CN112381904B - Limited angle CT image reconstruction method based on DTw-SART-TV iterative process - Google Patents
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Abstract
The invention provides a limited angle CT image reconstruction method based on DTw-SART-TV iteration process, in which SART is used as a basic value of each internal iteration and is constrained by using TV gradient descent. The reconstructed image performance is compared with that of SART, SART-TV and STH. Experimental results show that the linear combination of DTw-SART-TV based on IRS by using the first two iterative reconstruction results can effectively improve the convergence rate of CT image reconstruction. Furthermore, the details of the reconstructed image can be effectively improved by using TV constraints, and the overall convergence speed is further improved with dynamic relaxation factors. Generally, DTw-SART-TV, which has the same number of projection views, has faster convergence speed and better image reconstruction performance than SART, SART-TV, and STH in terms of several indexes MSE, PSNR, SSIM and TEI.
Description
Technical Field
The invention relates to the field of CT compression tomography, in particular to a limited-angle CT image reconstruction method based on DTw-SART-TV iterative process.
Background
CT compression tomography is a tomographic technique. It is based on the development of X-ray imaging, another revolution in medical imaging history. The basic principles of CT imaging dictate that each patient is required to receive multiple doses of X-ray radiation during the imaging process.
Currently, the radiation dose for CT examinations corresponds to 10 to 15X-rays. Despite the relatively high safety level of current CT doses, patients often have concerns about absorbing too much radiation when performing CT imaging examinations. In addition, CT imaging conditions are different due to the difference in structures of various parts of the human body, and certain parts of the human body cannot be scanned at all angles, that is, are captured at 180 degrees during CT scanning. Imaging procedures may lead to artifacts that affect the diagnosis of the physician.
To overcome the problems of large CT radiation dose and limited imaging angle, solutions can be divided into two categories: one is to reduce the X-ray flux, and the other is to reduce the amount of X-rays received by the human body. However, the former may cause excessive noise in the projection data, which is disadvantageous for forming high-quality image reconstruction; the latter would lead to incomplete projection data due to the small viewing angle, limited angle and internal CT, resulting in CT undersampling problems. The traditional filtering backward propagation algorithm and other analytic reconstruction algorithms have little influence on the sparse reconstruction problem, and cannot obtain high-quality reconstructed images. Therefore, the use of lower CT radiation doses and the use of algebraic reconstruction algorithms to obtain satisfactory imaging results during imaging is a major focus of research in current CT imaging techniques.
The conventional CT algebraic reconstruction algorithm mainly comprises an algebraic reconstruction technique (ART, simultaneous Iterative Reconstruction Technique (SIRT), simultaneous Algebraic Reconstruction Technique (SART), and the like, wherein in each iteration, the ART algorithm performs back projection on X-rays to solve the reconstruction problem and updates pixel data through which the X-rays pass.
However, when the projection data is highly undersampled without sufficient a priori information, the reconstructed image is difficult to fully converge, resulting in more artifacts and poor image detail. Accordingly, many researchers have focused and studied on eliminating CT image artifacts caused by the small viewpoint scan and proposed many novel algorithms such as iterative reconstruction re-projection (IRR), the Gerchberg-Papoulis method, singular Value Decomposition (SVD), projection Onto Convex Sets (POCS), and the like. Another new approach to solving the multi-view CT reconstruction problem is an iterative algorithm with a priori information. The a priori information includes boundaries, shapes, density ranges, etc. of the object. The more a priori information that is available to the image, the greater the likelihood that the algorithm used will reduce the wedge-shaped artifacts and achieve high reconstruction quality.
Disclosure of Invention
The invention provides a dynamic two-step SART-TV iteration method (DTw-SART-TV) so as to improve the iteration efficiency and the final reconstruction precision of small-view CT image reconstruction. The method is based on SART-TV and IRS, and we also use dynamic relaxation factors to improve the convergence efficiency of the iteration.
In order to achieve the above purpose, the specific technical scheme provided by the invention is as follows:
A limited angle CT image reconstruction method based on DTw-SART-TV iterative process is characterized by comprising two stages, wherein the first stage is to replace ART with SART in ART-TV, namely SART-TV; the second stage is to introduce IRS in the internal iteration, namely DTw-SART-TV.
The first stage comprises: step 1.1, reconstructing an initial CT image by using an SART method; step 1.2, applying non-negative limitation to the reconstructed CT image; step 1.3, solving the problem of TV specification reconstruction based on TV optimization; step 1.4, repeating steps 1.1 to 1.3 until global convergence is achieved.
In the step 1.1, the SART iterative formula is described as follows:
Where p i is projection data, w ij is an element of a system matrix, f= [ f 1,f2,…,fN]T ] is an image matrix, i and j represent rows and columns of the matrix, respectively, M is a number of rows of an image, and N is a number of columns of an image. W is the system matrix, W +j represents the sum of the j-th columns of the system matrix W, and W i+ represents the sum of the i-th rows of the system matrix W.
In the step 1.3, the result of the step 1.2 is used as an initial value to solve the following TV specification reconstruction problem based on TV optimization:
min||f||TVs.t.||Wf-P||≤ε，f≥0 (2)
Where i f TV is the total variation of image f, P is the CT projection data, and is the maximum allowable error.
The equation for solving the above equation using the gradient descent method is as follows:
d＝||fk-fk-1||2 (4)
where TV (f) is the total variation f of the image f, d η =d·η is the gradient descent step of each iteration update, d is the initial step, η is the relaxation factor.
The gradient of the total change of the image is defined as follows:
where epsilon usually takes a small value to prevent the denominator from being zero, usually epsilon=10 -8.
Repeating the steps (3) - (6) to obtain a solution of formula (2).
Steps 1.1 to 1.3 are repeated until global convergence is achieved.
Said step 1.3 is also called internal recycle. The iteration termination condition of the outer loop is determined by the rate of change of the residual values of the reconstructed CT image, which is defined as follows:
If G k(fk,fk-1) < σ, stopping the iteration; σ is a preset value, which can be set to 0.001.
SART-TV controls the gradient descent process by specifying the gradient descent relaxation factor eta and the step number, and adjusts the balance between the SART stage and the TV gradient descent stage; if the step size of the gradient descent is too large, the image will become too smooth and inconsistent with the projection data; if the step size of the gradient descent is too small, the effect of this method is almost the same as SART.
The second stage comprises: step 2.1: introducing IRS in the internal iteration; step 2.2: performing two-step iteration by using a TV gradient descent method; step 2.3: using the residual error of the CT projection value to reflect the iteration degree so as to dynamically adjust the relaxation factor of the inner ring to improve the convergence rate; step 2.4: dynamic termination conditions based on the rate of change of the total variation of the reconstructed CT image are used to determine termination of the inner loop.
The step 2.1 comprises two steps:
in the first step, a basic iterative method is used to solve the linear equation.
Secondly, the current iteration value is adjusted by using the linear weighted sum of the previous two iteration results, and a specific iteration formula is as follows:
f1＝Γλ(f0)
ft+1＝(1-α)ft-1+(α-β)ft+βΓλ(ft) (8)
Where α and β are parameters of the relaxation factor, calculated by the following formula. Γ (·) represents a specific noise reduction function.
Delta 1,δN is used to ensure the range of iterative processes, typically 0 < alpha < 2,0 < beta < 2 alpha
In equation (8), Γ (·) may be set to a different function depending on the case.
In the step 2.2, the complete two-step iterative process of the TV is as follows:
dη＝||fk-fk-1||2·η (11)
Thus, the result of the first two SART-TV iterations will be used to adjust the value of the next iteration.
The effective noise reduction performance of the TV and the two-step iterative acceleration can be effectively combined.
When δ 1 =1 and δ N =1 are in equation (9), equation (10) can be converted into a standard TV gradient descent method.
In the step 2.3 of the process described above,
Wherein,Representing the reconstructed value of the n-TV gradient descent iteration.
From the above equation, in the initial stage of the gradient descent iteration, the projection value of the difference between the reconstructed image and the target image is large, the value of the relaxation factor η is small, and the speed of convergence can be accelerated; when the projection value of the reconstructed image gradually approaches to the projection value of the target image, the relaxation factor becomes large, which can control the convergence speed of the iterative process and maintain the reconstruction performance of the CT image.
In the step 2.4, the dynamic termination condition based on the change rate of the total variation of the reconstructed CT image is used to determine the termination condition of the inner ring, which is defined as follows:
wherein, Is the maximum allowable error of the internal iteration.
The present invention introduces an iterative weighted contraction (IRS), a method that uses the results of previous iterations to accelerate convergence. The method combines the strategy with least square, and proposes iterative weighted least square, and the method is applied to the recovery of sparse signals. The iterative weighted contraction and the threshold contraction algorithm are combined to effectively increase the convergence speed of the threshold contraction algorithm and successfully apply the same to MR image reconstruction.
Drawings
Fig. 1 is a chest full angle sampling CT image diagram according to example 2;
fig. 2 is a windowed view of a chest full angle sampling CT image according to embodiment 2;
Fig. 3 is a chest CT image after limited angle reconstruction according to the method of embodiment 2.
Detailed Description
The invention will be further described with reference to the drawings and examples.
Example 1
A limited angle CT image reconstruction method based on DTw-SART-TV iterative process is characterized by comprising two stages, wherein the first stage is to replace ART with SART in ART-TV, namely SART-TV; the second stage is to introduce IRS in the internal iteration, namely DTw-SART-TV.
The first stage comprises: step 1.1, reconstructing an initial CT image by using an SART method; step 1.2, applying non-negative limitation to the reconstructed CT image; step 1.3, solving the problem of TV specification reconstruction based on TV optimization; step 1.4, repeating steps 1.1 to 1.3 until global convergence is achieved.
In the step 1.1, the SART iterative formula is described as follows:
Where p i is projection data, w ij is an element of a system matrix, f= [ f 1,f2,…,fN]T ] is an image matrix, i and j represent rows and columns of the matrix, respectively, M is a number of rows of an image, and N is a number of columns of an image. W is the system matrix, W +j represents the sum of the j-th columns of the system matrix W, and W i+ represents the sum of the i-th rows of the system matrix W.
In the step 1.3, the result of the step 1.2 is used as an initial value to solve the following TV specification reconstruction problem based on TV optimization:
min||f||TV s.t.||Wf-P||≤ε，f≥0 (2)
Where i f TV is the total variation of image f and P is the CT projection data.
The equation for solving the above equation using the gradient descent method is as follows:
where TV (f) is the total variation of the image f, d η =d·η is the gradient descent step size updated each iteration, d is the initial step size, η is the relaxation factor.
The gradient of the total change of the image is defined as follows:
where epsilon usually takes a small value to prevent the denominator from being zero, usually epsilon=10 -8.
Repeating the steps (3) - (6) to obtain a solution of formula (2).
Steps 1.1 to 1.3 are repeated until global convergence is achieved.
Said step 1.3 is also called internal recycle. The iteration termination condition of the outer loop is determined by the rate of change of the residual values of the reconstructed CT image, which is defined as follows:
if G k(fk,fk-1) < σ, stopping the iteration; sigma is a preset value.
SART-TV controls the gradient descent process by specifying gradient descent relaxation factor eta and step size, and adjusts the balance between SART stage and TV gradient descent stage; if the step size of the gradient descent is too large, the image will become too smooth and inconsistent with the projection data; if the step size of the gradient descent is too small, the effect of this method is almost the same as SART.
The second stage comprises: step 2.1: introducing IRS in the internal iteration; step 2.2: performing two-step iteration by using a TV gradient descent method; step 2.3: using the residual error of the CT projection value to reflect the iteration degree so as to dynamically adjust the relaxation factor of the inner ring to improve the convergence rate; step 2.4: dynamic termination conditions based on the rate of change of the total variation of the reconstructed CT image are used to determine termination of the inner loop.
The step 2.1 comprises two steps:
in the first step, a basic iterative method is used to solve the linear equation.
Secondly, the current iteration value is adjusted by using the linear weighted sum of the previous two iteration results, and a specific iteration formula is as follows:
f1＝Γλ(f0)
ft+1＝(1-α)ft-1+(α-β)ft+βΓλ(ft) (8)
Where α and β are parameters of the relaxation factor, calculated by the following formula. Γ (·) represents a specific noise reduction function.
Delta 1,δw is used to ensure the range of iterative processes, typically 0 < alpha < 2,0 < beta < 2 alpha
In equation (8), Γ (·) may be set to a different function depending on the case.
In the step 2.2, the complete two-step iterative process of the TV is as follows:
dη＝||fk-fk-1||2·η (11)
Thus, the result of the first two SART-TV iterations will be used to adjust the value of the next iteration.
The effective noise reduction performance of the TV and the two-step iterative acceleration can be effectively combined.
When δ 1 =1 and δ N =1 are in equation (9), equation (10) can be converted into a standard TV gradient descent method.
The basic ART-TV algorithm fixes the relaxation factor η to 0.2 during the iteration and the gradient drop decreases with a fixed step size. This does not apply to the entire iterative process and does not effectively balance the iterative speed and accuracy. In practice, we want to converge quickly in early iterations and reduce the contraction speed in later iterations to maintain intermediate reconstruction accuracy.
In the step 2.3 of the process described above,
Wherein,Representing the reconstructed value of the n-TV gradient descent iteration.
From the above equation, in the initial stage of the gradient descent iteration, the projection value of the difference between the reconstructed image and the target image is large, the value of the relaxation factor η is small, and the speed of convergence can be accelerated; when the projection value of the reconstructed image gradually approaches to the projection value of the target image, the relaxation factor becomes large, which can control the convergence speed of the iterative process and maintain the reconstruction performance of the CT image. As in ART-TV, η=0.2 is set as an initial value. ART-TV has a double iteration loop. The outer loop iteration stop condition is when a preset iteration number or a preset rate of change of the reconstructed residual is reached, but the iteration number of the inner loop is fixed. But a fixed number of iterations cannot optimally determine whether an inner loop iteration has completed.
In the step 2.4, the dynamic termination condition based on the change rate of the total variation of the reconstructed CT image is used to determine the termination condition of the inner ring, which is defined as follows:
wherein, For the maximum allowable error of the internal iteration, set to/>
As one possible implementation, the DTw-SART-TV complete iteration procedure is as follows:
Input:
W: system matrix
P: projection data
Maximum gradient descent number
Alpha, beta: parameters of relaxation coefficient
F: reconstructed CT image
Initializing:
k＝1,f0＝0,Ngrad＝20,μ＝0.9,ε＝10-8,σ＝10-5λ1＝0.001,λN＝1,η＝0.2,
When "G k(fk,fk-1) > σ", run;
updating f k by equation (1);
update f k by equation f k＝max(fk, 0);
When (when) " ", Run;
updating f k by equation (10);
Updating η by equation (12);
Updating d η by equation (11);
Updating τ by equation (13);
Ending;
k＝k+1；
Ending;
And (3) outputting: a reconstructed CT image f.
Example 2
In this embodiment, a windowed chest CT image (FIGS. 1-3) is used as the test image.
In the present embodiment, CT projection data is acquired by fan beam scanning.
The specific parameters are set as follows: the distance from the X-ray source to the rotation center is 40cm, the distance from the detector center to the rotation center is 40cm, and the length of the detector is 41.3cm.
Further, using a flat panel detector with 512 detection units, the reconstructed image was 256×256 pixels and the image array size was 20cm×20cm. The system projection matrix is obtained by the Siddon ray driven algorithm and uses 40 and 60 projection views of a 360 degree distribution.
During the experiment we first tested the performance of each improved aspect of the described method and then compared the proposed method with SART, SART-TV and STH to test CT images.
Finally, the reconstruction results of the four methods are evaluated using various evaluation parameters. The quality of the image reconstruction can be qualitatively and quantitatively analyzed by subjective and objective assessment, respectively.
Subjective assessment is based on the intuition of the observer, truly reflecting the visual perception of humans.
Objective assessment is based primarily on specific mathematical models and results based on digital calculations in some aspects that favor human subjective perception.
The present embodiment uses four evaluation parameters to evaluate the reconstruction effect of the CT image, including Mean Square Error (MSE), peak signal to noise ratio (PSNR), structural Similarity Index Measure (SSIM), and Transition Edge Information (TEI).
TEI is an evaluation parameter describing image edge information. The larger the value, the better the image reconstruction effect. The calculation formula is as follows:
Where Γ g、kg and σ g are constants that determine edge strength, and Γ a、ka and σ a are constants that determine edge direction. m and n are two images. G mn (i, j) and a mn (i, j) represent the relative edge strength and the relative edge direction, respectively, and are defined as follows:
After horizontal and vertical filtering using a Sobel filter, where g m (i, j) and g n (i, j) represent edge intensity information of the image, respectively; a m (i, j) and an (i, j) represent the direction information of the images m and n.
The foregoing has shown and described the basic principles, principal features and advantages of the invention. It will be understood by those skilled in the art that the present invention is not limited to the above-described embodiments, and that the above-described embodiments and descriptions are only preferred embodiments of the present invention, and are not intended to limit the invention, and that various changes and modifications may be made therein without departing from the spirit and scope of the invention as claimed. The scope of the invention is defined by the appended claims and equivalents thereof.
Claims (1)
1. A limited angle CT image reconstruction method based on DTw-SART-TV iterative process is characterized by comprising two stages, wherein the first stage is to replace ART with SART in ART-TV, namely SART-TV; the second stage is to introduce IRS in the internal iteration to carry out dynamic two-step iteration, namely DTw-SART-TV;
The first stage comprises: step 1.1, reconstructing an initial CT image by using an SART method; step 1.2, applying non-negative limitation to the reconstructed CT image; step 1.3, solving the problem of TV specification reconstruction based on TV optimization; step 1.4, repeating steps 1.1 to 1.3 until global convergence is achieved;
in the step 1.1, the SART iterative formula is described as follows:
Wherein p i is projection data, w ij is an element of a system matrix, f= [ f 1,f2,…,fN]T ] is an image matrix, i and j respectively represent rows and columns of the matrix, M is a row number of an image, and N is a column number of the image; w is a system matrix, W +j represents the sum of the j-th columns of the system matrix W, and W i+ represents the sum of the i-th rows of the system matrix W;
In the step 1.3, the result of the step 1.2 is used as an initial value to solve the following TV specification reconstruction problem based on TV optimization:
min||f||TV s.t.||Wf-P||≤ε,f≥0 (2)
wherein, f TV is the total variable component of the image f, P is CT projection data and is the maximum allowable error;
the above equation (2) is solved using the gradient descent method as follows:
d＝||fk-fk-1||2 (4)
where TV (f) is the total variable component of the image f, d η =d·η, is the gradient descent step size updated each iteration, d is the initial step size, η is the relaxation factor;
The gradient of the total change of the image is defined as follows:
where ε typically takes a small value to prevent the denominator from becoming zero;
Repeating (3) - (6) to obtain a solution of formula (2);
repeating steps 1.1 to 1.3 until global convergence is achieved;
said step 1.3 is also called internal recycle; the iteration termination condition of the outer loop is determined by the rate of change of the residual values of the reconstructed CT image, which is defined as follows:
If G k(fk,fk-1) < σ, stopping the iteration;
The second stage comprises: step 2.1: introducing IRS in the internal iteration; step 2.2: performing two-step iteration by using a TV gradient descent method; step 2.3: using the residual error of the CT projection value to reflect the iteration degree so as to dynamically adjust the relaxation factor of the inner ring to improve the convergence rate; step 2.4: a dynamic termination condition based on the rate of change of the total variation of the reconstructed CT image is used to determine termination of the inner loop;
The step 2.1 comprises two steps:
firstly, solving a linear equation by using a basic iteration method;
secondly, the current iteration value is adjusted by using the linear weighted sum of the previous two iteration results, and a specific iteration formula is as follows:
f1＝Γλ(f0)
ft+1＝(1-α)ft-1+(α-β)ft+βΓ(·)λ(ft) (8)
Wherein Γ (·) represents a specific noise reduction function, α and β are parameters of the relaxation factor, calculated by,
Wherein delta 1,δN is used to ensure the range of the iterative process, typically 0 < a < 2,0 < beta < 2a
In the equation (8) for the case of the optical fiber,It may be set to different functions according to circumstances;
in the step 2.2, the complete two-step iterative process of the TV is as follows:
dη＝||fk-fk-1||2·η (11)
the results of the previous two SART-TV iterations will be used to adjust the value of the next iteration; the effective noise reduction performance of the TV and the two-step iterative acceleration can be effectively combined;
When δ 1 =1 and δ N =1 in equation (9), equation (10) can be converted to a standard TV gradient descent method;
In the step 2.3 of the process described above,
Wherein,A reconstruction value representing the nth TV gradient descent iteration;
As can be seen from the above equation, in the initial stage of the gradient descent iteration, the projection value of the difference between the reconstructed image and the target image is large, the value of the relaxation factor is small, and the speed of convergence can be accelerated; when the projection value of the reconstructed image gradually approaches to the projection value of the target image, the relaxation factor becomes large, which can control the convergence rate of the iterative process and maintain the reconstruction performance of the CT image;
in the step 2.4, the dynamic termination condition based on the change rate of the total variation of the reconstructed CT image is used to determine the termination condition of the inner ring, which is defined as follows:
wherein, Is the maximum allowable error of the internal iteration.
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