WO2024092387A1 - 基于部分可分函数自适应动态磁共振快速成像方法及装置 - Google Patents
基于部分可分函数自适应动态磁共振快速成像方法及装置 Download PDFInfo
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- the present invention relates to the field of medical dynamic magnetic resonance imaging, and in particular to a method and device for adaptive dynamic magnetic resonance rapid imaging based on partially separable functions.
- Dynamic MRI can reveal both spatial and temporal information of the scanned object.
- the amount of data required for dynamic MRI is greatly increased, and its scanning time is extremely long, which is the main reason restricting the application of dynamic MRI.
- How to speed up dynamic MRI while ensuring the quality of MRI images is an important issue in current MRI research.
- Existing dynamic MRI methods include traditional parallel imaging methods and deep learning-based methods, among which:
- the existing deep learning-based magnetic resonance reconstruction method has a convolutional layer module that works as a black box, and it is difficult to explain its rationality.
- the embodiment of the present invention provides a method and device for adaptive dynamic magnetic resonance rapid imaging based on partially separable functions, so as to at least solve the technical problem of poor effect of existing dynamic magnetic resonance imaging.
- a method for adaptive dynamic magnetic resonance rapid imaging based on partially separable functions comprising the following steps:
- the dynamic MRI reconstruction model first transforms the input low-rank into a characterization of the image filter null space through the annihilation relation in the image domain;
- the dynamic magnetic resonance reconstruction model reuses the equivalent relationship between Hankel matrix product and convolution, expresses the low rank with a convolutional network, and expands its iterative solution into the convolutional network.
- constructing a dynamic magnetic resonance reconstruction model based on low-rank prior and sparse prior of dynamic magnetic resonance includes the following steps:
- M is the undersampling operator
- F is the Fourier operator
- Y is the undersampled original K-space data
- ⁇ is the image to be solved
- ⁇ ,Y ⁇ C n ⁇ m ⁇ T where the size of each image frame is n ⁇ m
- T is the number of time frames
- R( ⁇ ) is the prior regularization term.
- the low-rank input is transformed into a characterization of the null space of the image filter through the annihilation relation in the image domain, including:
- the magnetic resonance reconstruction problem is solved by the partially separable model, as shown in the following formula:
- ⁇ l (t) is a set of time bases
- c l (k) is the corresponding space basis
- ⁇ (r, t) is called L-order separable
- ⁇ (r, t) is the dynamic magnetic resonance image to be reconstructed, ⁇ C n ⁇ m ⁇ T ;
- the low rank of the magnetic resonance image is expressed by matrix multiplication, and the low rank is characterized by h[r, t] through the annihilation relationship.
- the low rank is expressed by a convolutional network, and its iterative solution is expanded into the convolutional network, including:
- the partially separable model ⁇ (k, t) reconstructed by dynamic magnetic resonance satisfies the annihilation relation, that is, there exists h[r, t] such that:
- Magnetic resonance images also satisfy the annihilation relation in the image domain, namely:
- ⁇ 1 and ⁇ 2 are the corresponding regularization parameters
- the dynamic magnetic resonance reconstruction model is implemented into the convolutional network.
- the convolution module used to solve the Z subproblem is a 5-layer 3D convolution with a convolution kernel size of 1x1x3;
- the convolution module of the sub-problem is a 5-layer 2D convolution, the convolution kernel size is 3x3, and the number of convolution channels is 32.
- the under-sampled K-space data is used as input and the fully-sampled image is used as the label.
- the input data is divided into two channels, real and imaginary, and input into the network.
- the loss function of the network is the minimum mean square error between the output image and the label.
- the batch size is set to 1, the initial value of the learning rate is 0.001, and it is set to exponential decay.
- the optimizer used by the network is Adam.
- the neural network is a supervised neural network.
- the method and device for adaptive dynamic magnetic resonance rapid imaging based on partially separable functions in the embodiment of the present invention first constructs a dynamic magnetic resonance reconstruction model based on the low-rank prior and sparse prior of dynamic magnetic resonance, first converts the input low-rank into a characterization of the image filter null space through the annihilation relationship in the image domain; then uses the equivalent relationship between Hankel matrix product and convolution to express the low-rank with a convolutional network, and expands its iterative solution into the convolutional network.
- the neural network proposed by the present invention can achieve a higher acceleration multiple of magnetic resonance reconstruction, achieve better reconstruction effect, and capture dynamic frames more accurately.
- FIG1 is a flow chart of a method for adaptive dynamic magnetic resonance rapid imaging based on partially separable functions according to the present invention
- FIG2 is a block diagram of a convolutional neural network in the present invention.
- FIG3 is a network structure diagram of the Adaptive Subspace Net in the present invention.
- FIG4 is a diagram showing the result of 8-fold under-mining in the present invention.
- FIG5 is a diagram showing the results of 12-fold under-mining in the present invention.
- FIG6 is a module diagram of the partially separable function-based adaptive dynamic magnetic resonance rapid imaging device of the present invention.
- a method for adaptive dynamic magnetic resonance rapid imaging based on partially separable functions is provided, referring to FIG1 , and comprising the following steps:
- S101 construct a dynamic magnetic resonance reconstruction model based on low-rank prior and sparse prior of dynamic magnetic resonance
- the dynamic magnetic resonance reconstruction model transforms the input low-rank annihilation relationship in the image domain into a characterization of the null space of the image filter.
- the dynamic magnetic resonance reconstruction model uses the equivalent relationship between Hankel matrix product and convolution to represent the low rank with a convolutional network and expands the iterative solution calculation process into the neural network.
- the partially separable function-based adaptive dynamic magnetic resonance fast imaging method in the embodiment of the present invention constructs a dynamic magnetic resonance reconstruction model based on the low-rank prior and sparse prior of dynamic magnetic resonance, converts the input low-rank into a characterization of the image filter null space through the annihilation relationship in the image domain; then uses the equivalent relationship between Hankel matrix product and convolution to express the low rank with a convolutional network, and expands its iterative solution into the convolutional network.
- the present invention reveals that the convolutional network module in the Adaptive Subspace Net is characterizing the null space, making the network capture dynamic frames more accurately.
