An improved espirit reconstruction method based on outer product efficiency and dictionary learning

Abstract

The application relates to an improved ESPIRiT reconstruction method based on outer product effectiveness and dictionary learning, and belongs to the technical field of magnetic resonance imaging. ESPIRiT is a parallel magnetic resonance imaging technology for estimating multiple sets of sensitivity maps to realize image reconstruction by using K-space calibration information. The application is based on an ESPIRiT model, combines an SOUPDIL regular term containing an L0 norm, and proposes an improved ESPIRiT reconstruction algorithm based on SOUPDIL, named SOUPDIL-ESPIRiT, uses FISTA technology for solving, and realizes parallel magnetic resonance imaging reconstruction through two steps of dictionary learning and image updating. Experimental results show that the application can better promote image sparsity, eliminate image reconstruction noise and artifacts, significantly improve the precision of the reconstructed image, and has the ability of better retaining image texture details and edge contour information.

Description

Technical Field

[0001] This invention relates to an improved ESPIRiT reconstruction method based on outer product efficiency and dictionary learning, belonging to the field of magnetic resonance imaging technology. Background Technology

[0002] Magnetic Resonance Imaging (MRI) technology, with its non-invasive and non-ionizing characteristics, has always been a research hotspot in the field of medical image processing. However, its long scan time affects the quality of reconstructed images. Therefore, in order to improve imaging speed while ensuring MRI quality, many effective acceleration schemes for MRI data acquisition have been proposed, which can be mainly divided into two aspects: Compressed Sensing (CS)-MRI and Parallel Imaging (PI).

[0003] CS theory utilizes the sparsity of signals to recover the original signal from undersampled data. In 2007, Lustig et al. applied CS theory to MRI, shortening scan time. PI allows for faster scans by reducing the number of phase increment steps and utilizing sensitivity encoding of multiple coils to recover unacquired data. Sensitivity encoding (SENSE) is a common PI method that solves for the target image by measuring the sensitivity information of the coils, yielding high-quality reconstructed images. However, obtaining high-precision sensitivity information in practical applications is difficult, hindering clinical diagnosis. Therefore, researchers proposed Iterative self-consistent parallel imaging reconstruction (SPIRiT), which estimates high-resolution reconstructed images from sampled autocalibrated signal lines. However, this method reconstructs images coil by coil, resulting in a long reconstruction time. Combining the advantages of the SENSE and SPIRiT models while overcoming their shortcomings, Uecker et al. proposed a new algorithm, iTerative Self-consistent Parallel Imaging Reconstruction using Eigenvector maps (ESPIRiT), which estimates multiple sets of sensitivity maps using K-space calibration information to achieve coil MRI reconstruction, resulting in higher image reconstruction quality.

[0004] The correct selection of regularization terms can effectively improve the quality of reconstruction algorithms. Existing algorithms such as L1-ESPIRiT (based on the L1 norm regularization term of the ESPIRiT model), LpJTV-ESPIRiT (combining the Lp pseudonorm joint total variation (LpJTV) regularization term), and LpTV-SENSE (based on the Lp pseudonorm total variation (LpTV) regularization term of the SENSE model can all complete image reconstruction, but the quality of image reconstruction still needs to be improved. Summary of the Invention

[0005] The purpose of this invention is to overcome the shortcomings of the prior art and provide an improved ESPIRiT reconstruction method based on outer product efficiency and dictionary learning, which can further improve the quality of magnetic resonance imaging reconstructed images.

[0006] The objective of this invention is achieved through the following technical solution: an improved ESPIRiT reconstruction method based on outer product efficiency and dictionary learning. Its characteristics include the following steps:

