CN108346167B - MRI image reconstruction method based on simultaneous sparse coding under orthogonal dictionary - Google Patents

MRI image reconstruction method based on simultaneous sparse coding under orthogonal dictionary Download PDF

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CN108346167B
CN108346167B CN201711463209.1A CN201711463209A CN108346167B CN 108346167 B CN108346167 B CN 108346167B CN 201711463209 A CN201711463209 A CN 201711463209A CN 108346167 B CN108346167 B CN 108346167B
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刘书君
沈晓东
曹建鑫
张新征
李勇明
张奎
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Shenzhen Zhongwu Photon Technology Co.,Ltd.
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Abstract

The invention discloses an MRI image reconstruction method based on simultaneous sparse coding under an orthogonal dictionary. Belongs to the technical field of digital image processing. The MRI image reconstruction method is an MRI image reconstruction method which utilizes an orthogonal dictionary to carry out sparse representation on an image and simultaneously carries out sparse coding optimization on sparse coefficients. Firstly, finding a similar image block set, namely a structure group, of a target image block, then based on an image reconstruction model established by simultaneous sparse coding of the structure group under an orthogonal dictionary, and finally solving the sparse coefficient of the structure group in the model by using a generalized soft threshold method and reconstructing an image; the invention can optimize the sparse representation performance of the structure group by carrying out sparse representation on the structure group through the orthogonal dictionary, and can carry out constraint and solution on the sparse coefficient by utilizing simultaneous sparse coding and a generalized soft threshold method, so that the sparse coefficient can be estimated more efficiently and accurately.

Description

MRI image reconstruction method based on simultaneous sparse coding under orthogonal dictionary
Technical Field
The invention belongs to the technical field of digital image processing, and particularly relates to an MRI image reconstruction method for performing sparse representation on a structural group and coefficient constraint on sparse coding simultaneously by using an orthogonal dictionary, which is used for high-quality recovery of medical images.
Background
Magnetic Resonance Imaging (MRI) is a method for drawing an image of the inside of an object by electromagnetic waves emitted from an external gradient magnetic field using the principle of nuclear magnetic resonance, and has been widely used in the fields of medical treatment, archaeology, petrochemical industry, and the like. Compared with the prior medical imaging technology, magnetic resonance imaging has the advantages of higher resolution ratio on soft tissues, no ionizing radiation damage on human bodies, higher imaging flexibility, more imaging parameters and the like, so that the magnetic resonance imaging becomes one of the most important medical imaging technologies at present. However, the magnetic resonance imaging technology has some problems to be solved, such as the imaging speed is slow, and the obtained images are often accompanied by artifact phenomena, which greatly interfere with the diagnostic information, so the main improvement on the magnetic resonance imaging aims to solve the two main problems.
In addition to improving the performance of hardware devices and the scanning technology of K space, along with the proposal of the compressed sensing theory, the theory can break through the limit of the traditional Nyquist sampling theorem, namely, the MRI image is reconstructed by utilizing the undersampled K space signals, thereby greatly reducing the scanning time. Therefore, how to better recover images from the sampled signals becomes a major research point. The traditional MRI image reconstruction method based on compressed sensing generally utilizes fixed transformation such as wavelet and the like to carry out sparse representation on an image and obtains a better result. In recent years, the similar characteristics inside the image are gradually emphasized, and the method is also applied to the fields of image noise reduction and the like, and the quality of the finally obtained image is greatly improved.
Disclosure of Invention
The invention aims to provide an MRI image reconstruction method based on simultaneous sparse coding under an orthogonal dictionary, aiming at the defects of the existing MRI image reconstruction method. According to the method, the sparsity of the image in a transform domain and the non-local similarity between image blocks are fully considered, the structural group obtained by similar image block sets is subjected to sparse representation by using an orthogonal dictionary, and the sparse coefficient is subjected to optimal estimation by using simultaneous sparse coding and a generalized soft threshold method, so that the estimation precision is improved. The method specifically comprises the following steps:
step one, obtaining of structure group
In order to realize the improvement of sparsity by utilizing simultaneous sparse coding and optimize the sparse coefficient of a similar image block set, namely a structure group under an orthogonal dictionary, a structure group corresponding to a target image block needs to be constructed, firstly, an image x after initial reconstruction is carried out(0)Extracting target image block xiThen, the Euclidean distance comparison method is used for comparing the target image block xiFinding corresponding similar image blocks for the search range of the center, and comparing the similar image blocks with the target image block xiConstructed as structural group Xi；
Step two, establishing a simultaneous sparse coding constraint model
Obtaining a structural group XiThen, in the process of simultaneous sparse coding, the non-convex norm pair structure group X is utilizediSparse coefficient set A under orthogonal dictionary DiCarrying out sparse constraint:
Figure BDA0001530717310000021
wherein p is more than 0 and less than 1, alphakRepresents a coefficient matrix AiOn the basis of the k line in (1), a constraint model about the image and the sparse coefficients is established:
Figure BDA0001530717310000022
wherein M is the number of structural groups, FuIn order to down-sample the fourier transform matrix,
Figure BDA0001530717310000023
extracting a matrix for the structure group;
step three, solving of sparse coefficient and reconstruction of MRI image
Solving the constraint model by using an alternating direction iterative algorithm, wherein the constraint model can be respectively solved by a sparse coefficient AiSolving for the optimization object with the reconstructed image x to be estimated, where the sub-questions about the sparse coefficientsThe question can be expressed as:
Figure BDA0001530717310000024
where β is a regularization parameter, in order to solve the sub-problem, one of the sub-problems may be solved
Figure BDA0001530717310000025
And (3) carrying out transformation:
Figure BDA0001530717310000026
wherein
Figure BDA0001530717310000027
Then transforming W by inequality transformationiAnd AiConverting into a diagonal matrix form, and further converting the subproblem into:
Figure BDA0001530717310000028
wherein ΛiSum-sigmaiAre diagonal arrays, and the size of each diagonal element is respectively equal to WiAnd AiThe two norms of each row coefficient in the system, and then the generalized soft threshold method is used to estimate sigmaiThe size of each diagonal element in the series, thereby obtaining an estimated sparse coefficient AiAnd substituted into a sub-problem with respect to reconstructed image x:
Figure BDA0001530717310000029
the least squares problem can be solved by conjugate gradient method to reconstruct the final image.
The invention is characterized in that the structure group is sparsely represented by an orthogonal dictionary by utilizing the local sparsity and non-local similarity of the image; the constraint of simultaneous sparse coding optimization on the sparse coefficient is applied, and the estimation precision of the sparse coefficient is further improved; the sparse coefficients are estimated using a generalized soft threshold method and applied to the reconstruction of Magnetic Resonance Images (MRI).
The invention has the beneficial effects that: the structure group is sparsely represented by the orthogonal dictionary, so that the sparse representation performance of the structure group is optimized; meanwhile, sparse coding is adopted to constrain the sparse coefficient, and a generalized soft threshold method is utilized to estimate the coefficient, so that the estimation precision of the coefficient is improved, the finally estimated image not only has good overall visual effect, but also retains a large amount of details in the image, and the whole estimation result is closer to a true value.
The invention mainly adopts a simulation experiment method for verification, and all steps and conclusions are verified to be correct on MATLAB 8.0.
Drawings
FIG. 1 is a workflow block diagram of the present invention;
FIG. 2 is an MRI cardiac image artwork used in the simulation of the present invention;
FIG. 3 is the result of reconstruction of MRI cardiac images with a 30% sampling rate using the RecPF method;
FIG. 4 shows the result of reconstructing an MRI cardiac image with a sampling rate of 30% using the PBDW method;
FIG. 5 is a result of a reconstruction with the PANO method for MRI cardiac images with a sampling rate of 30%;
figure 6 is the result of reconstruction of an MRI cardiac image with a sampling rate of 30% using the method of the present invention.
Detailed Description
Referring to fig. 1, the invention is an MRI image reconstruction method based on simultaneous sparse coding under an orthogonal dictionary, and the specific steps include the following:
step one, obtaining of structure group
Carrying out initial reconstruction on input original K space data y by using a total variation method to obtain an initial reconstruction image x(0)Then in the reconstructed image x(0)Extracting target image block xiThen with the target image block xiFor searching the center, one by one within the search rangeComparing image block with target image block xiThe smaller the Euclidean distance is, the more similar the Euclidean distance is, and the target image block x is searched through the matching of the similar image blocksiAnd extracting the matrix using the structure group
Figure BDA0001530717310000031
Obtaining the structural group corresponding to the target image block
Figure BDA0001530717310000032
Step two, establishing a simultaneous sparse coding constraint model
Obtaining a structural group XiThen, the sparse coefficient is constrained by using the non-convex norm of the formula (1), then a vertical (2) constraint model about the image and the sparse coefficient is constructed by combining the reconstruction of the image, and the constraint model is further rewritten into an unconstrained expression:
Figure BDA0001530717310000041
wherein λ and β are both regularization parameters;
step three, solving of sparse coefficient and reconstruction of MRI image
Solving constrained models of images and sparse coefficients by using an alternating direction iterative algorithm, firstly dividing the model into two sub-problems about sparse coefficients and image reconstruction, wherein the sub-problem about sparse coefficients is shown as a formula (3), and in order to solve the sub-problem, the sub-problem in the formula (3) can be solved
Figure BDA0001530717310000042
The term is equivalently transformed according to equation (4) and then further according to the cauchy-schwarz inequality, yielding:
Figure BDA0001530717310000043
the sub-problem of equation (3) can be converted to equation (5) since ∑ isiWhere each coefficient is uncorrelated with each other, ΛiIs a known quantity and therefore can continue to be converted to scalar form:
Figure BDA0001530717310000044
each coefficient is then estimated using a generalized soft threshold method:
Figure BDA0001530717310000045
wherein the expression of the threshold is:
Figure BDA0001530717310000046
after obtaining the sparse coefficient by the generalized soft threshold method, the sub-problem about image reconstruction is shown as formula (6), and the least square problem can be solved by a conjugate gradient method to obtain:
Figure BDA0001530717310000047
and obtaining a reconstructed image, and iterating the whole reconstruction process until the time difference between two adjacent reconstruction results is less than an iteration termination threshold, thereby obtaining the final reconstructed MRI image.
The effect of the invention can be further illustrated by the following simulation experiment:
experimental conditions and contents
The experimental conditions are as follows: a cartesian sampling model was used for the experiment; the experimental image was a real heart MRI image, as shown in fig. 2; the experimental result evaluation index objectively evaluates the reconstruction result by adopting the peak signal-to-noise ratio (PSNR), and the higher the PSNR value is, the better the reconstruction result is, and the reconstruction result is closer to a real image.
The experimental contents are as follows: under the above experimental conditions, the reconstruction results were compared with the method of the present invention using the RecPF method, PBDW method, and PANO method, which are currently representative in the field of MRI image reconstruction.
Experiment 1: the images sampled in the figure 2 are reconstructed by the method of the invention, a RecPF method, a PBDW method and a PANO method respectively. The RecPF method is a traditional method for performing l on the whole image by using wavelet transform and total variation1The norm sparse constraint method, the reconstruction result of which is shown in fig. 3; the PBDW method firstly finds the optimal direction wavelet transform of the image block and adopts l1Coefficient constraint is carried out on the norm to realize MRI image reconstruction, and the reconstruction result is shown in figure 4; the PANO method is a typical method for performing three-dimensional wavelet transform on a structural group and using1The reconstruction method of the norm constraint sparse coefficient has a reconstruction result shown in fig. 5. The method of the invention sets the size of the image block in the experiment
Figure BDA0001530717310000051
The number m of image blocks in the structure group is 32, the maximum iteration time T is 100, and the iteration termination threshold η is 5 × 10-8(ii) a The final reconstruction result is fig. 6.
As can be seen from the reconstruction results and the enlarged partial areas of the methods in fig. 3, 4, 5 and 6, comparing the RecPF method, the PBDW method, the PANO method and the method of the present invention, it can be seen that the method of the present invention is higher in the detail of the reconstruction results than the other comparative methods.
TABLE 1 PSNR indicators for different reconstruction methods
Image of a person RecPF method PBDW process PANO process The method of the invention
MRI human brain picture 34.11 34.40 34.61 35.36
Table 1 shows PSNR indexes of the reconstruction results of the methods, where higher PSNR values indicate better reconstruction effects; compared with other methods, the method has the advantages that the PSNR value is greatly improved, the reconstruction result of the method is closer to a real image, and the result is consistent with a reconstruction effect graph.
The experiment shows that the reconstruction method of the invention has obvious reduction effect, rich image content after reconstruction and higher objective evaluation index, thereby showing that the invention is effective for medical image reconstruction.

Claims (1)

1. An MRI image reconstruction method based on simultaneous sparse coding under an orthogonal dictionary is characterized by comprising the following steps:
step one, obtaining of structure group
In order to realize the improvement of sparsity by utilizing simultaneous sparse coding and optimize the sparse coefficient of a similar image block set, namely a structure group under an orthogonal dictionary, a structure group corresponding to a target image block needs to be constructed, firstly, an image x after initial reconstruction is carried out(0)Extracting target image block xiThen, the Euclidean distance comparison method is used for comparing the target image block xiFinding corresponding similar image blocks for the search range of the center, and comparing the similar image blocks with the target image block xiConstructed as structural group Xi；
Step two, establishing a simultaneous sparse coding constraint model
Obtaining a structural group XiThen, in the process of simultaneous sparse coding, the non-convex norm pair structure group X is utilizediSparse coefficient set A under orthogonal dictionary DiCarrying out sparse constraint:
Figure FDA0003186395600000011
wherein p is more than 0 and less than 1, alphakRepresents a coefficient matrix AiOn the basis of the k line in (1), a constraint model about the image and the sparse coefficients is established:
Figure FDA0003186395600000012
wherein M is the number of structural groups, FuIn order to down-sample the fourier transform matrix,
Figure FDA0003186395600000013
extracting a matrix for the structure group, wherein y is input original K space data, and x is a reconstructed image to be estimated;
step three, solving of sparse coefficient and reconstruction of MRI image
Solving the constraint model by using an alternating direction iterative algorithm, wherein the constraint model can be respectively solved by a sparse coefficient AiAnd solving the reconstructed image x needing to be estimated as an optimization object, wherein the sub-problem about the sparse coefficient can be represented as:
Figure FDA0003186395600000014
where β is a regularization parameter, in order to solve the sub-problem, one of the sub-problems may be solved
Figure FDA0003186395600000015
And (3) carrying out transformation:
Figure FDA0003186395600000016
wherein
Figure FDA0003186395600000017
Then transforming W by inequality transformationiAnd AiConverting into a diagonal matrix form, and further converting the subproblem into:
Figure FDA0003186395600000021
wherein ΛiSum-sigmaiAre diagonal arrays, and the size of each diagonal element is respectively equal to WiAnd AiThe two norms of each row coefficient in the system, and then the generalized soft threshold method is used to estimate sigmaiThe size of each diagonal element in the series, thereby obtaining an estimated sparse coefficient AiAnd substituted into a sub-problem with respect to reconstructed image x:
Figure FDA0003186395600000022
wherein lambda is a regularization parameter, and the least square problem can be solved by a conjugate gradient method, so that a final MRI image is reconstructed.
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