- constructing a dynamic magnetic resonance reconstruction model based on low-rank prior and sparse prior of dynamic magnetic resonance includes the following steps:
- M is the undersampling operator
- F is the Fourier operator
- Y is the undersampled original K-space data
- ⁇ is the image to be solved
- ⁇ ,Y ⁇ C n ⁇ m ⁇ T where the size of each image frame is n ⁇ m
- T is the number of time frames
- R( ⁇ ) is the prior regularization term.
- the low-rank input is converted into the characterization of the null space of the image filter through the annihilation relationship in the image domain, including:
- the magnetic resonance reconstruction problem is solved by the partially separable model, as shown in the following formula:
- ⁇ l (t) is a set of time bases
- c l (k) is the corresponding space basis
- ⁇ (r, t) is called L-order separable
- ⁇ (r, t) is the dynamic magnetic resonance image to be reconstructed, ⁇ C n ⁇ m ⁇ T ;
- the low rank of the magnetic resonance image is expressed by matrix multiplication, and the low rank is characterized by h[r, t] through the annihilation relationship.
- the partially separable model ⁇ (k, t) reconstructed by dynamic magnetic resonance satisfies the annihilation relation, that is, there exists h[r, t] such that:
- Magnetic resonance images also satisfy the annihilation relation in the image domain, namely:
- ⁇ 1 and ⁇ 2 are the corresponding regularization parameters
- the dynamic magnetic resonance reconstruction model is implemented into the convolutional network.
- the convolution module used to solve the Z subproblem is a 5-layer 3D convolution with a convolution kernel size of 1x1x3;
- the convolution module of the sub-problem is a 5-layer 2D convolution, the convolution kernel size is 3x3, and the number of convolution channels is 32.
- the under-sampled K-space data is used as input, and the fully sampled image is used as the label.
- the input data is divided into two channels, real and imaginary, and input into the network.
- the loss function of the network is the minimum mean square error between the output image and the label.
- the batch size is set to 1, the initial value of the learning rate is 0.001, and it is set to exponential decay.
- the optimizer used by the network is Adam.
- the neural network is a supervised neural network.
- the present invention relates to a method for fast imaging of dynamic magnetic resonance based on partially separable functions.
- the dynamic magnetic resonance reconstruction method based on deep learning can be divided into two categories: data-driven and model-driven.
- the present invention belongs to the category of model-driven.
- the model is constructed based on the properties of a specific task in a model-driven manner, such as using the properties of low-rank sparsity of images to construct a low-rank sparse model, and then expanding the traditional iterative algorithm into a neural network for solution.
- the present invention expands the partially separable model (PS model) into the neural network, which greatly reduces the hyperparameters that need to be adjusted in the model.
- the present invention uses the equivalent relationship between Hankel matrix and convolution to express low rank with a network, revealing that the convolutional network in Adaptive Subspace Net is characterizing the null space, which greatly enhances the interpretability of the network.
- the present invention expresses the low rank of the magnetic resonance image in the form of matrix multiplication, and converts the low rank into a characterization of the zero space of the image filter.
- the network learns the entire mapping process and indirectly and adaptively learns the low rank of the magnetic resonance image.
- the model proposed in the present invention is more accurate in capturing the image frames and has better effect of dynamic magnetic resonance reconstruction.
- the present invention proposes a convolutional neural network based on a partially separable model for magnetic resonance cardiac cine imaging.
- the block diagram of the scheme is shown in FIG2.
- the present invention specifically includes:
- Part I MRI reconstruction model
- the magnetic resonance reconstruction model based on K-space undersampling can be discretely expressed as:
- M is the undersampling operator
- F is the Fourier operator
- Y is the undersampled original K-space data
- ⁇ is the image to be solved
- ⁇ ,Y ⁇ C n ⁇ m ⁇ T where the size of each frame of the image is n ⁇ m
- T is the number of time frames; in this invention, only the model of single-channel reconstruction is discussed.
- R( ⁇ ) is the prior regularization term.
- ⁇ l (t) is a set of time basis
- c l (k) is the corresponding space basis
- ⁇ (r, t) is called L-order separable.
- ⁇ (r, t) is the dynamic magnetic resonance image to be reconstructed, ⁇ C n ⁇ m ⁇ T .
- the present invention can convert it into the following index form:
- the present invention can express the low rank of the magnetic resonance image by matrix multiplication, and characterize the low rank by using h[r, t] through the annihilation relationship.
- Magnetic resonance images also satisfy the annihilation relation in the image domain, namely:
- the final model can be:
- an iterative solution algorithm of the regularized model proposed in the present invention is constructed to solve different sub-problems according to different variables.
- the network structure of Adaptive Subspace Net is shown in Figure 3.
- the convolution module used to solve the Z sub-problem is a 5-layer 3D convolution with a convolution kernel size of 1x1x3;
- the convolution module of the sub-problem is a 5-layer 2D convolution, the convolution kernel size is 3x3, and the number of convolution channels is 32.
- the present invention adopts a supervised neural network, with under-sampled K-space data as input and fully-sampled images as labels.
- the input data is divided into two channels, real and imaginary, and input into the network.
- the loss function of the network is the minimum mean square error between the output image and the label.
- the batch size is set to 1, the initial value of the learning rate is 0.001, and it is set to exponential decay.
- the optimizer used by the network is Adam.
- the effectiveness of the present invention is verified through retrospective sampling simulation of Siemens cardiac cine imaging data set.
- the undersampling mode is cartesian, and the cardiac movie imaging data is reconstructed using the scheme proposed in the present invention, which achieves better reconstruction effect than other schemes.
- the first row in Figure 4 shows the full-sampled real image and the reconstruction result using the technical solution of the present invention
- the second row shows the enlarged view of the heart area framed by the dotted frame of the first row of images
- the third row shows the error map between the reconstructed image and the real image (display range [0, 0.09])
- the fourth row shows the time slice map of the 100th pixel along the y-axis time dimension
- the fifth row shows the error between the time slice map and the real image. From the error map in the third row, it can be seen that the Adaptive Subspace Net proposed by the present invention has better reconstruction effect than the existing methods.
- the present invention is applicable to other data sets of magnetic resonance dynamic imaging.