[0007] S0: Initialization, let k = 0, x 0 =(RFS) H y, z 0 =0,t 0 =0,

[0008] Where k is the loop variable, representing the k-th iteration of the data, and the superscript " 0 "Indicates the initial value, variable subscript" g " indicates the g-th component of the variable; This represents a multi-component image to be reconstructed, where N = N v ×N h N represents the total number of pixels in a single-coil amplitude image. v N h x represents the number of pixels in the vertical and horizontal directions of the image, respectively. g This represents the g-th component image of x, where g = 1...G represents the index of the sensitivity plot group, x1 and x... G These represent the first and Gth component images of the multi-component image x to be reconstructed, respectively, where G represents the total number of sensitivity maps. In this invention, G = 2. 0 This represents the initial value of x; This represents undersampled multi-coil K-space data, where M is the actual number of sampling points for single-coil K-space data, and Q represents the total number of receiving coils. H Representing the conjugate transpose of a matrix, M << N, the q-th coil data of y is represented by y q This indicates that q = 1...Q represents the coil index, and y1 and y2...Q Let y represent the first and Qth coil data points of the column-vectorized undersampled multi-coil K-space data y. 0 This represents the initial value of y; For coil-by-coil two-dimensional Fourier transform operators, U is a two-dimensional Fourier transform matrix. h and U v N h N v Point Fourier transform matrix, Indicates the Kronecker product; I represents the multi-coil K-space undersampling operator. Q It is a Q×Q identity matrix. This is a single-coil undersampling matrix that undersamples the K-space data to reduce scan time. It is the sensitivity map matrix, which can be obtained from the ESPIRiT model, where S qg S represents the sensitivity of the g-th group of the q-th coil of S. 11 This represents a set of sensitivity diagrams for the first coil of the S group. QG This represents the sensitivity diagram of the G group for the Q-th coil in S; and z represents an intermediate variable. 0 , t 0 and Let z, t and t represent respectively. The initial value.

[0009] X represents an intermediate variable. g To obtain the image x from a single coil g Extract N from p The size is The matrix formed by horizontally concatenating column vectorized image patches, P1x g and These represent the images x from a single coil. g The extracted first image patch column vector and the Nth p A column vector of image patches, express The initial value of X g The i-th column is the column vector P i x g , This represents the extraction matrix for the i-th overlapping image patch, derived from the single-coil image x. g Extract the size as Image patches are vectorized side-by-side into column vectors, N p This indicates the number of image patches.

[0010] matrix D represents the synthesized dictionary of image patches, where J is the number of transform coefficients. If J = n, then D is full rank; if J > n, then D is a redundant dictionary. g =[d g1 ......d gJ ] represents the g-th element of dictionary D, d g1 D represents the dictionary D g Column 1, d gJ D represents the dictionary D g The Jth column, d gj Dictionary D g The j-th column, d gj Initial values; matrix C g =[c g1 ......c gJ ] represents the g-th component of matrix C, c g1 Representing matrix C g Column 1, c gJ Representing matrix C g The Jth column, d gj D represents the dictionary D g The j-th column, c gj The initial value.

[0011] S1: Calculate the intermediate variable z in the (k+1)th iteration. k+1 The calculation formula is as follows:

[0012]

[0013] in, The intermediate variable representing the k-th iteration L i for The Lipschitz constant of the gradient;

[0014] S2: Initialize g = 1;

[0015] S3: Initialize j = 1;

[0016] S4: Calculate the matrix for the (k+1)th iteration. The calculation formula is as follows:

[0017]

[0018] When the variable index is "0", the number does not exist;

[0019] S5: Calculate the dictionary for the (k+1)th iteration. The calculation formula is as follows:

[0020]

[0021] S6: Calculate the intermediate variables in the (k+1)th iteration. The calculation formula is as follows:

[0022]

[0023] S7: Calculate the sparse coefficients for the (k+1)th iteration. The calculation formula is as follows:

[0024]

[0025] Among them, the hard threshold operator H λ (·) is defined as subscript " i " represents the vector element index, b i Let z represent the i-th element of vector b, L be a scalar, λ be a positive number controlling the overall sparsity, ⊙ denote element-wise multiplication, and z = min(a, L) represent vector b. The minimum value of each element in the complex number with respect to the scalar L; "∠" represents the phase of the complex number.

[0026] S8: Calculate the intermediate variables in the (k+1)th iteration. The calculation formula is as follows:

[0027]

[0028] S9: Calculate the dictionary atomic coefficients for the (k+1)th iteration. The calculation formula is as follows:

[0029]

[0030] Where ||·||2 represents the L2 norm, and v is the first column of the n×n identity matrix;

[0031] S10: Calculate the sparse coding matrix for the (k+1)th iteration. The calculation formula is as follows:

[0032]

[0033] The i-th column is represented as w gi That is, matrix W g The sparse coefficient vector of the i-th image patch;

[0034] S11: Determine if j = J. If j = J, proceed to step S12; otherwise, j = j + 1 and return to step S3.

