CN110533736A - Based on the dynamic magnetic resonance image reconstructing method for improving the principal component analysis of robust tensor - Google Patents

Based on the dynamic magnetic resonance image reconstructing method for improving the principal component analysis of robust tensor Download PDF

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CN110533736A
CN110533736A CN201910725866.1A CN201910725866A CN110533736A CN 110533736 A CN110533736 A CN 110533736A CN 201910725866 A CN201910725866 A CN 201910725866A CN 110533736 A CN110533736 A CN 110533736A
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蒋明峰
沈倩男
李杨
郑俊褒
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Zhejiang University of Technology ZJUT
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Abstract

The invention discloses based on the dynamic magnetic resonance image reconstructing method for improving the principal component analysis of robust tensor.The accuracy and speed of existing dynamic magnetic resonance image reconstructing method is to be improved.The method of the present invention: it indicates radially to sample obtained K space data using tensor, and constructs image reconstruction model using tensor Robust Principal Component Analysis tool, guarantee the spatial integrity of high dimensional data；It proposes that a kind of new tensor nuclear norm constrains low-rank part, guarantees to improve whole low-rank binding character while low-rank treatment effeciency；Threshold process is carried out again after carrying out time-frequency conversion to sparse part, improves reconstruction accuracy；The optimization problem that annual reporting law is finally solved using iteration soft-threshold contraction algorithm can effectively improve image reconstruction quality and accelerate reconstruct efficiency.The present invention can reconstruct high quality diagnostics division bit image within a short period of time, still obtain clear image under extremely low sample rate.

Description

Based on the dynamic magnetic resonance image reconstructing method for improving the principal component analysis of robust tensor
Technical field
The invention belongs to magnetic resonance medical imaging technology fields, and in particular to one kind is divided based on robust tensor principal component is improved The dynamic magnetic resonance image reconstructing method of analysis.
Background technique
Dynamic magnetic resonance imaging (dynamic Magnetic Resonance Imaging, dMRI) technology, can provide The indetectable tissue dynamic-change information in general MRI, therefore there is important clinical medicine application.But by It restricts it in the acquisition of slow data and image taking speed and further applies.Currently, compression sensing (Compressed Sensing it) has been successfully applied to dMRI reconstruction, has reduced data collecting quantity, has reduced patient and scans the time waited.In In CS-dMRI, reconstruction image is an ill-posed problem, is always magnetic resonance so improving picture quality and reconstructed velocity The hot spot of area research.
In most of existing CS-dMRI reconstructing methods, 3D rendering data are usually handled as a series of 2D matrix images Overlay model.Obviously, high order tensor (3D rendering) is remolded to a series of space that can ignore 3D data at (2D) matrixes or vector In intrinsic correlation and redundancy, to influence the quality of the dynamic image of reconstruct.Tensor is the most natural table of high dimensional data Show, technology has been applied in multidimensional processiug, and robust tensor principal component analytical method can effectively capture multidimensional number According to middle structural information, and solving background modeling, the problems such as subspace clustering, video compress sense in all obtain preferable effect Fruit.Therefore, in order to avoid destroying data and protection dMRI space-time structure, the processing side for being generally basede on matrix is replaced using tensor Method builds dMRI reconstruction model, and to accelerating, dynamic magnetic resonance is reconstructed and reconstruction accuracy is of great significance.
Summary of the invention
The purpose of the present invention is to solve above-mentioned dynamic magnetic resonance imaging accuracy and speed problems, provide one kind and are based on changing Into the dynamic magnetic resonance image reconstructing method of robust tensor principal component analysis, indicate radially to sample obtained K sky using tensor Between data, and construct image reconstruction model using tensor Robust Principal Component Analysis tool, guarantee that the space of high dimensional data is complete Whole property；It proposes that a kind of new tensor nuclear norm constrains low-rank part, guarantees to improve while low-rank treatment effeciency whole Low-rank it is restrictive；Threshold process is carried out again after carrying out time-frequency conversion to sparse part, improves reconstruction accuracy；Finally use iteration Soft-threshold contraction algorithm solves the optimization problem of annual reporting law, can effectively improve image reconstruction quality and accelerates reconstruct efficiency.
