CN112150570B - Compressed sensing magnetic resonance imaging method based on iterative p-threshold projection algorithm - Google Patents
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Abstract
The invention discloses a compressed sensing magnetic resonance imaging method based on an iterative p-threshold projection algorithm, which belongs to the field of magnetic resonance imaging and comprises the following steps of: performing Fourier transform on the T2 weighted brain map, transforming the T2 weighted brain map after Fourier transform into K space to obtain K space data of the T2 weighted brain map, and adopting the K space data of the T2 weighted brain map according to a nonlinear sampling template to obtain undersampled K space data of the T2 weighted brain map; performing redundancy transformation on the undersampled k-space data of the T2 weighted brain map, and performing sparse representation on the undersampled k-space data of the T2 weighted brain map; undersampling k-space data of a T2 weighted brain graph with sparse representation, carrying out image reconstruction based on an iterative p-threshold projection algorithm or a rapid iterative p-threshold projection algorithm, and flexibly changing p values to design a new sparse objective function so as to obtain a better reconstruction effect.
Description
Technical Field
The invention relates to the field of magnetic resonance imaging, in particular to a compressed sensing magnetic resonance imaging method based on an iterative p-threshold projection algorithm.
Background
Magnetic resonance imaging (Magnetic Resonance Imaging, MRI) plays an important role in clinical diagnostics, and although MRI can provide high quality images with excellent soft tissue contrast information, its imaging speed is not satisfactory, being mainly constrained by physical (e.g., gradient pulse amplitude and rate of change) and physiological (neural stimulation) constraints. The reconstruction model under the redundant system aims to recover the original data from incomplete observed data using a priori information. Currently, the related work is mainly based on two aspects: and reconstructing a decomposition type and a comprehensive type. For solution of the comprehensive model, an Iterative Soft Threshold Algorithm (ISTA) is widely used. However, the ISTA is an iterative operation in the transform domain, so the dimensions of the variables may be much higher than the dimensions of the image itself. The acceleration algorithm of the ISTA is named as the fast iterative soft threshold algorithm (fast iterative soft-thresholding algorithms, FISTA). To solve the problem of solving the decomposition model with the FISTA, smoothing with the Moreau envelope of the neighbor map, and then solving the smoothed approximation model with the FISTA, a Smoothed FISTA (SFISTA) is proposed. For solution of the decomposition-type model, the alternate direction multiplier method (Alternating direction multiplier method, ADMM) is often used, but ADMM is essentially an original-dual algorithm, so that both the image and the coefficients under the redundant system need to be stored in the iterative process. These all limit the application of redundant systems in CS-MRI reconstruction.
In order to be able to further solve the problem of magnetic resonance image reconstruction, expert students have been dedicated to study magnetic resonance imaging methods based on balanced reconstruction models. Liu Yunsong et al propose a simple and efficient compact frame-based compressed sensing magnetic resonance image reconstruction algorithm-a fast iterative soft threshold projection algorithm (pFISTA, projected fast ISTA), which references FISTA, and smart programming can enable pFISTA to avoid storing the representation coefficients under the whole redundancy system, so pFISTA can deal with the large-scale MRI reconstruction problem under the high redundancy system. And simultaneously, the reconstruction model and the decomposition model of the pFISTA are similar. However, the rapid iterative soft threshold projection algorithm uses a threshold function with low convergence speed and low precision, and cannot punish a small coefficient more, so that the final imaging error is high; the near-end mapping of penalty functions can solve the inverse regularization problem, so the soft threshold function is very effective as a near-end mapping of the corresponding penalty function.
Disclosure of Invention
According to the problems existing in the prior art, the invention discloses a compressed sensing magnetic resonance imaging method based on an iterative p-threshold projection algorithm, which comprises the following steps:
s1, carrying out Fourier transformation on a T2 weighted brain map, transforming the T2 weighted brain map after Fourier transformation into K space to obtain K space data of the T2 weighted brain map, and sampling the K space data of the T2 weighted brain map according to a nonlinear sampling template to obtain undersampled K space data of the T2 weighted brain map;
s2, performing redundancy transformation on the undersampled k-space data of the T2 weighted brain map, and performing sparse representation on the undersampled k-space data of the T2 weighted brain map;
and S3, undersampling k-space data of the sparsely represented T2 weighted brain graph, and carrying out image reconstruction based on an iterative p-threshold projection algorithm or a rapid iterative p-threshold projection algorithm.
