One-dimensional partial Fourier parallel magnetic resonance imaging method based on deep convolutional network
The present disclosure relates to a 1D partial Fourier parallel magnetic resonance imaging method with a deep convolutional network and belongs to the technical field of magnetic resonance imaging. The method includes steps of: creating a sample set and a sample label set for training; constructing an initial deep convolutional network model; inputting a training sample of the sample set to the initial deep convolutional network model for forward process, comparing an output result of the forward process with an expected result in the sample label set, and performing training with a gradient descent method until a parameter of each layer which enables consistency between the output result and the expected result to be maximum is obtained; creating an optimal deep convolutional network model by using the obtained parameter of the each layer; and inputting a multi-coil undersampled image sampled online to the optimal deep convolutional network model, performing the forward process on the optimal deep convolutional network model, and outputting a reconstructed single-channel full-sampled image. The present disclosure can well remove the noise of the reconstructed image, reconstruct a magnetic resonance image with a better visual effect, and has high practical value.
1. A one-dimensional partial Fourier parallel magnetic resonance imaging method based on a deep convolutional network, comprising:
creating, based on an existing undersampled multi-channel magnetic resonance image, a sample set and a sample label set for training;
constructing an initial deep convolutional network model comprising an input layer, L convolutional layers and an output layer which are sequentially connected;
inputting a training sample (x, y) of the sample set to the initial deep convolutional network model for forward process, comparing an output result of the forward process with an expected result in the sample label set, and training with a gradient descent method until a parameter of each layer which enables consistency between the output result and the expected result to be maximum is obtained;
creating an optimal deep convolutional network model by using the obtained parameter of the each layer; and
inputting a multi-coil undersampled image sampled online to the optimal deep convolutional network model, performing the forward process on the optimal deep convolutional network model, and outputting a reconstructed full-sampled image;
wherein the gradient descent method comprises:
for the training sample (x, y), calculating a gradient of a last convolutional layer C L according to the following formula:
δ
L
=
∂
J
∂
b
L
=
∂
J
∂
D
L
∂
D
L
∂
b
L
=
C
L
-
y
wherein
∂
D
l
∂
b
l
=
1
and
C
l
=
σ
(
D
l
)
;
updating a gradient of an lth-layer nonlinear mapping layer of δ l by the following formula:
δ
′
=
∂
J
∂
b
l
=
∂
J
∂
D
l
+
1
∂
D
l
+
1
∂
C
l
∂
C
l
∂
D
l
=
(
δ
l
+
1
*
W
l
+
1
)
•
∂
(
D
l
)
∂
D
l
wherein * denotes a cross-correlation operation, and ° denotes that array elements are sequentially multiplied;
obtaining a gradient of each of the L convolutional layers as:
{
∂
J
∂
W
l
=
∂
J
∂
D
l
∂
D
l
∂
W
l
=
δ
l
*
D
l
-
1
∂
J
∂
b
l
=
∂
J
∂
D
l
∂
D
l
∂
b
l
=
∂
J
∂
D
l
=
δ
l
updating a parameter of the each of the L convolutional layers based on the calculated gradient of the each of the L convolutional layers.
2. The one-dimensional partial Fourier parallel magnetic resonance imaging method based on the deep convolutional network according to claim 1 , wherein the training sample in the sample set is a coincident undersampled image extraction patch extracted from an existing offline multi-coil undersampled image, and a sample label in the label set is a square root of a sum of squares of a full-sampled multi-channel image extraction patch corresponding to the undersampled image extraction patch.
3. The one-dimensional partial Fourier parallel magnetic resonance imaging method based on the deep convolutional network according to claim 2 , wherein the undersampled image extraction patch used as the training sample is obtained according to the following formula:
arg
min
Θ
{
1
2
TN
∑
t
=
1
T
∑
n
=
1
N
C
(
x
t
,
n
;
Θ
)
-
y
t
,
n
2
2
}
wherein x is the undersampled image extraction patch, y is the corresponding full-sampled image extraction patch, C is an end-to-end mapping relationship estimated with a hidden layer parameter Θ={(W 1 ,b 1 ), . . . (W l ,b l ), . . . (W L ,b L )}, T is a number of samples extracted from an image, and N is a total number of images.
4. The one-dimensional partial Fourier parallel magnetic resonance imaging method based on the deep convolutional network according to claim 1 , wherein the existing offline multi-coil undersampled image is obtained by under sampling a K-space multi-coil full-sampled image.