- the design of the network is also effective if the number of convolution channels and convolution layers used in the present invention are slightly changed.
- a partially separable function-based adaptive dynamic magnetic resonance rapid imaging device comprising:
- a model building unit 201 is used to build a dynamic magnetic resonance reconstruction model based on a low-rank prior and a sparse prior of dynamic magnetic resonance;
- a conversion unit 202 used for the dynamic magnetic resonance reconstruction model to convert the low rank into a description of the null space of the image filter through the annihilation relationship
- the low-rank expression unit 203 is used to express the low rank using a convolutional network by utilizing the equivalent relationship between Hankel matrix product and convolution, and to expand its iterative solution into the convolutional network.
- the partially separable function-based adaptive dynamic magnetic resonance rapid imaging device in the embodiment of the present invention first constructs a dynamic magnetic resonance reconstruction model based on the low-rank prior and sparse prior of dynamic magnetic resonance, expresses the low rank of the magnetic resonance image by matrix multiplication, and converts the low rank into a characterization of the zero space of the image filter through the annihilation relationship; then, using the equivalent relationship between the Hankel matrix and the convolution, the low rank is expressed by a convolutional network, and the dynamic magnetic resonance reconstruction model is implemented in the convolutional network.
- the present invention proposes a model based on a partially separable function-based adaptive low rank, and uses it for dynamic magnetic resonance imaging, and based on the equivalence of the Hankel product and the convolution, the partially separable model is solved by a convolutional network, revealing that the convolutional network module in the Adaptive Subspace Net is characterizing the zero space, making the network capture dynamic frames more accurate.
- the present invention relates to an adaptive dynamic magnetic resonance fast imaging device based on partially separable functions.
- the dynamic magnetic resonance reconstruction method based on deep learning can be divided into two categories: data-driven and model-driven.
- the present invention belongs to the category of model-driven.
- the model is constructed based on the properties of a specific task in a model-driven manner, such as using the properties of low-rank sparsity of images to construct a low-rank sparse model, and then expanding the traditional iterative algorithm into a neural network for solution.
- the present invention expands the partially separable model (PS model) into the neural network, which greatly reduces the hyperparameters that need to be adjusted in the model.
- the present invention uses the equivalent relationship between Hankel matrix product and convolution to express low rank using a network, revealing that the convolutional network in Adaptive Subspace Net is characterizing the null space, greatly enhancing the interpretability of the network.
- the present invention expresses the low rank of the magnetic resonance image in the form of matrix multiplication, and converts the low rank into a characterization of the zero space of the image filter.
- the network learns the entire mapping process, and indirectly and adaptively learns the low rank of the magnetic resonance image.
- the model proposed in the present invention is more accurate in capturing image frames and has better effect of dynamic magnetic resonance reconstruction.
- the present invention proposes a convolutional neural network based on a partially separable model for magnetic resonance cardiac cine imaging.
- the block diagram of the scheme is shown in FIG2.
- the present invention specifically includes:
- Part I MRI reconstruction model
- the magnetic resonance reconstruction model based on K-space undersampling can be discretely expressed as:
- M is the undersampling operator
- F is the Fourier operator
- Y is the undersampled original K-space data
- ⁇ is the image to be solved
- ⁇ ,Y ⁇ C n ⁇ m ⁇ T where the size of each frame of the image is n ⁇ m
- T is the number of time frames; in this invention, only the model of single-channel reconstruction is discussed.
- R( ⁇ ) is the prior regularization term.
- ⁇ l (t) is a set of time basis
- c l (k) is the corresponding space basis
- ⁇ (r, t) is called L-order separable.
- ⁇ (r, t) is the dynamic magnetic resonance image to be reconstructed, ⁇ C n ⁇ m ⁇ T .
- the present invention can convert it into the following index form:
- the present invention can express the low rank of the magnetic resonance image by matrix multiplication, and characterize the low rank by using h[r, t] through the annihilation relationship.
- Magnetic resonance images also satisfy the annihilation relation in the image domain, namely:
- the final model can be:
- an iterative solution algorithm of the regularized model proposed in the present invention is constructed to solve different sub-problems according to different variables.
- the network structure of Adaptive Subspace Net is shown in Figure 3.
- the convolution module used to solve the Z sub-problem is a 5-layer 3D convolution with a convolution kernel size of 1x1x3;
- the convolution module of the sub-problem is a 5-layer 2D convolution, the convolution kernel size is 3x3, and the number of convolution channels is 32.
- the present invention adopts a supervised neural network, with under-sampled K-space data as input and fully-sampled images as labels.
- the input data is divided into two channels, real and imaginary, and input into the network.
- the loss function of the network is the minimum mean square error between the output image and the label.
- the batch size is set to 1, the initial value of the learning rate is 0.001, and it is set to exponential decay.
- the optimizer used in the network is Adam.
- the effectiveness of the present invention is verified through retrospective sampling simulation of Siemens cardiac cine imaging data set.
- the undersampling mode is cartesian, and the cardiac movie imaging data is reconstructed using the scheme proposed in the present invention, which achieves better reconstruction effect than other schemes.
- the first row in Figure 4 shows the full-sampled real image and the reconstruction result using the technical solution of the present invention
- the second row shows the enlarged view of the heart area framed by the dotted frame of the first row of images
- the third row shows the error map between the reconstructed image and the real image (display range [0, 0.09])
- the fourth row shows the time slice map of the 100th pixel along the y-axis time dimension
- the fifth row shows the error between the time slice map and the real image. From the error map in the third row, it can be seen that the Adaptive Subspace Net proposed by the present invention has better reconstruction effect than the existing methods.
- the present invention is applicable to other data sets of magnetic resonance dynamic imaging.
- the design of the network is also effective if the number of convolution channels and convolution layers used in the present invention are slightly changed.
- a storage medium stores a program file capable of implementing any one of the above-mentioned methods for adaptive dynamic magnetic resonance rapid imaging based on partially separable functions.
- a processor is used to run a program, wherein when the program is run, any one of the above-mentioned adaptive dynamic magnetic resonance rapid imaging methods based on partially separable functions is executed.
