METHOD FOR RECONSTRUCTING MAGNETIC RESONANCE SPECTRUM BASED ON DEEP LEARNING

Abstract

A new method for reconstructing a full spectrum from under-sampled magnetic resonance spectrum data by using a deep learning network. First, the exponential function is used to generate a time-domain signal of the magnetic resonance spectrum, and a zero-filling time-domain signal is obtained after the under-sampled operation is completed in the time domain. The zero-filling time-domain signal and the full spectrum corresponding to the full sampling are combined to form a training data set. Then, a data verification convolutional neural network model is established for magnetic resonance spectrum reconstruction, where the training data set is used to train neural network parameters to form a trained neural network. Finally, the under-sampled magnetic resonance time-domain signal is input to the trained data verification convolutional neural network, and the full magnetic resonance spectrum is reconstructed.

Claims

1. A method for reconstructing magnetic resonance spectrum based on deep learning, comprising: 1) generating a time-domain signal of the magnetic resonance spectrum using an exponential function; 2) constructing a training set comprising an under-sampled time-domain signal of the magnetic resonance spectrum and a corresponding full-sampling spectrum; 3) designing a convolutional neural network of a data verification convolutional neural network structure; 4) designing a bottleneck layer of the data verification convolutional neural network structure 5) designing a data verification layer of the data verification convolutional neural network structure; 6) designing data to verify a feedback function of the data verification convolutional neural network structure; 7) establishing the data verification convolutional neural network structure to function as a spectrum reconstruction model; 8) training parameters of the convolutional neural network; 9) reconstructing an under-sampled time-domain signal {tilde over (T)}.sub.u of a target magnetic resonance spectrum; and 10) using a fitting ability of the data verification convolutional neural network and a data verification ability of the data verification layer to complete a reconstruction from the under-sampled time-domain signal {tilde over (T)}.sub.u of the target magnetic resonance spectrum while performing an under-sampled operation in a time domain.

2. The method for reconstructing the magnetic resonance spectrum based on the deep learning according to claim 1, wherein in step 1): a data T.sup.n,m, in an n-th row and an-th column of a full-sampling magnetic resonance signal T∈.sup.N×M in time domain is generated according to an exponential function as follows: $\begin{array}{cc}{T}^{n,m}={.Math.}_{r=1}^R{a}_r{e}^{\left(i2\pi n\Delta t_1{f}_{1,r}-\frac{n\Delta t_1}{{\tau }_{1,r}}\right)}{e}^{\left(i2\pi m\Delta t_2{f}_{2,r}-\frac{m\Delta t_2}{{\tau }_{2,r}}\right)},& \left(1\right)\end{array}$ wherein H represents a set of complex numbers, N and M represent a number of rows and columns of a time signal, R represents a number of spectral peaks, a.sub.r represents a size of an amplitude, Δt.sub.1 and Δt.sub.2 represent time increases, f.sub.1,r and f.sub.2,r represent normalized frequencies, and τ.sub.1,r and τ.sub.2,r represent decay factors; and the expression (1) is also used to a full sampling signal of a one-dimensional free induction decay, when n=1, m>1, or m=1, n>1.

3. The method for reconstructing the magnetic resonance spectrum based on the deep learning according to claim 2, wherein in step 2), the constructing the training set of the under-sampled time-domain signal of the magnetic resonance spectrum and the full-sampling spectrum comprises: representing an under-sampled operation in a time domain by U, sampling a data point represented by white color, not sampling a data point represented by black color, wherein: Ω represents an index subset of an under-sampled template M, when an index (p, q) of a certain signal point appears in the index subset Ω, (p,q)∈Ω, and when the index (p, q) of the certain signal point does not appear in the index subset Ω, (p,q).Math.Ω, filling zeros to signal in all non-sampled positions to obtain a zero-filling time-domain signal T.sub.u according to the under-sampled template M, using Fourier transform to obtain a spectrum signal S.sub.u with aliasing from the zero-filling time-domain signal T.sub.u, using Fourier transform to obtain a full-sampling spectrum S from the hill-sampling signal T, and separately saving a real part and an imaginary part of the full-sampling spectrum S, that is, S ∈.sup.2×256×116, wherein represents a real number and both T.sub.u and S are combined to define the training set .

