CN112710975A - Magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition - Google Patents
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Abstract
The invention discloses a magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition, belongs to the technical field of digital image processing, and particularly relates to a magnetic resonance diffusion weighted imaging technology; the magnetic resonance diffusion image reconstruction method comprises the steps of firstly obtaining k-space undersampled data, calculating an initial reconstructed image, then constructing a magnetic resonance diffusion image compressed sensing reconstruction model based on sparse and local low-rank matrix decomposition, solving a magnetic resonance diffusion image background component by adopting a singular value soft threshold method, solving a sparse component by adopting a soft threshold algorithm, then updating the reconstructed image by adopting data consistency, and finally judging whether to continue iteration or obtain a final result of the reconstructed image according to whether a convergence condition is met or not; the magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition can retain the detail information of the image, realize the high-quality reconstruction of DW images in a plurality of different diffusion directions and accelerate the heart magnetic resonance diffusion imaging speed.
Description
Technical Field
The invention discloses a magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition, belongs to the technical field of digital image processing, and particularly relates to a magnetic resonance diffusion weighted imaging technology.
Background
The magnetic resonance diffusion imaging deduces the microstructure of the tissue by measuring the diffusion information of water molecules, has the advantages of non-invasion, no need of contrast agents and the like, and is the mainstream non-invasive method for detecting the microstructure information of the living tissue at present. Magnetic resonance Diffusion Tensor Imaging (DTI) is a magnetic resonance Diffusion Imaging method that describes the Diffusion movement of water molecules in biological tissues as a Diffusion Tensor. The Diffusion tensor for each voxel is computed by measuring more than 6 Diffusion Weighted (DW) images of different Diffusion gradient directions (not coplanar). The diffusion tensor reflects the diffusion anisotropy characteristics of water molecules in the voxel, and the fiber structure information of the tissue can be estimated. However, magnetic resonance diffusion imaging is slow and susceptible to motion. In the imaging process, the displacement of water molecules generated by respiratory motion and heart pulsation is far larger than the displacement caused by the self diffusion of the water molecules, so that a diffusion signal is additionally attenuated, the signal-to-noise ratio is reduced, and the reconstruction of myocardial fibers is seriously influenced.
Reducing the data volume is an effective method for improving the imaging speed, shortening the data acquisition time and reducing the motion influence. However, the conventional k-space undersampling scheme is difficult to realize high sparse sampling and high quality data reconstruction, limited by hardware conditions and theory. Compressed Sensing (CS) theory is a newly proposed nonlinear signal sampling theory, which indicates that a compressible signal can be nearly lossless restored to an original signal by a nonlinear reconstruction method using a priori knowledge of signal sparsity and the like, from a data amount far below that required by the Nyquist sampling theorem. Magnetic resonance diffusion tensor imaging acquires spectral data, i.e., k-space data, of DW images. The method comprises the steps of obtaining k-space sparse sampling data by performing down-sampling on original frequency spectrum data, and then reconstructing a magnetic resonance diffusion image from the k-space sparse sampling data through a nonlinear reconstruction method by utilizing prior knowledge such as signal sparsity. The reconstruction result influences the calculation of the diffusion tensor, and further influences the accuracy of the reconstruction of the myocardial fibers. How to utilize the essential characteristics of limited k-space data and a magnetic resonance diffusion image, solving the nonlinear reconstruction problem through prior condition constraint, inhibiting interference factors such as artifacts and noises, and reconstructing a high-quality magnetic resonance diffusion image is the key for accurately reconstructing a myocardial fiber structure.
In magnetic resonance diffusion imaging, magnetic resonance diffusion weighted images acquired from different diffusion directions image the same anatomical tissue, so that the images have relevance and correlation of physical information essentially. Due to the fact that the magnetic resonance diffusion weighting image has sparsity, and meanwhile, images in different diffusion gradient directions have low rank. Therefore, the Compressive sensing-based magnetic resonance Diffusion Imaging k-space undersampling reconstruction method usually adopts Low Rank constraint as prior information, and the common methods include a GLR (global Low Rank) Model introduced in an article "Phase-Constrained Low-Rank Model for Compressive Diffusion-Weighted MRI" and an article "Accelerated Heart Diffusion furnace Imaging Using Joint Low-Rank and sparse Constraints"; the LLR Model introduced in the article "influencing cardiac dispersion sensitivity Imaging combining local low-rank and 3D TV constraint", the L + S Model introduced in the article "Model-based compensated dispersion sensitivity Imaging" and the article "Parallel Imaging and compensated Combined frame for influencing High-Resolution dispersion sensitivity Imaging Using Inter-Image Correlation", etc.
However, these reconstruction algorithms do not sufficiently exploit sparsity of the magnetic resonance diffusion image and correlation information of the diffusion image in different diffusion gradient directions, and further improvement in the reconstructed image quality and the like is desired.
Disclosure of Invention
In order to solve the problem of low reconstruction effect of the existing reconstruction algorithm, the invention considers the characteristics of the cardiac magnetic resonance diffusion imaging, fully utilizes the sparsity and the correlation of magnetic resonance diffusion weighted images in different diffusion directions, and provides a magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition, so that the detail information of the images can be reserved, the high-quality reconstruction of DW images in a plurality of different diffusion directions is realized, and the cardiac magnetic resonance diffusion imaging speed is accelerated.
The purpose of the invention is realized as follows:
the magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition comprises the following steps:
step a, obtaining frequency domain data, namely k space data y, by adopting a random undersampling scheme; zero filling processing is carried out on the data which are not collected, inverse Fourier transform is carried out on the data which are subjected to the zero filling processing, and an initial reconstruction image X is obtained0；
B, constructing a magnetic resonance diffusion image compressed sensing reconstruction model based on sparse and local low-rank matrix decomposition;
step c, solving the background component L of the magnetic resonance diffusion image by adopting a singular value soft threshold method, wherein,
Nx，Nyrepresenting the number of pixels of the magnetic resonance diffusion image, NdThe magnetic resonance diffusion gradient direction number is shown, and the magnetic resonance diffusion image background component L in the k iteration process is shown as Lk；
Step d, solving the sparse component S by adopting a soft threshold algorithm, wherein,
the sparse component S in the kth iteration is represented as Sk；
And e, updating the reconstructed image X by adopting data consistency, wherein,
the reconstructed image X in the kth iteration is denoted Xk；
Step f, judging whether the convergence condition is met, if:
if not, returning to the step b;
then, the final result of the reconstructed image X is obtained.
The magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition comprises the following specific steps of:
in the formula:
min represents a minimization function;
Furepresents a random undersampling of k-space and has: fuPF, where P is a random undersampling scheme and F denotes fourier transform;
Rbrepresenting a local 3D image block extraction operation;
Ω represents a set of all local 3D image blocks;
Ψ represents a sparse transform;
|| ||2represents the l2 norm;
|| ||1is a norm of l 1;
||||*is a nuclear norm;
Σ denotes a summation symbol;
both τ and λ represent regularization parameters.
The magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition comprises the following specific steps of:
extraction operation R with local 3D image blocksbFor the image Lk-1＝Xk-1-Sk-1Extracting a series of overlapping image blocks, using Rb Lk-1Represents; extracted local 3D image block size of [ n ]x×ny,nd]I.e. Rb Lk-1Has a size of [ n ]x×ny,nd](ii) a Solving a low-rank matrix formed by local 3D image blocks by adopting an iterative singular value soft threshold method, wherein the low-rank matrix is expressed as [ u [ ]b,Σb,vb]＝SVD(RbLk-1) Obtaining a processed local 3D image block matrix ubΛτ(Σb)vb(ii) a Inverse operation R with local 3D image block extractionb TObtaining a background component L of the reconstructed magnetic resonance diffusion imagekIs shown as
Wherein, Λα(x) For soft threshold operation, denoted as
α is a threshold value.
In the magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition, in step d:
for soft threshold operation, one of two schemes is employed:
in the first scheme, all local image blocks adopt the same threshold;
according to the second scheme, different thresholds are adopted for each local image block; the threshold value is selected by adopting an empirical value.
The magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition comprises the following specific steps of:
the sparse component S is obtained by adopting a soft threshold algorithm and is expressed as Sk＝ΨT(Λα(Ψ(Xk-1-Lk-1) ))) wherein, Λ isα(x) For soft threshold operation, denoted as
α is a threshold value, Ψ is a sparse transform, ΨTThe inverse of it.
The sparse transform adopts an orthogonal transform basis.
The orthogonal transformation base is discrete wavelet transformation.
The magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition comprises the following specific steps of:
updating a reconstructed image X with data consistency, denoted Xk＝Lk+Sk-(Fu)T(Fu(Lk+Sk) -y) wherein FuFor k-space undersampling operations, (F)u)TIs FuThe inverse of (1).
In the magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition, the convergence condition is one of the following two conditions:
setting a maximum iteration number K under the condition one;
second, the difference of the quality of the reconstructed image is less than a certain specified threshold tol and is expressed as
Has the advantages that:
the invention provides a magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition, which comprises the steps of firstly adopting a random undersampling scheme to obtain frequency domain data, carrying out zero filling processing on data which are not acquired, and carrying out inverse Fourier transform on the data subjected to the zero filling processing to obtain an initial reconstruction image; then constructing a magnetic resonance diffusion image compressed sensing reconstruction model based on sparse and local low-rank matrix decomposition; then solving the background component of the magnetic resonance diffusion image by adopting a singular value soft threshold method, solving the sparse component by adopting a soft threshold algorithm, then updating the reconstructed image by adopting data consistency, and finally selecting continuous iteration or obtaining the final result of the reconstructed image according to whether the convergence condition is met. The steps are taken as a whole, the sparsity and the correlation of the magnetic resonance diffusion weighted images in different diffusion directions are fully utilized, further, the detail information of the images can be kept, the high-quality reconstruction of the DW images in a plurality of different diffusion directions is realized, and the heart magnetic resonance diffusion imaging speed is accelerated.
Drawings
Fig. 1 is a flow chart of the magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition of the present invention.
Fig. 2 is a diagram showing the number of pixels and the number of gradient directions in a magnetic resonance diffusion image.
Fig. 3 is a schematic diagram of a one-dimensional random variable density sampling scheme.
Fig. 4 is a schematic diagram of a two-dimensional random variable density sampling scheme.
Fig. 5 shows a schematic diagram of a partial 3D image block extraction operation.
Fig. 6 is an initial reconstructed image obtained using a one-dimensional random variable density sampling scheme.
Fig. 7 is a final reconstructed image obtained using a one-dimensional random variable density sampling scheme.
Detailed Description
The following describes embodiments of the present invention in further detail with reference to the accompanying drawings.
Detailed description of the invention
In the magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition according to the present embodiment, a flowchart is shown in fig. 1, and the method includes the following steps:
step a, obtaining frequency domain data, namely k space data y, by adopting a random undersampling scheme; zero filling processing is carried out on the data which are not collected, inverse Fourier transform is carried out on the data which are subjected to the zero filling processing, and an initial reconstruction image X is obtained0；
B, constructing a magnetic resonance diffusion image compressed sensing reconstruction model based on sparse and local low-rank matrix decomposition;
step c, solving the background component L of the magnetic resonance diffusion image by adopting a singular value soft threshold method, wherein,
Nx，Nyrepresenting the number of pixels of the magnetic resonance diffusion image, NdRepresenting the magnetic resonance diffusion gradient direction number, as shown in FIG. 2, the background component L of the magnetic resonance diffusion image during the k-th iteration is represented as Lk；
Step d, solving the sparse component S by adopting a soft threshold algorithm, wherein,
the sparse component S in the kth iteration is represented as Sk；
And e, updating the reconstructed image X by adopting data consistency, wherein,
the reconstructed image X in the kth iteration is denoted Xk；
Step f, judging whether the convergence condition is met, if:
if not, returning to the step b;
then, the final result of the reconstructed image X is obtained.
Detailed description of the invention
In the magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition in the present embodiment, on the basis of the first specific embodiment, a random undersampling scheme is further defined to select a one-dimensional random variable density sampling scheme, as shown in fig. 3.
Detailed description of the invention
In the magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition in the present embodiment, on the basis of the first specific embodiment, a random undersampling scheme is further defined to select a two-dimensional random variable density sampling scheme, as shown in fig. 4.
Detailed description of the invention
In the magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition in this embodiment, on the basis of the first specific embodiment, the second specific embodiment, or the third specific embodiment, the specific method in step b is further defined as follows:
in the formula:
min represents a minimization function;
Furepresents a random undersampling of k-space and has: fuPF, where P is a random undersampling scheme and F denotes fourier transform;
Rbrepresents a local 3D image block extraction operation, as shown in fig. 5;
Ω represents a set of all local 3D image blocks;
Ψ represents a sparse transform;
|| ||2represents the l2 norm;
|| ||1is a norm of l1；
||||*Is a nuclear norm;
Σ denotes a summation symbol;
both τ and λ represent regularization parameters.