Further, the iterative p-threshold projection algorithm image reconstruction process is as follows:
s3-1-1, determining the maximum iteration number M, reconstruction precision epsilon, a parameter threshold lambda and a step gamma;
s3-1-2, initializing: initializing an undersampled MRI image x to be reconstructed 0 Iteration number i, and i=0;
s3-1-3: performing p threshold projection algorithm iteration, and performing reconstruction by using formulas (1) and (2);
x k+1 ＝ΦT γλ (Φ H (x k +γF H U T (y-UFx k ))) (1)
T γλ (t)＝sgn(t i )max{0,|t i |-λ|t i | p-1 } (2)
wherein: y represents the fourier transform, y represents the sampling operator, y represents the undersampled k-space data, x k+1 Representing the image of the k+1 iteration, Φ H Representing a decomposition matrix, when phi H Phi=i, referring to phi as standard tight frame, gamma as step length, alpha k+1 Coefficients representing the k+1 iteration, F H Is inverse discrete Fourier transform, U T Is a transpose of U. T (T) γλ (t) is an iterative p-threshold function, λ represents a threshold, γ is a step size; sgn (alpha) i ) Representing a sign function, i.e. when alpha i >0 is 1 when alpha i <At 0, it is-1.
S3-1-4, iteration number i=i+1;
s3-1-5, iteration stop condition: if i>M is eitherEnding iteration, outputting an image x, otherwise, returning i=i+1 to S-1-3, and continuing iteration;
s3-1-6, obtaining an output reconstruction image x.
Further, the process of image reconstruction based on the fast iterative p-threshold projection algorithm is as follows:
s3-2-1: determining the maximum iteration number M, reconstruction precision epsilon, parameter threshold lambda and step gamma;
s3-2-2 initializing an undersampled MRI image x to be reconstructed 0 ，t 0 Number i of iterations is=1, and i=0;
s3-2-3, carrying out p threshold projection algorithm iteration, and carrying out reconstruction by utilizing formulas (1), (2), (3) and (4);
x k+1 ＝ΦT γλ (Φ H (x k +γF H U T (y-UFx k ))) (1)
T γλ (t)＝sgn(t i )max{0,|t i |-λ|t i | p-1 } (2)
wherein: y represents the fourier transform, y represents the sampling operator, y represents the undersampled k-space data, x k+1 Representing the image of the k+1 iteration, Φ H Representing a decomposition matrix, when phi H Phi=i, referring to phi as standard tight frame, gamma as step length, alpha k+1 Coefficients representing the k+1 iteration, F H Is inverse discrete Fourier transform, U T Is a transpose of U. T (T) γλ (t) is an iterative p-threshold function, λ represents a threshold, γ is a step size; sgn (alpha) i ) Representing a sign function, i.e. when alpha>0 is 1 when alpha<At 0, it is-1. t is t 0 ＝1,t k+1 Representing the direction of gradient descent for k +1 iterations,an image obtained by the (k+1) th time using the acceleration technique;
s3-2-4, iteration number i=i+1;
s3-2-5, judging iteration stop conditions, if i>M is eitherEnding iteration, outputting an image x, otherwise, returning i=i+1 to S-2-3 for iteration;
s3-2-6, obtaining an output reconstruction image x.
Further, the nonlinear sampling templates include a 30% gaussian sampling template and a 30% pseudo radiation template.
Further, the redundant transform includes a discrete wavelet transform or a Contourlets transform.
By adopting the technical scheme, the compressed sensing magnetic resonance imaging method based on the iterative p-threshold projection algorithm provided by the invention provides the iterative p-threshold projection algorithm (projected iterative p-thresholding algorithms, pIPTA) and the rapid iterative p-threshold projection algorithm (projected fast iterative p-thresholding algorithms, pPIPTA) of the accelerated version thereof, and the p-threshold function can be regarded as mapping of a penalty function with wider sparse constraint; aiming at the problem of poor shrinkage function of the soft threshold function, the soft threshold in pISTA is replaced by the p threshold function with better shrinkage performance, and the image edge noise is denoised, so that a better reconstructed image is obtained; a new sparse objective function is designed by flexibly changing the p value, so that a better reconstruction effect is obtained; the punishment of the p threshold value to the small coefficient is larger, so that the imaging speed and the imaging precision of the reconstructed MRI image are improved; the T2 weighted brain graph is adopted as a data set, the SIDWT is adopted as a tight frame in an experiment, and compared with various reconstruction algorithms, the experimental result shows that: the reconstruction error of pFIpTA is lower than that of other algorithms, and the RLNE value of pFIpTA is increased by about 0.01379 units over that of pfist.
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In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings that are required to be used in the embodiments or the description of the prior art will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments described in the present application, and other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a graph of p-threshold functions used in the present invention;
FIG. 2 (a) is a layer 7 view of a T2 weighted brain map of the present invention;
FIG. 2 (b) is a layer 10 view of the T2 weighted brain graph of the present invention;
FIG. 2 (c) is a 30% Gaussian sampling template diagram of the invention;
FIG. 2 (d) is a 30% pseudo radiation template diagram of the present invention;
FIG. 3 is a graph of the reconstruction result of the 7 th layer image of the T2 weighted brain map according to the present invention;
FIG. 4 is a graph of the reconstruction result of the image at layer 10 of the T2 weighted brain map according to the present invention;
FIG. 5 is a graph of convergence of the 7 th layer image of the T2 weighted brain map according to the present invention;
FIG. 6 is a graph of convergence of the 10 th layer image of the T2 weighted brain map according to the present invention;
FIG. 7 is a graph showing convergence under Contourlets' frame in accordance with the present invention;
FIG. 8 is a graph of reconstruction errors for verifying different p values in accordance with the present invention;
FIG. 9 is a graph of reconstruction errors corresponding to different p values under verification of SIDWT and Contourlets of the present invention.
Detailed Description
In order to make the technical scheme and advantages of the present invention more clear, the technical scheme in the embodiment of the present invention is clearly and completely described below with reference to the accompanying drawings in the embodiment of the present invention:
a compressed sensing magnetic resonance imaging method based on an iterative p-threshold projection algorithm is characterized in that: the method comprises the following steps:
s1, carrying out Fourier transformation on a T2 weighted brain map, transforming the T2 weighted brain map after Fourier transformation into a K space to obtain K space data of the T2 weighted brain map, and adopting the K space data of the T2 weighted brain map according to a nonlinear sampling template to obtain undersampled K space data of the T2 weighted brain map; brain map is an image obtained by scanning the head of a healthy volunteer using a 3T siemens Trio Tim magnetic resonance imager;
s2, performing redundancy transformation on the undersampled k-space data of the T2 weighted brain map, and performing sparse representation on the undersampled k-space data of the T2 weighted brain map;
and S3, undersampling k-space data of the sparsely represented T2 weighted brain graph, and carrying out image reconstruction based on an iterative p-threshold projection algorithm or a rapid iterative p-threshold projection algorithm.
Further, the iterative p-threshold projection algorithm (projected iterative p thresholding algorithms, plta) image reconstruction procedure is as follows:
s3-1-1, determining the maximum iteration number M, reconstruction precision epsilon, a parameter threshold lambda and a step gamma;
s3-1-2, initializing: initializing an undersampled reconstructed MRI image x to be reconstructed 0 Iteration number i, and i=0;
s3-1-3: performing p threshold projection algorithm iteration, and performing reconstruction by using formulas (1) and (2);
wherein: y represents the fourier transform, y represents the sampling operator, y represents the undersampled k-space data, x k+1 Representing the image of the k+1 iteration, Φ H Representing a decomposition matrix, when phi H Phi=i, referring to phi as standard tight frame, gamma as step length, alpha k+1 Coefficients representing the k+1 iteration, F H Is inverse discrete Fourier transform, U T Is a transpose of U. T (T) γλ (t) is an iterative p-threshold function, λ represents a threshold, γ is a step size; sgn (alpha) i ) Representing a sign function, i.e. when alpha i >0 is 1 when alpha i <-1 at 0;
s3-1-4, iteration number i=i+1;
s3-1-5, iteration stop condition: if i>M is eitherEnding iteration, outputting an image x, otherwise, returning i=i+1 to S-1-3, and continuing iteration;
s3-1-6, obtaining an output reconstruction image x.
Table I shows an iterative p-threshold projection algorithm in the invention
| Algorithm one: pIPTA |
| Parameters: lambda, gamma |
| Initializing: x is X 0 |
| Cycling until stopped: |
| x k+1 ＝ΦT γλ (Φ H (x k +γF H U T (y-UFx k ))) |
| T γλ (α)＝sgn(α i )max{0，|α i |-λ|α i |p-1} |
| and (3) outputting: x is x |
Further, the image reconstruction process based on the fast iterative p-threshold projection algorithm (projected fast iterative p-thresholding algorithms, pFIpTA) is as follows:
s3-2-1: determining the maximum iteration number M, reconstruction precision epsilon, parameter threshold lambda and step gamma;
s3-2-2: initializing an undersampled MRI image to be reconstructedt 0 Number i of iterations is=1, and i=0;
s3-2-3: performing p-threshold projection algorithm iteration, and performing reconstruction by using formulas (1), (2), (3) and (4);
x k+1 ＝ΦT γλ (Φ H (x k +γF H U T (y-UFx k ))) (1)
T γλ (t)＝sgn(t i )max{0，|t i |-λ|t i | p-1 } (2)
wherein: y represents the fourier transform, y represents the sampling operator, y represents the undersampled k-space data, x k+1 Representing the image of the k+1 iteration, Φ H Representing a decomposition matrix, when phi H Phi=i, referring to phi as standard tight frame, gamma as step length, alpha k+1 Coefficients representing the k+1 iteration, F H Is inverse discrete Fourier transform, U T Is a transpose of U. T (T) γλ (t) is an iterative p-threshold function, λ represents a threshold, γ is a step size; sgn (alpha) i ) Representing a sign function, i.e. when alpha i 1 when > 0, when alpha i And-1 when less than 0. t is t 0 ＝1，t k+1 Representing the direction of gradient descent for k +1 iterations,the (k+1) th time is an image obtained by using the acceleration technique.
S3-2-4: the iteration number i=i+1;
s3-2-5: judging the iteration stop condition if i is more than M orEnding iteration, outputting an image x, otherwise, returning i=i+1 to S-2-3 for iteration;
s3-2-6, obtaining an output reconstruction image x.
Wherein the relative L is defined by the following formula 2 The norm error serves as a digital indicator of the quality of the image reconstruction:
wherein, the liquid crystal display device comprises a liquid crystal display device,representing the reconstructed image and X represents the original image. The smaller the value, the higher the quality of the reconstruction of the magnetic resonance image.
An iterative p-threshold algorithm is designed for the under-tight-frame model reconstruction problem, the iterative p-threshold projection algorithm solves the under-tight-frame model reconstruction problem model, an iterative formula of the iterative soft-threshold projection algorithm can solve the balanced model reconstruction problem, the p-threshold is replaced by the soft-threshold and is integrated into a core formula of pIATA, and an iterative formula of pIATA is obtained:
x k+1 ＝ΦT γλ (Φ H (x k +γF H U T (y-UFx k ))) (1)
wherein T is γλ (t) is an iterative p-threshold function, λ represents a threshold, and γ is a step size.
Further, under the condition that the tight mark frame is the wavelet transformation and the contour wave which are not changed, verifying the reconstruction image errors corresponding to different p values, so that the minimum reconstruction error of the proposed algorithm under the condition that p is 0.7 is obtained; meanwhile, a Nesterov acceleration strategy is introduced, and the Nesterov method adopts a gradient which is forward from the previous iteration point to the previous iteration point, so that the convergence speed of the iterative p-threshold projection algorithm is greatly improved with little extra calculation amount, and the rapid iterative p-threshold projection algorithm pPIPTA is obtained.
Under the condition that the tight mark frame is the wavelet transformation and the contour wave which are not changed, the reconstruction image errors corresponding to different p values are verified, so that the reconstruction error of the proposed algorithm is minimum under the condition that p is 0.7. Meanwhile, a Nesterov acceleration strategy is introduced, and the Nesterov method adopts a gradient which is forward from the previous iteration point to the previous iteration point, so that the convergence speed of the iterative p-threshold projection algorithm is greatly improved with little extra calculation amount, and the rapid iterative p-threshold projection algorithm pPIPTA is obtained. Tables 1 and 2 are the pIPTA and pIPTA algorithms for MRI reconstruction under tight frame.
Further, the nonlinear sampling templates include a 30% gaussian sampling template and a 30% pseudo radiation template.
Further, the redundant transform includes a discrete wavelet transform or a Contourlets transform.
FIG. 2 (a) is a layer 7 view of a T2 weighted brain map of the present invention; FIG. 2 (b) is a layer 10 view of the T2 weighted brain graph of the present invention; the brain patterns in fig. 2 (a) and 2 (b) were obtained by scanning a healthy volunteer head using a 3T siemens Trio Tim magnetic resonance imager, using 32 coils, pulse sequences were T2 weighted Turbo spin echo sequences from layers 7, 10, respectively, where TR/te=6100/99 ms, fov=220×220mm 2 The thickness was 3mm.
FIG. 2 (c) is a 30% Gaussian sampling template diagram of the invention; FIG. 2 (d) is a 30% pseudo radiation template diagram of the present invention;
wherein fig. 2 (c) simulates two-dimensional phase encoding in three-dimensional imaging, and fig. 2 (d) simulates pseudo-radioactive line sampling in two-dimensional imaging, the sampling points of which are all the points closest to the true radioactive line sampling trajectory on a grid of cartesian coordinates.
FIG. 3 is a T2 weighted brain map 7 th layer image reconstruction result map, a first behavior original map and reconstructed images obtained by four algorithms; the first frame of the second row is a 30% pseudo radiation sampling template, and the remaining four frames are residual maps of four algorithms. RLNEs for the FIpTA, SFISTA, pFISTA, pFIpTA algorithm in the figure are 0.132426, 0.097078, 0.097029 and 0.083288, respectively.
Fig. 4 is a graph of the results of a weighted brain map layer 10 reconstruction, sampled using a 30% gaussian sampling template, with FIpTA, SFISTA, pFISTA, pFIpTA RLNEs 0.104166, 0.091703, 0.091573 and 0.069123, respectively. As can be seen from fig. 3, 4, the reconstructed images of FIpTA contain significant artifacts, while those artifacts in the reconstructed images of SFISTA and pFISTA are well suppressed, with minimal artifacts in pFIpTA. The reconstruction error of pFIpTA is lower than that of the other three algorithms. As fig. 5 and 6 show the corresponding convergence curves, the proposed pFIpTA is improved by about 0.01379 units over pFISTA, SFISTA.
In order to comprehensively evaluate pFIPTA, a contour wave (Contourlets) is selected as a tight frame in the experiment of this section, and local edge directivity of the image is applied to further sparsify the image by Contourlets, so that the reconstruction result can keep edge information as much as possible. And compared with a general algorithm ADMM for solving the decomposition model, the experimental results are shown in the following FIG. 7. The results in the graph show the same effect as using SIDWT as a tight frame. pFIpTA has lower reconstruction errors than others and converges faster. Therefore, the advantages of pPIPTA with pDISTA are not changed with the used tight frame, and the effect is better.
Since pFIpTA solves for a balanced model, not an exact decomposed model, we know the effect of the reconstruction result of pFIpTA and the reconstruction result of the decomposed model by comparing the general algorithm ADMM for solving the decomposed model. As can be seen from fig. 7, the ADMM of reconstruction errors of SFISTA and pFISTA are all substantially the same, but the pFIpTA algorithm has a faster convergence speed and higher accuracy.
We will verify from numerical experiments how the p-value affects the reconstruction speed and reconstruction error. In fig. 8, we choose the convergence curves with p values of 1, 0.8, 0.7, and 0.6, respectively, and p=1 is pFISTA algorithm. Fig. 9 shows RLNEs corresponding to different p values when the tight frame is SIDWT and Contourlets.
As can be seen from fig. 8 and 9, as the p value decreases, the convergence speed becomes faster, but RLNE does not follow this rule. It can be seen from fig. 8 that inflection points appear at 0.8, 0.7, and convergence accuracy of 0.8 and 0.7 is almost the same, but at 0.7, the speed is increased. The curve represented under Contourlets in FIG. 9 also shows that the reconstruction accuracy is a little better at 0.7. Considering together, we propose to set the value of p to 0.7.
The foregoing is only a preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art, who is within the scope of the present invention, should make equivalent substitutions or modifications according to the technical scheme of the present invention and the inventive concept thereof, and should be covered by the scope of the present invention.
Claims (3)
1. A compressed sensing magnetic resonance imaging method based on an iterative p-threshold projection algorithm is characterized in that: the method comprises the following steps:
s1: performing Fourier transform on the T2 weighted brain map, transforming the T2 weighted brain map after Fourier transform into K space to obtain K space data of the T2 weighted brain map, and sampling the K space data of the T2 weighted brain map according to a nonlinear sampling template to obtain undersampled K space data of the T2 weighted brain map;
s2, performing redundancy transformation on the undersampled k-space data of the T2 weighted brain map, and performing sparse representation on the undersampled k-space data of the T2 weighted brain map;
s3, undersampling k space data of a sparsely represented T2 weighted brain graph, and carrying out image reconstruction based on an iterative p-threshold projection algorithm or a rapid iterative p-threshold projection algorithm;
the iterative p-threshold projection algorithm image reconstruction process is as follows:
s3-1-1, determining the maximum iteration number M, reconstruction precision epsilon, a parameter threshold lambda and a step gamma;
s3-1-2, initializing: initializing an undersampled MRI image x to be reconstructed 0 Iteration number i, and i=0;
s3-1-3: performing p threshold projection algorithm iteration, and performing reconstruction by using formulas (1) and (2);
x k+1 ＝ΦT γλ (Φ H (x k +γF H U T (y-UFx k ))) (1)
T γλ (t)＝sgn(t i )max{0,|t i |-λ|t i | p-1 } (2)
wherein: y represents the fourier transform, y represents the sampling operator, y represents the undersampled k-space data, x k+1 Representing the image of the k+1 iteration, Φ H Representing a decomposition matrix, when phi H Phi=i, which is called a standard tight frame, gamma is the step length, alpha k+1 Coefficients representing the k+1 iteration, F H Is inverse discrete Fourier transform, U T Is the transposition of U; t (T) γλ (t) is an iterative p-threshold function, λ represents a threshold, γ is a step size; sgn (alpha) i ) Representing a sign function, i.e. when alpha i 1 when > 0, when alpha i -1 when less than 0;
s3-1-4, iteration number i=i+1;
s3-1-5, iteration stop condition: if i>M is eitherEnding iteration, outputting an image x, otherwise, returning i=i+1 to S-1-3, and continuing iteration;
s3-1-6, obtaining an output reconstruction image x;
the process of image reconstruction based on the fast iterative p-threshold projection algorithm is as follows:
s3-2-1: determining the maximum iteration number M, reconstruction precision epsilon, parameter threshold lambda and step gamma;
s3-2-2 initializing an undersampled MRI image x to be reconstructed 0 ，t 0 Number i of iterations, =1, and i=0;
s3-2-3, carrying out p threshold projection algorithm iteration, and carrying out reconstruction by utilizing formulas (1), (2), (3) and (4);
x k+1 ＝ΦT γλ (Φ H (x k +γF H U T (y-UFx k ))) (1)
T γλ (t)＝sgn(t i )max{0,|t i |-λ|t i | p-1 } (2)
wherein: y represents the fourier transform, y represents the sampling operator, y represents the undersampled k-space data, x k+1 Representing the image of the k+1 iteration, Φ H Representing a decomposition matrix, when phi H Phi=i, which is called a standard tight frame, gamma is the step length, alpha k+1 Coefficients representing the k+1 iteration, F H Is inverse discrete Fourier transform, U T Is the transposition of U; t (T) γλ (t) is an iterative p-threshold function, λ represents a threshold, γ is a step size; sgn (alpha) i ) Representing a sign function, i.e. when alpha i 1 when > 0, when alpha i -1 when less than 0; t is t 0 ＝1,t k+1 Representing the direction of gradient descent for k +1 iterations,an image obtained by the (k+1) th time using the acceleration technique;
s3-2-4, iteration number i=i+1;
s3-2-5, judging iteration stop conditions, if i>M is eitherEnding iteration, outputting an image x, otherwise, returning i=i+1 to S-2-3 for iteration;
s3-2-6, obtaining an output reconstruction image x.
2. The compressed sensing magnetic resonance imaging method based on the iterative p-threshold projection algorithm of claim 1, wherein: the nonlinear sampling templates include a 30% gaussian sampling template and a 30% pseudo radiation template.
3. The compressed sensing magnetic resonance imaging method based on the iterative p-threshold projection algorithm of claim 1, wherein: the redundant transforms include discrete wavelet transforms or Contourlets transforms.
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