[0035] S12: Calculate the image to be reconstructed in the (k+1)th iteration. The calculation formula is as follows:

[0036]

[0037] in, Representing the intermediate variable z k+1 The g-th component, superscript " T " indicates matrix transpose;

[0038] S13: Calculate the intermediate variables for the (k+1)th iteration. The calculation formula is as follows:

[0039]

[0040] S14: Determine if g = G. If g = G, proceed to step S15; otherwise, g = g + 1, return to step S2.

[0041] S15: For the reconstructed multi-component image x k+1 The reconstructed single-coil amplitude image is calculated using the square root of the sum of squares (SOS), as shown in the following formula:

[0042]

[0043] in, This represents the single-coil amplitude reconstruction image at the (k+1)th iteration;

[0044] S16: Update the intermediate variables of the (k+1)th iteration

[0045] S17: Update the intermediate variables in the (k+1)th iteration

[0046] S18: Determine if the maximum number of iterations has been reached. If it has, proceed to step S19; otherwise, k = k + 1 and return to S1.

[0047] S19: Output the final reconstructed single-coil amplitude image.

[0048] The beneficial effects of this invention are as follows: ESPIRIT is a parallel MRI reconstruction technique that reconstructs images by solving for encoded information about the sensitivity of multiple coils. This invention introduces EfficientSum of Outer Products Dictionary Learning with an L0 norm (SOUPDIL) into the ESPIRIT model, enhancing its reconstruction performance and improving the accuracy of magnetic resonance imaging. The proposed algorithm is mainly based on the Fast Iterative Shrinkage / Thresholding algorithm (FISTA), achieving MRI reconstruction through two steps: dictionary update and image update. Experimental results show that compared with other algorithms based on the ESPIRIT model (such as L1-ESPIRiT and LpJTV-ESPIRiT), the proposed new algorithm SOUPDIL-ESPIRiT can effectively suppress noise and remove image artifacts, improving image reconstruction quality. Compared with algorithms based on the SENSE model (such as LpTV-SENSE), this algorithm can accurately reconstruct FOV-limited data with better image reconstruction quality. Attached Figure Description

[0049] Figure 1 This is a flowchart of the method of the present invention;

[0050] Figure 2 To obtain brain images (i.e., dataset1) with a field of view (FOV) smaller than that of the subject, an eight-channel receiving coil was used for full sampling.

[0051] Figure 3 This is a two-dimensional Poisson disk undersampling mask with a 5x acceleration factor and a 24×24 centered full-sample self-calibration region;

[0052] Figures 4-7 The images are reconstructed from dataset1, which contains a 5x speedup factor and a 24×24 centered fully sampled self-calibrated region, using the L1-ESPIRiT algorithm, LpJTV-ESPIRiT algorithm, LpTV-SENSE algorithm, and SOUPDIL-ESPIRiT algorithm respectively, from a two-dimensional Poisson disk undersampling mask containing a 5x speedup factor.

[0053] Figures 8-11 They are respectively Figures 4-7 Error map corresponding to the reconstructed image (white dots represent errors, and the whiter the error map, the greater the error). Detailed Implementation

[0054] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings:

[0055] Example 1: This invention is an improved ESPIRiT reconstruction method based on outer product efficiency and dictionary learning, proposed based on the ESPIRiT framework.

[0056] 1) ESPIRiT framework:

[0057] Assumption This represents a multi-component image to be reconstructed, where N = N v ×N h N represents the number of pixels in a single-coil amplitude image. v N h x represents the number of pixels in the vertical and horizontal directions of the image, respectively. g This represents the g-th component image of x, where g = 1...G represents the index of the sensitivity plot group, x1 and x... G These represent the first and Gth component images of the multi-component image x to be reconstructed, respectively, where G represents the total number of sensitivity maps. In this invention, G = 2. The variable subscript " g " indicates the g-th component of the variable. This represents undersampled K-space data, where M is the actual number of sampling points for a single-coil K-space data set, and Q represents the total number of receiving coils. H Representing the conjugate transpose of a matrix, M << N, the q-th coil data of y is represented by y q This indicates that q = 1...Q represents the coil index, and y1 and y2... Q Let represent the first coil data and the Qth coil data of the column-vectorized undersampled multi-coil K-space data y.

