . Let denote the output of the subnet f t-1 , f t is recursively defined as follows:

[0012]

[0013]

[0014]

[0015] x′ in formula (1) = F -1 y′, cat(·, ·, …) represents the feature concatenation operation in the channel direction, and iCNN t represents the convolutional module of the subnet f t in the image domain, is the parameter set of iCNN t , and iCNN t is composed of D convolutional layers {conv t,1 , conv t,2 , …, conv t,D}. Except for conv t,1 , the other convolutional layers use the same-layer dense connection. Specifically, conv t,d+1 is defined as follows:

[0016]

[0017] Among them, H t,d is the feature map output by conv t,d , represents the convolution operation, σ(·) is the activation function, l = min(t - 1, δ - 1), 1 < δ ≤ T is the preset maximum number of dense connections in the same layer, represents the parameter set of conv t,d+1 ;

[0018] DC in formula (2) represents the Data Consistency layer, is the index of the k-space sampling position of the target modality, and the DC layer is defined as follows:

[0019]

[0020] Among them, [i, j] represents the specific position in the k-space,

[0021] In formula (3), kEL t is defined as follows:

[0022]

[0023] Among them, WS(·) calculates the weighted sum of the elements in, w t is the weight parameter, is the parameter set of the k-space convolution module kCNN t , and kCNN t consists of three convolutional layers, and the configuration of each convolutional layer is the same as that of conv t,d+1 in formula (4).

[0024] Furthermore, in step 3, the parameter set Θ of the depth concatenation network f Rec based on cross-subnet same-layer dense connection is obtained through the following steps:

[0025] Step 3.1 Randomly initialize the parameters of f Rec as Θ = Θ (0) , let the best reconstructed image quality the iteration number s = 0, the current training period e = 0, and E is the preset maximum number of training periods;

[0026] Step 3.2 Construct the training set under the target acceleration ratio u in the following way:

[0027]

[0028] Among them, the superscript i represents the index of the training sample of the target modality or the reference modality, xi = F -1 y i , where n is the number of training samples;

[0029] Step 3.3 Randomly further divide the training set D u into a training set and a validation set

[0030] Step 3.4 For all calculate the reconstructed image among them

[0031]

[0032] Step 3.5 For a given loss function l(·,·), optimize the following objective formula by gradient descent:

[0033]

[0034] Step 3.6 Evaluate the reconstructed image quality of the network f on the validation set Rec (·; Θ (s) ):

[0035]

[0036] where Q(·,·) is the quality evaluation function;

[0037] Step 3.7 If let save the parameter set Θ * = Θ (s) ;

[0038] Step 3.8 If e = E, stop the iteration; otherwise, let e = e + 1 and return to the above Step 3.2.

[0039] The beneficial effects of the present invention are as follows: Compared with the existing MRI-assisted reconstruction methods, the method provided by the present invention achieves a better reconstruction effect and has strong practicability. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Figure 1 is an example of a sampling mask matrix, where (a) is an example of a central undersampling mask matrix and (b) is an example of a Gaussian undersampling mask matrix.

[0041] Figure 2 is a schematic diagram of a deep cascaded network based on dense connections between the same layers of different subnets. Among them, (a) is the overall structural schematic diagram of the deep cascaded network based on dense connections between the same layers of different subnets proposed by the present invention: composed of f 1 、f2 …f T consisting of T subnets, x u is the u-fold undersampled MRI image of the input target modality, and the k-space data y of the target modality u and the fully sampled data (x′, y′) of the reference modality are used as auxiliary information to participate in the reconstruction of the target modality by each subnet. is the reconstructed image of the target modality finally output; (b) shows the specific configuration of the deep cascaded network based on cross-subnet same-layer dense connection proposed by the present invention: each subnet consists of three main modules, namely, a convolutional block based on same-layer dense connection, a data consistency layer (DC), and a k-space integrated learning module (kEL); the specific configuration of the iCNN t is given in the rounded rectangle box, and the specific way of cross-subnet same-layer dense connection is shown: the output of the previous l subnets and the previous convolutional layer of this subnet are concatenated in the way shown in formula (4) for each convolutional layer conv t,d (2 ≤ d ≤ D) of each subnet to form the input of conv t,d ; the data consistency layer makes full use of the acquired k-space data of the target modality to correct the reconstruction result of the network to ensure that this part of the data is not changed by the network; the k-space integrated learning module integrates the prediction results of all previous subnets and introduces the k-space data of the reference modality for auxiliary reconstruction.

