CN106934778B

CN106934778B - A kind of MR image rebuilding method based on small echo domain structure and non local grouping sparsity

Info

Publication number: CN106934778B
Application number: CN201710142054.5A
Authority: CN
Inventors: 施云惠; 霍莉杰; 齐娜; 丁文鹏; 尹宝才
Original assignee: Beijing University of Technology
Current assignee: Beijing University of Technology
Priority date: 2017-03-10
Filing date: 2017-03-10
Publication date: 2019-11-29
Anticipated expiration: 2037-03-10
Also published as: CN106934778A

Abstract

The present invention discloses a kind of MR image rebuilding method based on small echo domain structure and non local grouping sparsity, the following steps are included: step 1, first by image transform to Fourier carry out stochastical sampling, obtain sampled data y, later, it is initialized using basic compressed sensing reconstruction image principle；Step 2, iteration singular value threshold method solve low-rank matrix L_i；Step 3, alternating direction multipliers method (ADMM) solve image x.Using technical solution of the present invention, the reconstruction quality of image is improved.

Description

MR image reconstruction method based on wavelet domain structure and non-local grouping sparsity

Technical Field

The invention belongs to an image reconstruction method based on a compressed sensing principle, and particularly relates to an MR image reconstruction method based on a wavelet domain structure and non-local grouping sparsity, which combines two characteristics of a coefficient structure of an image in a wavelet domain and a non-local low rank of the image.

Background

Nuclear magnetic resonance is a nuclear physical phenomenon, and describes a process in which atoms with non-zero magnetic moments undergo spin energy level splitting under the action of an external magnetic field, and resonance absorbs radio frequency energy of a certain frequency. Initially, Bloch, Purcell et al discovered magnetic resonance in 1946. The nuclear magnetic resonance imaging method is an emerging discipline that nuclear magnetic resonance emerges by utilizing the discovery of spectroscopy to explore the microstructure of a substance.

Continuous progress of Magnetic Resonance Imaging (MRI) technology by means of magnet technology, low-temperature superconducting technology, electronic and computer technology and other related technologies has become an important means in the field of modern medical Imaging, and has been developed in the fields of physics, chemistry, biotechnology, clinical diagnosis and treatment. Particularly, the technology becomes an important member in the modern medical imaging technology by virtue of the advantages of high imaging definition, multiple imaging angles, no ionizing radiation and the like. Particularly, in medical application, the magnetic resonance imaging has high detection sensitivity on soft tissues and high tissue imaging contrast, can reflect the anatomical structure information of a human body and the physiological and chemical change process in the tissues, provides a very valuable diagnosis basis for clinical medicine, and promotes the development of medical diagnosis technology.

However, in practical applications, there are some problems, such as the imaging time of the magnetic resonance image is composed of two aspects, that is, the scanning time for acquiring the magnetic resonance data on one hand, and the time for reconstructing the acquired magnetic resonance data on the other hand, and the imaging time of the magnetic resonance is often long. Especially, in early mri, the scanning time of one imaging procedure often required several minutes, which required the patient to be completely still, otherwise the quality of imaging would be reduced due to motion artifacts. Some children or involuntary patients often cannot remain immobile during this process. Therefore, the lengthy scanning time becomes a problem that must be solved in the technical development. There are two main approaches to solve this problem, one is to improve the data acquisition speed, and the other is to improve the reconstruction algorithm to achieve the purpose of reconstructing an image as accurate as possible with less sampling data amount.

An image reconstruction method based on a compressive sensing theory. The compressed sensing theory developed in recent years can break through the Nyquist data sampling rate, and well solves the problem of undersampling recovery of data. The theory uses the prior knowledge that the signal can be expressed sparsely in a certain transform domain to provide that the original signal can be completely recovered with high probability through data obtained by partial sampling. In 2006, Candes theoretically proves that the original signal can be accurately reconstructed by using the local Fourier transform coefficient, and a compressed sensing theory is provided. Can utilize sampling data quantity far less than that required by traditional sampling method to reconstruct original signal, and break through Shannon sampling determinationNeck of the kitchen^[1-2]。

The MR image signals can meet the requirement of compressed sensing reconstruction by carrying out sparse transformation on the MR image signals, and for some images with sparsity in an image domain, sampling reconstruction such as vascular magnetic resonance imaging can be directly carried out without sparse transformation. Therefore, the CS theory has good application advantage in the field of MRI reconstruction.