- the present invention proposes an adaptive low-rank model based on partially separable functions and uses it for dynamic magnetic resonance imaging. Based on the equivalence of Hankel product and convolution, the partially separable model is solved by a convolutional network, revealing that the convolutional network module in the Adaptive Subspace Net is characterizing the null space, making the network capture dynamic frames more accurate.
- the beneficial effects of the present invention are at least:
- the network model is based on the PS Model, which uses matrix multiplication to express low rank. It can capture the relationship between frames more accurately, achieve higher acceleration multiples, and have better reconstruction effects than existing technologies.
- the convolutional network is used to solve sparse items, and the convolutional network works as a black box.
- the present invention uses the equivalent relationship between the Hankel matrix and the convolution to express the low rank with the network, revealing that the convolutional network module in the Adaptive Subspace Net is characterizing the null space, thereby enhancing the interpretability of the network.
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Abstract
一种基于部分可分函数自适应动态磁共振快速成像方法及装置，基于动态磁共振的低秩先验和稀疏先验构建动态磁共振重建模型(S101)，动态磁共振重建模型首先将输入的低秩通过图像域的湮没关系转化为对图像滤波器零空间的刻画，动态磁共振重建模型再利用Hankel矩阵乘积与卷积的等价关系，将低秩用卷积网络表达，并将其迭代求解展开到卷积网络中(S103)。神经网络可以到达更高的磁共振重建加速倍数，达到更好的重建效果，对动态帧的捕捉更为准确。
Description
本发明涉及医学动态磁共振成像领域，具体而言，涉及一种基于部分可分函数自适应动态磁共振快速成像方法及装置。
动态磁共振成像可以同时揭示扫描对象的空间维度信息和时间维度信息，但是由于动态磁共振成像需要的数据量大大增加，其扫描时间极长，是制约动态磁共振成像应用的主要原因。如何在保证磁共振图像质量的前提下，加快动态磁共振成像的速度是当下磁共振研究的一个重要问题。
现有的动态磁共振成像方法包括传统的并行成像方法和基于深度学习的方法，其中：
现有传统动态磁共振重建缺点如下：
1.目前基于低秩模型的传统动态磁共振重建方法，有很多参数需要调整，不同的参数选择对重建结果影响极大，传统算法迭代计算耗时较长，对于参数调整十分不便。
2.现有传统动态磁共振重建算法大多是基于奇异值分解求解的迭代计算过程，耗时很长，限制了磁共振的进一步发展。
现有基于深度学习的磁共振重建方法的缺点如下：
1.现有基于深度学习的磁共振重建方法通常需要大量的训练数据和很长的训练时间。
2.现有基于深度学习的磁共振重建方法，其卷积层模块是黑箱工作，很难解释其合理性。
3.现有基于模型驱动的深度学习重建方法，多是将低秩表达为核范数的形式，而网络训练得到的奇异值阈值只是一个定值，并不能很好刻画磁共振图像 低秩的特性。
4.现有的深度学习模型对心脏收缩期重建的效果均不佳，存在将上一帧图像带入下一帧的情况。
发明内容
本发明实施例提供了一种基于部分可分函数自适应动态磁共振快速成像方法及装置，以至少解决现有动态磁共振成像效果差的技术问题。
根据本发明的一实施例，提供了一种基于部分可分函数自适应动态磁共振快速成像方法，包括以下步骤：
基于动态磁共振的低秩先验和稀疏先验构建动态磁共振重建模型；
动态磁共振重建模型首先将输入的低秩通过图像域的湮没关系转化为对图像滤波器零空间的刻画；
动态磁共振重建模型再利用Hankel矩阵乘积与卷积的等价关系，将低秩用卷积网络表达，并将其迭代求解展开到卷积网络中。
进一步地，基于动态磁共振的低秩先验和稀疏先验构建动态磁共振重建模型包括以下步骤：
基于K空间欠采样的磁共振重建模型离散表示为：
MFγ＝Y
其中M是欠采样算子，F是傅里叶算子，Y是欠采样的原始K空间数据，γ是待求解的图像，γ，Y∈C
n×m×T，其中图像每帧的大小为n×m，T为时间帧数；
通过低秩先验信息，将上述反问题转化为一个无约束的优化问题：
其中R(γ)为先验正则项。
进一步地，将输入的低秩通过图像域的湮没关系转化为对图像滤波器零空间的刻画包括：