The present invention includes the following steps:
1, lack sampling is carried out using K space data of the higher-dimension radial direction sample track mode to dynamic nuclear magnetic resonance (DNMR) image, obtained To lack sampling data, lack sampling data are indicated using tensor γ；
2, zero padding is carried out to collected lack sampling data γ, then carries out inversefouriertransform and obtains initial pictures χ0, and according to robust tensor principal component analysis, by initial pictures χ0It is decomposed into the initial value l of low-rank tensor l0With sparse tensor ξ's Initial value ξ0, wherein low-rank tensor l and sparse tensor ξ meets A (l+ ξ)=γ, encoding operation accord with A be after Fourier transform again Execute lack sampling；
3, according to the principal component analysis of robust tensor, to l0It adds nuclear norm constraint and regularization, process is as follows:
3.1 couples of l0Matrix nuclear norm is added, and is mapped to tensor and obtains the tensor after addition matrix nuclear norm:
3.1.1 rightIt carries out tensor singular value decomposition and obtains core tensor E, tensor singular value decomposition is public Formula is [U, E, V]=t-SVD (l0), t-SVD indicates that tensor singular value decomposition function, R indicate real number, and U, V are coefficient tensor, N1、N2And N3Respectively indicate l0The first, second, and third dimension size；Then core tensor E is arranged as nuclear matrix form Unfold (E), unfold operation expression map-germ tensor E to size are N1N3×N2Nuclear matrix；
3.1.2 threshold process is carried out to singular value again after carrying out Singular Value Decomposition Using to nuclear matrix, obtains addition matrix Nuclear matrix after nuclear norm, i.e. svtλ(unfold (E)), svtλThreshold value is carried out to singular value again after representing matrix singular value decomposition Processing；
3.1.3 by svtλ(unfold (E)) is mapped to tensor, after then obtaining addition matrix nuclear norm as the following formula Amount: l'=U*fold (svtλ(unfold(E)))*VH, the inverse operation of fold expression unfold, H is transposition symbol；
3.2 couples of tensor l' add tensor nuclear norm, obtain low-rank tensor lΘ, it is specific as follows:
3.2.1 tensor l ' carry out Fourier transformation is obtained into frequency domain tensor formIt willInOn position Front slice successively carry out Singular Value Decomposition Using and singular value threshold process；
3.2.2 will through step 3.2.1 treated slice carry out conjugate transposition, then according toSequence It is assigned to respectivelyAll slices on position；
3.2.3 all slice Fourier inversions step 3.2.2 obtained obtain low-rank tensor l into time domainΘ。
4, according to the principal component analysis of robust tensor, first to ξ0Fourier transformation is carried out, then to ξ0Add l1Norm；To ξ0Add Add l1Norm and the process of regularization are as follows: to the ξ after Fourier transformation0Carry out threshold process, then inverse transformation into time domain, Obtain sparse tensor ξΘ。
5, sparse tensor ξΘAdd low-rank tensor lΘAfterwards, by subtracting K spatial redundancy A*(A(lΘ+ξΘ)-γ), it is reconstructed Image χ, χ=lΘ+ξΘ-A*(A(lΘ+ξΘ)-γ), A*For the inverse operation of A；
6, when number of repetition reaches predetermined number of times or iteration error χ-χ0When less than designated value, stop circulation, output weight Structure dynamic magnetic resonance image χ, otherwise, assignment χ again0=χ, l0=lΘ, ξ0=ξΘ, repeat step 3,4,5.