Lusting^[3]In 2007, the compressive sensing theory is applied to the problem of reconstructing a magnetic resonance image, and on the premise that the image can be sparsely represented in a certain transform domain, aliasing artifacts appearing in the image reconstructed by using K-space downsampling data are incoherent, so that an original image can be completely reconstructed with high probability by a nonlinear reconstruction method, and a classical reconstruction model is provided as shown in formula (1). The feasibility and the effectiveness of the method are verified through experiments, and the result shows that the method can greatly reduce the K space data volume required by MR image reconstruction^[3]。

This translates the reconstruction of the image into solving a constrained optimization problem. In the reconstruction model, the observed data is represented as y ═ F_ux + n, wherein F_uRepresenting a local fourier transform, x representing the original image, and n representing white gaussian noise.Representing a reconstructed magnetic resonance image, Ψ representing a sparse transformation, | | x | | luminance_TVRepresenting the full variation constraint of the image, α and β are two non-negative parameters. The domestic scholars' flexor waves and the like obtain better reconstruction effects by introducing the curvelet waves, directional wavelets and other transformations into the MRI reconstruction process. The method has the advantages that good results are obtained in the aspect that images are represented through wavelet sparseness, a directional wavelet MRI reconstruction algorithm based on image blocks is provided, image sub-blocks are trained to obtain self-adaptive directional wave sparseness representation, and sparse constraint reconstruction is carried out on a compressed sensing reconstruction model through directional wavelets^[4]。

Because the total variation regularization term is based on that the image is a priori with smooth blocks, the model has poor reconstruction effect on abundant textures in the image. In recent years, the non-local similarity of images plays a great role in the field of image reconstruction, and the non-local grouping sparsity is particularly important for modeling and texture recovery parts of natural signals. Non-locally similar blocks are not only sparse in the transform domain, but their non-zero elements share a joint sparse pattern. By correlation between the modeled sparse coefficients, the resulting more accurate reconstruction can be substantially reduced^[5-6]Uncertainty of the unknown signal. Document [7 ]]A non-local low rank Normalization (NLR) method is proposed for structured sparsity and applied to reconstruction of compressed sensing MRI. And a good reconstruction effect is achieved.

Therefore, the process of reconstructing the original image by using partial K space data essentially utilizes some prior knowledge existing in the image to solve an equation through prior condition constraint so as to achieve accurate reconstruction. Therefore, what prior conditions are used and how these prior constraints are used directly relate to the quality of the reconstructed image, and the research on the reconstruction model and the prior constraints becomes the key point of the research on the reconstruction algorithm.

Reference to the literature

[1]Donoho D L.Compressed sensing[J].IEEE Transactions on Information Theory,2006,52(4):1289-1306

[2]Candes E J,Romberg J,Tao T.Robust uncertainty principles:Exact signal reconstruction from highly incomplete frequency information[J].IEEE Transactions on Information Theory,2006.52(2):489-509

[3]Lusting M,Donoho D,pauly J M.Sparse MRI:The application of compressed sensing for rapid MR imaging[J].Magn Reson Med,2007,58(6):1182-1195.[4]Qu X B,Guo D,Ning B D.et al.Undersampled MRI reconstruction with patch-based directional wavelets[J].Magnetic Resonance Imaging,2012,30(7):964-977

[5]A.Buades,B.Coll,and J.M.Morel,“A review of image denoising algorithms,with a new one,”Multiscale Model.Simul.,vol.4,no.2,pp.490–530,2005.

