CN109188327B - Magnetic resonance image fast reconstruction method based on tensor product complex small compact framework - Google Patents

Magnetic resonance image fast reconstruction method based on tensor product complex small compact framework Download PDF

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CN109188327B
CN109188327B CN201811124319.XA CN201811124319A CN109188327B CN 109188327 B CN109188327 B CN 109188327B CN 201811124319 A CN201811124319 A CN 201811124319A CN 109188327 B CN109188327 B CN 109188327B
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蒋明峰
陆亮
沈益
吴龙
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Abstract

The invention discloses a magnetic resonance image fast reconstruction method based on a tensor product complex small compact frame, and provides a new research method for magnetic resonance fast imaging. According to the invention, the K space data is subjected to undersampling by utilizing a Cartesian sampling track mode, so that the scanning speed of the equipment is greatly improved; the image is decomposed from multiple directions based on sparse transformation of a tensor product complex small compact frame, so that the precision of magnetic resonance imaging is improved; the convex optimization problem in the image reconstruction is solved by using a projection fast iterative soft threshold algorithm, so that the reconstruction speed of the magnetic resonance image is increased; in the projection fast iterative soft threshold algorithm, regularization parameters are calculated in a self-adaptive mode by adopting a bivariate contraction-based method, so that the process of blindly selecting the parameters is omitted, and the reconstruction quality of the image is improved.

Description

Magnetic resonance image fast reconstruction method based on tensor product complex small compact framework
Technical Field
The invention belongs to the technical field of magnetic resonance imaging, and particularly relates to a magnetic resonance image fast reconstruction method based on a tensor product complex wavelet compact framework.
Background
Magnetic Resonance Imaging (MRI) has the characteristics of no ionizing radiation, multi-angle Imaging, no damage to human tissues and the like, so that the MRI becomes an important detection means in medical clinic and medical scientific research; however, MRI has the disadvantages of slow imaging speed and low time resolution of dynamic MRI; meanwhile, a single patient has long scanning time and high cost, and further popularization of MRI is limited. In order to increase the scanning speed and improve the scanning efficiency, Parallel Imaging (PI) technology and compressive sensing theory have become a research hotspot in recent years.
The parallel magnetic resonance imaging technology adopts a multi-channel sampling coil to simultaneously receive magnetic resonance signals, and reduces the number of phase encoding times required when compared with the use of a single coil, thereby reducing the signal acquisition time and accelerating the imaging speed. Currently, two methods are widely used to reconstruct images from acquired partially parallel data, one of which uses coil sensitivity information, such as sense (sensitivity encoding); another approach uses correlations between multi-channel K-space data, such as GRAPPA (Generalized Auto-calibrating Parallel acquisition). However, the acceleration factor of parallel magnetic resonance is still limited due to the performance limitations of the severely ill-conditioned system matrix and reconstruction algorithms.
The compressed sensing theory holds that as long as the signals are sparse, or sparse in a certain transform domain, we can break through the limitation of Nyquist sampling rate and use less data to recover a large amount of signals.
Disclosure of Invention
The invention aims to overcome the defects of the method and provides a magnetic resonance image fast reconstruction method based on a tensor product complex small compact frame, the method takes the tensor product complex small compact frame as sparse transformation, and decomposes signals from multiple directions, thereby improving the precision of magnetic resonance imaging; the image reconstruction problem is solved by using a projection fast iterative soft threshold algorithm, so that the speed of magnetic resonance imaging reconstruction is improved; and the regularization parameters are calculated in a self-adaptive manner based on a bivariate shrinkage method, so that the reconstruction quality of the image is further improved.
The method comprises the following specific steps:
1) and performing undersampling on the magnetic resonance K space data by utilizing a Cartesian sampling track to obtain the K space undersampled data.
2) And performing zero filling on the K space undersampled data, and then obtaining an initial magnetic resonance image through Fourier inverse transformation.
3) Taking the initial magnetic resonance image as an input image, carrying out sparse decomposition by using a tensor product complex wavelet compact frame, and obtaining a decomposition coefficient expression after decomposition as follows:
Figure BDA0001811988650000021
wherein α is the decomposition coefficient, x is the input image, and T is the transposed symbol;
Figure BDA0001811988650000022
a complex wavelet tightening framework is a tensor product;
4) and (3) performing minimum solution on the objective function of the magnetic resonance image by adopting a projection fast iterative soft threshold algorithm, wherein the optimization problem is expressed as follows:
Figure BDA0001811988650000031
the N-dimensional complex vector x and the M-dimensional complex vector y respectively represent a magnetic resonance image to be reconstructed and K-space undersampled data, the N × N complex matrix F represents a discrete Fourier transform matrix, the M × N real matrix U represents an undersampled matrix when the K-space data are undersampled, and the M × N real matrix U represents an undersampled matrix when the K-space data are undersampled<N; λ is the regularization parameter, | ·| non-woven phosphor2Is a two-norm operation, | | ·| non-conducting phosphor1Is a norm operation.
The iteration equation is as follows:
μ＝xk+F*UT(y-UFxk)
Figure BDA0001811988650000032
Figure BDA0001811988650000033
Figure BDA0001811988650000034
wherein x iskRepresenting the reconstructed image at the k-th iteration; c. mu and sk+1Are all intermediate transition vectors, F*And UTRespectively representing discrete Fourier transformsTransposing of transform and sampling matrices;
Figure BDA0001811988650000035
is a soft threshold function for shrinking the wavelet coefficients; t is tkStep size for the k-th iteration.
And when the iteration times reach a set value, outputting the reconstructed magnetic resonance image.
Further, each layer α in decomposition coefficient αqThe corresponding λ is evaluated using a bivariate contraction function, as follows:
Figure BDA0001811988650000041
wherein, αqIs the current scaling factor, αpIs αqCoefficient of the parent scale, σαIs within the current sub-band (current layer) q at α q3 × 3 window centered prediction variance, noise variance
Figure BDA0001811988650000042
Using a median estimator to estimate:
Figure BDA0001811988650000043
α thereinhIs the coefficient of the high-frequency sub-band in the first layer decomposition, and the mean is the operator for taking the median.
The invention discloses a magnetic resonance image fast reconstruction method based on a tensor product complex small compact framework, and provides a new research method for magnetic resonance fast imaging. According to the invention, the K space data is subjected to undersampling by utilizing the Cartesian sampling track, so that the scanning speed of the equipment is greatly improved; the image is decomposed from multiple directions based on sparse transformation of a tensor product complex small compact frame, so that the precision of magnetic resonance imaging is improved; the convex optimization problem in the image reconstruction is solved by using a projection fast iterative soft threshold algorithm, so that the reconstruction speed of the magnetic resonance image is increased; and in the projection fast iterative soft threshold algorithm, regularization parameters are calculated in a self-adaptive mode by adopting a bivariate contraction-based method, so that the reconstruction quality of the image is further improved.
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FIG. 1 is a schematic flow chart of the present invention.
Fig. 2 is a diagram of a cartesian sampling trajectory pattern.
Fig. 3(a) is a shoulder magnetic resonance image.
Fig. 3(b) is a chest magnetic resonance image.
Fig. 3(c) shows a brain magnetic resonance image.
FIG. 4(a) is a graph showing the peak signal-to-noise ratio PSNR of the magnetic resonance image of the brain compared with the currently popular D L MRI, PBDWS, PANO and GBRWT algorithms under different acceleration factors.
Fig. 4(b) is a diagram comparing the present invention with the currently popular D L MRI, PBDWS, PANO and GBRWT algorithms to measure SSIM on structural similarity of magnetic resonance images of the brain under different acceleration factors.
Detailed Description
In order to more specifically describe the present invention, the following detailed description is provided for the technical solution of the present invention with reference to the accompanying drawings and the specific embodiments.
As shown in fig. 1, the method for rapidly reconstructing a magnetic resonance image based on a tensor product complex wavelet compact framework of the present invention comprises the following specific steps:
1. undersampling magnetic resonance K space data based on a Cartesian sampling track shown in FIG. 2 to obtain K space undersampled data, wherein a white pixel point value is 1 and represents a sampling point; the black pixel point value is 0, indicating a non-sampling point.
2. And (4) performing zero filling on the K-space undersampled data (setting non-sampling point values as 0), and obtaining an initial magnetic resonance image through Fourier inversion.
3. Taking the initial magnetic resonance image as an input image, carrying out sparse decomposition by using a tensor product complex wavelet compact frame, and obtaining a decomposition coefficient expression after decomposition as follows:
Figure BDA0001811988650000051
wherein α isDecomposition coefficients (also called wavelet coefficients), x being the input image and T being the transposed symbol;
Figure BDA0001811988650000052
for the tensor product complex small compact framework, the construction method adopts the prior art, such as Han B, Zhao Z.Tensor product complete frame with creating direction, which is published in 7 months in 2014 in the Journal on ImagingSciences.
4. And (3) minimizing and solving an objective function of the magnetic resonance image by adopting a projection fast iterative soft threshold algorithm (pFISTA), wherein the optimization problem is expressed as follows:
Figure BDA0001811988650000053
where the N-dimensional complex vector x and the M-dimensional complex vector y represent the magnetic resonance image to be reconstructed and the K-space undersampled data, respectively, the N × N complex matrix F represents a discrete fourier transform matrix, the M × N (M × N)<N) the real number matrix U represents an undersampling matrix when undersampling K space data; λ is the regularization parameter, | ·| non-woven phosphor2Is a two-norm operation, | | ·| non-conducting phosphor1Is a norm operation.
The iteration equation is as follows:
μ＝xk+F*UT(y-UFxk)
Figure BDA0001811988650000061
Figure BDA0001811988650000062
Figure BDA0001811988650000063
wherein x iskRepresenting the reconstructed image at the k-th iteration; c. mu and sk+1Are all intermediate transition vectors, F*And UTRespectively representing inverse discrete Fourier transform and transposition of a sampling matrix;
Figure BDA0001811988650000064
is a soft threshold function for shrinking the wavelet coefficients; t is tkStep size for the k-th iteration.
Wherein each layer α of decomposition coefficients α is evaluated using a bivariate contraction functionqThe corresponding λ is specifically as follows:
Figure BDA0001811988650000065
wherein, αqIs the current scaling factor, αpIs αqCoefficient of the parent scale, σαIs within the current sub-band (current layer) q at α q3 × 3 window centered prediction variance, noise variance
Figure BDA0001811988650000066
Using a median estimator to estimate:
Figure BDA0001811988650000067
α thereinhIs the coefficient of the high-frequency sub-band in the first layer decomposition, and the mean is the operator for taking the median.
And when the iteration times reach a set value, outputting the reconstructed magnetic resonance image.
In order to verify the reconstruction performance of the magnetic resonance image fast reconstruction method (TPCTF-BS) based on the tensor product complex wavelet compact framework, a simulation experiment is performed by using three fully sampled magnetic resonance images, as shown in fig. 3(a), 3(b), and 3(c), which are respectively the magnetic resonance image data of the shoulder, the chest, and the brain, fig. 2 is a cartesian sampling matrix with an acceleration factor of 4 (equivalent to a sampling rate of 25%), which is used to undersample data on K space, the method of the present invention is compared with the currently popular magnetic resonance reconstruction algorithm, which mainly includes D L MRI, PBDWS, PANO, and GBRWT, and the reconstruction effects of different algorithms are measured by using peak signal-to-noise ratio (signal-to-noise ratio), and Structural Similarity (SSIM) as objective evaluation indexes to verify the reconstruction performance of the method, two objective evaluation indexes are defined as:
Figure BDA0001811988650000071
Figure BDA0001811988650000072
wherein x is the reconstructed image,
Figure BDA0001811988650000073
for images under full sampling, W represents images x and
Figure BDA0001811988650000074
h denotes the images x and
Figure BDA0001811988650000075
height of (d), muxAnd
Figure BDA0001811988650000076
are x and
Figure BDA0001811988650000077
the average value of (a) of (b),
Figure BDA0001811988650000078
and
Figure BDA0001811988650000079
are x and
Figure BDA00018119886500000710
the variance of (a) is determined,
Figure BDA00018119886500000711
is the covariance of the two images, C1And C2Is a constant used to maintain stability.
Tables 1 and 2 record that under-sampling 25% K-space data under Cartesian under-sampling matrix, and different algorithms are used for reconstructing PSNR and SSIM of a magnetic resonance image, so that the TPCTF-BS has the best reconstruction quality.
Table 3 shows the average reconstruction time of the magnetic resonance image under different algorithms, and it can be found that the TPCTF-BS of the present invention has the fastest reconstruction speed.
Fig. 4(a) and fig. 4(b) are PSNR and SSIM of magnetic resonance images reconstructed from magnetic resonance images of a brain under different acceleration factors, respectively, and it can be seen that as the acceleration factor increases, i.e. the data of a sampling K space decreases, the reconstruction quality of all algorithms decreases, but under the same acceleration factor, the TPCTF-BS of the present invention still has the highest PSNR and SSIM values, indicating that the method has applicability under any reconstruction condition.
TABLE 1
Image \ algorithm DLMRI PBDWS PANO GBRWT TPCTF-BS
Shoulder image 35.69 37.49 37.37 36.64 37.69
Chest image 26.36 26.33 26.69 26.03 28.50
Brain image 28.55 30.84 29.35 28.92 31.33
TABLE 2
Image \ algorithm DLMRI PBDWS PANO GBRWT TPCTF-BS
Shoulder image 0.9291 0.9433 0.9436 0.9376 0.9484
Chest image 0.7390 0.8285 0.7829 0.7998 0.8620
Brain image 0.8488 0.9079 0.8704 0.8890 0.9145
TABLE 3
Algorithm DLMRI PBDWS PANO GBRWT TPCTF-BS
Reconstruction time(s) 208.7 92.6 54.6 95.6 45.8
The above-mentioned embodiments are provided to further explain the objects, technical solutions and advantages of the present invention in detail, and it should be understood that the above-mentioned embodiments are only examples of the present invention and are not intended to limit the scope of the present invention. It should be understood that any modifications, equivalents, improvements and the like, which come within the spirit and principle of the invention, may occur to those skilled in the art and are intended to be included within the scope of the invention.

Claims (2)

1. A magnetic resonance image fast reconstruction method based on a tensor product complex small compact frame is characterized in that: the method comprises the following specific steps:
1) performing undersampling on the magnetic resonance K space data by utilizing a Cartesian sampling track to obtain K space undersampled data;
2) performing zero filling on the K space undersampled data, and then obtaining an initial magnetic resonance image through Fourier inverse transformation;
3) taking the initial magnetic resonance image as an input image, carrying out sparse decomposition by using a tensor product complex wavelet compact frame, and obtaining a decomposition coefficient expression after decomposition as follows:
Figure FDA0002494280740000011
wherein α is the decomposition coefficient, x is the input image, and T is the transposed symbol;
Figure FDA0002494280740000012
a complex wavelet tightening framework is a tensor product;
4) and (3) performing minimum solution on the objective function of the magnetic resonance image by adopting a projection fast iterative soft threshold algorithm, wherein the optimization problem is expressed as follows:
Figure FDA0002494280740000013
the N-dimensional complex vector x and the M-dimensional complex vector y respectively represent a magnetic resonance image to be reconstructed and K-space undersampled data, the N × N complex matrix F represents a discrete Fourier transform matrix, the M × N real matrix U represents an undersampled matrix when the K-space data are undersampled, and the M × N real matrix U represents an undersampled matrix when the K-space data are undersampled<N; λ is the regularization parameter, | ·| non-woven phosphor2Is a two-norm operation, | | ·| non-conducting phosphor1Is a norm operation;
the iteration equation is as follows:
μ＝xk+F*UT(y-UFxk)
Figure FDA0002494280740000014
Figure FDA0002494280740000021
Figure FDA0002494280740000022
wherein x iskRepresenting the reconstructed image at the k-th iteration; c. mu and sk+1Are all intermediate transition vectors, F*And UTRespectively representing inverse discrete Fourier transform and transposition of a sampling matrix;
Figure FDA0002494280740000023
is a soft threshold function for shrinking the wavelet coefficients; t is tkStep length of the k iteration;
and when the iteration times reach a set value, outputting the reconstructed magnetic resonance image.
2. The method for rapidly reconstructing magnetic resonance image based on tensor product complex small compact framework as claimed in claim 1, wherein each layer α in decomposition coefficient αqThe corresponding λ is evaluated using a bivariate contraction function, as follows:
Figure FDA0002494280740000024
wherein, αqIs the current scaling factor, αpIs αqCoefficient of the parent scale, σαIs within the current sub-band (current layer) q at αq3 × 3 window centered prediction variance, noise variance
Figure FDA0002494280740000025
Using a median estimator to estimate:
Figure FDA0002494280740000026
α thereinhIs the coefficient of the high-frequency sub-band in the first layer decomposition, and the mean is the operator for taking the median.
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