4. The method for reconstructing the magnetic resonance spectrum based on the deep learning according to claim 1, wherein in step 3): a module of the convolutional neural network comprises L convolutional layers, each convolutional layer of the L convolutional layers comprises I filters, the convolutional layers are densely connected, an input of each convolutional layer is an union of outputs of all previous convolutional layers in the module, in all convolutional layers, sizes of the convolution kernels are k, an input signal S.sub.l of an l.sup.th layer (1≤l≤L) passes through the convolutional neural network to obtain an output signal S.sub.cnn,l due to the module of the convolutional neural network, where
S.sub.cnn,l=f(S.sub.l|θ)  (2), wherein θ represents a training parameter of the convolutional neural network and f(S.sub.l|θ) represents a non-linear mapping from S.sub.l to S.sub.cnn,l of the training.

5. The method for reconstructing the magnetic resonance spectrum based on the deep learning according to claim 4, wherein in step 4), the designing the bottleneck layer of the data verification convolutional neural network structure comprises: changing a number of feature maps in the data verification convolutional neural network structure using the bottleneck layer; and disposing the bottleneck layer before and after the module of the convolutional neural network, wherein: the signal passes through the bottleneck layer of K.sub.i filters to increase the number of feature maps before entering the module of the convolutional neural network, and an output signal of the module of the convolutional neural network also passes through the bottleneck layer of k, filters to reduce the number of feature maps.

6. The method for reconstructing the magnetic resonance spectrum based on the deep learning according to claim 5, wherein in step 5), the designing the data verification layer of the data verification convolutional neural network structure comprises: verifying the data by the data verification layer in the data verification convolutional neural network structure, inputting an output signal S.sub.cnn,l from an i-th convolutional neural network, converting the output signal S.sub.cnn,l back to the time domain using inverse Fourier transform F.sup.H to obtain signal T.sub.l, wherein a formula is as follows:
T.sub.l=F.sup.HS.sub.cnn,l  (3), an expression of the data verification layer is as follows: $\begin{array}{cc}{\hat{T}}_l^{n,m}=\{\begin{array}{c}{T}_l^{n,m},\mathrm{if}\left(n,m\right).Math.\Omega \\ \frac{T_l^{n,m}+\lambda T_u^{n,m}}{1+\lambda },\mathrm{if}\left(n,m\right)\in \Omega \end{array}.& \left(4\right)\end{array}$ finally outputting reconstructed spectrum Ŝ.sub.l=F{circumflex over (T)}.sub.l, wherein: at last time, a spectrum ŜL of an L-th layer (L>1) defines an output Ŝ of an entire deep learning network.

7. The method for reconstructing the magnetic resonance spectrum based on the deep learning according to claim 2, wherein in step 6): the feedback function enables an output of each module of a combination of the convolutional neural network and the data verification layer to be close to a full-sampling spectrum signal S−FT in the data verification convolutional neural network structure and enables an input of a next module to be more interpretable, and the designing data to verify the feedback function of the data verification convolutional neural network structure comprises: comparing an output of each data verification layer with the full-sampling spectrum signal S=FT to feed back to each module to update parameters, wherein: T represents a full-sampling time-domain signal in formula (1), and F represents Fourier transform.

8. The method for reconstructing the magnetic resonance spectrum based on the deep learning according to claim 6, wherein in step 7): the data verification convolutional neural network structure cascades with multiple modules of combinations of the convolutional neural network and the data verification layer, establishing the data verification convolutional neural network structure to function as the spectrum reconstruction model comprises: inputting the under-sampled magnetic resonance time-domain signal T.sub.u, and outputting a reconstructed magnetic resonance spectrum signal Ŝ, thereby constituting an end-to-end deep neural network structure, a loss function of the data verification the convolutional neural network structure is as follows:
(θ)=∥S−Ŝ∥.sub.F.sup.2  (5), wherein represents the training set, ∥⋅∥.sub.F represents F-norm (Frobenius norm) of a matrix, Ŝ=f(T.sub.u|θ, λ), θ represents a training parameter of the convolutional neural network, λ represents a data verification parameter of the data verification layer, and both parameters θ and λ need to be trained.