5. The one-dimensional partial Fourier parallel magnetic resonance imaging method based on the deep convolutional network according to claim 1 , wherein the L convolutional layers of the initial deep convolutional network model are created in the following manner:
{
C
0
=
x
C
1
=
σ
(
W
1
*
x
+
b
1
)
C
l
=
σ
(
W
l
*
C
l
-
1
+
b
l
)
,
l
∈
2
,
…
,
L
-
1
C
L
=
σ
(
W
L
*
C
L
-
1
+
b
L
)
wherein C denotes a convolutional layer and x denotes an input sample; in a formula, W 1 is a convolution operator of a first convolutional layer C 1 and is equal to c×M 1 ×M 1 ×n 1 , b 1 is an element-related n 1 -dimensional offset, c is a number of image channels, M 1 is a filter size and n 1 is a number of filters; W l is a convolution operator of a l-th convolutional layer C 1 and is equal to n l-1 ×M l ×M l ×n l , b l is an element-related n l -dimensional offset, M l is a filter size and n l is a number of filters; W L is a convolution operator of a last convolutional layer C L and is equal to n L-1 ×M L ×M L ×c, wherein b L is an element-related n L -dimensional offset, c is a number of image channels, M L is a filter size, and n L is a number of filters.
6. The one-dimensional partial Fourier parallel magnetic resonance imaging method based on the deep convolutional network according to claim 5 , wherein the initial deep convolutional network model further comprises activation layers connected to one or more of the L convolutional layers.
7. The one-dimensional partial Fourier parallel magnetic resonance imaging method based on the deep convolutional network according to claim 6 , wherein the activation layers use a ReLu activation function.
8. The one-dimensional partial Fourier parallel magnetic resonance imaging method based on the deep convolutional network according to claim 5 , wherein the initial deep convolutional network model comprises the input layer, a first convolutional layer, a second convolutional layer, a third convolutional layer and the output layer, wherein the first convolutional layer is connected to a first activation layer and the second convolutional layer is connected to a second activation layer.
9. The one-dimensional partial Fourier parallel magnetic resonance imaging method based on the deep convolutional network according to claim 8 , wherein the output layer uses an EuclideanLoss function.
10. A one-dimensional partial Fourier parallel magnetic resonance imaging apparatus based on a deep convolutional network, comprising:
a module configured to create, based on an existing undersampled multi-channel magnetic resonance image, a sample set and a sample label set for training;
a module configured to construct an initial deep convolutional network model comprising an input layer, L convolutional layers and an output layer which are sequentially connected;
a module configured to input a training sample (x, y) of the sample set to the initial deep convolutional network model for forward process, compare an output result of the forward process with an expected result in the sample label set, and perform training with a gradient descent method until a parameter of each layer which enables consistency between the output result and the expected result to be maximum is obtained;
a module configured to create an optimal deep convolutional network model by using the obtained parameter of the each layer; and
a module configured to input a multi-coil undersampled image sampled online to the optimal deep convolutional network model, perform the forward process on the optimal deep convolutional network model, and output a reconstructed full-sampled image;
wherein the gradient descent method comprises:
for the training sample (x, y), calculating a gradient of a last convolutional layer C L according to the following formula:
δ
L
=
∂
J
∂
b
L
=
∂
J
∂
D
L
∂
D
L
∂
b
L
=
C
L
-
y
wherein
∂
D
l
∂
b
l
=
1
and
C
l
=
σ
(
D
l
)
;
updating a gradient of an lth-layer nonlinear mapping layer of δ l by the following formula:
δ
′
=
∂
J
∂
b
l
=
∂
J
∂
D
l
+
1
∂
D
l
+
1
∂
C
l
∂
C
l
∂
D
l
=
(
δ
l
+
1
*
W
l
+
1
)
•
∂
(
D
l
)
∂
D
l
wherein * denotes a cross-correlation operation, and denotes that array elements are sequentially multiplied;
obtaining a gradient of each of the L convolutional layers as:
{
∂
J
∂
W
l
=
∂
J
∂
D
l
∂
D
l
∂
W
l
=
δ
l
*
D
l
-
1
∂
J
∂
b
l
=
∂
J
∂
D
l
∂
D
l
∂
b
l
=
∂
J
∂
D
l
=
δ
l
updating a parameter of the each of the L convolutional layers based on the calculated gradient of the each of the L convolutional layers.