由部分可分模型来求解磁共振重建问题，见下式：
其中φ
l(t)是一组时间基，c
l(k)是对应的空间基，称γ(r，t)为L阶可分离，γ(r，t)是待重建的动态磁共振图像，γ∈C
n×m×T；
将其转化为以下指数形式：
r和t是独立的，由prony’s results可知其满足湮没关系，即存在h[r，t]，使得：
将磁共振图像的低秩用矩阵相乘来表达，通过湮没关系用h[r，t]来刻画低秩。
进一步地，利用Hankel矩阵乘积与卷积的等价关系，将低秩用卷积网络表达，并将其迭代求解展开到卷积网络中包括：
动态磁共振重建的部分可分模型γ(k，t)满足湮没关系，即存在h[r，t]，使得：
上述卷积关系被等价表达为Hankel矩阵的乘积，H(γ)h＝0；
磁共振图像还满足图像域的湮没关系，即：
其中γ表示图像，r表示图像空间坐标，
表示图像的梯度，由卷积定理可得：
上式可等价表达为Hankel矩阵的乘积，
最终可建模为：
引入辅助变量
z＝γ，则上式可等价表述为：
依赖半二次方分裂法(HQS)，构造正则化模型的迭代求解算法，根据不同变量求解不同子问题；
关于
子问题：
关于Z子问题：
关于γ子问题：
通过共轭梯度下降法求解，上述三个子问题解得：
由Hankel乘积与卷积的等价关系，将动态磁共振重建模型实现到卷积网络中。
进一步地，用于求解Z子问题的卷积模块为5层3D卷积，卷积核大小为1x1x3；用于求解
子问题的卷积模块为5层2D卷积，卷积核大小为3x3，卷积通道数均为32层。
进一步地，欠采的K空间数据作为输入，全采的图像作为标签，将输入数 据分为实部和虚部两个通道输入网络，网络的损失函数为输出图像与标签的最小均方误差，batch size设置为1，学习率初始值为0.001，并将其设置为指数衰减，网络采用的优化器为Adam。
进一步地，神经网络为采用有监督的神经网络。
本发明实施例中的基于部分可分函数自适应动态磁共振快速成像方法及装置，首先基于动态磁共振的低秩先验和稀疏先验构建动态磁共振重建模型，首先将输入的低秩通过图像域的湮没关系转化为对图像滤波器零空间的刻画；再利用Hankel矩阵乘积与卷积的等价关系，将低秩用卷积网络表达，并将其迭代求解展开到卷积网络中。本发明提出的神经网络可以到达更高的磁共振重建加速倍数，达到更好的重建效果，对动态帧的捕捉更为准确。
此处所说明的附图用来提供对本发明的进一步理解，构成本申请的一部分，本发明的示意性实施例及其说明用于解释本发明，并不构成对本发明的不当限定。在附图中：
图1为本发明基于部分可分函数自适应动态磁共振快速成像方法的流程图；
图2为本发明中卷积神经网络的方案框图；
图3为本发明中Adaptive Subspace Net的网络结构图；
图4为本发明中8倍欠采结果图；
图5为本发明中12倍欠采结果图；
图6为本发明基于部分可分函数自适应动态磁共振快速成像装置的模块图。
为了使本申请的目的、技术方案及优点更加清楚明白，以下结合附图及实施例，对本申请进行进一步详细说明。应当理解，此处所描述的具体实施例仅 用以解释本申请，并不用于限定本申请。
实施例1
根据本发明一实施例，提供了一种基于部分可分函数自适应动态磁共振快速成像方法，参见图1，包括以下步骤：
S101:基于动态磁共振的低秩先验和稀疏先验构建动态磁共振重建模型；
S102:动态磁共振重建模型将输入的低秩通过图像域的湮没关系转化为对图像滤波器零空间的刻画。
S103:动态磁共振重建模型利用Hankel矩阵乘积与卷积的等价关系，将低秩用卷积网络表示，并将迭代求解的计算过程展开到神经网络中。
本发明实施例中的基于部分可分函数自适应动态磁共振快速成像方法，基于动态磁共振的低秩先验和稀疏先验构建动态磁共振重建模型，将输入的低秩通过图像域的湮没关系转化为对图像滤波器零空间的刻画；再利用Hankel矩阵乘积与卷积的等价关系，将低秩用卷积网络表达，并将其迭代求解展开到卷积网络中。本发明揭示了在Adaptive Subspace Net中的卷积网络模块是在刻画零空间，使得网络对动态帧的捕捉更为准确。
其中，基于动态磁共振的低秩先验和稀疏先验构建动态磁共振重建模型包括以下步骤：
基于K空间欠采样的磁共振重建模型离散表示为：
MFγ＝Y
其中M是欠采样算子，F是傅里叶算子，Y是欠采样的原始K空间数据，γ是待求解的图像，γ，Y∈C
n×m×T，其中图像每帧的大小为n×m，T为时间帧数；
通过低秩先验信息，将上述反问题转化为一个无约束的优化问题：
其中R(γ)为先验正则项。
其中，将输入的低秩通过图像域的湮没关系转化为对图像滤波器零空间的 刻画包括：
由部分可分模型来求解磁共振重建问题，见下式：
其中φ
l(t)是一组时间基，c
l(k)是对应的空间基，称γ(r，t)为L阶可分离，γ(r，t)是待重建的动态磁共振图像，γ∈C
n×m×T；
将其转化为以下指数形式：
r和t是独立的，由prony’s results可知其满足湮没关系，即存在h[r，t]，使得：
将磁共振图像的低秩用矩阵相乘来表达，通过湮没关系用h[r，t]来刻画低秩。
其中，利用Hankel矩阵乘积与卷积的等价关系，将低秩用卷积网络表达，并将其迭代求解展开到卷积网络中包括：
动态磁共振重建的部分可分模型γ(k，t)满足湮没关系，即存在h[r，t]，使得：
上述卷积关系被等价表达为Hankel矩阵的乘积，H(γ)h＝0；
磁共振图像还满足图像域的湮没关系，即：
其中γ表示图像，r表示图像空间坐标，
表示图像的梯度，由卷积定理可得：
上式可等价表达为Hankel矩阵的乘积，
最终可建模为：
引入辅助变量
z＝γ，则上式可等价表述为：
依赖半二次方分裂法(HQS)，构造正则化模型的迭代求解算法，根据不同变量求解不同子问题；
关于
子问题：
关于Z子问题：
关于γ子问题：
通过共轭梯度下降法求解，上述三个子问题解得：
由Hankel乘积与卷积的等价关系，将动态磁共振重建模型实现到卷积网络中。
其中，用于求解Z子问题的卷积模块为5层3D卷积，卷积核大小为1x1x3；用于求解
子问题的卷积模块为5层2D卷积，卷积核大小为3x3，卷积通道数 均为32层。