9. The method for reconstructing the magnetic; resonance spectrum based on the deep learning according to claim 8, wherein in step 8), the training parameters of the convolutional neural network comprises: training parameters of the spectrum reconstruction model in step 7) using Adam algorithm to obtain value {circumflex over (θ)} and {circumflex over (λ)} of the spectrum reconstruction model.

10. The method for reconstructing the magnetic resonance spectrum based on the deep learning according to claim 9, wherein in step 9), the reconstructing from under-sampled time-domain signal {tilde over (T)}.sub.u of the target magnetic resonance spectrum comprises: inputting the under-sampled time-domain signal {tilde over (T)}.sub.u to the spectrum reconstruction model, and reconstructing the reconstructed magnetic resonance spectrum signal {tilde over (S)} after a forward propagation of the spectrum reconstruction model, where:
{tilde over (S)}=f({tilde over (T)}.sub.u|{tilde over (θ)}, {tilde over (λ)})  (6).

11. The method for reconstructing the magnetic resonance spectrum based on the deep learning according to claim 7, wherein in step 7): the data verification convolutional neural network structure cascades with multiple modules of combinations of the convolutional neural network and the data verification layer, establishing the data verification convolutional neural network structure to function as the spectrum reconstruction model comprises: inputting the under-sampled magnetic resonance time-domain signal T.sub.u, and outputting a reconstructed magnetic resonance spectrum signal Ŝ, thereby constituting an end-to-end deep neural network structure, a loss function of the data verification the convolutional neural network structure is as follows:
(θ)=∥S−Ŝ∥.sub.F.sup.2  (5), wherein represents the training set, ∥⋅∥.sub.F represents F-norm (Frobenius norm) of a matrix, Ŝ=f(T.sub.u|θ, λ), θ represents a training parameter of the convolutional neural network, λ represents a data verification parameter of the data verification layer, and both parameters θ and λ need to be trained.

12. The method for reconstructing the magnetic resonance spectrum based on the deep learning according to claim 11, wherein in step 8), the training parameters of the convolutional neural network comprises: training parameters of the spectrum reconstruction model in step 7) using Adam algorithm to obtain value {circumflex over (θ)} and {circumflex over (λ)} of the spectrum reconstruction model.

13. The method for reconstructing the magnetic resonance spectrum based on the deep learning according to claim 12, wherein in step 9), the reconstructing from under-sampled time-domain signal {tilde over (T)}.sub.u of the target magnetic resonance spectrum comprises: inputting the under-sampled time-domain signal {tilde over (T)}.sub.u to the spectrum reconstruction model, and reconstructing the reconstructed magnetic resonance spectrum signal {tilde over (S)} after a forward propagation of the spectrum reconstruction model, where:
{tilde over (S)}=f({tilde over (T)}.sub.u|{circumflex over (θ)}, {circumflex over (λ)})  (6)

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0035] FIG. 1 illustrates a data verification convolutional neural network structure for magnetic resonance spectrum reconstruction.

[0036] FIG. 2 illustrates an under-sampled template.

[0037] FIGS. 3a and 3b illustrate a full-sampling spectrum and a reconstructed spectrum obtained after reconstruction of under-sampled magnetic resonance time-domain data of Embodiment 1. FIG. 3a illustrates the full-sampling spectrum, and FIG. 3b illustrates the reconstructed spectrum of Embodiment 1 of the present disclosure.

[0038] FIG. 4 illustrates a correlation between a peak intensity of the full-sampling spectrum and a peak intensity of the reconstructed spectrum.

DETAILED DESCRIPTION OF THE EMBODIMENTS

[0039] The present disclosure will be further described in combination with the accompanying embodiments and drawings.

[0040] In an embodiment of the present disclosure, an exponential function is used to generate a training network for a magnetic resonance signal, and then a two-dimensional magnetic resonance spectrum is reconstructed from an under-sampled magnetic resonance time domain signal. The detailed process is as follows.