[0042] Figure 3 This is the k-space data of some subjects (here, 5 subjects are selected) collected by Philips Ingenia 3T for this invention patent. Among them, subfigure (a) is the k-space data of the T1-weighted image (T1WI), and subfigure (b) is the k-space data of the T2-weighted image (T2WI); each row represents a subject, and each column represents the k-space data of a certain slice position; from left to right, the indexes of the selected slice positions are: 60, 78, 98, 125, and 140.

[0043] Figure 4 is composed of Figure 3 The MRI images of T1WI and T2WI obtained by inverse Fourier transform of the k-space data in.

[0044] Figure 5 This is the benchmark network used in this invention patent: a typical deep cascaded network.

[0045] Figure 6 is Figure 5 The deep cascaded network with cross-subnet same-layer dense connection applied to each subnet of the benchmark network in.

[0046] Figure 7 is Figure 5A densely connected depth cascaded network is applied inside each subnet of the reference network in

[0047] Figure 8 are the visualization results of different methods for reconstructing 12-fold undersampled T2WI. Subgraphs above the dotted line: from top to bottom and from left to right are the ground truth image, zero-filled image, reconstructed image of UNet, reconstructed image of DenseUNet, reconstructed image of RefineGAN, reconstructed image of D5C5, reconstructed image of D5C5 based on dense connection between the same layers of different subnets, reconstructed image of D5C5 based on dense connection between the same layers of different subnets and k-space ensemble learning; Subgraphs below the dotted line: from left to right are the ground truth image, zero-filled image, reconstructed image of UNet, reconstructed image of DenseUNet, reconstructed image of RefineGAN, reconstructed image of D5C5, reconstructed image of D5C5 based on dense connection between the same layers of different subnets, and the magnified images in three regions of interest. Detailed implementation manners

[0048] The present invention will be further described below with reference to the accompanying drawings.

[0049] Refer to Figures 1 to 8 , a fast magnetic resonance imaging method based on a depth cascaded network with dense connection between the same layers of different subnets, comprising the following steps:

[0050] Step 1 Input the u-fold undersampled k-space MRI data of the target modality and the fully sampled k-space MRI data of the reference modality Here, represents the k-space domain,

[0051] Step 2 Map y u and y' to the image domain : Here, F -1 is the inverse Fourier transform matrix, M u ∈{0,1} M×N represents the mask matrix for u-fold undersampling of the fully sampled k-space MRI data y of the target modality. Commonly used mask matrices include the central undersampling matrix and the Gaussian undersampling matrix, as shown in Figure 1 (a) and Figure 1 (b) respectively;

[0052] Step 3 Reconstruct the MRI image of the target by the depth cascaded network model f Rec (·; Θ) Figure 2 (a) shows the overall network structure of f Rec , Figure 2(b) gives the specific configuration of f Rec .

[0053] In step 3, the deep cascaded network f Rec (·; Θ) consists of T subnets {f 1 (·; Θ 1 ), f 2 (·; Θ 2 ), …, f T (·; Θ T )} in cascade. Here, Θ = {Θ 1 , Θ 2 , …, Θ T} is the parameter set of f Rec , and Θ t is the parameter set of the subnet f t . Figure 2 (a) gives the specific cascading method of the T subnets. The k-space data y u of the target modality and the fully sampled data (x′, y′) of the reference modality participate in the reconstruction of the target modality by each subnet as auxiliary information. Let denote the output of the subnet f t-1 . The subnet f t is recursively defined as follows:

[0054]

[0055]

[0056]

[0057] In formula (1), x′ = F -1 y′, cat(·, ·, …) represents the feature concatenation operation in the channel direction, and iCNN t represents the convolutional module of the subnet f t in the image domain. is the parameter set of iCNN t , and iCNN t consists of D convolutional layers {conv t,1 , conv t,2 , …, conv t,D}. Except for conv t,1 , the other convolutional layers use same-layer dense connections. Specifically, conv t,d+1 is defined as follows:

