CN111696166A - FDK (finite Difference K) type preprocessing matrix-based circumferential cone beam CT (computed tomography) fast iterative reconstruction method - Google Patents

FDK (finite Difference K) type preprocessing matrix-based circumferential cone beam CT (computed tomography) fast iterative reconstruction method Download PDF

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CN111696166A
CN111696166A CN202010523774.8A CN202010523774A CN111696166A CN 111696166 A CN111696166 A CN 111696166A CN 202010523774 A CN202010523774 A CN 202010523774A CN 111696166 A CN111696166 A CN 111696166A
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刘华锋
王婷
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Zhejiang University ZJU
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Abstract

本发明公开了一种基于FDK型预处理矩阵的圆周锥束CT快速迭代重建算法，其通过利用低剂量和稀疏视角情况下的CT图像重建模型，结合了解析重建算法速度快和迭代重建算法效果好的优势，有效的提升了图像重建的质量和速度，在对该模型进行求解的过程中引入基于FDK算法的预处理矩阵来对三维的迭代重建过程进行加速，并使用交替投影接近算法来对其进行求解，反复迭代直至终止条件被满足；与现有重建算法的实验对比表明，本发明能取得较好的重建效果。

Figure 202010523774

The invention discloses a fast iterative reconstruction algorithm for circular cone beam CT based on an FDK type preprocessing matrix, which combines the high speed of the analytical reconstruction algorithm and the effect of the iterative reconstruction algorithm by using the CT image reconstruction model under the condition of low dose and sparse viewing angle. Good advantage, it effectively improves the quality and speed of image reconstruction. In the process of solving the model, a preprocessing matrix based on the FDK algorithm is introduced to accelerate the three-dimensional iterative reconstruction process, and an alternate projection approach algorithm is used to It solves and iterates repeatedly until the termination condition is satisfied; the experimental comparison with the existing reconstruction algorithm shows that the present invention can achieve better reconstruction effect.

Figure 202010523774

Description

基于FDK型预处理矩阵的圆周锥束CT快速迭代重建方法Fast iterative reconstruction method of circular cone beam CT based on FDK-type preprocessing matrix

技术领域technical field

本发明属于CT成像技术领域，具体涉及一种基于FDK型预处理矩阵的圆周锥束CT快速迭代重建算法。The invention belongs to the technical field of CT imaging, and in particular relates to a fast iterative reconstruction algorithm for circular cone beam CT based on an FDK type preprocessing matrix.

背景技术Background technique

X光CT作为一种无损的检测技术，已被广泛的应用于医疗诊断、安全检查、工业无损检测以及产品质量检测等各种领域。在CT领域，目前已有很多种不同的扫描方式来进行投影数据的测量，主要可分为平行光束扫描、扇形光束扫描和锥形光束扫描等；其中锥形光束是X光源的自然形状，与平行光束和扇形光束相比，锥形光束的一次投影能够获得的数据量要大得多，因此锥束扫描结构更有利于提高扫描速度和重建图像质量，但其对应的三维图像重建算法相比于平行光束和扇形光束扫描结构对应的二维图像重建算法要复杂得多，且计算量大。锥束CT使用的探测器结构主要有平面探测器和柱面探测器两种，二者在竖直方向上探测器单元均为等间隔排列，但在水平方向上，前者探测器单元为等间隔排列，后者为等角度排列；锥束CT根据光源的扫描轨迹，可进一步分为圆周锥束CT和螺旋锥束CT等。As a non-destructive testing technology, X-ray CT has been widely used in various fields such as medical diagnosis, safety inspection, industrial non-destructive testing and product quality testing. In the field of CT, there are currently many different scanning methods to measure projection data, which can be mainly divided into parallel beam scanning, fan beam scanning and cone beam scanning, etc. The cone beam is the natural shape of the X light source, and the Compared with the parallel beam and the fan beam, the amount of data that can be obtained by one projection of the cone beam is much larger, so the cone beam scanning structure is more conducive to improving the scanning speed and reconstructed image quality, but the corresponding 3D image reconstruction algorithm is more The two-dimensional image reconstruction algorithms corresponding to parallel beam and fan beam scanning structures are much more complicated and computationally intensive. The detector structures used in cone beam CT mainly include plane detectors and cylindrical detectors. Both detector units are arranged at equal intervals in the vertical direction, but in the horizontal direction, the former detector units are equally spaced. According to the scanning trajectory of the light source, cone beam CT can be further divided into circular cone beam CT and spiral cone beam CT.

基于Radon变换的FBP算法在平行光束和扇形光束扫描结构对应的二维图像重建中有着广泛的应用，1984年Feldcamp、Davis和Kress等人在此基础上提出了一种适用于平面圆周扫描结构的锥束投影重建算法，被称为FDK算法；FDK是一种近似的重建算法，其重建原理是将锥形光束分解成多个倾斜的扇形光束，然后单独对其中每一个倾斜的扇形光束进行滤波和反投影操作，将所有的反投影结果进行相加即可得到三维重建图像。FDK算法利用的是二维滤波和二维反投影，数学理论较为简单，计算量也较小，已得到了实际的使用，当锥角小于10°时重建图像中的伪影较小；除了近似重建算法之外，还有针对锥束CT的精确重建算法，但其往往计算复杂，重建速度慢。The FBP algorithm based on Radon transform has a wide range of applications in the reconstruction of two-dimensional images corresponding to parallel beam and fan beam scanning structures. In 1984, Feldcamp, Davis and Kress et al. Cone beam projection reconstruction algorithm, known as FDK algorithm; FDK is an approximate reconstruction algorithm whose reconstruction principle is to decompose a cone beam into multiple oblique fan beams, and then filter each of the oblique fan beams individually And back-projection operation, adding all the back-projection results to get a 3D reconstructed image. The FDK algorithm uses two-dimensional filtering and two-dimensional back-projection. The mathematical theory is relatively simple and the amount of calculation is small. It has been used in practice. When the cone angle is less than 10°, the artifacts in the reconstructed image are small; In addition to the reconstruction algorithm, there are also accurate reconstruction algorithms for cone beam CT, but they are often computationally complex and the reconstruction speed is slow.

FDK等解析重建算法对于投影数据的要求较高，因此在低剂量和稀疏视角CT重建问题中其表现往往较差，且会出现较多明显的伪影。迭代重建算法在这些情况下更具优势，其将重建问题建模成线性系统，然后使用线性代数方法对其进行求解，如以代数重建方法(Algebraic Reconstruction Technique，ART)为代表的Kaczmarz家族的一系列算法和期望最大化(Expectation Maximization，EM)方法等。为了提升迭代重建算法的效果，很多研究者在重建过程中加入各种先验约束，如图像的各种全变分(Total Variation，TV)、图像的低秩特性、稀疏编码等；因为能够融合噪声模型并利用各种先验信息，迭代重建算法在低剂量和稀疏视角CT中往往具有比解析重建算法更好的重建效果，多年的研究和实践也证明了这一点；但是，由于迭代重建算法在实施过程中往往需要较多次的迭代运算，其重建速度依赖于算法的设计和设备的运算能力。Analytic reconstruction algorithms such as FDK have high requirements for projection data, so their performance is often poor in low-dose and sparse-view CT reconstruction problems, and there will be more obvious artifacts. Iterative reconstruction algorithms are more advantageous in these cases, which model the reconstruction problem as a linear system, and then solve it using linear algebraic methods, such as the Algebraic Reconstruction Technique (ART) represented by the Kaczmarz family of one of the family. A series of algorithms and expectation maximization (Expectation Maximization, EM) methods, etc. In order to improve the effect of the iterative reconstruction algorithm, many researchers add various prior constraints in the reconstruction process, such as various Total Variation (TV) of images, low-rank characteristics of images, sparse coding, etc.; Noise model and using various prior information, iterative reconstruction algorithm often has better reconstruction effect than analytical reconstruction algorithm in low-dose and sparse view CT, which has been proved by years of research and practice; however, due to the iterative reconstruction algorithm In the implementation process, more iterations are often required, and the reconstruction speed depends on the design of the algorithm and the computing power of the device.