[0041] 1) The exponential function is used to generate a time domain signal of a magnetic resonance spectrum

[0042] In this embodiment, 5200 free induction decay signals are generated. The data T.sup.n,m, in the n-th row and m-th column of the full sampling magnetic resonance signal T∈.sup.N×M in time domain, can be generated according to the exponential function as follows:

[00003]$\begin{array}{cc}{T}^{n,m}={.Math.}_{r=1}^R{a}_r{e}^{\left(i2\pi n\Delta t_1{f}_{1,r}-\frac{n\Delta t_1}{{\tau }_{1,r}}\right)}{e}^{\left(i2\pi m\Delta t_2{f}_{2,r}-\frac{m\Delta t_2}{{\tau }_{2,r}}\right)}.& \left(1\right)\end{array}$

[0043] Wherein represents a set of complex numbers, N and M represent a number of rows and columns of a time signal, R represents a number of spectral peaks, a.sub.r represents a size of an amplitude, Δt.sub.1 and Δt.sub.2 represent time increases, f.sub.1,r and f.sub.2,r epresent normalized frequencies, and τ.sub.1,r and τ.sub.2,r represent decay factors. In this embodiment, N=256 and M=116, and the number of the spectral peaks R is 2 to 52. With respect to fixed spectral peaks, 200 free induction decay signals with various amplitudes, frequencies, and decay factors will be generated. A range of the amplitude a.sub.r is 0.05≤a.sub.r≤1, ranges of the normalized frequencies f.sub.1,r and f.sub.2,r are 0.05≤f.sub.1,r and f.sub.2,r≤1, and ranges of the decay factors τ.sub.1,r and τ.sub.2,r are 19.2≤τ.sub.1,r and τ.sub.2,r≤179.2.

[0044] 2) A training set including the under-sampled time-domain signal and a corresponding full-sampling spectrum is established

[0045] U represents an under-sampled operation in the time domain. FIG. 2 illustrates a schematic diagram of an under-sampled template M. In the under-sampled template M, a data point represented by white color is sampled, and a data point represented by black color is not sampled. 30% of data points are sampled in total. Ω represents an index subset of the under-sampled template M. When an index (p, q) of a certain signal point appears in the index subset Ω, (p,q)∈Ω. When the index (p, q) of the certain signal point does not appear in the index subset Ω, (p,q).Math.Ω. Filling zeros to signal in all non-sampled positions to obtain a zero-filling time-domain signal T.sub.u according to the under-sampled template M, and a spectrum signal S.sub.u with aliasing is obtained from the zero-filling time-domain signal T.sub.u using Fourier transform. A full-sampling spectrum S is obtained from the full-sampling signal T using Fourier transform, and a real part and an imaginary part of the full-sampling spectrum S are separately saved, that is, S∈.sup.2×256×116, wherein represents a real number. The training set comprises the zero-filling time-domain signal T.sub.u and the full-sampling spectrum S.

[0046] 3) A convolutional neural network of a data verification convolutional neural network structure is designed

[0047] A module of the convolutional neural network comprises 8 convolutional layers, and each convolutional layer comprises 12 filters. The convolutional layers are densely connected, and an input of each convolutional layer is a union of outputs of all previous convolutional layers in the module. In all convolutional layers, the size of convolution kernels in each of the convolution layers is 3×3. An input signal S.sub.l of an l.sup.th layer (1≤l≤L) passes through the convolutional neural network to obtain an output signal S.sub.cnn,l due to the module of the convolutional neural network. A definition of the output signal S.sub.cnn,l is as follows:

S.sub.cnn,l=f(S.sub.l|θ)  (2),

[0048] wherein θ represents a training parameter of the convolutional neural network, and f(S.sub.l|θ) represents a non-linear mapping from S.sub.l to S.sub.cnn,l of the training.

[0049] 4) A bottleneck layer of the data verification convolutional neural network structure is designed

[0050] The bottleneck layer is mainly used to change a number of feature maps in the data verification convolutional neural network structure. The bottleneck layer is disposed before and after the module of the convolutional neural network. The signal will pass through the bottleneck layer of 16 filters to increase the number of feature maps before entering into the module of the convolutional neural network, and an output signal of the module of the convolutional neural network will also pass through the bottleneck layer of 2 filters to reduce the number of feature maps.