[6]W.Dong,G.Shi,X.Li,L.Zhang,and X.Wu,“Image reconstruction with locally adaptive sparsity and nonlocal robust regularization,”SignalProcess.,Image Commun.,vol.27,no.10,pp.1109–1122,2012.

[7]Weisheng Dong,Guangming Shi,Xin Li,Yi Ma,Feng Huang:Compressive Sensing via Nonlocal Low-Rank Regularization.IEEE Trans.Image Processing 23(8):3618-3632(2014)。

Disclosure of Invention

The technical problem to be solved by the invention is to provide an MR image reconstruction based on wavelet domain structure and non-local grouping sparsity, which utilizes the structural sparsity of coefficients of an image in a wavelet domain to be beneficial to capturing the implicit structural and detail information of an original signal, and particularly, when the structural sparsity of the image is matched with the structural characteristics of the signal, the reconstruction quality of the image signal can be improved to a greater extent. Meanwhile, strong space-time correlation exists in the image signals, the image signals have non-local similarity, and image details can be well reserved. By sparsely unifying the structure and the grouping into the same frame, the reconstruction quality of the image is finally improved. The method is mainly applied to reconstruction of medical nuclear magnetic resonance images, and the method is improved to a certain extent in subjectivity and objectivity.

In order to achieve the purpose, the invention adopts the following technical scheme:

1. a MR image reconstruction method based on wavelet domain structure and non-local grouping sparsity is characterized by comprising the following steps:

step 1, firstly, transforming an image into a Fourier domain to carry out random sampling to obtain sampling data y, and then initializing the sampling data y by using a basic compressed sensing image reconstruction principle;

step 2, solving the low-rank matrix L by using an iterative singular value threshold method_i

For the whole initialized image, a series of overlapped blocks are extractedIndicating, for each extracted block, within a local windowFinding the first m blocks similar to the k neighbor method, thenThen, a low-rank matrix L is solved through a singular value threshold method_i；

Step 3, solving the image x by an Alternative Direction Multiplier Method (ADMM)

Introduction of two auxiliary variables into equation (10)And z ∈ R^NThe expression (10) is rewritten as follows:

applying an ADMM algorithm to the formula (11), wherein the augmented Lagrangian function is as follows:

wherein, mu is equal to R^NAnd gamma. epsilon.R^NIs the Lagrange multiplier, beta₁,β₂> 0 is the constraint x-z andthe penalty parameter (12) can be rewritten as:

the optimization for equation (13) involves the following iterations:

decomposing the image into 3 subproblems by using an ADMM algorithm to solve the image x, wherein the specific process is as follows:

1. for solving z^(l+1)The formula (14) itself contains a closed solution, which can be solved in one step by the least square method;

2. for solving for x^(l+1)The formula (16) also contains a closed solution, and can be solved by a least square method in one step;

3. for solvingWhen the wavelet coefficient structure in the formula (15) is constrained by high frequency and low frequency (WL2-L1), the formula (28) is put in the wavelet domain to discuss and solve, namely

When the wavelet coefficient structure in equation (15) is the parent-child Group (Group) constraint, the solution is discussed by putting equation (32) in the wavelet domain, that is,

preferably, for solvingThe specific process is as follows:

a) when inEquation (15) can be written as:

when solving, the formula (28) is put in the wavelet domain to discuss the solution, and the solution is carried outAndall the coefficients after wavelet transform, the high frequency coefficients and the low frequency coefficients, that is, (28) the optimized solution of formula is:

according toSame principle b^(l)Can also be divided intoAndtherefore (29) can be divided into 2 problems to solve respectively:

the formulae (30) and (31) can be easily understood by the proximal method, and finally,the inverse wavelet transform may be obtained by an inverse wavelet transform, i.e.,

b) the same principle is usedEquation (15) can be written as:

as with the process of a), putting equation (32) in the wavelet domain for discussion will be:

will find the vector b andis expanded intoAndthen equation (33) can be written as:

wherein,is expanded intoThe matrices G and H are such thatAndin the same wayAndequation (34) can be obtained by the SPAMS algorithm, which, in the same way,the inverse wavelet transform may be obtained by an inverse wavelet transform, i.e.,

drawings

FIG. 1. compressed sensing framework of MR images based on wavelet domain structure sparseness and non-local grouping sparseness;

FIG. 2(a) is a Brain image of an MRI test;

FIG. 2(b) is a Chest image of an MRI test;

FIG. 2(c) is a Heart image of an MRI test;

FIG. 2(d) is an Arterm image of an MRI test;

FIG. 3. an observation matrix;

fig. 4 is an overall flow chart of the present invention.