[0058]

[0059] where, H t,d is convt,d The output feature map represents a convolutional operation, σ(·) is the activation function, l = min(t - 1, δ - 1), and 1 < δ ≤ T is the preset maximum number of dense connections in the same layer, represents conv t,d+1 the parameter set of Figure 2 (b) The rounded rectangle box gives the specific configuration of iCNN t and shows the specific way of cross-subnet same-layer dense connection: Each convolutional layer conv t,d (2 ≤ d ≤ D) of each subnet concatenates the outputs of the previous l subnets and the previous convolutional layer of this subnet in the way shown in formula (4) to form the input of conv t,d ; The maximum number of dense connections in the same layer δ is determined by the number of subnets T cascaded by f Rec For example, when T = 5, the recommended value of δ is 4;

[0060] DC in formula (2) represents the Data Consistency layer, and is the index of the k-space sampling position of the target modality. The DC layer is specifically defined as follows:

[0061]

[0062] where [i, j] represents the specific position in the k-space,

[0063] In formula (3), kEL t is specifically defined as follows:

[0064]

[0065] where, WS(·) calculates the weighted sum of the elements in X t and w t is the weight parameter. is the parameter set of the k-space convolutional module kCNN t and kCNN t consists of three convolutional layers, and the configuration of each convolutional layer is the same as that of conv t,d+1 in formula (4), Figure 2 (b) The bottom shows the integration method of the reconstruction results of kEL t for all previous subnets.

[0066] In step 3 described above, the parameter set Θ of the depth cascaded network f Rec based on cross-subnet same-layer dense connection is trained through the following steps:

[0067] Step 3.1 Randomly initialize the parameters Θ of f Rec to Θ (0) , where the initial parameter Θ (0) can be sampled from a Gaussian distribution with a mean of 0 and a variance of 1; Let the best reconstructed image quality Set the iteration count s = 0, the current training epoch e = 0, and E is the preset maximum number of training epochs. In a specific implementation, E can be set to 100;

[0068] Step 3.2 Construct a training set under the target speedup ratio u in the following way:

[0069]

[0070] where the superscript i represents the index of the training sample of the target modality or reference modality, and x i = F -1 y i , and n is the number of training samples;

[0071] Step 3.3 Further randomly divide the training set D u into a training set and a validation set

[0072] Step 3.4 For all calculate the reconstructed image where

[0073]

[0074] Step 3.5 For a given loss function l(·,·), solve the following objective through an optimization method:

[0075]

[0076] Here, the loss function l(·,·) can use the l 1 norm, l 2 norm, or the Charbonnier penalty function, where the Charbonnier penalty function is defined as: where x 1 , x 2 are the reconstructed image and the original image respectively, and ∈ is a small constant, which can be set to ∈ = 0.001; The optimization method can be selected as: Stochastic Gradient Descent, Adaptive Gradient Algorithm, RMSprop, Adaptive Moment Estimation (Adam); For the method proposed in this invention patent, it is recommended to use the Charbonnier penalty function and the Adam optimization method;

[0077] Step 3.6 Evaluate the network f on the validation set Rec (·; Θ (s) ) for the reconstructed image quality:

[0078]

[0079] where Q(·, ·) is the quality evaluation function. In specific implementation, peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) can be used as the quality evaluation functions;

[0080] Step 3.7 If let save the parameter set Θ * = Θ (s) ;

[0081] Step 3.8 If e = E, stop the iteration; otherwise, let e = e + 1 and return to the above Step 3.2.

[0082] Next, the beneficial effects of the deep cascaded network based on cross-subnet same-layer dense connection proposed by the present invention are verified through experiments.

[0083] Data acquisition: 15 subjects were scanned using Philips Ingenia 3T to obtain MRI data of two modalities for each subject: T1-weighted magnetic resonance image (T1WI) and T2-weighted magnetic resonance image (T2WI). Each modality contains 260 slices in the axial direction with a resolution of 336×261. Figure 3 shows some of the acquired k-space data, Figure 4 and shows the corresponding MRI images. The training set and validation set were constructed according to the methods described in Steps 3.2 and 3.3 of the invention. Specifically, the MRI images of 13 randomly selected subjects were used as the training set, and the MRI images of the remaining two subjects were used as the validation set and test set respectively; T1WI was used as the auxiliary modality and T2WI was used as the target modality.