[0058] Then, the ESPIRiT reconstruction model is obtained as follows:

[0059] y = RFSx (1)

[0060] in, For coil-by-coil two-dimensional Fourier transform operators, U is a two-dimensional Fourier transform matrix. h and U v N h N v Point Fourier transform matrix, It represents the Kronecker product. I represents the multi-coil K-space undersampling operator. Q It is a Q×Q identity matrix. This is a single-coil undersampling matrix that undersamples the K-space data to reduce scan time. It is the sensitivity map matrix, which can be obtained from the ESPIRiT model, where Sqg S represents the sensitivity of the g-th group of the q-th coil of S. 11 This represents a set of sensitivity diagrams for the first coil of the S group. QG This represents the sensitivity diagram of the G group of the Q-th coil in S.

[0061] In the ESPIRiT model, the formula for calculating the sensitivity operator is:

[0062]

[0063] in, Let r be the sensitivity vector at position r, where r represents the K-space position index; The convolution of the semidefinite matrix values ​​at each position r is defined as follows: It is a test operator with a matrix value kernel, defined as:

[0064]

[0065] Among them, V || From the K-space calibration matrix, if the calibration matrix is ​​A, then the singular value decomposition of matrix A is:

[0066] A=UΣV H (4)

[0067] Where Λ is the singular value diagonal matrix of A, and U and V are unitary matrices, whose columns are the left and right singular vectors of the singular values, respectively. Decomposing V yields the null space span V of A. ⊥ The row space span V of A || The columns of V are the basis of the rows of matrix V, and all self-calibration information is located in V. || middle.

[0068] 2) Algorithm Derivation:

[0069] Although the ESPIRiT algorithm, which combines L1 and LpJTV regularization methods, can improve MRI reconstruction quality to some extent, it still has significant shortcomings. To further improve MRI reconstruction quality, this invention introduces outer product efficiency with L0 norm and dictionary learning (SOUPDIL) into the ESPIRiT model, proposing the SOUPDIL-ESPIRiT algorithm. The resulting optimization problem is expressed as follows:

[0070]

[0071] Where, N p This indicates the number of image blocks into which a single coil image is segmented. This represents the extraction matrix for the i-th overlapping image patch, derived from a single coil image x. gExtract the size as The image patches are vectorized into column vectors. Matrix D represents the synthesized dictionary of image patches (J is the number of coefficients; if J = n, then D is full rank; if J > n, then D is a redundant dictionary), D g Let d represent the g-th element of dictionary D. gj D represents g The j-th column; W g Representation matrix The g-th component, w gi Let W represent the sparse coefficient vector of the i-th image patch. g The i-th column, ||w gi ||0 represents the L0 norm of the sparse coefficient vector. The positive number λ is the weight that controls the overall sparsity, and L>0.

[0072] Introducing the FISTA technique, the optimization problem (5) can be decomposed into the following problem:

[0073]

[0074]

[0075]

[0076]

[0077] In equations (6)-(9), the variables are superscripted " k+1 "and k Let k+1 and k be the iterations of the algorithm. and L represents an intermediate variable. i for The Lipschitz constant of the gradient is, for ease of description, represented by parameters μ and λ in the derivation below. 2 Indicates μ / L i and λ 2 / L i z k+1 , t k+1 and z is an intermediate variable in the (k+1)th iteration. k , t k and It is an intermediate variable in the k-th iteration. x k Let x represent the image to be reconstructed in the k-th iteration. k+1 This represents the image to be reconstructed in the (k+1)th iteration.

[0078] The optimization problem (7) can be decomposed into the following subproblems:

[0079]

[0080]

[0081] Dictionary learning: Let as well as Problem (10) can then be represented as the following subproblem:

[0082]

[0083] make and c gj It is C g The j-th column. For ease of description, λ will be used in the following derivation. 2 Represents 2λ 2 If we have μ, then we can derive the following SOUPDIL formula:

[0084]

[0085] The matrix in equation (13) Represented as the sum of (sparse) rank one matrices or outer products Problem (13) can then be expressed as:

[0086]

[0087] Problem (14) is decomposed into two subproblems: sparse coding and dictionary atomic update.

[0088]

[0089]

[0090] Sparse coding: Keeping other variables constant during the process, we solve problem (15) for c. gj .