Detailed Description

The compressed sensing theory mainly researches how to reconstruct a signal by obtaining a few sampling observed values and utilizing the sparsity priori knowledge of the signal. Different from the traditional signal acquisition mode, the compressed sensing theory acquires signals through sparse representation, projection measurement and reconstruction algorithm. The sampling rate can be far lower than the Nyquist sampling frequency, the acquisition and transmission cost of signals is greatly reduced, and the original signals can be accurately reconstructed with high probability through a reconstruction algorithm.

The signal is reconstructed based on the compressed sensing theory, and the reconstructed model is subjected to constraint solution by utilizing the characteristic that the image can be sparsely represented and a plurality of prior knowledge existing in the image, and the prior information is included in the reconstructed model of the signal in the form of a regular term. The image signal is usually structured and can be sparsely represented under a certain substrate. Meanwhile, the invention combines two kinds of prior information, namely the coefficient structure sparsity of the image in the wavelet domain and the non-local similar grouping sparsity of the image, achieves the optimal reconstruction effect through alternate optimization, and has the following characteristics:

1. model frame

The compressed sensing theory shows that: signals which are sparse in a certain transform domain can be sparsely reconstructed by a small amount of observation data by utilizing an optimization method. The non-adaptive compressed sampling condenses the information contained in the signal on a small amount of observation data, greatly reduces the number of samples required for accurately reconstructing the original signal, and then reconstructs the signal by utilizing a plurality of prior knowledge existing in the image. The invention provides a method for reconstructing an original image by utilizing two kinds of prior information, namely the coefficient structure characteristic of the image in a wavelet domain and the non-local low-rank characteristic of the image per se, in the wavelet transform domain.

1) Image non-local group sparseness

There is more or less a certain relation between pixels in the image, which is called the similarity of the image. Taking a window neighborhood (or an image block) with a pixel point as a center as an example, except that a center pixel is likely to be similar to its surrounding pixels, the neighborhood may also be similar to some neighborhood pixels at other positions in the image (structural features are similar), so that the pixel points at different positions in the image often show strong correlation, which is often referred to as non-local similarity of the image.

The image block similar to the current image block is searched in the whole image (usually in a relatively large search window in order to reduce the calculation amount) by the image block matching method, and then the found similar image blocks are weighted, so that the texture details of the image can be well maintained.

2) Sparse wavelet domain structure

The multi-resolution nature of the wavelet transform results in the image wavelet domain exhibiting the following characteristics: (1) the energy of the wavelet coefficients is concentrated in the lowest frequency part: the lowest sub-band carries information of a spatial domain statistical correlation region, and therefore contains the main energy of the image; (2) the high frequency coefficients are sparse: the high-frequency sub-bands of each level mainly bear the boundary information of edges and objects in the image, and the visual information borne by the part greatly contributes to the whole image beyond the energy of the whole image, so that the structure of the image is described. (3) A zero tree structure: each coefficient node is associated with four higher-level subband coefficient nodes, i.e., the parent and child nodes are typically non-zero (or zero) at the same time.

According to the characteristics of the wavelet domain coefficient, the method is divided into two grouping situations of low frequency + high frequency and low frequency + high frequency. The invention utilizes the 2 grouping modes and takes the 2 grouping characteristics as another constraint of the traditional sparse representation, thereby not only further improving the sparse representation capability of different sparse bases and thinning the sparse constraints of different images, but also making the overall sparse constraint on the whole image.