其中，欠采的K空间数据作为输入，全采的图像作为标签，将输入数据分为实部和虚部两个通道输入网络，网络的损失函数为输出图像与标签的最小均方误差，batch size设置为1，学习率初始值为0.001，并将其设置为指数衰减，网络采用的优化器为Adam。
其中，神经网络为采用有监督的神经网络。
下面以具体实施例，并参见图2-5，对本发明的基于部分可分函数自适应动态磁共振快速成像方法进行详细说明：
本发明涉及一种基于部分可分函数自适应动态磁共振快速成像方法。基于深度学习的动态磁共振重建方法可细分为数据驱动与模型驱动两大类，本发明属于模型驱动的范畴，基于模型驱动的方式通过特定任务的性质来构建模型，比如利用图像低秩稀疏等性质来构造低秩稀疏模型，再将传统的迭代算法展开到神经网络中求解。
本发明提出的基于部分可分函数低秩网络化的模型存在以下优势：
1.本发明将部分可分模型(PS model)展开到神经网络中，模型需要调整的超参数大大减少。
2.本发明利用Hankel矩阵与卷积的等价关系，将低秩用网络表达，揭示了Adaptive Subspace Net中的卷积网络是在刻画零空间，大大增强了网络的可解释性。
3.本发明将磁共振图像的低秩用矩阵相乘的形式表达，将低秩转化为对图像滤波器零空间的刻画，网络学习的是整个映射的过程，间接且自适应学习了磁共振图像的低秩。
4.本发明提出的模型对于图像帧与帧之间的捕捉更加准确，动态磁共振重建的效果更好。
本发明提出了一种基于部分可分模型的卷积神经网络，用于磁共振心脏电影成像。其方案框图如图2所示。本发明具体包括：
第一部分：磁共振重建模型
1)动态磁共振重建模型
基于K空间欠采样的磁共振重建模型可以离散表示为：
MFγ＝Y
其中M是欠采样算子，F是傅里叶算子，Y是欠采样的原始K空间数据，γ是待求解的图像，γ，Y∈C
n×m×T，其中图像每帧的大小为n×m，T为时间帧数；在本本发明中，仅论述单通道重建的模型。
通过低秩等先验信息，可将上述反问题转化为一个无约束的优化问题：
其中R(γ)为先验正则项。
2)子空间模型——基于部分可分模型
2007年，Liang提出可由部分可分模型来求解磁共振重建问题，见下式：
其中φ
l(t)是一组时间基，c
l(k)是对应的空间基，称γ(r，t)为L阶可分离，在这里，γ(r，t)就是待重建的动态磁共振图像，γ∈C
n×m×T。
本发明可将其转化为以下指数形式：
在这里r和t是独立的，由prony’s results可知其满足湮没关系，即存在h[r，t]，使得：
因此在这里本发明可以将磁共振图像的低秩用矩阵相乘来表达，通过湮没关系用h[r，t]来刻画低秩。
第二部分：本发明提出的Adaptive Subspace Net
由上可知动态磁共振重建的部分可分模型γ(k，t)满足湮没关系，即存在h[r，t]，使得：
上述卷积关系可被等价表达为Hankel矩阵的乘积，H(γ)h＝0。
磁共振图像还满足图像域的湮没关系，即：
其中γ表示图像，r表示图像空间坐标，
表示图像的梯度，由卷积定理可得：
同理可得，上式可等价表达为Hankel矩阵的乘积，
因此，最终可建模为：
这里
代表稀疏先验正则项，
代表部分可分关系的正则项，λ
1和λ
2是对应的正则化参数。
引入辅助变量
z＝γ，则上式可等价表述为：
依赖半二次方分裂法(HQS)，构造本发明所提正则化模型的迭代求解算法，根据不同变量求解不同子问题。
关于
子问题：
关于Z子问题：
关于γ子问题：
上述三者均属于二次规划问题，可通过共轭梯度下降法求解，上述三个子问题解得：
由Hankel乘积与卷积的等价关系，可将上述模型实现到卷积网络中，Adaptive Subspace Net算法伪代码见下所示：
Adaptive Subspace Net网络结构图如图3所示。图3中，用于求解Z子问题的卷积模块为5层3D卷积，卷积核大小为1x1x3；求解
子问题的卷积模块为5层2D卷积，卷积核大小为3x3，卷积通道数均为32层。
本发明采用有监督的神经网络，欠采的K空间数据作为输入，全采的图像作为标签，将输入数据分为实部和虚部两个通道输入网络，网络的损失函数为输出图像与标签的最小均方误差，batch size设置为1，学习率初始值为0.001，并将其设置为指数衰减，网络采用的优化器为Adam。
通过西门子心脏电影成像数据集回顾性采样仿真，验证了本发明的有效性。
具体地，在欠采样倍数为8倍和12倍的情况下，欠采样模式为cartesian，利用本发明所提方案重建心脏电影成像数据，相比于其他方案获得更好的重建效果。
8倍欠采结果如图4所示，在同一数据集上分别与现有的几种方法比较(DCCNN，CRNN，kt-SLR和SLR-Net)。
图4中第一行展示了全采的真实图像与采用本发明技术方案的重建结果图，第二行展示了由第一行图像虚线框框起的心脏区域放大视图，第三行表示重建图像与真实图像的误差图(显示范围[0，0.09])，第四行表示沿y轴时间维度第100个像素的时间切片图，第五行表示时间切片图与真实图像的误差。从第三行的误差图可以看到本发明提出的Adaptive Subspace Net比现有的方法重建效果都要好。
12倍欠采结果如图5所示，误差图表明Adaptive Subspace Net在高加速倍数下的重建优势比现有方法更大。
关于替代方案的选择：
1.本发明适用于磁共振动态成像的其他数据集。
2.求解算法的设计，除了本发明使用的半二次方分裂法，其他的使用梯度下降算法同样有效。
3.网络的设计，对本发明使用的卷积通道数和卷积层数稍加更改同样有效。
实施例2
根据本发明的一实施例，提供了一种基于部分可分函数自适应动态磁共振快速成像装置，参加图6，包括：
模型构建单元201，用于基于动态磁共振的低秩先验和稀疏先验构建动态磁共振重建模型；
转化单元202，用于动态磁共振重建模型通过湮没关系将低秩转化为对图像滤波器零空间的刻画；
低秩表达单元203，用于利用Hankel矩阵乘积与卷积的等价关系，将低秩用卷积网络表达，并将其迭代求解展开到卷积网络中。
本发明实施例中的基于部分可分函数自适应动态磁共振快速成像装置，首先基于动态磁共振的低秩先验和稀疏先验构建动态磁共振重建模型，将磁共振图像的低秩用矩阵相乘来表达，通过湮没关系将低秩转化为对图像滤波器零空间的刻画；再利用Hankel矩阵与卷积的等价关系，将低秩用卷积网络表达，将动态磁共振重建模型实现到卷积网络中。本发明提出基于部分可分函数自适应低秩的模型，并将其用于动态磁共振成像，并基于Hankel乘积与卷积的等价性，将部分可分模型用卷积网络求解，揭示了在Adaptive Subspace Net中的卷积网络模块是在刻画零空间，使得网络对动态帧的捕捉更为准确。
下面以具体实施例，并参见图2-5，对本发明的基于部分可分函数自适应动态磁共振快速成像装置进行详细说明：
本发明涉及一种基于部分可分函数自适应动态磁共振快速成像装置。基于深度学习的动态磁共振重建方法可细分为数据驱动与模型驱动两大类，本发明属于模型驱动的范畴，基于模型驱动的方式通过特定任务的性质来构建模型，比如利用图像低秩稀疏等性质来构造低秩稀疏模型，再将传统的迭代算法展开到神经网络中求解。
本发明提出的基于部分可分函数低秩网络化的模型存在以下优势：
1.本发明将部分可分模型(PS model)展开到神经网络中，模型需要调整的超参数大大减少。
2.本发明利用Hankel矩阵乘积与卷积的等价关系，将低秩用网络表达，揭示了Adaptive Subspace Net中的卷积网络是在刻画零空间，大大增强了网络的可解释性。
3.本发明将磁共振图像的低秩用矩阵相乘的形式表达，将低秩转化为对图像滤波器零空间的刻画，网络学习的是整个映射的过程，间接且自适应学习了磁共振图像的低秩。
4.本发明提出的模型对于图像帧与帧之间的捕捉更加准确，动态磁共振重 建的效果更好。
本发明提出了一种基于部分可分模型的卷积神经网络，用于磁共振心脏电影成像。其方案框图如图2所示。本发明具体包括：
第一部分：磁共振重建模型
1)动态磁共振重建模型
基于K空间欠采样的磁共振重建模型可以离散表示为：
MFγ＝Y
其中M是欠采样算子，F是傅里叶算子，Y是欠采样的原始K空间数据，γ是待求解的图像，γ，Y∈C
n×m×T，其中图像每帧的大小为n×m，T为时间帧数；在本本发明中，仅论述单通道重建的模型。
通过低秩等先验信息，可将上述反问题转化为一个无约束的优化问题：
其中R(γ)为先验正则项。
2)子空间模型——基于部分可分模型
2007年，Liang提出可由部分可分模型来求解磁共振重建问题，见下式：
其中φ
l(t)是一组时间基，c
l(k)是对应的空间基，称γ(r，t)为L阶可分离，在这里，γ(r，t)就是待重建的动态磁共振图像，γ∈C
n×m×T。
本发明可将其转化为以下指数形式：
在这里r和t是独立的，由prony’s results可知其满足湮没关系，即存在h[r，t]，使得：
因此在这里本发明可以将磁共振图像的低秩用矩阵相乘来表达，通过湮没关系用h[r，t]来刻画低秩。
第二部分：本发明提出的Adaptive Subspace Net
由上可知动态磁共振重建的部分可分模型γ(k，t)满足湮没关系，即存在h[r，t]，使得：
上述卷积关系可被等价表达为Hankel矩阵的乘积，H(γ)h＝0。
磁共振图像还满足图像域的湮没关系，即：
其中γ表示图像，r表示图像空间坐标，
表示图像的梯度，由卷积定理可得：
同理可得，上式可等价表达为Hankel矩阵的乘积，
因此，最终可建模为：
这里
代表稀疏先验正则项，
代表部分可分关系的正则项，λ
1和λ
2是对应的正则化参数。
引入辅助变量
z＝γ，则上式可等价表述为：
依赖半二次方分裂法(HQS)，构造本发明所提正则化模型的迭代求解算法，根据不同变量求解不同子问题。
关于
子问题：
关于Z子问题：
关于γ子问题：
上述三者均属于二次规划问题，可通过共轭梯度下降法求解，上述三个子问题解得：
由Hankel乘积与卷积的等价关系，可将上述模型实现到卷积网络中，Adaptive Subspace Net算法伪代码见下所示：
Adaptive Subspace Net网络结构图如图3所示。图3中，用于求解Z子问题的卷积模块为5层3D卷积，卷积核大小为1x1x3；求解
子问题的卷积模块为5层2D卷积，卷积核大小为3x3，卷积通道数均为32层。
本发明采用有监督的神经网络，欠采的K空间数据作为输入，全采的图像作为标签，将输入数据分为实部和虚部两个通道输入网络，网络的损失函数为输出图像与标签的最小均方误差，batch size设置为1，学习率初始值为0.001， 并将其设置为指数衰减，网络采用的优化器为Adam。
通过西门子心脏电影成像数据集回顾性采样仿真，验证了本发明的有效性。
具体地，在欠采样倍数为8倍和12倍的情况下，欠采样模式为cartesian，利用本发明所提方案重建心脏电影成像数据，相比于其他方案获得更好的重建效果。
8倍欠采结果如图4所示，在同一数据集上分别与现有的几种方法比较(DCCNN，CRNN，kt-SLR和SLR-Net)。
图4中第一行展示了全采的真实图像与采用本发明技术方案的重建结果图，第二行展示了由第一行图像虚线框框起的心脏区域放大视图，第三行表示重建图像与真实图像的误差图(显示范围[0，0.09])，第四行表示沿y轴时间维度第100个像素的时间切片图，第五行表示时间切片图与真实图像的误差。从第三行的误差图可以看到本发明提出的Adaptive Subspace Net比现有的方法重建效果都要好。
12倍欠采结果如图5所示，误差图表明Adaptive Subspace Net在高加速倍数下的重建优势比现有方法更大。
关于替代方案的选择：
1.本发明适用于磁共振动态成像的其他数据集。
2.求解算法的设计，除了本发明使用的半二次方分裂法，其他的使用梯度下降算法同样有效。
3.网络的设计，对本发明使用的卷积通道数和卷积层数稍加更改同样有效。
实施例3
一种存储介质，存储介质存储有能够实现上述任意一项基于部分可分函数自适应动态磁共振快速成像方法的程序文件。
实施例4
一种处理器，处理器用于运行程序，其中，程序运行时执行上述任意一项 的基于部分可分函数自适应动态磁共振快速成像方法。
上述本发明实施例序号仅仅为了描述，不代表实施例的优劣。
在本发明的上述实施例中，对各个实施例的描述都各有侧重，某个实施例中没有详述的部分，可以参见其他实施例的相关描述。
本发明提出基于部分可分函数自适应低秩的模型，并将其用于动态磁共振成像；基于Hankel乘积与卷积的等价性，将部分可分模型用卷积网络求解，揭示了在Adaptive Subspace Net中的卷积网络模块是在刻画零空间，使得网络对动态帧的捕捉更为准确。
现有技术相比，本发明的有益效果至少在于：
1.网络模型基于PS Model，利用矩阵相乘的形式来表达低秩，对帧之间的关系捕捉更加准确，可以做到更高的加速倍数，重建效果要好于现有技术。
2.现有技术的模型中，卷积网络用来求解稀疏项，卷积网络是黑箱工作；本发明利用Hankel矩阵与卷积的等价关系，将低秩用网络表达，揭示了在Adaptive Subspace Net中的卷积网络模块是在刻画零空间，增强了网络的可解释性。
3.并不需要用奇异值分解算法(svd)分解求解低秩，svd算法的时间复杂度为O(n^3)级别，Adaptive Subspace Net的时间复杂度更低。
对所公开的实施例的上述说明，使本领域专业技术人员能够实现或使用本发明。对这些实施例的多种修改对本领域的专业技术人员来说将是显而易见的，本发明中所定义的一般原理可以在不脱离本发明的精神或范围的情况下，在其它实施例中实现。因此，本发明将不会被限制于本发明所示的这些实施例，而是要符合与本发明所公开的原理和新颖特点相一致的最宽的范围。
Claims (10)
- 一种基于部分可分函数自适应动态磁共振快速成像方法，其特征在于，包括以下步骤：基于动态磁共振的低秩先验和稀疏先验构建动态磁共振重建模型；所述动态磁共振重建模型首先将输入的低秩通过图像域的湮没关系转化为对图像滤波器零空间的刻画；所述动态磁共振重建模型再利用Hankel矩阵乘积与卷积的等价关系，将低秩用卷积网络表达，并将其迭代求解展开到卷积网络中。
- 根据权利要求1所述的基于部分可分函数自适应动态磁共振快速成像方法，其特征在于，所述基于动态磁共振的低秩先验和稀疏先验构建动态磁共振重建模型包括以下步骤：基于K空间欠采样的磁共振重建模型离散表示为：MFγ＝Y其中M是欠采样算子，F是傅里叶算子，Y是欠采样的原始K空间数据，γ是待求解的图像，γ，Y∈C n×m×T，其中图像每帧的大小为n×m，T为时间帧数；通过低秩先验信息，将上述反问题转化为一个无约束的优化问题：
其中R(γ)为先验正则项。 - 根据权利要求2所述的基于部分可分函数自适应动态磁共振快速成像方法，其特征在于，将输入的低秩通过图像域的湮没关系转化为对图像滤波器零空间的刻画包括：由部分可分模型来求解磁共振重建问题，见下式：
其中φ l(t)是一组时间基，c l(k)是对应的空间基，称γ(r，t)为L阶可分离， γ(r，t)是待重建的动态磁共振图像，γ∈C n×m×T；将其转化为以下指数形式：
r和t是独立的，由prony’s results可知其满足湮没关系，即存在h[r，t]，使得：
将磁共振图像的低秩用矩阵相乘来表达，通过湮没关系用h[r，t]来刻画低秩。 - 根据权利要求3所述的基于部分可分函数自适应动态磁共振快速成像方法，其特征在于，所述利用Hankel矩阵乘积与卷积的等价关系，将低秩用卷积网络表达，并将其迭代求解展开到卷积网络中包括：动态磁共振重建的部分可分模型γ(k，t)满足湮没关系，即存在h[r，t]，使得：
上述卷积关系被等价表达为Hankel矩阵的乘积，H(γ)h＝0；磁共振图像还满足图像域的湮没关系，即：
其中γ表示图像，r表示图像空间坐标，表示图像的梯度，由卷积定理可得：
上式可等价表达为Hankel矩阵的乘积，
最终可建模为：
代表稀疏先验正则项，
代表部分可分关系的正则项，λ 1和λ 2是对应的正则化参数；
引入辅助变量z＝γ，则上式可等价表述为：
依赖半二次方分裂法(HQS)，构造正则化模型的迭代求解算法，根据不同变量求解不同子问题；关于子问题：
关于Z子问题：
关于γ子问题：
通过共轭梯度下降法求解，上述三个子问题解得：
由Hankel乘积与卷积的等价关系，将动态磁共振重建模型实现到卷积网络中。 - 根据权利要求4所述的基于部分可分函数自适应动态磁共振快速成像方法，其特征在于，用于求解Z子问题的卷积模块为5层3D卷积，卷积核大小为1x1x3；用于求解子问题的卷积模块为5层2D卷积，卷积核大小为3x3，卷积通道数均为32层。
- 根据权利要求5所述的基于部分可分函数自适应动态磁共振快速成像方法，其特征在于，欠采的K空间数据作为输入，全采的图像作为标签，将输入数据分为实部和虚部两个通道输入网络，网络的损失函数为输出图像与标签的 最小均方误差，batch size设置为1，学习率初始值为0.001，并将其设置为指数衰减，网络采用的优化器为Adam。
- 根据权利要求6所述的基于部分可分函数自适应动态磁共振快速成像方法，其特征在于，神经网络为采用有监督的神经网络。
- 一种基于部分可分函数自适应动态磁共振快速成像装置，其特征在于，包括：模型构建单元，用于基于动态磁共振的低秩先验和稀疏先验构建动态磁共振重建模型；转化单元，用于动态磁共振重建模型将输入的低秩通过图像域的湮没关系转化为对图像滤波器零空间的刻画；低秩表达单元，用于动态磁共振重建模型利用Hankel矩阵乘积与卷积的等价关系，将低秩用卷积网络表达，并将其迭代求解展开到卷积网络中。
- 一种存储介质，其特征在于，所述存储介质存储有能够实现权利要求1至7中任意一项所述基于部分可分函数自适应动态磁共振快速成像方法的程序文件。
- 一种处理器，其特征在于，所述处理器用于运行程序，其中，所述程序运行时执行权利要求1至7中任意一项所述的基于部分可分函数自适应动态磁共振快速成像方法。
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