近年来随着计算机技术尤其是GPU(Graphical Processing Unit)加速技术的飞速发展，迭代重建算法受到了越来越广泛的关注；针对其速度慢的问题，很多人在如何对其进行加速方面进行研究并已取得了很多成果，其中一个主要的研究方向是使用各种加速方法来实现迭代算法的快速收敛，如快速迭代阈值收缩算法(Fast Iterative Shrinkage-ThresholdingAlgorithm，FISTA)和有序子集(Ordered Subset，OS)方法等；另外一个主要的研究方向是将迭代算法与并行计算方法相结合来对算法进行加速，如可以对投影数据、图像像素点或者系统矩阵进行划分，然后对每个子集进行并行运算，或者可以将优化问题分解成多个子问题进行求解，如针对凸问题求解的交替方向乘子法(AlternatingDirection Method of Multiplier，ADMM)等。In recent years, with the rapid development of computer technology, especially the GPU (Graphical Processing Unit) acceleration technology, the iterative reconstruction algorithm has received more and more attention. In view of its slow speed, many people have studied how to accelerate it. And has achieved many results, one of the main research directions is to use various acceleration methods to achieve fast convergence of iterative algorithms, such as Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) and Ordered Subset (Ordered Subset) , OS) method, etc.; another main research direction is to combine iterative algorithms with parallel computing methods to accelerate algorithms, such as dividing projection data, image pixels or system matrices, and then parallelizing each subset. The optimization problem can be decomposed into multiple sub-problems for solving, such as the Alternating Direction Method of Multiplier (ADMM) for solving convex problems.

发明内容SUMMARY OF THE INVENTION

鉴于上述，本发明提出了一种基于FDK型预处理矩阵的圆周锥束CT快速迭代重建算法，涵盖了低剂量和稀疏视角两种情况，该算法结合了解析重建算法和迭代重建算法的优势，能够以较少的迭代次数得到高质量的重建图像。In view of the above, the present invention proposes a fast iterative reconstruction algorithm for circular cone beam CT based on FDK-type preprocessing matrix, covering both low-dose and sparse viewing angles. The algorithm combines the advantages of the analytical reconstruction algorithm and the iterative reconstruction algorithm, High-quality reconstructed images can be obtained with fewer iterations.

一种基于FDK型预处理矩阵的圆周锥束CT快速迭代重建算法，包括如下步骤：A fast iterative reconstruction algorithm for circular cone beam CT based on FDK-type preprocessing matrix, comprising the following steps:

(1)利用圆周锥束CT系统在不同角度方向采集获取CT图像的投影数据，组成投影数据集

Figure BDA0002532999630000031
若在测量时采集了m个角度方向的投影数据，则投影数据集
Figure BDA0002532999630000032
的维度为m×N，N为圆周锥束CT系统中的探测器单元数量，所述投影数据集
Figure BDA0002532999630000033
由对应m个角度方向下所采集得到的投影数据向量组成，所述投影数据向量维度为N且向量中每一元素值为对应探测器单元测得的投影数据；(1) Use the circular cone beam CT system to collect and obtain the projection data of CT images in different angular directions to form a projection data set
Figure BDA0002532999630000031
If the projection data of m angular directions are collected during the measurement, the projection data set
Figure BDA0002532999630000032
The dimension is m×N, where N is the number of detector units in a circular cone beam CT system, the projection dataset
Figure BDA0002532999630000033
It is composed of projection data vectors collected in corresponding m angular directions, the dimension of the projection data vector is N, and the value of each element in the vector is the projection data measured by the corresponding detector unit;

(2)分别建立低剂量和稀疏视角情况下的CT图像重建模型；(2) Establish CT image reconstruction models under low-dose and sparse viewing angles, respectively;