By unifying the grouping sparsity and the wavelet domain structure sparsity into the same framework, the MRI image compressed sensing reconstruction based on the structure and the grouping sparsity is proposed, as shown in FIG. 1.

The objective function of the model of the invention is as follows:

wherein, y frequency domain sampling data, phi is an observation matrix, x is an image to be reconstructed, and let x be_i＝R_ix，x_iIndicating a size extracted from the i position of the image asImage block of x_i∈CⁿWherein R is_iIs a matrix whose function is to extract the image block x from the i position in the image x_iLet us order Each column in the matrix is x_iLike blocks of (1) toHas a low rank property, whereinD_iIs an over-complete dictionary and is,is a sparse coefficient matrix and Ω (-) is the sparse constraint of the wavelet domain structure. η, α, θ are regular term coefficients.

Due to the fact thatHas a low rank property because of the fact thatIs susceptible to noise pollution, is free from low rank constraints, and therefore passes through the low rank matrix L_i∈C^n×mN is equal to or less than mLow rank matrix L_iSolving by optimizing the following formula:

instead of the convex kernel criterion, a non-convex log logdet (x) is used as a smooth proxy function for the rank. For a semi-positive definite matrix X ∈ R^n×nThe rank minimization optimization problem can be approximated as the sum of the logs of matrix singular values:

for non-semi-positive definite low rank matrix L_i∈C^n×mThe rank minimum optimization problem can be solved by performing singular value decomposition on it:

where epsilon is a small constant. Σ is a diagonal matrix, the diagonal element in the matrix is the matrix L_iL_i ^TThe characteristic value of (2). Sigma^1/2Is also a diagonal matrix, the diagonal element in the matrix is the matrix L_iThe singular value of (a).

By choosing an appropriate λ value, equation (3) can be rewritten as:

the overall objective function (2) can be rewritten as:

for the sparse constraint that omega (·) is a wavelet domain structure, the invention utilizes two wavelet domain coefficient structures, which are introduced in the research content part. The first is high-frequency + low-frequency structure sparsity, Ω (·) ═ ω | | | P_LΨx||₂+(1-ω)||P_HΨx||₁In which P is_LAnd P_HRespectively representing the low-frequency coefficient and the high-frequency coefficient of the extracted image x after the wavelet transform domain, and omega is the weight coefficient of the high-frequency coefficient and the low-frequency coefficient. The second is sparse low frequency + high frequency intra-parent-child packet structure,g is a set of different grouped forms of variables, a vectorRepresenting a certain group G in the set G_iCoefficient (2) of (1). Omega_iRepresenting the weights of the different packets.

2. Model optimization

Because the model (6) comprises a fidelity term and two regular constraint terms, and the constraint terms are all non-smooth, the solving is difficult. The invention provides an optimization method, which is used for cooperatively optimizing three items, namely a fidelity item, a wavelet domain sparse constraint item and a non-local grouping sparse constraint item. The optimization problem is decomposed into two subproblems, and the subproblems are solved through an alternative iteration singular value threshold method and an alternative direction multiplier method.

1) Singular value threshold method for solving low-rank matrix L_i

According to^[7]The article of (a) is a paper of (b),(7) the formula can be written as follows:

where τ ═ λ/2 η, k denotes the number of iterations, n₀＝min{n,m}，Representation with weightsThe weighted nuclear norm of (a) of (b),σ_jis L_iThe (j) th singular value of (a),the solution can be solved by:

wherein,is X_iSingular value decomposition of (x)₊Max (x,0), we initialize w ═ 1,1]^TThus, the first iteration is equivalent to solving a common minimization of the kernel norm.

2) Solving image x by alternative direction multiplier method

The Alternating Direction Multiplier Method (ADMM) is a very popular method for dealing with large scale optimization problems. Therefore we first introduce two auxiliary variablesAnd z ∈ R^N. Rewrite equation (10) to the following equation:

the optimization for equation (13) involves the following iterations:

where ρ > 1 is a constant and l represents the number of iterations for which several variablesCan be solved by dividing into 3 subproblems.