Further, higher-dimension radial direction sample track mode is specifically: radial sampling plate being placed in first frame dynamic nuclear-magnetism first In resonance image, is compared when then radial sampling plate is placed on the i-th frame dynamic nuclear magnetic resonance (DNMR) image and be placed in first frame dynamic nuclear-magnetism It is had rotated when resonance image angle k (i-1), i value is 2,3 ..., N3, the frame number of dynamic nuclear magnetic resonance (DNMR) image is N3。
The beneficial effect of the present invention compared with the prior art is:
The present invention indicates radially to sample obtained K space data using tensor, and uses tensor robust principal component point Analysis tool constructs image reconstruction model, guarantees the spatial integrity of high dimensional data；It is proposed the new tensor nuclear norm of one kind to low Order part is constrained, and guarantees to improve whole low-rank binding character while low-rank treatment effeciency；Time-frequency is carried out to sparse part Threshold process is carried out after transformation again, improves reconstruction accuracy；The optimization of annual reporting law is finally solved using iteration soft-threshold contraction algorithm Problem can effectively improve image reconstruction quality and accelerate reconstruct efficiency.The present invention can reduce interference in terms of artifact, as patient is sweeping Interference caused by involuntary movement or breathing during retouching, while more detailed information are retained, more high definition is presented Picture material；Have in time aspect and significantly reduce, greatly improves reconstructed image efficiency；It is answered in the data of more higher-dimension In, compared to other reconstructing methods, the present invention can still obtain clear image under extremely low sample rate, have very big It can practicability.The present invention applies in clinical medicine, when sampled data output is greatly decreased to reduce the sampling time, also High quality diagnostics division bit image can be reconstructed within a short period of time, reduce the time that patient checks and waits result, have Important practical application meaning.
Detailed description of the invention
Fig. 1 is the flow chart of the method for the present invention.
Fig. 2 (a), Fig. 2 (b), Fig. 2 (c) and Fig. 2 (d) are that K spatial radial sample track mode down-sampling plate respectively rotates angle Schematic diagram.
Fig. 3-1 (a) is the MR image of the fully sampled K space data of heart perfusion experiment.
Fig. 3-1 (b-1)~Fig. 3-1 (f-1) is respectively heart perfusion experiment in the case where accelerated factor is 8, IRTPCA of the present invention The magnetic resonance reconstructed image of algorithm and k-t SLR, HOSVD, k-t RPCA and RTPCA algorithm.
Fig. 3-1 (b-2)~Fig. 3-1 (f-2) be respectively heart perfusion experiment using IRTPCA algorithm of the present invention, k-t SLR, The residual image of HOSVD, k-t RPCA and RTPCA algorithm.
Fig. 3-2 (a) is the MR image of the fully sampled K space data of brain perfusion experiment.
Fig. 3-2 (b-1)~Fig. 3-2 (f-1) is respectively brain perfusion experiment in the case where accelerated factor is 8, and IRTPCA of the present invention is calculated The magnetic resonance reconstructed image of method and k-t SLR, HOSVD, k-t RPCA and RTPCA algorithm.
Fig. 3-2 (b-2)~Fig. 3-2 (f-2) be respectively brain perfusion experiment using IRTPCA algorithm of the present invention, k-t SLR, The residual image of HOSVD, k-t RPCA and RTPCA algorithm.
Fig. 3-3 (a) is the first frame MR image of the fully sampled K space data of 4D dynamic heart magnetic resonance.
Fig. 3-3 (b-1)~Fig. 3-3 (e-1) is respectively the magnetic resonance of 4D dynamic heart in the case where accelerated factor is 11, the present invention The magnetic resonance reconstructed image of IRTPCA algorithm and k-t SLR, HOSVD, k-t RPCA and RTPCA algorithm.
Fig. 3-3 (b-2)~Fig. 3-3 (e-2) is respectively that 4D dynamic heart magnetic resonance uses IRTPCA algorithm of the present invention, k-t The residual image of SLR, HOSVD, k-t RPCA and RTPCA algorithm.
Fig. 4 (a) is the 4th frame MR image of the fully sampled K space data of 4D dynamic heart magnetic resonance；
Fig. 4 (b), Fig. 4 (c) and Fig. 4 (d) are respectively 4D of the IRTPCA algorithm of the present invention under accelerated factor 5,17 and 33.8 Cardiac image reconstructs schematic diagram.
Specific embodiment
In order to more specifically describe the present invention, technical solution of the present invention is described in detail with reference to the accompanying drawing.