[0084] The network structure proposed by the present invention (as shown in Figure 2 (b)) was constructed by adding two modules, "Cross-subnet Dense Connection (CDC)" and "k-space Ensemble Learning (kEL)", to the benchmark network as shown in Figure 5 .

[0085] First, verify the beneficial effects brought by "Cross-subnet Dense Connection (CDC)". Use "~" to represent Figure 5the reference network in, and let D = 5, T = 5 therein; use "~ + CDC" to represent the network obtained by adding only CDC to "~", as Figure 6 shown; currently, the existing methods mainly use "Intra-subnet DenseConnection (IDC)", and use "~ + IDC" to represent the network obtained by adding only IDC to "~", obtaining a network structure as Figure 7 shown. When performing dense connection, the connection method of feature maps from different convolutional layers and the maximum connection length δ will both affect the network performance. The optional feature map connection methods are as follows: a) cat: directly splice the feature maps; b) conv + cat: perform convolution first and then splice; c) ReLU + conv + cat: activate first and then perform convolution and splicing; let the number of subnets T = 5, then δ can take: 2, 3, 4, 5. Taking the feature map connection method and the maximum connection length δ as two parameters of IDC / CDC, for 8-fold undersampled T2WI (i.e., u = 8), Table 1 compares the impacts of different dense connection methods on the network performance (PSNR and SSIM). It can be seen from the first three rows of Table 1 that using CDC is better than the reference network without dense connection and IDC, and it is not easy to cause performance degradation; it can be seen from the 5th row of Table 1 that using ReLU + conv + cat for feature map connection is very easy to cause performance degradation; it can be seen from the 4th, 6th, 7th, and 8th rows of Table 1 that as the maximum connection length δ increases, the reconstruction performance of the network gradually improves, and when δ = 4, the performance tends to saturation; when δ = 5, the performance may locally decline compared to the performance when δ = 4.

[0086]

[0087]

[0088] Table 1

[0089] Further verify the beneficial effects brought by "k-space Ensemble Learning (kEL)". Use "~ + CDC(conv + cat, 4)" as the reference network, and use "~~" to represent this reference network. From formula (4), kEL has the following 5 variants: a) only use WS(X): only perform weighted summation on the outputs X of each subnet; directly perform k-space convolution on the output of iCNN ; For Perform k-space convolution on the k-space data y' of the auxiliary modality; d) kCNN(WS(X)): Perform k-space convolution on WS(X); e) kCNN(WS(X), y'): Perform k-space convolution on WS(X) and y', which is kEL defined in formula (4) of the present invention. Reconstruct the 8-fold undersampled T2WI (i.e., u = 8) using 5 variants of "~~" and "~~" + kEL, and three cross-validation experiments were conducted. Table 2 reports the relevant experimental results. It can be seen from the reported results that compared with the 4 variants of kEL from (a) to (d), the kEL method proposed by the present invention (i.e., the (e)th variant of kEL) can achieve stable and optimal reconstruction results.

[0090]

[0091] Table 2

[0092] Further compare the performance of the method proposed by the present invention with the benchmark zero-filling method (ZeroFilling) and four advanced deep neural network-based methods, namely UNet, DenseUNet, RefineGAN, and D5C5. Reconstruct the 4-fold (u = 4), 8-fold (u = 8), and 12-fold (u = 12) undersampled T2WI in sequence, and 5-fold cross-validation was conducted. Table 3 statistics the reconstruction performance under two metric indicators of PSNR and SSIM: the mean ± variance of PSNR / SSIM. It can be seen that the method proposed by the present invention has achieved optimal reconstruction performance in different scenarios, especially at a higher acceleration ratio, the performance advantage of the method proposed by the present invention is more significant. Figure 8 Further show the visualization results of each method for reconstructing the 12-fold undersampled T2WI, and magnify and show 3 selected regions of interest. It can be clearly seen that the method proposed by the present invention better restores the anatomical details than other methods.

[0093]

[0094]

[0095] Table 3

[0096] The content described in the embodiments of this specification is only a list of implementation forms of the inventive concept and is only for illustrative purposes. The protection scope of the present invention should not be regarded as limited to the specific forms stated in this embodiment. The protection scope of the present invention also extends to equivalent technical means that can be conceived by those of ordinary skill in the art based on the inventive concept of the present invention.

Claims

1. A fast magnetic resonance imaging method based on a deep cascaded network with cross-subnet same-layer dense connections, characterized in that, the method comprises the following steps: Step 1: Input the u-fold undersampled k-space MRI data of the target modality and the fully sampled k-space MRI data of the reference modality Here represents the k-space domain Step 2 maps y u and y′ to the image domain as follows: Here, F -1 is the inverse Fourier transform matrix, and M u ∈ {0, 1} M×N denotes the mask matrix that undersamples the fully sampled k-space MRI data y of the target modality by a factor of u; Step 3 reconstructs the target MRI image by the deep cascaded network model f Rec (·; Θ) In step 3, the deep cascaded network \(f\) based on cross-subnet same-layer dense connection Rec \((·;\Theta)\) is composed of \(T\) subnets \(\{f\) 1 (·;\Theta 1 ), \(f\) 2 (·;\Theta 2 ), …, \(f\) T (·;\Theta T )\}\) cascaded together. Here, \(\Theta=\{\Theta 1 ,\Theta 2 , …, \Theta T \}\) is the parameter set of \(f\) Rec , and \(\Theta t \) is the parameter set of the subnet \(f\) t . Let denote the output of the subnet \(f\) t-1 . \(f\) t is recursively defined as follows: x' in formula (1) = F -1 y', cat(·, ·, …) represents the feature concatenation operation in the channel direction, and iCNN t represents the subnet f t convolution module in the image domain, is the parameter set of iCNN t , and iCNN t consists of D convolutional layers {conv t,1 , conv t,2 , …, conv t,D}, and except for conv t,1 , the other convolutional layers all use same-layer dense connections. Specifically, conv t,d+1 is defined as follows: Among them, H t,d is the feature map output by conv t,d , denotes the convolution operation, σ(·) is the activation function, l = min(t - 1, δ - 1), 1 < δ ≤ T is the preset maximum number of dense connections in the same layer, denotes the parameter set of conv t,d+1 . In formula (2), DC represents the data consistency layer, which is the index of the k-space sampling position of the target modality. The DC layer is defined as follows: where [i, j] represents a specific position in the k-space, In formula (3), kEL t is defined as follows: Among them, WS(·) calculates the weighted sum of the elements, w t is the weight parameter, is the parameter set of the k-space convolution module kCNN t kCNN t consists of three convolutional layers, and the configuration of each convolutional layer is the same as that of conv t,d+1 in formula (4).

2. The fast magnetic resonance imaging method based on a deep cascaded network with cross-subnet same-layer dense connections according to claim 1, characterized in that, In step 3, the parameter set Θ of the deep cascaded network f based on cross-subnet same-layer dense connection Rec is obtained through the following steps of training: Step 3.1 Randomly initialize the parameters Θ = Θ of f Rec and set the best reconstruction image quality Q (0) = 0, the number of iterations s = 0, the current training epoch e = 0, and E is the preset maximum number of training epochs; max ​ step 3.2 constructs a training set under a target acceleration ratio u in the following manner: where the superscript \(i\) represents the index of the training samples of the target modality or the reference modality, \(x\) i = F -1 y i , and \(n\) is the number of training samples; Step 3.3 Split the training set D u further randomly into a training set and a validation set Step 3.4 For all calculate among them the reconstructed image step 3.5 for a given loss function l(·,·), optimizes the following objective formula by the gradient descent method: Step 3.6 Evaluate the network f on the validation set Rec (·; Θ (s) ) for the reconstructed image quality: where Q(·,·) is a quality evaluation function; Step 3.7 If Q (s) >Q max , let Q max =Q (s) , save the parameter set Θ * =Θ (s) ; step 3.8 if e = E, stop the iteration; otherwise, let e = e + 1, and return to the above step 3.2.

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