A. Solving for z^(l+1)

The closed form of equation (14) can be solved in one step as follows:

wherein,is a diagonal matrix of the grid,the representation represents the average of similar blocks.

B. Solving for x^(l+1)

Similarly for equation (16), the solution is:

where Φ is a partial fourier transform matrix, Φ ═ DF, and D and F denote a downsampled matrix and a fourier transform matrix, respectively. Substituting Φ ═ DF into (22) to obtain:

obtaining by solution:

C. solving for

For equation (15), solve by the proximal method, let h (x) be a convex function, and the proximal operator of h (x) is defined as:

this function has a unique minimum for each u. For l₁Normal form h (x) | | x | | non-conducting phosphor₁Case (2), convergence operator prox_thComprises the following steps:

for l₂Normal form h (x) | | x | | non-conducting phosphor₂，prox_thComprises the following steps:

in the invention, two wavelet domain coefficient structures are provided, wherein each wavelet domain coefficient structure is

And

a) when in useEquation (15) can be written as:

at the time of solution, we place the equation (28) in the wavelet domain to discuss the solution, since the low and high frequency coefficients have no overlap and the wavelet transform is invertible. Order toAndall the coefficients after wavelet transform, high frequency coefficients and low frequency coefficients are represented separately. That is to say that the first and second electrodes, (28) the optimized solution of formula is:

the formulae (30) and (31) can be easily understood by the proximal method. Finally, the process is carried out in a batch,the inverse wavelet transform may be obtained by an inverse wavelet transform, i.e.,

b) all the same asEquation (15) can be written as:

since the grouping sparse constraint term has overlapping groupings, it may cause that the calculation result of a certain coefficient to be solved in each iterative solution process has a difference because the coefficient to be solved is in a different grouping, and therefore the vectors to be solved b and b are combinedIs expanded intoAndthen equation (33) can be written as:

wherein,is expanded intoThus, there are matrices G and H such thatAndin the same wayAndequation (34) is a common optimization problem and can be found by the spam algorithm. In the same way, the method for preparing the composite material,the inverse wavelet transform may be obtained by an inverse wavelet transform, i.e.,

as shown in fig. 4, the MR image reconstruction method based on wavelet domain structure and non-local grouping sparsity of the present invention specifically includes the following steps:

step 1, firstly, transforming the image to a Fourier domain for random sampling to obtain sampling data y. Then, the method for reconstructing the image by using the basic compressed sensing (the DCT method is adopted by the invention) is initialized.

For the entire initialized image, a series of overlapped blocks of size 6 x 6 is extractedIt shows that for each extracted block, the k neighbor method is used to find the first 43 blocks similar to the extracted block in the local window, and thenWherein m is 43. Then, a low-rank matrix L is solved through a singular value threshold method_i。

Introduction of two auxiliary variables into equation (10)And z ∈ R^NDecomposing the problem into 3 subproblems by using an ADMM algorithm to solve; the image x is solved by an Alternating Direction Multiplier Method (ADMM), and the specific process is as follows:

1. for solving z^(l+1)Equation (14) itself contains a closed solution, which can be solved in one step by the least squares method.

2. For solving for x^(l+1)Equation (16) also contains a closed-form solution, which can be solved in one step by the least squares method.

3. For solvingWhen the wavelet coefficient structure in equation (15) is constrained by high and low frequencies (WL2-L1), we will put equation (28) in the wavelet domain to discuss solving, because there is no overlap between the low and high frequency coefficients and the wavelet transform is invertible.Andrespectively representing all the coefficients after wavelet transformation, high frequency coefficients and low frequency coefficients, whereinThen the high frequency l is applied₁Normal and low frequency l₂The normal forms are solved by corresponding proximal methods respectively. Finally, the process is carried out in a batch,can be obtained by inverse wavelet transform, i.e.When the wavelet coefficient structure in the formula (15) is a parent-child Group (Group) constraint, similarly, the formula (32) is put in a wavelet domain for discussion solution, and since there are overlapping groups in the Group sparse constraint term, the value of a certain coefficient to be solved in each iteration solution process may be different because of different groups, and the calculation result may be different, so that the vector b to be solved and the Group b to be solved are combinedIs expanded intoAndand (4) obtaining the target through SPAMS algorithm. In the same way, the method for preparing the composite material,the inverse wavelet transform may be obtained by an inverse wavelet transform, i.e.,

