CN110942495A - CS-MRI image reconstruction method based on analysis dictionary learning - Google Patents

CS-MRI image reconstruction method based on analysis dictionary learning Download PDF

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CN110942495A
CN110942495A CN201911276763.8A CN201911276763A CN110942495A CN 110942495 A CN110942495 A CN 110942495A CN 201911276763 A CN201911276763 A CN 201911276763A CN 110942495 A CN110942495 A CN 110942495A
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刘书君
蒲宁杰
曹建鑫
卢宏伟
张奎
李东
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Huanghu Science And Technology Co ltd
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Chongqing University
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Abstract

The invention discloses a CS-MRI image reconstruction method based on analysis dictionary learning, and belongs to the technical field of digital image processing. The method is a method for improving the image sparse representation capability by utilizing the analysis dictionary learning, solves the problem of insufficient adaptivity of the traditional fixed transformation, and does not increase the complexity of sparse coding. Firstly, an over-complete analysis dictionary learning model based on tight frame constraint is established, then an MRI reconstruction model is established by taking image block coefficients as objects, and finally, a model is solved by adopting an alternating direction multiplier method. When the method adopts the alternative direction multiplier method to solve the model, the analysis dictionary, the sparse coefficient and the reconstructed image are continuously updated, and the reconstructed image retains a large amount of detail information and obtains higher reconstruction performance, so that the method can be used for restoring the medical image to solve the problem of poor self-adaptability of the traditional fixed transformation.

Description

CS-MRI image reconstruction method based on analysis dictionary learning
Technical Field
The invention belongs to the technical field of digital image processing, and particularly relates to a method for realizing CS-MRI image reconstruction by utilizing analysis dictionary learning and improving the reconstruction quality of images.
Background
Compressed Sensing (CS), an emerging sampling theory, can accurately reconstruct sparse signals from fewer random measurements than the conventional nyquist sampling theorem. Since Magnetic Resonance Imaging (MRI) requires a long scan time to acquire image spectra, i.e., K-space data, not only is the risk of patient discomfort increased, but motion artifact phenomena may appear in the resulting images. Therefore, to shorten the scan time, the CS theory is applied to MRI to reconstruct an image using randomly undersampled K-space data, thereby speeding up the imaging speed.
The traditional CS-MRI reconstruction method utilizes the sparsity of the coefficient of the whole image under fixed transformation and adopts l1The norm is used as a sparse regularization term, and although a specific image texture structure can be recovered, the fixed transformation is difficult to adapt to diversified image characteristics, so that the performance of MRI image reconstruction is greatly limited. To overcome this drawback, a learning dictionary sparse representation based on image blocks is used in CS-MRI reconstruction. In the reconstruction process, the learning dictionary obtained from the image block training in the target reconstruction image can adaptively and sparsely represent the characteristic structure in the reconstruction image, and lower sparse representation errors are realized, so that the MRI image reconstruction performance is improved. Although the learning dictionary is more adaptive than the fixed transformation, the dictionary training process and the corresponding sparse coding are higher in complexity, and the dictionary atoms have larger correlation.
Disclosure of Invention
The invention aims to provide a CS-MRI image reconstruction method based on analysis dictionary learning by utilizing the adaptivity of a learning dictionary to an image. The method can improve the expression capability of the dictionary to the image without increasing the complexity of sparse coding; the method comprises the steps of firstly establishing an image reconstruction model based on analysis dictionary learning, and then effectively solving the model by adopting an alternating direction multiplier method. The method specifically comprises the following steps:
(1) establishing a learning model of an over-complete analysis dictionary based on tight frame constraint:
Figure BDA0002315769520000011
wherein
Figure BDA0002315769520000012
Is an over-complete analysis dictionary to be trained,
Figure BDA0002315769520000013
is a matrix of a fourier code and is,
Figure BDA0002315769520000014
is a matrix that is under-sampled,
Figure BDA0002315769520000015
is a matrix of a fourier transform and is,
Figure BDA0002315769520000016
representing a complex space, K is the base number of an analysis dictionary, N is the number of pixel points contained in an image block, M is the number of frequency points after encoding, N is the number of pixel points contained in the whole image, x is the image to be reconstructed,
Figure BDA0002315769520000021
representing the image block extraction matrix, λ is a regularization parameter,
Figure BDA0002315769520000022
represents the square of the two-norm of the vector, | · | | non-woven phosphor1Representing a norm of a vector, ΨHIs the conjugate transpose of Ψ, I is the unit matrix;
(2) introducing image block coefficients αi＝ΨRix, establish coefficients α for image blocksiImage reconstruction model of
Figure BDA0002315769520000023
Figure BDA0002315769520000024
Solving the reconstruction model by adopting an alternating direction multiplier method, and firstly establishing an augmented Lagrange function corresponding to the reconstruction model
Figure BDA0002315769520000025
Wherein
Figure BDA0002315769520000026
Is a function of the lagrange multiplier and,
Figure BDA0002315769520000027
is biThe method comprises the following steps of (1) conjugate transposing, wherein mu is more than 0 and is a punishment parameter, each optimization variable is alternately solved, and the Lagrange multiplier and the punishment parameter are updated, and the method can be converted into the following solving steps:
(2a) to solve for the analysis dictionary Ψ in t +1 iterations(t+1)The image block coefficients obtained in the t-th iteration may be used
Figure BDA0002315769520000028
Image x(t)And lagrange multiplier
Figure BDA0002315769520000029
By substituting an augmented Lagrangian function, i.e.
Figure BDA00023157695200000210
(2b) To solve the image block coefficient in t +1 iterations
Figure BDA00023157695200000211
The analysis dictionary Ψ obtained in the t +1 th iteration may be divided into(t+1)And the image x obtained in the t-th iteration(t)And lagrange multiplier
Figure BDA00023157695200000212
By substituting an augmented Lagrangian function, i.e.
Figure BDA00023157695200000213
(2c) To solve for image x in t +1 iterations(t+1)The analysis dictionary Ψ obtained in the t +1 th iteration may be(t+1)Image block coefficients
Figure BDA00023157695200000214
And lagrange multiplier obtained in the t-th iteration
Figure BDA00023157695200000215
By substituting an augmented Lagrangian function, i.e.
Figure BDA00023157695200000216
(2d) Updating lagrange multipliers
Figure BDA00023157695200000217
The following were used:
Figure BDA00023157695200000218
(2e) updating the penalty parameter mu(t+1)The following were used:
μ(t+1)＝ρμ(t)
where ρ > 1 is an increasing factor of μ.
(2f) And (4) repeating the steps (2a) to (2e) until the obtained estimated image meets the condition or the iteration number reaches a preset upper limit.
The invention has the innovation point that a CS-MRI reconstruction model based on over-complete analysis dictionary learning is established by utilizing the self-adaptive sparse representation advantage of the learning dictionary. The model can train a proper analysis dictionary aiming at different target reconstruction images, and the correlation among bases in the learning analysis dictionary is effectively inhibited by adopting a tight frame constraint. Therefore, the precision of sparse representation and the sparsity of image coefficients are improved, and meanwhile, the quality of a reconstructed image is greatly improved. Finally, aiming at the proposed reconstruction model, an alternative direction multiplier method is adopted for effective solution.
The invention has the beneficial effects that: an over-complete analysis dictionary is learned from the image block, so that the adaptivity of sparse representation of the image block is improved; the analysis dictionary is constrained by a tight frame, so that trivial solution of the analysis dictionary is avoided, and correlation between bases in the dictionary is inhibited; and solving the reconstructed model by adopting an alternating direction multiplier method, and realizing continuous and quick updating of the analysis dictionary, the sparse coefficient and the reconstructed image. Therefore, the finally output reconstructed image keeps more texture structures, better removes the smooth region artifacts and has higher reconstruction performance.
The invention mainly adopts a simulation experiment method for verification, and all steps and conclusions are verified to be correct on MATLAB 8.0.
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FIG. 1 is a flow chart of the present invention;
FIG. 2 is a human brain MRI artwork used in the simulation of the present invention;
FIG. 3 shows the result of different methods for reconstructing MRI images of the human brain at a sampling rate of 30%;
fig. 4 is an error of the reconstructed result of the MRI image of the human brain corresponding to the different methods with a sampling rate of 30%.
Detailed Description
Referring to fig. 1, the invention relates to a CS-MRI image reconstruction method based on analytical dictionary learning, which comprises the following specific steps:
step 1, establishing a CS-MRI image reconstruction model based on analysis dictionary learning.