Detailed description of the invention
In the magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition in this embodiment, on the basis of the first specific embodiment, the second specific embodiment, or the third specific embodiment, the specific method in step c is further defined as follows:
extraction operation R with local 3D image blocksbFor the image Lk-1＝Xk-1-Sk-1Extracting a series of overlapping image blocks, using Rb Lk-1Represents; extracted local 3D image block size of [ n ]x×ny,nd]I.e. Rb Lk-1Has a size of [ n ]x×ny,nd](ii) a Solving a low-rank matrix formed by local 3D image blocks by adopting an iterative singular value soft threshold method, wherein the low-rank matrix is expressed as [ u [ ]b,Σb,vb]＝SVD(RbLk-1) Obtaining a processed local 3D image block matrix ubΛτ(Σb)vb(ii) a Inverse operation R with local 3D image block extractionb TObtaining a background component L of the reconstructed magnetic resonance diffusion imagekIs shown as
Wherein, Λα(x) For soft threshold operation, denoted as
α is a threshold value.
Detailed description of the invention
In this embodiment, the magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition is further defined in step d on the basis of the first embodiment, the second embodiment, or the third embodiment:
for soft threshold operation, one of two schemes is employed:
in the first scheme, all local image blocks adopt the same threshold;
according to the second scheme, different thresholds are adopted for each local image block; the threshold value is selected by adopting an empirical value.
Detailed description of the invention
In the magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition in this embodiment, on the basis of the first specific embodiment, the second specific embodiment, or the third specific embodiment, the specific method of step d is further defined as follows:
the sparse component S is obtained by adopting a soft threshold algorithm and is expressed as Sk＝ΨT(Λα(Ψ(Xk-1-Lk-1) ))) wherein, Λ isα(x) For soft threshold operation, denoted as
α is a threshold value, Ψ is a sparse transform, ΨTThe inverse of it.
Detailed description of the invention
In the magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition in the present embodiment, on the basis of the seventh specific embodiment, it is further limited that the sparse transform employs an orthogonal transform basis.
Detailed description of the invention
In the magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition in the present embodiment, on the basis of the eighth specific embodiment, the orthogonal transformation basis is further limited to discrete wavelet transformation.
Detailed description of the preferred embodiment
In the magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition in this embodiment, on the basis of the first specific embodiment, the second specific embodiment, or the third specific embodiment, the specific method of step e is further defined as follows:
updating a reconstructed image X with data consistency, denoted Xk＝Lk+Sk-(Fu)T(Fu(Lk+Sk) -y) wherein FuFor k-space undersampling operations, (F)u)TIs FuThe inverse of (1).
Detailed description of the invention
In the magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition according to this embodiment, on the basis of the first embodiment, the second embodiment, or the third embodiment, the convergence condition is further defined as one of the following two conditions:
setting a maximum iteration number K under the condition one;
second, the difference of the quality of the reconstructed image is less than a certain specified threshold tol and is expressed as
Detailed description of the invention
In the magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition in the present embodiment, for specifically displaying the image for executing the method of the present invention, an initial reconstructed image obtained by using the one-dimensional random variable density sampling scheme shown in fig. 3 is shown in fig. 6, and a final result of the reconstructed image X obtained by using the reconstruction algorithm of the present invention is shown in fig. 7.
It should be noted that in the above embodiments, permutation and combination can be performed without any contradictory technical solutions, and since a person skilled in the art can exhaust the results of all permutation and combination according to the mathematical knowledge of permutation and combination learned in high-school stages, the results are not listed in this application, but it should be understood that each permutation and combination result is described in this application.
It should be noted that the above embodiments are only illustrative for the patent, and do not limit the protection scope thereof, and those skilled in the art can make modifications to the parts thereof without departing from the spirit of the patent.
Claims (9)
1. The magnetic resonance diffusion image reconstruction method based on sparse and local low-rank matrix decomposition is characterized by comprising the following steps of:
step a, obtaining frequency domain data, namely k space data y, by adopting a random undersampling scheme; zero filling processing is carried out on the data which are not collected, inverse Fourier transform is carried out on the data which are subjected to the zero filling processing, and an initial reconstruction image X is obtained0；
B, constructing a magnetic resonance diffusion image compressed sensing reconstruction model based on sparse and local low-rank matrix decomposition;
step c, solving the background component L of the magnetic resonance diffusion image by adopting a singular value soft threshold method, wherein,
Nx，Nyrepresenting the number of pixels of the magnetic resonance diffusion image, NdThe magnetic resonance diffusion gradient direction number is shown, and the magnetic resonance diffusion image background component L in the k iteration process is shown as Lk；
Step d, solving the sparse component S by adopting a soft threshold algorithm, wherein,
the sparse component S in the kth iteration is represented as Sk；
And e, updating the reconstructed image X by adopting data consistency, wherein,
the reconstructed image X in the kth iteration is denoted Xk；
Step f, judging whether the convergence condition is met, if:
if not, returning to the step b;
then, the final result of the reconstructed image X is obtained.
2. The sparse and local low-rank matrix decomposition-based magnetic resonance diffusion image reconstruction method according to claim 1, wherein the specific method of step b is as follows:
in the formula:
min represents a minimization function;
Furepresents a random undersampling of k-space and has: fuPF, where P is a random undersampling scheme and F denotes fourier transform;
Rbrepresenting a local 3D image block extraction operation;
Ω represents a set of all local 3D image blocks;
Ψ represents a sparse transform;
||||2represents the l2 norm;
||||1is a norm of l 1;
||||*is a nuclear norm;
Σ denotes a summation symbol;
both τ and λ represent regularization parameters.
3. The sparse-and-local-low-rank-matrix-decomposition-based magnetic resonance diffusion image reconstruction method according to claim 1, wherein the specific method of the step c is as follows:
extraction operation R with local 3D image blocksbFor the image Lk-1＝Xk-1-Sk-1Extracting a series of overlapping image blocks, using RbLk-1Represents; extracted local 3D image block size of [ n ]x×ny,nd]I.e. RbLk-1Has a size of [ n ]x×ny,nd](ii) a Solving a low-rank matrix formed by local 3D image blocks by adopting an iterative singular value soft threshold method, wherein the low-rank matrix is expressed as [ u [ ]b,Σb,vb]＝SVD(RbLk -1) Obtaining a processed local 3D image block matrix ubΛτ(Σb)vb(ii) a Inverse operation R with local 3D image block extractionb TObtaining the background component of the reconstructed magnetic resonance diffusion imageLkIs shown as
Wherein, Λα(x) For soft threshold operation, denoted as
α is a threshold value.
4. The sparse and local low rank matrix decomposition based magnetic resonance diffusion image reconstruction method of claim 1, wherein in step d:
for soft threshold operation, one of two schemes is employed:
in the first scheme, all local image blocks adopt the same threshold;
according to the second scheme, different thresholds are adopted for each local image block; the threshold value is selected by adopting an empirical value.
5. The sparse and local low-rank matrix decomposition-based magnetic resonance diffusion image reconstruction method according to claim 1, wherein the specific method of step d is as follows:
the sparse component S is obtained by adopting a soft threshold algorithm and is expressed as Sk＝ΨT(Λα(Ψ(Xk-1-Lk-1) ))) wherein, Λ isα(x) For soft threshold operation, denoted as
α is a threshold value, Ψ is a sparse transform, ΨTThe inverse of it.
6. The sparse-and-local-low-rank-matrix-decomposition-based magnetic resonance diffusion image reconstruction method according to claim 5, wherein the sparse transform employs an orthogonal transform basis.
7. The sparse-and local-low-rank matrix decomposition-based magnetic resonance diffusion image reconstruction method of claim 6, wherein the orthogonal transformation basis is a discrete wavelet transform.
8. The sparse and local low-rank matrix decomposition-based magnetic resonance diffusion image reconstruction method according to claim 1, wherein the specific method of step e is as follows:
updating a reconstructed image X with data consistency, denoted Xk＝Lk+Sk-(Fu)T(Fu(Lk+Sk) -y) wherein FuFor k-space undersampling operations, (F)u)TIs FuThe inverse of (1).
9. The sparse-and local-low-rank matrix decomposition-based magnetic resonance diffusion image reconstruction method of claim 1, wherein the convergence condition is one of the following two conditions:
setting a maximum iteration number K under the condition one;
second, the difference of the quality of the reconstructed image is less than a certain specified threshold tol and is expressed as
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