[0091] Assumption If it is a fixed matrix based on the latest values ​​of all other atoms and coefficients, then problem (15) can be transformed into:

[0092]

[0093] If and only if vector When there are no elements with magnitude λ, (17) has a unique minimum. and The global minimum of the sparse coding problem (17) is obtained by the following truncation hard thresholding operation:

[0094]

[0095] Here, ⊙ denotes element-wise multiplication. Assuming L > λ holds, for a vector... z = min(a, L) represents the minimum value of each element in a relative to the scalar L, where "∠" represents the phase of the complex number. Hard thresholding operator H λ (·) is defined as:

[0096]

[0097] in, Its subscript " i "Used to index vector entries, using b" i Let represent the i-th element of vector b. Introduce an intermediate variable. The optimization problem (18) can then be transformed into:

[0098]

[0099]

[0100] The intermediate variable b represents the (k+1)th iteration. g , X represents the intermediate variable X in the k-th iteration. g , The dictionary atomic coefficient d represents the number of iterations in the k-th iteration. gj , c represents the sparse coefficients in the k-th iteration. gj , c represents the sparse coefficients in the (k+1)th iteration. gj , C represents the variable matrix of the (k+1)th iteration. g , The dictionary D representing the variable in the (k+1)th iteration g When the variable index is "0", the number does not exist.

[0101] Dictionary Atomic Update: Keeping other variables fixed, solve for dictionary atom d. gj Using the preceding expressions, question (16) can be transformed into the following question:

[0102]

[0103] Therefore, a global minimum of problem (22) is:

[0104]

[0105] Here, v is set as the first column of the n×n identity matrix if and only if c gj When ≠0, it has a unique solution. An intermediate variable h is introduced.g =E gj c gj The optimization problem can then be transformed into:

[0106]

[0107]

[0108] In equations (24) and (25) This represents the intermediate variable in the (k+1)th iteration. The dictionary atomic coefficient d represents the (k+1)th iteration. gj .

[0109] Then the sparse coding matrix of the (k+1)th iteration From the following formula, we can obtain:

[0110]

[0111] The i-th column is represented as w gi That is, matrix W g The sparse coefficient vector of the i-th image patch.

[0112] Image Update: Since the variable x in problem (11) is independent of the variables of other components, each variable can be solved individually:

[0113]

[0114] in, Represents variable z k+1 Let the g-th component, with other variables unchanged, be... Since the derivative is 0, we have:

[0115]

[0116] because Since it is a diagonal matrix, its inverse is easily obtained, thus the analytical solution to problem (28) can be obtained as follows:

[0117]

[0118] For the reconstructed multi-component image x k+1 The reconstructed single-coil amplitude image is calculated using the square root of the sum of squares (SOS). have to:

[0119]

[0120] In summary, a novel parallel MRI reconstruction algorithm with SOUPDIL regularization terms, known as the SOUPDIL-ESPIRiT algorithm, was obtained.

[0121] The detailed flowchart is as follows: Figure 1 As shown, the steps are as follows:

[0122] S0: Initialization, let k = 0, x 0 =(RFS) H y, t 0 =0,

[0123] Among them, the superscript " 0 " indicates the initial value; This represents a multi-component image to be reconstructed. d gj initial value, c gj The initial value of X g For a single coil image x g Extract N from p The size is The matrix formed by horizontally concatenating column vectorized image patches, P1x g and These represent the images x from a single coil. g The extracted first image patch column vector and the Nth p A column vector of image patches, express The initial value of z 0 , t 0 and It corresponds to the intermediate variables z, t and The initial value.

[0124] S1: Calculate the intermediate variable z in the (k+1)th iteration. k+1 The calculation formula is as shown in (6);

[0125] S2: Initialize g = 1;

[0126] S3: Initialize j = 1;

[0127] S4: Calculate the matrix for the (k+1)th iteration.

[0128] S5: Calculate the dictionary for the (k+1)th iteration.

[0129] S6: Calculate the intermediate variables in the (k+1)th iteration. The calculation formula is as shown in (20);

[0130] S7: Calculate the sparse coefficients for the (k+1)th iteration. The calculation formula is as shown in (21);

[0131] S8: Calculate the intermediate variables in the (k+1)th iteration. The calculation formula is as follows (24);

[0132] S9: Calculate the dictionary atomic coefficients for the (k+1)th iteration. The calculation formula is as shown in (25);

[0133] S10: Calculate the sparse coding matrix for the (k+1)th iteration. The calculation formula is as follows (26);

[0134] S11: Determine if j = J. If j = J, proceed to step S12; otherwise, j = j + 1 and return to step S3.

[0135] S12: Calculate the image to be reconstructed in the (k+1)th iteration. The calculation formula is as shown in (29);

[0136] S13: Calculate the intermediate variables for the (k+1)th iteration.