(1a) Establishing a learning model of an over-complete analysis dictionary based on tight frame constraint:
Figure BDA0002315769520000041
in the formula (1), the reaction mixture is,
Figure BDA0002315769520000042
is an over-complete analysis dictionary to be trained,
Figure BDA0002315769520000043
is a matrix of a fourier code and is,
Figure BDA0002315769520000044
is a matrix that is under-sampled,
Figure BDA0002315769520000045
is a matrix of a fourier transform and is,
Figure BDA0002315769520000046
representing a complex space, K is the base number of an analysis dictionary, N is the number of pixel points of an image block, M is the number of frequency points after encoding, N is the number of pixel points contained in the whole image, x is the image to be reconstructed,
Figure BDA0002315769520000047
is the image block extraction matrix, λ is the regularization parameter,
Figure BDA00023157695200000417
represents the square of the two-norm of the vector, | · | | non-woven phosphor1Representing a norm of a vector, ΨHIs the conjugate transpose of Ψ, and I is the unit matrix.
(1b) Introducing image block coefficients αi＝ΨRix, establish coefficients α for image blocksiThe image reconstruction model of (1):
Figure BDA0002315769520000048
and 2, solving the reconstruction model in the formula (2) by adopting an alternative direction multiplier method.
(2a) Establishing an augmented Lagrangian function of a reconstruction model in the step (2):
Figure BDA0002315769520000049
in the formula (3), the reaction mixture is,
Figure BDA00023157695200000410
is a function of the lagrange multiplier and,
Figure BDA00023157695200000411
is biAnd (3) conjugate transpose, wherein mu is more than 0 and is a penalty parameter, and each variable is alternately solved and the Lagrange multiplier and the penalty parameter are updated.
(2b) To solve for the analysis dictionary Ψ in t +1 iterations(t+1)The image block coefficients obtained in the t-th iteration may be used
Figure BDA00023157695200000412
Image x(t)And lagrange multiplier
Figure BDA00023157695200000413
In formula (3), namely:
Figure BDA00023157695200000414
the equation (4) can be solved by using a singular value decomposition method, and for simplifying the expression, the iterative superscript in the equation (4) is omitted in the following solving steps:
(2b1) the subproblem is an optimization problem with equality constraints, whose Lagrangian function is
Figure BDA00023157695200000415
Wherein the content of the first and second substances,
Figure BDA00023157695200000416
is a Lagrange multiplier, due to the equality constraint ΨHΨ ═ I is the hermitian matrix equation, so Ω must be the hermitian matrix, i.e., Ω ═ ΩH. According to the Karush-Kuhn-Tucker condition, the optimal solution of the subproblem should satisfy the following conditions:
Figure BDA0002315769520000051
wherein the content of the first and second substances,
Figure BDA0002315769520000052
is a Lagrangian function
Figure BDA0002315769520000053
With respect to the gradient of Ψ, (R)ix)HIs RiConjugate transpose of x, left-multiplying Ψ to the first equation in equation (6)HAnd substituted into the second equation ΨHΨ ═ I, available
Figure BDA0002315769520000054
Substituting formula (7) for formula (6) and increasing the constraint condition Ω ═ ΩHThe following conditions can be simplified:
Figure BDA0002315769520000055
(2b2) calculating the singular value decomposition of the matrix on the right side of the first equation in equation (8) yields:
Figure BDA0002315769520000056
wherein the content of the first and second substances,
Figure BDA0002315769520000057
and
Figure BDA0002315769520000058
for corresponding unitary matrices, P, after singular value decompositionHIs the conjugate transpose of P, VHIs the conjugate transpose of V,
Figure BDA0002315769520000059
is a matrix of singular values. The singular value decomposition result is substituted into the third equation in formula (8), and can obtain:
ΨHPΔVH＝VΔPHpsi type (10)
When t isHP ═ V or ΨHWhen P is 0, expression (10) is established. Then, the singular value decomposition result is substituted into (A)8) In the first equation, we can get:
ΨΨHp is P type (11)
Since P is a non-zero matrix, ΨHP-0 is not true, i.e. only ΨHP ═ V can make equations (10) and (11) hold simultaneously, and substituting it into equation (11) yields the Ψ -closed solution form as follows:
Ψ＝PVHformula (12)
Since P and V are unitary matrices, ΨHΨ＝VPHPVHI, i.e., formula (8), are satisfied.
(2c) To solve the image block coefficient in t +1 iterations
Figure BDA00023157695200000510
The analysis dictionary Ψ obtained in the t +1 th iteration(t+1)And the image x obtained in the t-th iteration(t)And lagrange multiplier
Figure BDA00023157695200000511
Substitution formula (3), namely:
Figure BDA0002315769520000061
the subproblem in equation (13) can be solved by using a soft threshold method, and for simplifying expression, iterative superscript in equation (13) is omitted in the following solving steps:
(2c1) the sub-problem is an unconstrained convex optimization problem with an order of optimization condition of
Figure BDA0002315769520000062
Wherein the content of the first and second substances,
Figure BDA0002315769520000063
is αiA sub-differential of a norm whose k-th dimension element can be expressed as
Figure BDA0002315769520000064
Wherein [ -1,1 [ ]]Is shown as (α)i)kThe k-th dimension element of the secondary differential can take any number between-1 and 1;
(2c2) first order optimization conditions according to equation (14) and
Figure BDA0002315769520000065
αiIs solved as the subtended quantity Ψ Rix-biMu dimension elements are soft-thresholded to 1/mu, αiThe solution is in the form:
Figure BDA0002315769520000066
where max (·, ·) is a maximum function.
(2d) To solve for image x in t +1 iterations(t+1)The analysis dictionary Ψ obtained in the t +1 th iteration may be(t+1)Coefficient of image block
Figure BDA0002315769520000067
And lagrange multiplier obtained in the t-th iteration
Figure BDA0002315769520000068
Substitution formula (3), namely:
Figure BDA0002315769520000069
equation (17) is a least squares problem, and for simplicity of expression, the iterative superscript in equation (17) is omitted in the following solving step:
(2d1) the sub-problem optimal solution may be obtained by solving the following normal equations:
Figure BDA00023157695200000610
wherein the content of the first and second substances,
Figure BDA00023157695200000611
is FuThe conjugate transpose of (a) is performed,
Figure BDA00023157695200000612
is Ri，jThe conjugate transpose of (1);
(2d2) the image blocks can be extracted according to the period boundary condition to meet the requirement
Figure BDA00023157695200000613
The closed solution for x can be expressed as:
Figure BDA0002315769520000071
in the formula (19), FHIs the conjugate transpose of F, UHIs the conjugate transpose of U, and c > 0 is an overlap factor.
(2e) Updating lagrangian multipliers and penalty parameters:
Figure BDA0002315769520000072
in the formula (20), ρ > 1 is an increasing factor of μ.
And 3, repeating the processes from (2b) to (2e) until the obtained estimated image meets the condition or the iteration number reaches a preset upper limit.
The effect of the invention can be further illustrated by the following simulation experiment:
experimental conditions and contents
The experimental conditions are as follows: the experiment used a random sampling matrix; the experimental image is shown in fig. 2 by using a real human brain MRI image; the evaluation index of the experimental result adopts peak signal to noise ratio (PSNR), which is defined as:
Figure BDA0002315769520000073
in formula (21), x and
Figure BDA0002315769520000074
are respectively full sampling chartsImages and reconstructed images, a higher value of which indicates a better performance of the reconstructed image. Another evaluation index used was Structural Similarity (SSIM), defined as:
Figure BDA0002315769520000075
in formula (22), x and
Figure BDA0002315769520000076
respectively a fully sampled image and a reconstructed image, muxAnd
Figure BDA0002315769520000077
respectively represent x and
Figure BDA0002315769520000078
mean value of (a)xAnd
Figure BDA0002315769520000079
respectively represent x and
Figure BDA00023157695200000712
the standard deviation of (a) is determined,
Figure BDA00023157695200000710
is x and
Figure BDA00023157695200000711
covariance of (C)1And C2Are two constants that avoid instability. A higher SSIM value indicates a higher reconstruction quality of the image.
The experimental contents are as follows: under the conditions, the RecPF method, the DLMRI method and the PBDW method which are at the leading level in the field of MRI reconstruction are adopted to compare with the method of the invention.
Experiment 1: the MRI images shown in FIG. 2 are reconstructed under the same conditions by the method of the present invention and the RecPF method, DLMRI method and PBDW method, respectively. The RecPF method combines wavelet and total variation regularization terms and adopts an operator separation algorithm to solve a model, the reconstruction result is shown in figure 3(a), and the reconstruction error is shown in figure 3FIG. 4 (a); the DLMRI method is a typical integrated dictionary learning method, a redundant integrated dictionary is learned for an image block by using a K-SVD method, the reconstruction result is shown in fig. 3(b), and the reconstruction error is shown in fig. 4 (b); the PBDW method is to train direction wavelet and use wavelet coefficient l for image block1The norm minimization method, the reconstruction result of which is fig. 3(c), and the reconstruction error of which is fig. 4 (c); in the experiment, the image block size n is set to 8 × 8 for all the methods, and the fidelity term regularization parameter λ is set to 106Setting the regularization parameter to 10 for total variation in RecPF-2The overlap factor for PBDW and the method is set to c 64, the number of dictionary atoms or bases for DLMRI and the method is set to K128, and the other parameter for the method of the invention is set to μ(0)128, ρ 1.2, the final reconstruction result of the method is shown in fig. 3(d), and the reconstruction error is shown in fig. 4 (d).
In fig. 3, the image in the small square is the selected magnified region and the image in the large square is the magnified image thereof. As can be seen from the reconstruction result and the enlarged partial view of fig. 3, the reconstruction result of the RecPF method has a more serious artifact phenomenon; the DLMRI method has the advantages that the texture structure of an amplified region is blurred, and the loss of detail information is serious; the PBDW method has obvious block artifacts in an enlarged area; the reconstruction result of the method shows that the contrast of the detail information and the overall reconstruction effect are superior to those of other reconstruction methods. The same conclusion can be drawn from the reconstruction error result of fig. 4, and the reconstruction error of the method is significantly smaller than that of the RecPF method, the DLMRI method and the PBDW method, so that the reconstruction effect of the method is the best.
TABLE 1 PSNR indicators for different reconstruction methods
Image of a person RecPF method DLMRI method PBDW process The method of the invention
Human brain picture 27.43 31.58 31.67 33.27
The PSNR indexes of the reconstruction results of the methods are shown in table 1, and it can be seen that the PSNR values of the method of the present invention are greatly improved compared with those of other methods, which indicates that the reconstruction performance of the method of the present invention is the highest, and the result is consistent with the reconstruction effect graph.
TABLE 2 SSIM index for different reconstruction methods
Image of a person RecPF method DLMRI method PBDW process The method of the invention
Human brain picture 0.5559 0.6972 0.8507 0.8574
Table 2 shows the SSIM of the reconstruction result of each method, and it can be seen that the SSIM value corresponding to the method of the present invention is the highest, which indicates that the image information is completely protected, and the result matches with the reconstruction effect map.
The experiments show that the reconstructed image obtained by the invention has complete detail information and better visual effect and objective evaluation index, so that the invention is effective to the reconstruction of the medical image.

Claims (4)

1. A CS-MRI image reconstruction method based on analysis dictionary learning comprises the following steps:
(1) establishing learning model based on over-complete analysis dictionary under tight frame constraint
Figure FDA0002315769510000011
Wherein
Figure FDA0002315769510000012
Is an over-complete analysis dictionary to be trained,
Figure FDA0002315769510000013
is a matrix of a fourier code and is,
Figure FDA0002315769510000014
is a matrix that is under-sampled,
Figure FDA0002315769510000015
is a matrix of a fourier transform and is,
Figure FDA0002315769510000016
representing a complex space, K is the base number of an analysis dictionary, N is the number of pixel points contained in an image block, M is the number of frequency points after encoding, N is the number of pixel points contained in the whole image, x is the image to be reconstructed,
Figure FDA0002315769510000017
representing the image block extraction matrix, λ is a regularization parameter,
Figure FDA0002315769510000018
represents the square of the two-norm of the vector, | · | | non-woven phosphor1Representing a norm of a vector, ΨHIs the conjugate transpose of Ψ, I is the unit matrix;
(2) introducing image block coefficients αi＝ΨRix, establish coefficients α for image blocksiThe image reconstruction model of (1):
Figure FDA0002315769510000019
Figure FDA00023157695100000110
solving the reconstruction model by adopting an alternating direction multiplier method, and firstly establishing an augmented Lagrange function corresponding to the reconstruction model
Figure FDA00023157695100000111
Wherein
Figure FDA00023157695100000112
Is a function of the lagrange multiplier and,
Figure FDA00023157695100000113
is biThe method comprises the following steps of (1) conjugate transposing, wherein mu is more than 0 and is a punishment parameter, each optimization variable is alternately solved, and the Lagrange multiplier and the punishment parameter are updated, and the method can be converted into the following solving steps:
(2a) to solve for the analysis dictionary Ψ in t +1 iterations(t+1)The image block coefficients obtained in the t-th iteration are compared
Figure FDA00023157695100000114
Image x(t)And lagrange multiplier
Figure FDA00023157695100000115
By substituting an augmented Lagrangian function, i.e.
Figure FDA00023157695100000116
(2b) To solve the image block coefficient in t +1 iterations
Figure FDA00023157695100000117
The analysis dictionary Ψ obtained in the t +1 th iteration may be divided into(t+1)And the image x obtained in the t-th iteration(t)And lagrange multiplier
Figure FDA00023157695100000118
By substituting an augmented Lagrangian function, i.e.
Figure FDA00023157695100000119
(2c) To solve for image x in t +1 iterations(t+1)The analysis dictionary Ψ obtained in the t +1 th iteration may be(t+1)Coefficient of image block
Figure FDA0002315769510000021
And lagrange multiplier obtained in the t-th iteration
Figure FDA0002315769510000022
By substituting an augmented Lagrangian function, i.e.
Figure FDA0002315769510000023
(2d) Updating lagrange multipliers
Figure FDA0002315769510000024
As follows
Figure FDA0002315769510000025
(2e) Updating the penalty parameter mu(t+1)As follows
μ(t+1)＝ρμ(t)
Where ρ > 1 is an increasing factor of μ;
(2f) and (4) repeating the steps (2a) to (2e) until the obtained estimated image meets the condition or the iteration number reaches a preset upper limit.
2. The CS-MRI image reconstruction method based on analysis dictionary learning as claimed in claim 1, wherein the sub-problem concerning the analysis dictionary Ψ in the step (2a) can be solved by singular value decomposition, and for simplifying the expression, the iterative superscript in the step (2a) is omitted in the following solving step, and the closed solution form of the sub-problem is
Ψ＝PVH
Wherein
Figure FDA0002315769510000026
And
Figure FDA0002315769510000027
is a unitary matrix corresponding to the decomposed following singular values
Figure FDA0002315769510000028
Wherein
Figure FDA0002315769510000029
Is a singular value matrix, since P and V are unitary matrices, ΨHΨ＝VPHPVH＝Ι，PHIs the conjugate transpose of P, VHIs the conjugate transpose of V, i.e. the solution to Ψ satisfies the tight frame condition.
3. The CS-MRI image reconstruction method based on analytical dictionary learning according to claim 1, characterized in that the coefficients for the image blocks in (2b) can be reconstructed by soft threshold method
Figure FDA00023157695100000210
The sub-problem solution of (2b) is omitted in the following solution step for simplifying the expression, and the form of the sub-problem solution is as follows
Figure FDA00023157695100000211
Where max (·, ·) is a maximum function.
4. The CS-MRI image reconstruction method based on analytical dictionary learning as claimed in claim 1, wherein the sub-problem about the image x in (2c) is a least square problem, and for simplifying the expression, the iterative superscript in (2c) is omitted in the following solving step
(2c1) The optimal solution to this sub-problem can be obtained by solving the following normal equations:
Figure FDA0002315769510000031
wherein the content of the first and second substances,
Figure FDA0002315769510000032
is FuThe conjugate transpose of (a) is performed,
Figure FDA0002315769510000033
is Ri，jThe conjugate transpose of (c).
(2c2) To simplify the calculation, the image blocks may be extracted according to a period boundary condition to satisfy
Figure FDA0002315769510000034
Where c is an overlap factor, the closed solution for image x can be expressed as:
Figure FDA0002315769510000035
wherein, FHIs the conjugate transpose of F, UHIs the conjugate transpose of U.
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