11. A non-transitory computer readable medium, which is configured to store programs, wherein the programs are computer-executable and cause a computer to perform processing comprising:
creating, based on an existing undersampled multi-channel magnetic resonance image, a sample set and a sample label set for training;
constructing an initial deep convolutional network model comprising an input layer, L convolutional layers and an output layer which are sequentially connected;
inputting a training sample (x, y) of the sample set to the initial deep convolutional network model for forward process, comparing an output result of the forward process with an expected result in the sample label set, and performing training with a gradient descent method until a parameter of each layer which enables consistency between the output result and the expected result to be maximum is obtained;
creating an optimal deep convolutional network model by using the obtained parameter of each layer; and
inputting a multi-coil undersampled image sampled online to the optimal deep convolutional network model, performing the forward process on the optimal deep convolutional network model, and outputting a reconstructed full-sampled image;
wherein the gradient descent method comprises:
for the training sample (x, y), calculating a gradient of a last convolutional layer C L according to the following formula:
δ
L
=
∂
J
∂
b
L
=
∂
J
∂
D
L
∂
D
L
∂
b
L
=
C
L
-
y
wherein
∂
D
l
∂
b
l
=
1
and
C
l
=
σ
(
D
l
)
;
updating a gradient of an lth-layer nonlinear mapping layer of δ l by the following formula:
δ
l
=
∂
J
∂
b
l
=
∂
J
∂
D
l
+
1
∂
D
l
+
1
∂
C
l
∂
C
l
∂
D
l
=
(
δ
l
+
1
*
W
l
+
1
)
•
∂
(
D
l
)
∂
D
l
wherein * denotes a cross-correlation operation, and ° denotes that array elements are sequentially multiplied;
obtaining a gradient of each of the L convolutional layers as:
{
∂
J
∂
W
l
=
∂
J
∂
D
l
∂
D
l
∂
W
l
=
δ
l
*
D
l
-
1
∂
J
∂
b
l
=
∂
J
∂
D
l
∂
D
l
∂
b
l
=
∂
J
∂
D
l
=
δ
l
updating a parameter of the each of the L convolutional layers based on the calculated gradient of the each of the L convolutional layers.
12. The computer readable medium according to claim 11 , wherein a training sample in the sample set is a coincident undersampled image extraction patch extracted from an existing offline multi-coil undersampled image, and a sample label in the label set is a square root of a sum of squares of a full-sampled multi-channel image extraction patch corresponding to the undersampled image extraction patch.
13. The computer readable medium according to claim 12 , wherein the undersampled image extraction patch as the training sample is obtained according to the following formula:
arg
min
θ
{
1
2
TN
∑
i
=
1
T
∑
n
=
1
N
C
(
x
t
,
n
;
Θ
)
-
y
t
,
n
2
2
}
wherein x is the undersampled image extraction patch, y is the corresponding full-sampled image extraction patch, C is an end-to-end mapping relationship estimated with a hidden layer parameter Θ={(W 1 ,b 1 ), . . . (W l ,b l ), . . . (W L ,b L )}, T is a number of samples extracted from an image, and N is a total number of images.
14. The computer readable medium according to claim 11 , wherein the existing offline multi-coil undersampled image is obtained by under sampling a K-space multi-coil full-sampled image.
15. The computer readable medium according to claim 11 , wherein the L convolutional layers of the initial deep convolutional network model are created in a following manner:
{
C
0
=
x
C
1
=
σ
(
W
1
*
x
+
b
1
)
C
l
=
σ
(
W
l
*
C
l
-
1
+
b
l
)
,
l
∈
2
,
…
,
L
-
1
C
L
=
σ
(
W
L
*
C
L
-
1
+
b
L
)
wherein C denotes a convolutional layer and x denotes an input sample; in a formula, W 1 is a convolution operator of a first convolutional layer C 1 and is equal to c×M 1 ×M 1 ×n 1 , b 1 is an element-related n 1 -dimensional offset, c is a number of image channels, M 1 is a filter size and n 1 is a number of filters; W l is a convolution operator of a l-th convolutional layer C l and is equal to n l-1 ×M l ×M l ×n l , b l is an element-related n l -dimensional offset, M l is a filter size and n l is a number of filters; W L is a convolution operator of a last convolutional layer C L and is equal to n L-1 ×M L ×M L ×c, wherein b L is an element-related n L -dimensional offset, c is a number of image channels, M L is a filter size, and n L is a number of filters.
16. The computer readable medium according to claim 15 , wherein the initial deep convolutional network model further comprises activation layers connected to one or more of the L convolutional layers.
17. The computer readable medium according to claim 16 , wherein the activation layers use a ReLu activation function, and the output layer uses an EuclideanLoss function.
18. The computer readable medium according to claim 15 , wherein the initial deep convolutional network model comprises the input layer, a first convolutional layer, a second convolutional layer, a third convolutional layer and the output layer, wherein the first convolutional layer is connected to a first activation layer and the second convolutional layer is connected to a second activation layer.