Mri reconstruction method based on structural group vectorization learning and log ratio constraint

Abstract

The application discloses an MRI reconstruction method based on structural group vectorization learning and log ratio constraint, and belongs to the field of digital image processing. The method is an MRI image reconstruction method which takes structural group vectorization form as a processing object, establishes a sparse representation model based on transformation learning, and constructs a log ratio function as a regular term. Firstly, pre-reconstruction is performed on MRI image undersampling data to obtain a similar block set, then the structural group is classified through a k-means method, and a structural group vectorization transformation matrix is learned by using a classification result, so that the limitation of sparse representation capability caused by a fixed base in structural group 3D transformation is broken through. Meanwhile, the application defines a log ratio regular term to realize coefficient sparsity constraint, and solves the model through an alternating direction multiplier method. Not only can the method highly sparsely represent an image, but also the log ratio function more closely approximates an l0 norm, so that a reconstruction result retains a large amount of details and obtains high reconstruction performance, and therefore the method can be used for medical image recovery.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of digital image processing, and particularly relates to constructing a structural group vectorization learning and logarithmic ratio constraint model by using image sparsity and non-local similarity to realize MRI image reconstruction for high-quality recovery of medical images. BACKGROUND

[0002] As a non-invasive and non-radiation imaging technique, magnetic resonance imaging (MRI) can provide clear anatomical information and high soft tissue contrast. However, the relatively slow scanning speed can lead to reduced spatial resolution and motion artifacts, limiting the application of MRI. As a popular and fundamental method, the theory of compressed sensing (CS) has been proven to be able to accurately reconstruct sparse signals from measurements below the Nyquist sampling rate. By applying the CS theory to MRI, a new imaging technique called CS-MRI is born, which can greatly speed up the imaging speed of MRI images.

[0003] In CS-MRI, the commonly used MRI image prior is the sparsity of the image. The higher the sparsity, the smaller the error in the reconstructed image. Therefore, finding a suitable sparse representation of the MRI image is a crucial step in CS-MRI. Traditional CS-MRI methods choose fixed wavelets, finite differences, and contourlets as sparse transforms, but they can only sparsely represent a few types of image features, resulting in suboptimal reconstruction quality. To adaptively represent different images, local image blocks are used as the basic unit of sparse representation. By learning a redundant dictionary from local image blocks, each image block can be linearly represented by a small number of internal elements, achieving sparse coding of the image. However, independent sparse representation of image blocks ignores the correlation between image blocks, limiting the quality of the reconstructed image. To solve this problem and further enhance the sparsity of the image representation coefficients, the set of similar image blocks can be first classified, and then a structural group vectorization learning model can be constructed for different classification results. By utilizing the non-local similarity of the image, the adaptive sparse representation of the set of similar image blocks can have better sparsity, and then a logarithmic ratio function can be used as a regularization term to further constrain and enhance the sparsity of the sparse representation coefficients, resulting in a higher quality of the reconstructed result. SUMMARY

[0004] The purpose of the present application is to combine adaptive sparse transform and non-local similarity of images, and propose an MRI reconstruction method based on structural group vectorization learning and log ratio constraint. The method constructs a corresponding structural group vectorization transform learning model for each classified structural group, greatly enhances the sparsity of the structural group sparse representation coefficient, and defines a log ratio function as the regularization term of the MRI reconstruction model. Compared with the existing approximation function of l0 norm, the function more accurately approximates the l0 norm, further describes the sparsity of the structural group sparse representation coefficient, and in order to accelerate the solution of the model of the present application, an iterative formula with convergence guarantee is proposed, so that the final reconstruction performance is greatly improved. Specifically, the method comprises the following steps:

[0005] 1. An MRI reconstruction method based on structural group vectorization learning and log ratio constraint, comprising the following steps:

[0006] (1) inputting k-space sampling data y of an MRI image, pre-reconstructing y by using a traditional method to obtain an initial reconstruction image x (0) ;

[0007] (2) in order to fully utilize the correlation between image blocks, taking the vectorization form of a similar image block set, i.e., a structural group, as an object, a sparse representation model based on structural group vectorization transform learning is established:

[0008] (2a) finding similar image blocks in the initial reconstruction image, and constructing an i-th structural group with m most similar image blocks wherein n is the number of pixels in the image block, x is the image to be reconstructed, is an image block extraction matrix, represents a complex space, and N is the number of pixel points contained in the whole image;

[0009] (2b) dividing all structural groups obtained into K categories by using a k-means method, and establishing a sparse representation model under structural group vectorization transform learning for the structural groups X i in the r-th category classification result

[0010]

[0011] wherein represents the vectorization result of the structural group X i , G r represents a transform matrix learned for the structural groups X i in the r-th category classification result, and each category classification result corresponds to a G r , is the sparse representation coefficient of the structural group vectorization ;

[0012] (3) in order to enhance the sparsity of the sparse coefficient To address the sparsity of MRI, a log-ratio function is defined and used as a regularization term to establish a vectorized transformation learning and log-ratio-constrained MRI reconstruction model for structure groups.

[0013]

[0014] Where λ>0 is a constant. I represents the square of the l2 norm of a matrix. nm C represents an identity matrix of size nm × nm. r This represents the index set of the structure group in the classification results of the r-th class. The logarithmic ratio function is defined as follows: Where log(·) represents the natural logarithm function, k>0 is a constant, and j is a function of the natural logarithm. The element index number, α i,j express The j-th element in the array, |·| represents the absolute value, ε1 and ε2 are constants, and ε1>0, ε2>1, ε1<ε2. Indicates any, It is G r The conjugate transpose of the function is used; the alternating direction multiplier method is employed to solve the above reconstructed model. First, the augmented Lagrangian function of the reconstructed model is established:

[0015]

[0016] Where d i Let represent the Lagrange multiplier, <·> represent the matrix inner product, and μ>0 be the penalty parameter. Alternatingly solving for each optimization variable and updating the Lagrange multipliers and penalty parameter can be transformed into the following solution steps:

[0017] (3a) Obtained in the t-th iteration Given x, solve for variable G in the (t+1)th iteration. r The problem can be transformed into a least-squares subproblem with equality constraints using the augmented Lagrangian function.

[0018]

[0019] (3b) Obtain x in the t-th iteration and G in the (t+1)-th iteration. r In the case of solving the variable in the (t+1)th iteration The problem can be transformed into a convex optimization problem using the augmented Lagrangian function.

[0020]

[0021] (3c) Obtained in the (t+1)th iteration and G rIn the case, the problem of solving the variable x in the t+1 iteration can be transformed into a typical least square problem by the augmented Lagrange function

[0022]

[0023] (3d) updating the Lagrange multiplier d in the t+1 iteration i , first multiply the obtained μ in the t iteration and the obtained in the t+1 iteration to obtain Then, the sum of the obtained d i in the t iteration and can realize the update of d i ;

[0024] (3e) updating the penalty parameter μ in the t+1 iteration can be realized by multiplying the constant c>1 and the obtained μ in the t iteration;

[0025] (3f) repeating steps (3a)-(3e) until the obtained estimated image satisfies the condition or the iteration number reaches the preset upper limit.

[0026] The innovation point of the application proposes a MRI reconstruction method based on structure group vectorization learning and log ratio constraint, the structure group vectorization result is transformed and learned, the sparse representation ability is improved, and more accurate sparse coefficients are obtained through the log ratio constraint. Compared with the previous CS-MRI model using learning dictionary or transformation, the method proposed in the application not only fully utilizes the non-local similarity between images but also realizes adaptive learning of the structure group transformation matrix of different images. The log ratio regular term proposed in the application can realize better approximation of the l0 norm than the existing approximation function of the l0 norm, so as to strengthen the sparsity of the sparse representation coefficient. Finally, in order to speed up the solution of the model, an iterative algorithm with convergence guarantee is proposed to solve the log ratio regular term. For the reconstruction model, the alternating direction multiplier method is used to alternately solve each variable, so as to realize fast solution of the model.

[0027] The application has the beneficial effects that the adaptive sparse transformation of structure group vectorization and the log constraint are applied to the CS-MRI technology to improve the image reconstruction quality; the established structure group vectorization transformation learning model can be adapted to different images, so that the reconstruction result has better visual effect, the sparse representation coefficient obtained through the structure group vectorization transformation has better sparsity, and the log ratio function constraint is applied to the sparse representation coefficient, so that the sparsity can be better described. Therefore, the finally obtained reconstructed image not only has good overall visual effect, but also improves the contrast of the detail texture structure, and is closer to the real image.

[0028] The present application mainly adopts the method of simulation experiment to verify, all steps, conclusions are verified correct on MATLAB8.0. BRIEF DESCRIPTION OF DRAWINGS

[0029] Figure 1 It is the flow chart of the present application.

[0030] Figure 2 It is the MRI original drawing of human brain used in simulation of the present application.

[0031] Figure 3 It is the reconstruction result of human brain MRI image with sampling rate of 25% using different methods.

[0032] Figure 4 It is the error of reconstruction result of human brain MRI image with sampling rate of 25% using different methods. DETAILED DESCRIPTION

[0033] Reference Figure 1 , the present application is a kind of MRI reconstruction method based on structure group vectorization learning and log ratio constraint, specific steps are as follows:

[0034] Step 1, input the k-space sampling data y of an MRI image, pre-reconstruction is carried out to y using traditional method, and initial reconstruction image x is obtained (0) .

[0035] Step 2, in order to make full use of the correlation within and between MRI image blocks, the vectorization form of similar image block set, i.e. structure group, is taken as object, and the sparse representation model based on structure group vectorization transformation learning is established:

[0036] (2a) find similar image blocks for initial reconstruction image, and the i th structure group is constructed with m most similar image blocks Wherein n is the number of pixels in image block, x is the MRI image to be reconstructed, is image block extraction matrix, represents complex space, and N is the number of pixel points contained in the whole image;

[0037] (2b) all structure groups obtained are classified into K categories using k-means method, and the structure group X i of r th classification result is established under the sparse representation model of structure group vectorization transformation learning

[0038]

[0039] Wherein represents the vectorization result of structure group X i , G r represents the structure group X iThe learned transformation matrix, each classification result corresponds to a G r , Sparse representation coefficients of the structure group .

[0040] Step 3, to enhance the sparsity of the enhancement coefficients , a log-ratio function is defined and the MRI reconstruction model of structure group vectorization transformation learning and log-ratio constraint is established by taking the function as a regularization term:

[0041]

[0042] Wherein λ>0 is a constant, represents the square of the matrix l2 norm, I nm represents an nm×nm size identity matrix, C r represents the index set of the structure group in the rth classification result, represents the log-ratio function, which is defined as Wherein log(·) represents the natural logarithm function, k>0 is a constant, j is the element index number in i,j , α represents the jth element in , |·| represents the absolute value, ε1, ε2 are constants, and ε1>0, ε2>1, ε1<ε2, is the conjugate transpose of G r ; the above reconstruction model is solved by using the alternating direction multiplier method, and first the augmented Lagrange function of the reconstruction model is established

[0043]

[0044] Wherein d i represents the Lagrange multiplier, <·> represents the matrix inner product, μ>0 is the penalty parameter

[0045] (3a) in the case of obtaining and x in the tth iteration, the problem of solving the variable G r in the (t+1) th iteration can be transformed into a constrained least squares problem by using the augmented Lagrange function

[0046]

[0047] The problem can be solved by the method of singular value decomposition:

[0048] (3a1) calculate the singular value decomposition

[0049]

[0050] Wherein and are the left and right singular matrices, respectively, denotes the conjugate transpose of V r is a singular value matrix, denotes the conjugate transpose of

[0051]

[0052] (3b) Given x r and G in the tth iteration, the problem of finding

[0053]

[0054] The optimal solution of (7) can be obtained by transforming it into a scalar optimization problem with respect to the elements a i,j of :

[0055] (3b1) Let (3b) can be transformed into

[0056]

[0057] (3b2) The optimization problem in (3b1) can be further decomposed into a scalar optimization problem with respect to a i,j :

[0058]

[0059] where γ i,j denotes the jth element of the vector γ i :

[0060] (3b3) According to the Cauchy-Schwarz inequality and letting the scalar optimization problem with respect to a i,j can be further simplified as

[0061]

[0062] (3b4) Based on the above equation, the iterative formula for a i,j can be obtained by using the gradient descent method as

[0063]

[0064] ​​where superscripts l and l-1 are iteration numbers, p > 0 is the step size of gradient descent, f'(a i,j ) is the first derivative of f(a i,j ); after iteration of equation (11) , since 0 is also a local optimal solution of equation (10), further comparison is needed to obtain the global optimal solution of equation (10), the optimal solution of a i,j is

[0065]

[0066] (3c) Given x and G r in the t+1th iteration, solving the problem of variable x in the t+1th iteration can be transformed into a typical least square problem by using the augmented Lagrangian function

[0067]

[0068] The optimal solution of equation (13) can be obtained by solving the corresponding normal equation, but it involves high complexity large-scale non-diagonal matrix inversion, in order to effectively solve it, the alternating direction multiplier method is used to solve equation (13):

[0069] (3c1) Transform the objective function in equation (13):

[0070]

[0071] (3c2) Introduce an intermediate variable s and add the constraint condition x = s, equation (14) can be transformed into

[0072]

[0073] Equation (15) is a least square optimization problem with equality constraints, write its augmented Lagrangian function

[0074]

[0075] where d is the Lagrange multiplier, d H is the conjugate transpose of d, τ > 0 is the penalty parameter; in order to improve the calculation speed, the single iteration alternating direction multiplier method is used to calculate the approximate solution of x; the initial x is assigned to the value of x obtained in the tth iteration, and d = 0, first fix x, solve the sub-problem about s

[0076]

[0077] The closed solution of equation (17) is

[0078]

[0079] Where τ>0 is a small constant, I N It is an N×N identity matrix, where N is the number of pixels in the entire image. It is R i,j The conjugate transpose of equation (18) Since it is a diagonal matrix, the inversion operation can be directly achieved by taking the reciprocals of the diagonal elements; after obtaining the optimal solution for s, fix s and solve the subproblems about x.

[0080]

[0081] The closed-form solution of equation (19) is

[0082]

[0083] Where F is the Fourier transform matrix, F H It is the conjugate transpose of F, and U is the undersampling matrix. H It is the conjugate transpose of U; thus, an approximate solution for reconstructing the image x is obtained;

[0084] (3d) Update the Lagrange multiplier d in the (t+1)th iteration. i First, combine the μ obtained in the t-th iteration and the μ obtained in the (t+1)-th iteration. Multiply to get Then calculate d obtained in the t-th iteration. i and The sum of d can then be used to achieve the desired result for d. i Update;

[0085] (3e) In the (t+1)th iteration, the penalty parameter μ can be updated by multiplying the constant c>1 by the μ obtained in the tth iteration.

[0086] Step 4: Repeat steps (3a) to (3e) until the estimated image meets the conditions or the number of iterations reaches the preset upper limit.

[0087] The effects of this invention can be further illustrated by the following simulation experiments:

[0088] I. Experimental Conditions and Content

[0089] Experimental conditions: The experiment used a Cartesian sampling matrix; the experimental images were real human brain MRI images, such as... Figure 2 As shown; the peak signal-to-noise ratio (PSNR) and structural similarity score (SSIM) are used to objectively evaluate the reconstruction results. PSNR and SSIM are defined as follows:

[0090]

[0091] Where x and are the full-sampling image and the reconstructed image, respectively, μ x and are the mean of x and , σ x and are the standard deviation of x and , C1 and C2 are two constants to avoid instability. The higher the PSNR and SSIM values are, the higher the quality of the reconstructed image is. are the covariance of x and

[0092] Experimental content: Under the above experimental conditions, the FDLCP method, the PANO method, the NLR method, the M-GSOD method and the method of the present application were compared.