the model of the invention is implemented using true two-dimensional Magnetic Resonance (MR) images. The 4 images are a Brain image of 210 × 210, a Chest image of 220 × 220, a Heart image of 192 × 192, and an artist image of 220 × 220, respectively, as shown in fig. 2(a), fig. 2(b), fig. 2(c), and fig. 2 (d).

In the observation setting of the image, a partial Fourier observation method is adopted. The observation matrix is shown in fig. 3.

Assuming that the MR image x has n pixels, the partial fourier transform matrix R is composed of randomly selected m rows of an n × n fourier transform matrix. And the observation values corresponding to the randomly selected m rows form an observation value vector y. The sampling rate is defined as m/n. If the sampling rate is small, the MR image scan time is short.

In order to verify the effectiveness of the above mentioned nuclear magnetic resonance image reconstruction scheme based on compressed sensing, several commonly used medical images are reconstructed, and the objective quality is compared, wherein the objective quality is mainly measured by Peak Signal to Noise Ratio (PSNR) and is measured in decibels (dB). The calculation formula is as follows:

the mean square error MSE of the two images of size m × n is defined as follows:

where I and J denote the original image and the reconstructed image, respectively, and I (x, y), J (x, y) are pixel values corresponding to a position (x, y), the smaller the mean square error, the higher psnr, the higher the quality of the reconstructed image.

To compare the quality of image reconstruction, table 1 gives the PSNR comparison results for the reconstructed grayscale images:

TABLE 1 reconstruction of MRI images at different sampling rates

In table 1, the last two columns are the models proposed by the present invention, where NLR + WL1-L2 is the model when the wavelet domain coefficient constraint selects high frequency + low frequency, and NLR + Group is the model when the wavelet domain coefficient constraint selects low frequency + parent-child grouping. The NLR model is a compressed sensing reconstruction model based on non-local low-rank constraint. In order to effectively illustrate the effect of different models on the image reconstruction quality, the specific comparative experiment is divided into two parts: (1) comparing the NLR model reconstruction result based on non-local low-rank constraint with the NLR + WL1-L2 model reconstruction; (2) and (3) comparing the NLR model reconstruction result based on the non-local low-rank constraint with the NLR + Group model reconstruction.

As shown in Table 1, the model of the invention has obvious advantages, and is improved to a certain extent compared with NLR method. Although the advantages of the NLR + Group model herein are smaller than those of the NLR + WL2-L1 model at each sampling rate, it is sufficient to prove that the model presented herein is superior to the NLR model.

Claims

Extracting a series of overlapped blocks from the initialized whole imageShowing that for each extracted block, the top m blocks similar to the extracted block are searched in a local window by using a k neighbor method, and thenThen, a low-rank matrix L is solved through a singular value threshold method_i；R_iIs a matrix whose function is to extract the image block x from the i position in the image x_i，Each column in the matrix is x_iLike blocks of (1) toHas low rank properties;

the optimization for equation (13) involves the following iterations:

s1, solving for z^(l+1)The formula (14) itself contains a closed solution, which can be solved in one step by the least square method;

s2, solving for x^(l+1)The formula (16) also contains a closed solution, and is solved by a least square method in one step;

s3 solvingWhen the wavelet coefficient structure in the formula (15) is constrained by high frequency and low frequency (WL2-L1), the formula (28) is put in the wavelet domain to discuss and solve, namely

2. the method for reconstructing an MR image based on wavelet domain structure and non-local sparsity grouping according to claim 1, wherein the solution is performedThe specific process is as follows:

a) when inEquation (15) can be written as:

b) the same principle is usedEquation (15) can be written as: