CN112529980B - A Multi-target Finite-Angle CT Image Reconstruction Method Based on Minimization - Google Patents

Abstract

The invention relates to a multi-target limited-angle CT image reconstruction method based on minimization, and belongs to the field of image reconstruction. The method includes the following steps: S1: collecting projection data; S2: establishing a multi-target finite-angle CT image reconstruction model; S3: finite-angle CT iterative reconstruction; S4: outputting a reconstructed image. Taking advantage of the multi-scale, multi-resolution decomposition and artifact directional characteristics of non-subsampling contourlet transform, two optimization objectives, namely, the minimization of the L0 norm of the non-subsampled contourlet transform and the L0 norm of the image gradient, are established. Objective-optimized reconstruction of the model. By adopting this method, the artifacts and noise of the reconstructed image can be effectively suppressed and the image boundary can be protected, thereby improving the quality of the CT reconstructed image.

Description

A multi-target limited-angle CT image reconstruction method based on maximin minimization

technical field

The invention belongs to the field of image reconstruction, and relates to a maximin-based multi-target limited-angle CT image reconstruction method.

Background technique

In some specific applications or scanning scenarios, such as: C-arm CT (Computed Tomography), in-service detection of long-sized objects, etc., projection data can only be collected within a certain range of angles. The reconstruction problem of this scanning method is called limited angle. CT reconstruction problem. Since the collected data does not meet the conditions for accurate reconstruction, the traditional filtered back-projection reconstruction algorithm will cause many artifacts to appear in the reconstructed image, resulting in the loss or concealment of some important structural information, which greatly affects the accuracy of nondestructive testing or doctors. diagnosis. Therefore, how to reconstruct a high-quality image that meets the doctor's diagnostic requirements or non-destructive testing standards has great practical significance. The traditional algebraic reconstruction algorithm causes a lot of artifacts and distortions in the reconstructed image. The image reconstruction algorithm based on TV (TotalVariation) can effectively deal with the sparse-angle CT reconstruction problem, but when it is applied to a small limited-angle scan, the reconstructed image will also have certain artifacts and distortion. In order to suppress artifacts and protect boundaries, Yu Wei introduced image gradient L0 regularization, which can suppress artifacts to a certain extent and further improve the quality of reconstructed images. Wang Chengxiang and others used the multi-scale and multi-resolution characteristics of wavelet transform, and used the L0 quasi-norm of image wavelet transform as regularization to deal with limited-angle CT reconstruction problems. This method can improve the quality of reconstructed images to a certain extent.

The publication number is CN 107978005A Patent Application Disclosure "A Limited Angle CT Image Reconstruction Algorithm Based on Boundary Preserving Diffusion and Smoothing". This method mainly performs gradient L0 boundary-preserving diffusion correction on the x and y directions of the reconstructed image to further improve the quality of the reconstructed image. The method described in the above patent application can protect the boundary and eliminate linear artifacts. However, the following defects still exist: (1) The method described in the above-mentioned patent application only considers the image x and y axes to perform gradient L0 boundary-preserving diffusion correction, and performs image x and y-axis gradient L0 regularization constraints after data fidelity constraints, in order to Suppressing artifacts may cause the image to be too smooth, resulting in the loss of image details corresponding to the lack of scanning angle; (2) the gradient transformation of the image in the method described in the above patent application only has high-frequency information in the high-frequency part of the image, and lacks regularization for the low-frequency part (3) From the perspective of optimization, the method described in the above-mentioned patent application is a single-objective optimization method, and does not consider optimizing multiple index.

The publication number is CN 110717959A Patent Application Disclosure "X-ray Limited Angle CT Image Reconstruction Method and Device Based on Curvature Constraint". This method mainly performs image gradient L0 regularization sparse constraints on the first reconstructed image, and then performs curvature constraints on the results of the sparse constraints. Although the method described in the above-mentioned patent application considers two optimization indexes, it can further overcome the blurred boundary or the step effect in the existing limited-angle CT reconstruction algorithm. However, there are still the following defects: (1) The gradient transformation and curvature constraints of the image in the method described in the above patent application only have the high-frequency information of the high-frequency part of the image, lack of regularization constraints on the low-frequency part, and do not take into account the artifacts. Multi-directional features; (2) From an optimization point of view, although the method described in the above patent application considers two optimization indicators (image gradient L0 constraint and curvature constraint), the single-objective optimization method is used to process these two optimizations respectively indicators, and did not directly consider optimizing these two indicators from the perspective of multi-objective optimization; (3) this single-objective method may cause the image to be too smooth due to the image gradient L0 constraint, and the image details corresponding to the lack of scanning angle are lost, so that the following Curvature-constrained metrics cannot recover lost details.

The publication number is CN 109697691A Patent Application Disclosure "Based on a Finite Angle Projection Reconstruction Method Based on Double Regular Term Optimization Based on L0 Norm and Singular Value Threshold Decomposition". In this method, the image gradient L0 regularization sparse constraint is firstly performed on the reconstructed image, and then the kernel norm constraint is performed on the image. Although the method described in the above patent application can restore the CT image contour and reduce limited angle artifacts. However, there are still the following defects: (1) The gradient transformation and curvature constraints of the image in the method described in the above patent application only have the high-frequency information of the high-frequency part of the image, lack of regularization constraints on the low-frequency part, and do not take into account the artifacts. Multi-directional features; (2) From an optimization point of view, this method is a double-regularized single-objective optimization method, and does not directly consider the optimization of image gradient L0 constraints and image kernel norms from the perspective of multi-objective optimization. index. (3) This single-target method may cause the image to be too smooth due to the constraint of the image gradient L0, and the details of the image corresponding to the lack of scanning angle are lost, so that the subsequent image kernel norm index cannot recover the lost details.

Most of the existing limited-angle CT optimal reconstruction methods do not take into account the multi-directional characteristics of artifacts. They all use single-objective optimization methods to optimize multiple indicators, and single-objective methods are also used to solve multi-regularized optimization models. The invention considers the sparsity of images under non-subsampled contourlet transform and image gradient transform, utilizes the multi-scale, multi-resolution decomposition and artifact directionality characteristics of non-subsampled contourlet transform, and sets up the non-subsampled contourlet transform The minimization of the L0 norm and the L0 norm of the image gradient are two optimization objectives, and the maximinization method in the multi-objective optimization is used to optimize these two objectives, so as to improve the quality of the CT reconstruction image.

Contents of the invention

In view of this, the object of the present invention is to provide a multi-target limited-angle CT image reconstruction method based on maximin minimization.

To achieve the above object, the present invention provides the following technical solutions:

A method for reconstructing multi-target limited-angle CT images based on maximin and minimization, the method comprising the following steps:

S1: Collect projection data:

S2: Establish a multi-target limited-angle CT image reconstruction model;

S3: Limited-angle CT iterative reconstruction;

S4: Output the reconstructed image.

Optionally, the S1 specifically includes: rotating the ray source around the rotation center along the scanning orbit for a limited angle to obtain incomplete projection data.

Optionally, the specific S2 is: when the discrete model is used for reconstruction, f(x, y) corresponding to the coordinates (x, y) first needs to be rearranged into a 1-dimensional vector f according to the dimension of y, and its dimension is N×1, where N=n ₁ ×n ₂ , n ₁ is the dimension of f(x,y) in the x direction, n ₂ is the dimension of f(x,y) in the y direction; then, all The coordinates (a, s) corresponding to the projection data g ^δ (a, s) are rearranged into a 1-dimensional vector g ^δ according to the dimension of s, and the dimension of the column vector g ^δ is M×1, where M=m ₁ ×m ₂ , m ₁ is the dimension of g ^δ (a, s) in the a direction, m ₂ is the dimension of g ^δ (a, s) in the s direction; according to the present invention, it is considered to use non-subsampling contourlet transform to decompose the reconstructed image into low-frequency part and high-frequency part, so that the image can be decomposed into multi-scale and multi-resolution;

In order to suppress noise and artifacts and avoid loss of some details caused by over-smoothing, L0 sparse regularization constraints are performed on the coefficients of non-subsampled contourlet transformation; for image smoothness, L0 regularization of image gradients is used to constrain;

The model established from the multi-objective perspective is as follows:

where A∈R ^M×N is the finite-angle CT system matrix, f∈R ^N×1 is the image to be reconstructed, g ^δ ∈ R ^M×1 is the finite-angle CT projection data, and Ω is a convex set (Ω:={f| f≥0}); λ is the relaxation parameter, W is the non-subsampled contourlet transform, i is the index of the subband; ||β|| ₀ is the number of non-zero elements of statistics β, ▽β=(▽ _x β, ▽ _y β), the component form of ▽ _x β,▽ _y β is (▽ _x β) _i′,j′ =β _i′,j′ -β _i′-1,j′ ,(▽ _y β) _{i′ , j′} =β _i′,j′ -β _i′,j′-1 ; when performing non-subsampling contourlet transformation, rearrange f into a 2-dimensional matrix f(x,y), and then The non-subsampled contourlet transformation is performed on the three-dimensional matrix. After the non-subsampled contourlet inverse transformation is completed, f(x, y) is rearranged into a 1-dimensional vector f according to the dimension of y.

Optionally, the S3 is specifically:

According to the established model (1), the model (1) is solved by the method of maximization and minimization;

The specific process of step S3 finite-angle CT iterative reconstruction is as follows:

First, the model (1) is transformed into a single-objective optimization model through the maximin method, and its form is as follows:

Then, in order to effectively solve the single-objective optimization model, take λ as two special values. First, λ=1 to solve the corresponding optimization problem; then λ=0, and then solve the corresponding optimization problem; in order to make the final solution converge, repeat The above value process of λ; so the single-objective optimization model is simplified as follows:

Model (3) is equivalent to the following form:

3) When λ=1, in order to solve the first optimization model with constraints in (4), it is converted into the following form:

且对所有i′＝1,2,...,M，

γ_i是正则化参数，再采用交替方向迭代法(ADMM)求解模型(5)，其迭代格式如下：Where ||x|| _D ＝＜Dx,x>; D is a diagonal matrix, and its diagonal elements are

And for all i'=1,2,...,M,

γ _i is the regularization parameter, and then the model (5) is solved by the alternating direction iterative method (ADMM), and the iterative format is as follows:

t is a parameter introduced by the ADMM algorithm;

In order to avoid the inverse of the system matrix A in the sub-problem about the first variable f or to use the iterative method to solve the sub-problem, the idea of adjacent alternating linearization is embedded, and the iterative format (6) is converted into ADMM of adjacent alternating linearization The iterative format, as follows:

是临近线性化引入的松弛参数；(7)中

为ART迭代更新格式；通过临近交替线性化巧妙地将经典的ART迭代算法融入其中；In iterative format (7)

is the relaxation parameter introduced by the proximity linearization; in (7)

Updated format for ART iteration; subtly incorporates classic ART iteration algorithm through adjacent alternating linearization;

Then find the optimal solution to the subproblem of the iterative format (7), and the iterative format is as follows:

where W ^T represents the non-subsampled contourlet inverse transform,

4) When λ=0, in order to solve the second optimization model with constraints in (4), it is converted into the following form:

且对所有i′＝1,2,...,M，

μ是正则化参数，采用临近线性化的思想将问题转化为：where D is a diagonal matrix whose diagonal elements are

And for all i'=1,2,...,M,

μ is a regularization parameter, using the idea of near linearization to transform the problem into:

用L0最小化方法求解(10)式，其迭代格式如下：in

Using the L0 minimization method to solve equation (10), the iterative format is as follows:

的迭代形式如下：对所有图像坐标i′,j′

The iterative form of is as follows: For all image coordinates i′, j′

表示傅里叶变换，

表示傅里叶反变换，

表示傅里叶变换的复共轭，▽_x,▽_y分别表示x,y的梯度算子；β控制

相似性的参数，κ(κ＞1)表示控制β增长速度的参数；n表示

算法的迭代次数，

算法的停机标准是β大于迭代前预设的参数β_max，当

算法的停机后输出图像为f^k+1；in

represents the Fourier transform,

represents the inverse Fourier transform,

Represents the complex conjugate of Fourier transform, ▽ _x , ▽ _y represent the gradient operators of x and y respectively; β controls

The parameter of similarity, κ (κ>1) represents the parameter that controls the growth rate of β; n represents

the number of iterations of the algorithm,

The stop criterion of the algorithm is that β is greater than the preset parameter β _max before iteration, when

The output image after the shutdown of the algorithm is f ^k+1 ;

Assume that the stoppage criteria corresponding to iteration formats (8) and (11) reach the preset iteration times N _ite1 and N _{ite2 respectively} , set the total number of iterations of the entire algorithm as N _Tot , and tt is the index of the total number of iterations, then iteration The process is as follows:

Step S3 limited-angle CT iterative reconstruction includes the following two sub-steps:

S31. L0 regularized image reconstruction based on non-subsampled contourlet transform;

S32. L0 regularized image reconstruction based on gradient transformation;

Wherein S31 includes the following 3 sub-steps:

S311. ART iterative reconstruction and non-subsampled contourlet inverse transformation are combined to obtain a preliminary reconstruction result;

S312. Perform hard threshold processing on the high-frequency part and low-frequency part in the non-subsampled contourlet transform domain to the obtained preliminary results to suppress artifacts and noise, and at the same time perform boundary protection; S313. Update the dual variable v;

Wherein S32 includes the following two sub-steps:

S321. ART iterative reconstruction, wherein the initial image is the output result of S31;

最小化光滑处理，来进一步抑制伪影和噪声，并进行平滑处理；当达到一定的迭代次数N_Tot时停止迭代，否则重复步骤S31-S32。S322. Carry out the ART result

Minimize smoothing to further suppress artifacts and noise, and perform smoothing; stop iteration when a certain number of iterations N _Tot is reached, otherwise repeat steps S31-S32.

Optionally, said S4 is specifically: when the iterative algorithm in step S3 stops iterating, outputting the reconstructed image.

The beneficial effects of the present invention are: the present invention considers the sparsity of the image under non-subsampling contourlet transformation and image gradient transformation, and utilizes the multi-scale, multi-resolution decomposition and artifact directionality characteristics of non-subsampling contourlet transformation , setting up two optimization objectives of the non-subsampled contourlet transform L0 norm minimization and the L0 norm of the image gradient, and established a multi-objective optimization reconstruction model. In order to suppress noise, directional artifacts and protect boundaries, the present invention performs hard threshold processing on the high-frequency and low-frequency parts of the non-subsampled contourlet transform domain on the basis of the ART algorithm; in order to further suppress artifacts and noise and make the reconstructed image It becomes smooth, and the image gradient L0 is minimized on the basis of the ART algorithm. The multi-objective limited-angle CT image reconstruction method of maximization and minimization disclosed by the present invention establishes a multi-objective optimization model from the perspective of multi-objectives, and uses the maximization and minimization method to solve the multi-objective model, which includes ART iteration, Hard thresholding in the non-subsampled contourlet transform domain, L0 minimization of image gradients. Through this maximin-minimum multi-object method, while protecting the boundaries of CT images, limited-angle artifacts and noises are effectively suppressed, thereby improving the quality of CT reconstruction images to a great extent.

Other advantages, objects and features of the present invention will be set forth in the following description to some extent, and to some extent, will be obvious to those skilled in the art based on the investigation and research below, or can be obtained from It is taught in the practice of the present invention. The objects and other advantages of the invention may be realized and attained by the following specification.

Description of drawings

In order to make the purpose of the present invention, technical solutions and advantages clearer, the present invention will be described in detail below in conjunction with the accompanying drawings, wherein:

Fig. 1 is a schematic diagram of the geometric structure of a multi-target limited-angle CT image reconstruction method based on maximin minimization;

Fig. 2 is a flowchart of a multi-target limited-angle CT image reconstruction method based on maximin minimization.

Detailed ways

Embodiments of the present invention are described below through specific examples, and those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific implementation modes, and various modifications or changes can be made to the details in this specification based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the diagrams provided in the following embodiments are only schematically illustrating the basic concept of the present invention, and the following embodiments and the features in the embodiments can be combined with each other in the case of no conflict.

Wherein, the accompanying drawings are for illustrative purposes only, and represent only schematic diagrams, rather than physical drawings, and should not be construed as limiting the present invention; in order to better illustrate the embodiments of the present invention, some parts of the accompanying drawings may be omitted, Enlargement or reduction does not represent the size of the actual product; for those skilled in the art, it is understandable that certain known structures and their descriptions in the drawings may be omitted.

In the drawings of the embodiments of the present invention, the same or similar symbols correspond to the same or similar components; , "front", "rear" and other indicated orientations or positional relationships are based on the orientations or positional relationships shown in the drawings, which are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the referred devices or elements must It has a specific orientation, is constructed and operated in a specific orientation, so the terms describing the positional relationship in the drawings are for illustrative purposes only, and should not be construed as limiting the present invention. For those of ordinary skill in the art, the understanding of the specific meaning of the above terms.

Figure 1 is a schematic diagram of the geometric structure of a multi-target limited-angle CT image reconstruction method based on maximin minimization. As shown in the figure: establish a right-handed rectangular coordinate system O-xy from the ray source S to the rotation center O, the y-axis passes through the line between the rotation center O and the ray source S, and the direction from the rotation center O to the ray source S is the positive direction, x The axis is perpendicular to the y-axis and to the right is its positive direction. (x, y) is the coordinate of the reconstructed pixel J, and SL represents a ray passing through the reconstructed point J. The a-axis represents the corresponding position of the detection unit of the detector and the positive direction is consistent with the x-axis, and (a, s) represents the coordinate formed by the projection viewing angle index s and the detector position a.

Fig. 2 is the flowchart of the multi-objective limited-angle CT image reconstruction method based on maximization, as shown in the figure: the multi-objective limited-angle CT image reconstruction method based on maximum minimization comprises the following steps:

S1. Acquisition of projection data: rotate the ray source around the rotation center along the scanning track for a limited angle to obtain incomplete projection data;

S2. Establish a multi-target limited-angle CT image reconstruction model: when using a discrete model for reconstruction, first of all, it is necessary to rearrange f(x, y) corresponding to all coordinates (x, y) shown in Figure 1 according to the dimension of y into A 1-dimensional vector f, its dimension is N×1, where N=n ₁ ×n ₂ , n ₁ is the dimension of f(x,y) in the x direction, n ₂ is the dimension of f(x,y) in y The dimensionality of the direction. Then, rearrange the projection data g ^δ (a, s) corresponding to all coordinates (a, s) into a 1-dimensional vector g ^δ according to the dimension of s, and the dimension of the column vector g ^δ is M×1, where M=m ₁ ×m ₂ , m ₁ is the dimension of g ^δ (a, s) in the a direction, m ₂ is the dimension of g ^δ (a, s) in the s direction. According to the present invention, non-subsampling contourlet transformation is considered to decompose the reconstructed image into low-frequency part and high-frequency part, so that the image can be decomposed into multi-scale and multi-resolution. In order to suppress noise and artifacts and avoid loss of some details caused by over-smoothing, the L0 sparse regularization constraint is applied to the coefficients of the non-subsampled contourlet transform. Since the reconstructed image may be undersmooth in order to preserve the details, in order to make the image smooth, the L0 regularization of the image gradient is used for constraints. The model that the present invention sets up from multi-objective angle is as follows:

where A∈R ^M×N is the finite-angle CT system matrix, f∈R ^N×1 is the image to be reconstructed, g ^δ ∈ R ^M×1 is the finite-angle CT projection data, and Ω is a convex set (Ω:={f| f≥0}). λ is the relaxation parameter, W is the non-subsampled contourlet transform, and i is the subband index. ||β|| ₀ is the number of non-zero elements of statistical β, ▽β=(▽ _x β,▽ _y β), the component form of ▽ _x β,▽ _y β is (▽ _x β) _i′,j′ =β _i',j' -β _i'-1,j' ,(▽ _y β) _i',j' =β _i',j' -β _i',j'-1 . When doing non-subsampled contourlet transformation, rearrange f into a 2-dimensional matrix f(x,y), and then perform non-subsampled contourlet transformation on the 2-dimensional matrix. After the non-subsampled contourlet inverse transformation is completed , rearrange f(x,y) into a 1-dimensional vector f according to the dimension of y.

S3. Limited-angle CT iterative reconstruction: according to the established model (1), the model (1) is solved by the method of maximization and minimization;

The specific process of step S3 finite-angle CT iterative reconstruction is as follows:

First, the model (1) is transformed into a single-objective optimization model through the maximin method, and its form is as follows:

Then, in order to effectively solve the single-objective optimization model, λ is taken as two special values. First, λ=1 to solve the corresponding optimization problem; then λ=0 to solve the corresponding optimization problem. In order to make the final solution converge, repeat the above process of λ value selection. So the single-objective optimization model is simplified as follows:

Model (3) is equivalent to the following form:

5) When λ=1, in order to solve the first optimization model with constraints in (4), it is converted into the following form:

且对所有i′＝1,2,...,M，

γ_i是正则化参数，再采用交替方向迭代法(ADMM)求解模型(5)，其迭代格式如下：where ||x|| _D =<Dx,x>. D is a diagonal matrix whose diagonal elements are

And for all i'=1,2,...,M,

γ _i is the regularization parameter, and then the model (5) is solved by the alternating direction iterative method (ADMM), and the iterative format is as follows:

t is a parameter introduced by the ADMM algorithm.

In order to avoid the deficiency of seeking the inverse of the system matrix A in the subproblem about the first variable f or adopting the iterative method to solve the subproblems, the present invention embeds the idea of adjacent alternating linearization, and converts the iterative format (6) into adjacent alternating linearization The ADMM iteration format is as follows:

是临近线性化引入的松弛参数。(7)中

为ART迭代更新格式。通过临近交替线性化巧妙地将经典的ART迭代算法融入其中。In iterative format (7)

is the slack parameter introduced near linearization. (7)

Update format for ART iterations. The classic ART iterative algorithm is subtly integrated into it by adjacent alternating linearization.

Then find the optimal solution to the subproblem of the iterative format (7), and the iterative format is as follows:

where (W ^T represents the non-subsampled contourlet inverse transform),

6) When λ=0, in order to solve the second optimization model with constraints in (4), it is converted into the following form:

且对所有i′＝1,2,...,M，

μ是正则化参数，采用临近线性化的思想将问题转化为：where D is a diagonal matrix whose diagonal elements are

And for all i'=1,2,...,M,

μ is a regularization parameter, using the idea of near linearization to transform the problem into:

用L0最小化方法求解(10)式，其迭代格式如下:in

Using the L0 minimization method to solve equation (10), the iterative format is as follows:

的迭代形式如下：对所有图像坐标i′,j′

The iterative form of is as follows: For all image coordinates i′, j′

表示傅里叶变换，

表示傅里叶反变换，

表示傅里叶变换的复共轭，▽_x,▽_y分别表示x,y的梯度算子；β控制

相似性的参数，κ(κ＞1)表示控制β增长速度的参数；n表示

算法的迭代次数，

算法的停机标准是β大于迭代前预设的参数β_max，当

算法的停机后输出图像为f^k+1。in

represents the Fourier transform,

represents the inverse Fourier transform,

Represents the complex conjugate of Fourier transform, ▽ _x , ▽ _y represent the gradient operators of x and y respectively; β controls

The parameter of similarity, κ (κ>1) represents the parameter that controls the growth rate of β; n represents

the number of iterations of the algorithm,

The stop criterion of the algorithm is that β is greater than the preset parameter β _max before iteration, when

The output image after the stoppage of the algorithm is f ^k+1 .

Assuming that the shutdown criteria corresponding to iteration formats (8) and (11) are respectively reaching the preset iteration times N _ite1 and N _ite2 , and setting the total number of iterations of the entire algorithm as N _Tot (tt is the index of the total number of iterations), then The iteration process is as follows:

最小化光滑处理，来进一步抑制伪影和噪声，并进行平滑处理。当达到一定的迭代次数N_Tot时停止迭代，否则重复步骤S31-S32。According to the above method, step S3 finite-angle CT iterative reconstruction includes the following two sub-steps: S31. L0 regularized image reconstruction based on non-subsampled contourlet transformation; S32. L0 regularized image reconstruction based on gradient transformation. Among them, S31 includes the following three sub-steps: S311. ART iterative reconstruction and non-subsampled contourlet inverse transformation combination to obtain preliminary reconstruction results; Hard threshold processing to suppress artifacts and noise, and at the same time perform boundary protection; S313. Dual variable v update. Wherein S32 includes the following two sub-steps: S321. ART iterative reconstruction, wherein the initial image is the output result of S31; S322.

Minimize smoothing to further suppress artifacts and noise and smooth. When a certain number of iterations N _Tot is reached, the iteration is stopped, otherwise steps S31-S32 are repeated.

S4. Outputting the reconstructed image. When the iterative algorithm in step S3 stops iterating, the reconstructed image is output.

Finally, it is noted that the above embodiments are only used to illustrate the technical solutions of the present invention without limitation. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that the technical solutions of the present invention can be carried out Modifications or equivalent replacements, without departing from the spirit and scope of the technical solution, should be included in the scope of the claims of the present invention.

Claims (2)

1. A multi-target limited-angle CT image reconstruction method based on maximin minimization, characterized in that: the method may further comprise the steps: S1: Collect projection data: S2: Establish a multi-target limited-angle CT image reconstruction model; S3: Limited-angle CT iterative reconstruction; S4: output the reconstructed image; The specific S2 is: when the discrete model is used for reconstruction, f(x, y) corresponding to the coordinates (x, y) first needs to be rearranged into a 1-dimensional vector f according to the dimension of y, and its dimension is N×1 , where N=n ₁ ×n ₂ , n ₁ is the dimension of f(x,y) in the x direction, n ₂ is the dimension of f(x,y) in the y direction; then, all the coordinates (a, s) The corresponding projection data g ^δ (a,s) is rearranged into a 1-dimensional vector g ^δ according to the dimension of s, and the dimension of the column vector g ^δ is M×1, where M=m ₁ ×m ₂ , and m ₁ is The dimension of g ^δ (a, s) in the a direction, m ₂ is the dimension of g ^δ (a, s) in the s direction; the reconstructed image is decomposed into low-frequency part and high-frequency part by non-subsampling contourlet transform, Make the image decompose in multi-scale and multi-resolution; In order to suppress noise and artifacts and avoid loss of some details caused by over-smoothing, L0 sparse regularization constraints are performed on the coefficients of non-subsampled contourlet transformation; for image smoothness, L0 regularization of image gradients is used to constrain; The model established from the multi-objective perspective is as follows:

where A∈R ^M×N is the finite-angle CT system matrix, f∈R ^N×1 is the image to be reconstructed, g ^δ ∈ R ^M×1 is the finite-angle CT projection data, Ω is a convex set, Ω:={f| f≥0}; λ is the relaxation parameter, W is the non-subsampled contourlet transform, i is the index of the subband; ||β|| ₀ is the number of non-zero elements of the statistical β,

The component form of

When doing non-subsampled contourlet transformation, rearrange f into a 2-dimensional matrix f(x,y), and then perform non-subsampled contourlet transformation on the 2-dimensional matrix. After the non-subsampled contourlet inverse transformation is completed , rearrange f(x,y) into a 1-dimensional vector f according to the dimension of y; The S3 is specifically: According to the established model (1), the model (1) is solved by the method of maximization and minimization; The specific process of step S3 finite-angle CT iterative reconstruction is as follows: First, the model (1) is transformed into a single-objective optimization model through the maximin method, and its form is as follows:

Then, in order to effectively solve the single-objective optimization model, take λ as two special values. First, λ=1 to solve the corresponding optimization problem; then λ=0, and then solve the corresponding optimization problem; in order to make the final solution converge, repeat The above value process of λ; so the single-objective optimization model is simplified as follows:

Model (3) is equivalent to the following form:

1) When λ=1, in order to solve the first optimization model with constraints in (4), it is converted into the following form:

Where ||x|| _D ＝<Dx,x>; D is a diagonal matrix whose diagonal elements are

And for all i'=1,2,...,M,

γ _i is the regularization parameter, and the model (5) is solved by using the alternating direction iterative method ADMM, and the iterative format is as follows:

t is a parameter introduced by the ADMM algorithm; In order to avoid the inverse of the system matrix A in the sub-problem about the first variable f or to use the iterative method to solve the sub-problem, the idea of adjacent alternating linearization is embedded, and the iterative format (6) is converted into ADMM of adjacent alternating linearization The iterative format, as follows:

In the iterative format (7), ρ is the relaxation parameter introduced by the proximity linearization,

(7)

Updated format for ART iteration; subtly incorporates classic ART iteration algorithm through adjacent alternating linearization; Then find the optimal solution to the subproblem of the iterative format (7), and the iterative format is as follows:

where W ^T represents the non-subsampled contourlet inverse transform,

2) When λ=0, in order to solve the second optimization model with constraints in (4), it is converted into the following form:

where D is a diagonal matrix whose diagonal elements are

And for all i'=1,2,...,M,

μ is a regularization parameter, using the idea of near linearization to transform the problem into:

in

Using the L0 minimization method to solve equation (10), the iterative format is as follows:

The iterative form of is as follows: For all image coordinates i′, j′

in

represents the Fourier transform,

represents the inverse Fourier transform,

represents the complex conjugate of the Fourier transform,

represent the gradient operator of x and y respectively; τ controls

The parameter of similarity, η>1 means the parameter that controls the growth rate of τ; n means

the number of iterations of the algorithm,

The stopping criterion of the algorithm is that τ is greater than the preset parameter τ _max before the iteration, when

The output image after the shutdown of the algorithm is f ^k+1 ; Assume that the stoppage criteria corresponding to iteration formats (8) and (11) reach the preset iteration times N _ite1 and N _{ite2 respectively} , set the total number of iterations of the entire algorithm as N _Tot , and tt is the index of the total number of iterations, then iteration The process is as follows:

Step S3 limited-angle CT iterative reconstruction includes the following two sub-steps: S31. L0 regularized image reconstruction based on non-subsampled contourlet transform; S32. L0 regularized image reconstruction based on gradient transformation; Wherein S31 includes the following 3 sub-steps: S311. ART iterative reconstruction and non-subsampled contourlet inverse transformation are combined to obtain a preliminary reconstruction result; S312. Perform hard threshold processing on the high-frequency part and low-frequency part in the non-subsampled contourlet transform domain to the obtained preliminary results to suppress artifacts and noise, and at the same time perform boundary protection; S313. Update the dual variable v; Wherein S32 includes the following two sub-steps: S321. ART iterative reconstruction, wherein the initial image is the output result of S31; S322. Carry out the ART result

Minimize smoothing processing to further suppress artifacts and noise, and perform smoothing processing; stop iteration when a certain number of iterations N _Tot is reached, otherwise repeat steps S31 to S32; Said S4 is specifically: when the iterative algorithm in step S3 stops iterating, outputting the reconstructed image. 2. A multi-target limited-angle CT image reconstruction method based on maximin and minimization according to claim 1, characterized in that: said S1 is specifically: the ray source is rotated limitedly around the rotation center along the scanning orbit angle to obtain incomplete projection data.

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