[0137] S14: Determine if g = G. If g = G, proceed to step S15; otherwise, g = g + 1, return to step S2.

[0138] S15: For the reconstructed multi-component image x k+1 The square root of the sum of squares (SOS) is used to calculate the reconstructed single-coil amplitude image, as shown in formula (30).

[0139] S16: Update the intermediate variable t in the (k+1)th iteration. k+1 The calculation formula is as shown in (8);

[0140] S17: Update the intermediate variables in the (k+1)th iteration The calculation formula is as shown in (9);

[0141] S18: Determine if the maximum number of iterations has been reached. If it has, proceed to step S19; otherwise, k = k + 1 and return to S1.

[0142] S19: Output the final reconstructed single-coil amplitude image.

[0143] Experimental results:

[0144] In the following experiments, to test the reconstruction performance of the SOUPDIL-ESPIRiT algorithm, this invention compares the proposed SOUPDIL-ESPIRiT algorithm with the L1-ESPIRiT, LpJTV-ESPIRiT, and LpTV-SENSE algorithms, respectively. All algorithms in the experiments were implemented in Matlab (MathWorks, Natick, MA). The laptop was configured with an Intel Core i7-6500U@3.1GHz processor, 8GB of RAM, and a Windows 10 operating system (64-bit).

[0145] To compare the reconstruction performance of different algorithms, this invention uses a single slice of a human brain for simulation experiments, named dataset1. Figure 2 As shown, dataset1 is a brain MRI dataset with a field of view (FOV) smaller than that of the subjects. A two-dimensional Poisson disk undersampling mask with an acceleration factor of AF=5 and a 24×24 centrally fully sampled self-calibrated region is used, as shown... Figure 3 As shown, a test dataset is generated by manually sampling the fully sampled dataset.

[0146] First, a visual comparison is performed on the four reconstruction algorithms in dataset1. This invention selects to compare the reconstructed images when the speedup factor is 5 times (undersampling rate of 20%), such as... Figures 4-7 As shown. Figures 4-7 The reconstructed images are shown in turn by the L1-ESPIRiT algorithm, LpJTV-ESPIRiT algorithm, LpTV-SENSE algorithm, and SOUPDIL-ESPIRiT algorithm. Figure 4 The image was reconstructed using the L1-ESPIRiT algorithm. It is clear that the image has a large number of artifacts and severe loss of details. Figure 5 The image is reconstructed using the LpJTV-ESPIRiT algorithm, which can further remove image artifacts. Figure 6 The image was reconstructed using the LpTV-SENSE algorithm. The image contains a large number of overlapping artifacts, resulting in a large reconstruction error. It is not suitable for reconstructing images with a limited field of view. Figure 7 The image is reconstructed using the SOUPDIL-ESPIRiT algorithm. This algorithm has the fewest reconstruction artifacts, preserves details that are missed by other algorithms, and the reconstructed image is closer to the original image. It can also accurately reconstruct images with limited field of view.

[0147] To further illustrate that the reconstruction performance of the novel algorithm SOUPDIL-ESPIRiT proposed in this invention is superior to that of algorithms L1-ESPIRiT, LpJTV-ESPIRiT, and LpTV-SENSE, Figures 8-11 Error maps of dataset 1 under four algorithms are shown (more white dots on the error map indicate a larger error). The error maps show that L1-ESPIRiT and LpJTV-ESPIRiT algorithms have more white dots on their error maps, indicating poorer image reconstruction quality. The LpTV-SENSE algorithm has the most white dots and the worst reconstruction quality, while SOUPDIL-ESPIRiT has the fewest white dots and the smallest reconstruction error. Therefore, the proposed algorithm has better reconstruction performance.

[0148] In summary, experiments were conducted on the selected dataset (dataset1) using different reconstruction algorithms. The SOUPDIL-ESPIRiT algorithm demonstrated good visual performance, and the reconstructed image quality was significantly better than that of the L1-ESPIRiT, LpJTV-ESPIRiT, and LpTV-SENSE algorithms.

[0149] The specific embodiments of the present invention have been analyzed and described in detail above with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Within the scope of knowledge possessed by those skilled in the art, various changes can be made without departing from the spirit of the present invention.