As shown in Figure 1, based on the dynamic magnetic resonance image reconstructing method for improving the principal component analysis of robust tensor, specific steps It is as follows:
1, radial sampling plate is radially sampled into rail according to the higher-dimension as shown in Fig. 2 (a), Fig. 2 (b), Fig. 2 (c) and Fig. 2 (d) Mark mode samples the K space data of dynamic nuclear magnetic resonance (DNMR) image, obtains lack sampling data, and lack sampling data use tensor γ is indicated；Higher-dimension radial direction sample track mode is specifically: radial sampling plate being placed in first frame dynamic nuclear magnetic resonance (DNMR) figure first As upper, compared when then radial sampling plate is placed on the i-th frame dynamic nuclear magnetic resonance (DNMR) image and be placed in first frame dynamic nuclear magnetic resonance (DNMR) figure As when have rotated angle k (i-1), i value be 2,3 ..., N3, the frame number of dynamic nuclear magnetic resonance (DNMR) image is N3, ensure that hits Irrelevance between.
2, zero padding is carried out to collected lack sampling data, then carries out inversefouriertransform and obtains initial pictures χ0, According to tensor Robust Principal Component Analysis by initial pictures χ0It is decomposed into the initial value l of low-rank tensor l0With the initial value of sparse tensor ξ ξ0。
In tensor Robust Principal Component Analysis, the image reconstruction model for constructing lack sampling data is as follows:
Wherein, s.t. indicate " so that ", | | | |*Indicate tensor nuclear norm；||||lFor l1Norm, indicate element absolute value it With；Encoding operation symbol A is to execute lack sampling again after executing Fourier transform；L and ξ respectively indicates the sparse tensor of low-rank tensor sum；
3, according to the principal component analysis of robust tensor, to l0It adds nuclear norm constraint and regularization, process is as follows:
3.1 couples of l0Matrix nuclear norm is added, and is mapped to tensor and obtains the tensor after addition matrix nuclear norm:
3.1.1 rightIt carries out tensor singular value decomposition (t-SVD) and obtains core tensor E, tensor is unusual Value decomposition formula is [U, E, V]=t-SVD (l0), U, V are coefficient tensor, N1、N2And N3Respectively indicate l0The first, second He The size of third dimension；Then core tensor E is arranged as nuclear matrix form unfold (E), unfold operation indicates map-germ Measure E to size be N1N3×N2Nuclear matrix；
3.1.2 threshold process is carried out to singular value again after carrying out Singular Value Decomposition Using to nuclear matrix, obtains addition matrix Nuclear matrix after nuclear norm, i.e. svtλ(unfold (E)), svtλThreshold value is carried out to singular value again after representing matrix singular value decomposition Processing；
3.1.3 by svtλ(unfold (E)) is mapped to tensor, after then obtaining addition matrix nuclear norm as the following formula Amount: l'=U*fold (svtλ(unfold(E)))*VH, the inverse operation of fold expression unfold, H is transposition symbol；
3.2 couples of tensor l' add tensor nuclear norm, obtain low-rank tensor lΘ, it is specific as follows:
3.2.1 tensor l ' progress Fourier transformation is obtained into frequency domain tensor formIt willInOn position Front slice successively carry out Singular Value Decomposition Using and singular value threshold process；
3.2.2 will through step 3.2.1 treated slice carry out conjugate transposition, then according toSequence It is assigned to respectivelyAll slices on position；
3.2.3 all slice Fourier inversions step 3.2.2 obtained obtain low-rank tensor l into time domainΘ。
It is restrictive not only to increase whole low-rank for the tensor nuclear norm that step 3.2 adds tensor l', but also its foundation is frequently Computation complexity is greatly lowered in domain tensor product form of Definition；
4, according to the principal component analysis of robust tensor, first to ξ0Fourier transformation is carried out, then to ξ0Add l1Norm；To ξ0Add Add l1Norm and the process of regularization are as follows: to the ξ after Fourier transformation0Carry out threshold process, then inverse transformation into time domain, Obtain sparse tensor ξΘ.Sparse tensor ξ itself has sparsity in the time domain, adds l by being fourier transformed into frequency domain1Model Number, keeps whole sparse constraint stronger, increases the accuracy of reconstructed image.
5, in order to keep data consistency, sparse tensor ξΘAdd low-rank tensor lΘAfterwards, by subtracting K spatial redundancy A*(A (lΘ+ξΘ)-γ), obtain reconstructed image χ, χ=lΘ+ξΘ-A*(A(lΘ+ξΘ)-γ), A*For the inverse operation of A；
6, when number of repetition reaches predetermined number of times or iteration error χ-χ0When less than designated value, stop circulation, output weight Structure dynamic magnetic resonance image χ, otherwise, assignment χ again0=χ, l0=lΘ, ξ0=ξΘ, repeat step 3,4,5, solve dynamic The optimization problem of magnetic resonance image reconstruct.
It is imaged based on the tensor Robust Principal Component Analysis method for improving tensor nuclear norm in dynamic nuclear magnetic resonance (DNMR) to verify The reconstruction property of method (IRTPCA) carries out simulated experiments using three groups of fully sampled dynamic magnetic resonance images, as Fig. 3-1 (a), Shown in 3-2 (a) and 3-3 (a), they be respectively heart perfusion (size be 190 × 190 × 70), brain be perfused (size is 128 × 128 × 60) and 4D cardiac magnetic resonance images data (size be 256 × 256 × 10 × 20).By the method for the present invention and 2D mode weight Tensor robust principal component before structure method k-t SLR, k-t RPCA, 3D pattern refactoring algorithm HOSVD and improvement nuclear norm Analysis method (RTPCA) compares, to verify the reconstruction property of the method for the present invention.Meanwhile with Y-PSNR (PSNR), Reconstitution time of structural similarity (SSIM) and every frame image etc. measures the reconstruct effect of algorithms of different as index is objectively evaluated Fruit.Wherein:
Y-PSNR
Structural similarity
Wherein, ψres、ψfullA respectively frame reconstructed image and the fully sampled dynamic magnetic resonance image of a frame, | | | |FIt represents F norm, μresIt is the data mean value of a frame reconstructed image, μfullIt is the data mean value of the fully sampled dynamic magnetic resonance image of a frame, σresIt is the mean square deviation of a frame reconstructed image data, σfullIt is the mean square deviation of the fully sampled dynamic magnetic resonance image data of a frame, σfully resIt is the covariance of a frame reconstructed image data and the fully sampled dynamic magnetic resonance image data of a frame, C1、C2Be for Maintain stable constant.
Fig. 3-1 (a) is the MR image of the fully sampled K space data of heart perfusion experiment, and image is void in solid white frame The enlarged drawing of image in line white box；Fig. 3-1 (b-1)~Fig. 3-1 (f-1) be respectively heart perfusion experiment accelerated factor be 8 Under, the magnetic resonance reconstructed image of IRTPCA algorithm of the present invention and k-t SLR, HOSVD, k-t RPCA and RTPCA algorithm, Fig. 3-1 (b-1) image is to put in each figure with dotted line white box corresponding position in Fig. 3-1 (a) in solid white frame in~Fig. 3-1 (f-1) Big figure.Fig. 3-1 (b-2)~Fig. 3-1 (f-2) be respectively heart perfusion experiment using IRTPCA algorithm of the present invention, k-t SLR, The residual image of HOSVD, k-t RPCA and RTPCA algorithm.Fig. 3-2 (a) is the MR of the fully sampled K space data of brain perfusion experiment Image, image is the enlarged drawing of image in dotted line white box in solid white frame；Fig. 3-2 (b-1)~Fig. 3-2 (f-1) is respectively Brain perfusion experiment is in the case where accelerated factor is 8, IRTPCA algorithm of the present invention and k-t SLR, HOSVD, k-t RPCA and RTPCA algorithm Magnetic resonance reconstructed image, image is in each figure and in Fig. 3-2 (a) in solid white frame in Fig. 3-2 (b-1)~Fig. 3-2 (f-1) The enlarged drawing of dotted line white box corresponding position.Fig. 3-2 (b-2)~Fig. 3-2 (f-2) is respectively that brain perfusion experiment uses the present invention The residual image of IRTPCA algorithm, k-t SLR, HOSVD, k-t RPCA and RTPCA algorithm.Fig. 3-3 (a) is 4D dynamic heart magnetic The first frame MR image of the fully sampled K space data of resonance.Fig. 3-3 (b-1)~Fig. 3-3 (e-1) is respectively 4D dynamic heart magnetic Resonance is in the case where accelerated factor is 11, the magnetic of IRTPCA algorithm of the present invention and k-t SLR, HOSVD, k-t RPCA and RTPCA algorithm Resonate reconstructed image.Fig. 3-3 (b-2)~Fig. 3-3 (e-2) is respectively that 4D dynamic heart magnetic resonance is calculated using IRTPCA of the present invention The residual image of method, k-t SLR, HOSVD, k-t RPCA and RTPCA algorithm.As it can be seen that IRTPCA method of the present invention can be rebuild Height lack sampling MR image out has great similarity with original image, and has clearer boundary, and in terms of reducing pseudomorphism Some improvement are provided, retain more detailed information than other methods.
Fig. 4 (a) is the 4th frame MR image of the fully sampled K space data of 4D dynamic heart magnetic resonance；Fig. 4 (b), Fig. 4 (c) It is respectively that 4D cardiac image of the IRTPCA algorithm of the present invention under accelerated factor 5,17 and 33.8 reconstructs schematic diagram with Fig. 4 (d).It can See, when handling High Dimensional Data Set, the present invention can still obtain good reconstruction quality, even if in high accelerated factor Under, IRTPCA of the present invention still can produce satisfactory visual results.
Table 1 have recorded heart perfusion and brain perfusion experiment in, under 9 times of accelerated factors, IRTPCA algorithm of the present invention with The PSNR (unit dB) and SSIM value mean value and PSNR of k-t SLR, HOSVD, RPCA and RTPCA algorithm reconstructed magnetic resonance image With the mean square deviation (" ± " number back in table 1) of SSIM value, it is found that evaluation index of the present invention obtains peak.In addition, using The mean square deviation variation of the PSNR and SSIM of IRTPCA method of the present invention are less than other methods, it means that IRTPCA method of the present invention Stable reconstruction performance can be provided.
Table 2 have recorded 4D cardiac magnetic resonance experiment in, under different accelerated factors, using IRTPCA algorithm of the present invention with The PSNR value mean value of HOSVD, k-t RPCA and RTPCA algorithm reconstructed magnetic resonance image, it is found that IRTPCA of the present invention has Best reconstruction quality.
Table 3 is average reconstitution time (the unit s), Ke Yifa of three groups of dynamic magnetic resonance images all frames under algorithms of different It is existing, reconstitution time is substantially reduced based on tensor Robust Principal Component Analysis method RTPCA, only in brain perfusion and 4D dynamic heart K-t RPCA is inferior in average reconstitution time.The present invention increases due to adding necessary additional low-rank constraint, time compared to RTPCA Greatly, but reconstruction accuracy but much surmounts RTPCA as seen from Table 2.Experiment for 4D, the present invention are even more to have and can not substitute Superiority and adaptability.
Table 1
Table 2
Table 3
The above specific experiment application has carried out the purpose of the present invention, technical scheme and beneficial effects further detailed Explanation, it should be understood that the above description is only a specific example of the present invention, is not intended to limit the scope of protection of the present invention. It particularly points out, to those skilled in the art, all within the spirits and principles of the present invention, any modification for being made, etc. With replacement, improvement etc., should all be included in the protection scope of the present invention.

Claims (2)

1. based on the dynamic magnetic resonance image reconstructing method for improving the principal component analysis of robust tensor, it is characterised in that: this method tool Body is as follows:
Step 1 carries out lack sampling using K space data of the higher-dimension radial direction sample track mode to dynamic nuclear magnetic resonance (DNMR) image, obtains To lack sampling data, lack sampling data are indicated using tensor γ；
Step 2 carries out zero padding to collected lack sampling data γ, then carries out inversefouriertransform and obtains initial pictures χ0, And according to robust tensor principal component analysis, by initial pictures χ0It is decomposed into the initial value l of low-rank tensor l0At the beginning of sparse tensor ξ Initial value ξ0, wherein low-rank tensor l and sparse tensor ξ meets A (l+ ξ)=γ, and encoding operation accords with A to hold again after Fourier transform Row lack sampling；
Step 3, foundation robust tensor principal component analysis, to l0It adds nuclear norm constraint and regularization, process is as follows:
3.1 couples of l0Matrix nuclear norm is added, and is mapped to tensor and obtains the tensor after addition matrix nuclear norm:
3.1.1 rightIt carries out tensor singular value decomposition and obtains core tensor E, tensor singular value decomposition formula For [U, E, V]=t-SVD (l0), t-SVD indicates that tensor singular value decomposition function, R indicate real number, and U, V are coefficient tensor, N1、 N2And N3Respectively indicate l0The first, second, and third dimension size；Then core tensor E is arranged as nuclear matrix form Unfold (E), unfold operation expression map-germ tensor E to size are N1N3×N2Nuclear matrix；
3.1.2 threshold process is carried out to singular value again after carrying out Singular Value Decomposition Using to nuclear matrix, obtains addition matrix core model Nuclear matrix after number, i.e. svtλ(unfold (E)), svtλThreshold process is carried out to singular value again after representing matrix singular value decomposition；
3.1.3 by svtλ(unfold (E)) is mapped to tensor, the tensor after then obtaining addition matrix nuclear norm as the following formula: l' =U*fold (svtλ(unfold(E)))*VH, the inverse operation of fold expression unfold, H is transposition symbol；
3.2 couples of tensor l' add tensor nuclear norm, obtain low-rank tensor lΘ, it is specific as follows:
3.2.1 tensor l ' carry out Fourier transformation is obtained into frequency domain tensor formIt willInOn position just Face slice successively carries out the threshold process of Singular Value Decomposition Using and singular value；
3.2.2 will through step 3.2.1 treated slice carry out conjugate transposition, then according toSequence point It is not assigned toAll slices on position；
3.2.3 all slice Fourier inversions step 3.2.2 obtained obtain low-rank tensor l into time domainΘ；
Step 4, foundation robust tensor principal component analysis, first to ξ0Fourier transformation is carried out, then to ξ0Add l1Norm；To ξ0Addition l1Norm and the process of regularization are as follows: to the ξ after Fourier transformation0Threshold process is carried out, then inverse transformation is obtained into time domain Obtain sparse tensor ξΘ；
Step 5, sparse tensor ξΘAdd low-rank tensor lΘAfterwards, by subtracting K spatial redundancy A*(A(lΘ+ξΘ)-γ), obtain reconstruct image As χ, χ=lΘ+ξΘ-A*(A(lΘ+ξΘ)-γ), A*For the inverse operation of A；
Step 6 reaches predetermined number of times or iteration error χ-χ when number of repetition0When less than designated value, stop circulation, output reconstruct Dynamic magnetic resonance image χ, otherwise, assignment χ again0=χ, l0=lΘ, ξ0=ξΘ, repeat step 3,4,5.
2. the dynamic magnetic resonance image reconstructing method according to claim 1 based on improvement robust tensor principal component analysis, It is characterized by: higher-dimension radial direction sample track mode is specifically: radial sampling plate being placed in first frame dynamic nuclear magnetic resonance (DNMR) first On image, is compared when then radial sampling plate is placed on the i-th frame dynamic nuclear magnetic resonance (DNMR) image and be placed in first frame dynamic nuclear magnetic resonance (DNMR) It is had rotated when image angle k (i-1), i value is 2,3 ..., N3, the frame number of dynamic nuclear magnetic resonance (DNMR) image is N3。
CN201910725866.1A 2019-08-07 2019-08-07 Based on the dynamic magnetic resonance image reconstructing method for improving the principal component analysis of robust tensor Withdrawn CN110533736A (en)

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CN111640298A (en) * 2020-05-11 2020-09-08 同济大学 Traffic data filling method, system, storage medium and terminal
CN114708349A (en) * 2022-04-11 2022-07-05 朱心歌 Dynamic MRI construction system based on non-convex low rank
US11553139B2 (en) 2020-09-29 2023-01-10 International Business Machines Corporation Video frame synthesis using tensor neural networks

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111640298A (en) * 2020-05-11 2020-09-08 同济大学 Traffic data filling method, system, storage medium and terminal
US11553139B2 (en) 2020-09-29 2023-01-10 International Business Machines Corporation Video frame synthesis using tensor neural networks
CN114708349A (en) * 2022-04-11 2022-07-05 朱心歌 Dynamic MRI construction system based on non-convex low rank

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Application publication date: 20191203