[0051] 5) A data verification layer of the data verification convolutional neural network structure is designed

[0052] The data verification layer is mainly used to verify data in the data verification convolutional neural network structure. The output signal S.sub.cnn,l from the l.sup.th convolutional neural network functions as an input, and the output signal S.sub.cnn,l (which becomes an input signal) is converted back to the time domain using inverse Fourier transform F.sup.H to obtain a signal T.sub.l. The formula is as follows:

T.sub.l=F.sup.HS.sub.cnn,l  (3)

[0053] An expression of the data verification layer is as follows:

[00004]$\begin{array}{cc}{\hat{T}}_l^{n,m}=\{\begin{array}{c}{T}_l^{n,m},\mathrm{if}\left(n,m\right).Math.\Omega \\ \frac{T_l^{n,m}+\lambda T_u^{n,m}}{1+\lambda },\mathrm{if}\left(n,m\right)\in \Omega \end{array}.& \left(4\right)\end{array}$

[0054] A reconstructed spectrum signal Ŝ.sub.l=F{circumflex over (T)}.sub.l is finally output, wherein the last time, that is, a spectrum Ŝ.sub.8 of an 8.sup.th layer, defines an output Ŝ of an entire deep learning network.

[0055] 6) A feedback function of the data verification convolutional neural network structure is designed

[0056] The feedback function enables an output of each module of a combination of the convolutional neural network and the data verification layer to be close to a full-sampling spectrum signal S=FT in the convolutional neural network structure and enables an input of a next module to be more interpretable. An output of each data verification layer is compared with the full-sampling spectrum signal S=FT to feed back to each module to update parameters, wherein T represents a full-sampling time-domain signal in formula (1), and F represents Fourier transform.

[0057] 7) The data verification convolutional neural network structure is established to function as the spectrum reconstruction model

[0058] The data is used to verify the convolutional neural network structure that is cascaded with multiple modules of the combinations of the convolutional neural networks and the data verification layers. The under-sampled magnetic resonance time-domain signal T.sub.u is inputted, and a reconstructed magnetic resonance spectrum signal Ŝ is outputted, thereby constituting an end-to-end deep neural network structure. A loss function of the data verification convolutional neural network structure is defined as follows:

(θ)=∥S−Ŝ∥.sub.F.sup.2  (5),

wherein represents the training set, ∥⋅∥.sub.F represents F-norm (Frobenius norm) of a matrix, Ŝ=f(T.sub.u|θ, λ), θ represents a training parameter of the convolutional neural network, λ represents a data verification parameter of the data verification layer, and both parameters θ and λ need to be trained. FIG. 1 illustrates a final design of the data verification convolutional neural network structure.

[0059] 8) Parameters of the convolutional neural network are trained and optimized

[0060] Adam algorithm that is conventional in deep learning is used to train the parameters of the spectrum reconstruction model in step 7) to obtain optimal value {circumflex over (θ)} and {circumflex over (λ)} of the spectrum reconstruction model.

[0061] 9) The under-sampled time-domain signal {tilde over (T)}.sub.u of a target magnetic resonance spectrum {tilde over (S)} is reconstructed

[0062] The under-sampled magnetic resonance time-domain signal {tilde over (T)}.sub.u is input to a spectrum reconstruction model. After a forward propagation of the spectrum reconstruction model, the constructed magnetic resonance spectrum {tilde over (S)} is reconstructed. The formula is as follows:

{tilde over (S)}=f({tilde over (T)}.sub.u|{circumflex over (θ)}, {circumflex over (λ)})  (6).

[0063] FIG. 3b illustrates the spectrum obtained using reconstruction from the under-sampled magnetic resonance time-domain data in the embodiment. Compared with the full-sampling spectrum illustrated in FIG. 3a, it can be seen that a designed data verification convolutional neural network and a network trained by a synthetic simulation magnetic resonance spectrum are used to obtain a high-quality magnetic resonance spectrum. FIG. 4 illustrates a correlation between a peak intensity of the full-sampling spectrum and a peak intensity of the reconstructed spectrum.