Claims

1. An improved ESPIRiT reconstruction method based on outer product efficiency and dictionary learning, characterized in that: Includes the following steps: S0: Initialization, let , , , , , , , ; in, k It is a loop variable, representing the first iteration of the data. k The next iteration, superscript " "Indicates the initial value, variable subscript" " indicates the first" of the variable. One component; This represents a multi-component image to be reconstructed. This represents the total number of pixels in a single-coil amplitude image. , These represent the number of pixels in the vertical and horizontal directions of the image, respectively. express The One component image, Indicates the index of the sensitivity plot group. and These represent the multi-component images to be reconstructed. The first component and the second One component image, This indicates the total number of sensitivity plots. , express The initial value; This represents undersampled multi-coil K-space data. This represents the actual number of sampling points for single-coil K-space data. This indicates the total number of receiving coils. Represents the conjugate transpose of a matrix. , The Each coil data is used express, Indicates coil index, and These represent column-vectorized undersampled multi-coil K-space data, respectively. The first coil data and the second Data for each coil, express The initial value; For coil-by-coil two-dimensional Fourier transform operators, It is a two-dimensional Fourier transform matrix. and They are respectively , Point Fourier transform matrix, Indicates the Kronecker product; This represents the multi-coil K-space undersampling operator. for The identity matrix, This is a single-coil undersampling matrix, which undersamples the K-space data to reduce scan time. It is the sensitivity map matrix, which can be obtained from the ESPIRiT model, where express The The first coil Group sensitivity, express No. One set of sensitivity diagrams for each coil. express No. The first coil Group sensitivity plot; , and Indicates intermediate variables. , and They represent , and The initial value; Indicates intermediate variables. For images from a single coil Extraction The size is The matrix is ​​formed by horizontally concatenating column vectors of image patches. and These represent images from a single coil. The first image patch column vector extracted and the second A column vector of image patches, express initial value, The Columns are column vectors , Indicates the first Extraction matrix of overlappable image patches from single-coil images Extract the size as The image patches are vectorized into column vectors. Indicates the number of image patches; matrix A composite dictionary representing image patches. J Let be the number of transformation coefficients, if J = n ,but D For full rank; if J > n ,but D This is a redundant dictionary. Dictionary The One portion, Dictionary Column 1 Dictionary The List, Dictionary The List, express Initial values; matrix , Representation matrix The One portion, Representation matrix Column 1 Representation matrix The List, Dictionary The List, express The initial value; S1: Calculate the... Intermediate variables of the next iteration The calculation formula is as follows: ； in, Indicates the first Intermediate variables of the next iteration , for The Lipschitz constant of the gradient; S2: Initialization ; S3: Initialization ; S4: Calculate the... The matrix of the next iteration The calculation formula is as follows: ； When the variable index is " When the number does not exist, it is not found. S5: Calculate the... dictionary of the next iteration The calculation formula is as follows: ； S6: Calculate the... Intermediate variables of the next iteration The calculation formula is as follows: ； S7: Calculate the... The sparse coefficients of the next iteration The calculation formula is as follows: ； Among them, the hard threshold operator Defined as , subscript " " represents the vector element index, Representing vectors The One element, L A scalar, a positive number To control the overall sparsity weights, This indicates element-wise multiplication. " indicates the phase of a complex number; S8: Calculate the... Intermediate variables of the next iteration The calculation formula is as follows: ； S9: Calculate the... dictionary atomic coefficients of the next iteration The calculation formula is as follows: ； in, Represents the L2 norm. for The first column of the identity matrix; S10: Calculate the... The sparse coding matrix of the next iteration The calculation formula is as follows: ； The Column representation , That is, a matrix No. A sparse coefficient vector of image patches; S11: Determine if ,when Proceed to step S12 if the condition is met; otherwise... Return to step S3; S12: Calculate the... The image to be reconstructed in the next iteration The calculation formula is as follows: ； in, Indicate intermediate variables The Each component, superscript " " indicates matrix transpose; S13: Calculate the... Intermediate variables of the next iteration The calculation formula is as follows: ； S14: Determine if ,when Proceed to step S15 if the condition is met; otherwise... Return to step S2; S15: Reconstructed multi-component images Using the square root of the sum of squares SOS The formula for calculating the reconstructed single-coil amplitude image is as follows: ； in, Indicates the first The image is reconstructed from the amplitude of a single coil in the next iteration; S16: Update Intermediate variables in iteration ; S17: Update Intermediate variables in iteration ; S18: Determine if the maximum number of iterations has been reached. If it has, proceed to step S19; otherwise... and return to S1; S19: Output the final reconstructed single-coil amplitude image. .

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