[0093] Experiment 1: The method of the present application and the FDLCP method, the PANO method, the NLR method and the M-GSOD method were used to reconstruct the MRI image shown in Figure 2 under the same conditions. The FDLCP method takes the image block as a sparse representation unit, and uses a learned multi-class orthogonal dictionary and an l1 norm to construct an MRI reconstruction model, and the reconstruction result is as shown in Figure 3 (a), and the reconstruction error is as shown in Figure 4 (a); the PANO method uses the sparse l1 norm of the 3D Haar wavelet of the structure group as a regularization term, and uses the sparsity of the structure group under the 3D wavelet transform, and the reconstruction result is as shown in Figure 3 (b), and the reconstruction error is as shown in Figure 4 (b); the NLR method uses the low-rank property of the image block as prior knowledge, and uses the logdet function as a constraint regularization term of the low-rank property, and the reconstruction result of the NLR method is as shown in Figure 3 (c), and the reconstruction error is as shown in Figure 4 (c); the M-GSOD method enhances the low-rank structure of the structure group through the Schatten-p norm, and the reconstruction result is as shown in Figure 3 (d), and the reconstruction error is as shown in Figure 4 (d); in the experiment, the similar image block size of all methods was set to n = 8 x 8, and the regularization parameter λ = 10 6 , the other parameters of the comparison method were set to the best performance state, and the other parameters of the method of the present application were set to: K = 32, k = 10, ε1 = 1, ε2 = 1.2, L = 5, c = 1.2, τ = 0.01, and the final reconstruction result of the method is as shown in Figure 3 (e), and the reconstruction error is as shown in Figure 4 (e).

[0094] Figure 3 ​The image in the small square is the selected enlarged region, and the image in the large square is the enlarged image thereof. Figure 3 It can be seen from the reconstruction results that the reconstruction result of the FDLCP method has relatively rough image details and has a blocking effect; the PANO method has artifacts in the smooth region; the NLR method suppresses the visible artifacts but loses many image details; the M-GSOD method has blurred textures; compared with the above methods, the method of the application suppresses most of the artifacts and retains more image details; from Figure 4 It can also be seen from the reconstruction result error that the reconstruction result error of the method of the application is obviously smaller than that of the FDLCP method, the PANO method, the NLR method and the M-GSOD method.

[0095] Table 1 PSNR index of different reconstruction methods

[0096]

[0097] Table 1 shows the PSNR index of the reconstruction results of each method, wherein the higher the PSNR value is, the better the reconstruction effect is, and the highest PSNR value is marked in bold, and it can be seen from Table 1 that the method of the application has a greater improvement than other methods, and this result is consistent with the visual effect of the reconstruction result.

[0098] Table 2 SSIM index of different reconstruction methods

[0099]

[0100] Table 2 shows the SSIM of the reconstruction results of each method, wherein the higher the SSIM value is, the closer the reconstruction result is to the real image, and the highest SSIM value is marked in bold, and it can be seen that the SSIM value corresponding to the method of the application is the highest, and the reconstruction result is closer to the original image, and this result is consistent with the reconstruction effect diagram.

[0101] The above experiments show that the reconstruction image obtained by the application not only suppresses most of the artifacts but also retains more image details, and the visual effect and the objective evaluation index are good, and thus it can be seen that the reconstruction of the MRI image by the application is effective.

Claims

1. A method for MRI reconstruction based on structured group vectorization learning and log-ratio constraint, comprising the following steps: (1) input the k-space sampling data y of an MRI image, pre-reconstruct y by using the traditional method to obtain an initial reconstructed image x (0) ; (2) To make full use of the correlation between the blocks in the MRI image, the sparse representation model based on structured group vectorization transformation learning is established for the vectorization form of the similar image block set, i.e. the structured group: (2a) finding similar image blocks to the initial reconstructed image to construct the i-th structure group with m most similar image blocks where n is the number of pixels in the image block, x is the MRI image to be reconstructed, (j = 1, 2, …, m) is the image block extraction matrix, represents the complex space, and N is the number of pixels contained in the entire image. (2b) using k-means method to classify all the structures into K categories, and the structure set X in the rth category i Sparse representation model under the establishment of structure set vectorization transformation learning: wherein represents the vectorization result of the structure group X i G r represents the vectorization result of the structure group X i in the rth classification result, each classification result corresponds to a G r , is the sparse representation coefficient of the structure group vectorization ; (3) is the sparse coefficient To enhance the sparsity of the coefficient, a log-ratio function is defined and the structural group vectorization transform learning and log-ratio constrained MRI reconstruction model is established by taking the function as a regularization term where λ > 0 is a constant, denotes the square of the vector l2 norm, I nm denotes an identity matrix of size nm x nm, C r denotes an index set of the structure group in the rth classification result, denotes a log-ratio function, defined as where log(·) denotes a natural logarithm function, k > 0 is a constant, j is an element index number in, and a i,j denotes the jth element in, |·| denotes an absolute value, ε1, ε2 are constants, and ε1 > 0, ε2 > 1, ε1 < ε2, denotes any, is the conjugate transpose of G r ; solving the above reconstruction model by using an alternating direction multiplier method, first an augmented Lagrangian function of the reconstruction model is established where d i denotes the Lagrange multiplier, <·> denotes the matrix inner product, μ>0 is the penalty parameter; solving each optimization variable alternately and updating the Lagrange multiplier and the penalty parameter can be converted into the following solving steps: (3a) Obtain at iteration t and x, solve for variable G at iteration t+1 r The problem can be transformed into a least squares subproblem with equality constraints by means of an augmented Lagrangian function: (3b) Obtain x at iteration t and G at iteration t + 1 r In the case of solving the variable The problem can be transformed into a convex optimization problem by an augmented Lagrangian function: (3c) Obtain x at iteration t + 1 and G r In case of, the problem of solving for the variables x at iteration t + 1 can be transformed into a typical least squares problem by means of an augmented Lagrangian function: (3d) update the Lagrange multiplier d in the (t+1)th iteration i : first multiply the μ obtained in the tth iteration by the d obtained in the (t+1)th iteration to obtain Then multiply the d obtained in the tth iteration by the μ obtained in the (t+1)th iteration i to obtain The sum of the two results is the updated d i in the (t+1)th iteration. (3e) The update of the penalty parameter μ in the t+1 iteration can be realized by multiplying the constant c>1 with the μ obtained in the t iteration; (3f) Repeat steps (3a)-(3e) until the estimated image meets the condition or the iteration number reaches the preset upper limit.

2. The method for MRI reconstruction based on multiple structured group transformation learning and log-ratio regularization term according to claim 1, wherein the problem in (3a) can be solved by singular value decomposition: (3a1) Calculate the singular value decomposition wherein, and are the left and right singular matrices, respectively, denotes the conjugate transpose of V r is the singular value matrix, denotes the conjugate transpose of ​ 3. The MRI reconstruction method based on multi-group transform learning and log-ratio regularization term according to claim 1, wherein, The problem in (3b) can be solved according to the following steps: (3b1) Let The problem in (3b) can be converted into (3b2) further decomposing the optimization problem in (3b1) into scalar optimization problems with respect to a i,j and where γ i,j denotes the jth element in the vector γ i . (3b3) According to Cauchy-Schwarz inequality and let The scalar optimization problem about a i,j can be further simplified to 4. The method for MRI reconstruction based on structured group vectorization learning and log-ratio constraint according to claim 1, wherein the least squares sub-problem in (3c) can be solved according to the following steps: (3c1) Deform the objective function in (3c) (3c2) To obtain the approximate solution of x, first calculate the auxiliary variable s where τ > 0 is a small constant, I N is an N x N identity matrix, N is the number of pixels in the whole image, is the conjugate transpose of R i,j and the approximate solution of x can be further calculated as x = F H (λU H U + μτI N ) -1 (λU H y + μτFs) where F is the Fourier transform matrix, F H is the conjugate transpose of F, U is an under-sampling matrix, U H is the conjugate transpose of U.

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