(3)对上述两种情况下的CT图像重建方程进行预处理，进而利用拉格朗日对偶方法将预处理后的CT图像重建方程转换成鞍点问题，得到对应的目标函数；(3) Preprocess the CT image reconstruction equation in the above two cases, and then use the Lagrangian dual method to convert the preprocessed CT image reconstruction equation into a saddle point problem to obtain the corresponding objective function;

(4)根据实际情况选择对应的目标函数，对目标函数进行优化求解以重建得到CT图像。(4) Select the corresponding objective function according to the actual situation, and optimize and solve the objective function to reconstruct the CT image.

进一步地，所述步骤(2)中低剂量情况下的CT图像重建模型表达如下：Further, the CT image reconstruction model under the low-dose situation in the step (2) is expressed as follows:

Figure BDA0002532999630000034
Figure BDA0002532999630000034

稀疏视角情况下的CT图像重建模型表达如下：The CT image reconstruction model in the case of sparse viewing angles is expressed as follows:

Figure BDA0002532999630000035
Figure BDA0002532999630000035

其中：

Figure BDA0002532999630000036
为CT图像数据集且
Figure BDA0002532999630000037
其中每一元素值为待重建CT图像中对应像素点处的X光吸收系数，J为待重建CT图像的像素点数量，T表示转置，A为系统矩阵，
Figure BDA0002532999630000038
为用来对目标函数进行约束的惩罚项，β为给定的权重系数，|| ||表示2范数。in:
Figure BDA0002532999630000036
is a CT image dataset and
Figure BDA0002532999630000037
Each element is the X-ray absorption coefficient at the corresponding pixel in the CT image to be reconstructed, J is the number of pixels in the CT image to be reconstructed, T is the transposition, A is the system matrix,
Figure BDA0002532999630000038
is the penalty term used to constrain the objective function, β is the given weight coefficient, || || represents the 2 norm.

进一步地，所述步骤(3)中对于低剂量情况下的CT图像重建方程，首先引入一个额外的变量

Figure BDA0002532999630000039
将式(1)改写成如下形式：Further, in the step (3), for the CT image reconstruction equation in the case of low dose, an additional variable is first introduced
Figure BDA0002532999630000039
Rewrite formula (1) into the following form:

Figure BDA00025329996300000310
Figure BDA00025329996300000310

其中：变量

Figure BDA00025329996300000311
表示为对向量
Figure BDA00025329996300000312
进行正投影计算得到的投影数据；where: variable
Figure BDA00025329996300000311
represented as a pair vector
Figure BDA00025329996300000312
The projection data obtained by the orthographic projection calculation;

然后，引入非奇异矩阵Pcone，将式(3)进一步改写为如下形式：Then, a non-singular matrix P cone is introduced, and Equation (3) is further rewritten into the following form:

Figure BDA00025329996300000313
Figure BDA00025329996300000313

式(4)对应的拉格朗日方程

Figure BDA00025329996300000314
定义如下：The Lagrange equation corresponding to formula (4)
Figure BDA00025329996300000314
Defined as follows:

Figure BDA0002532999630000041
Figure BDA0002532999630000041

其中：

Figure BDA0002532999630000042
为拉格朗日向量；in:
Figure BDA0002532999630000042
is a Lagrangian vector;

进而将式(4)转换成如下鞍点问题：Then formula (4) can be transformed into the following saddle point problem:

Figure BDA0002532999630000043
Figure BDA0002532999630000043

最后通过对变量

Figure BDA0002532999630000044
进行极小化计算可以将其从鞍点问题中消去，得到如下目标函数：Finally by pairing the variable
Figure BDA0002532999630000044
It can be eliminated from the saddle point problem by performing a minimization calculation, and the following objective function is obtained:

Figure BDA0002532999630000045
Figure BDA0002532999630000045

其中：

Figure BDA0002532999630000046
为
Figure BDA0002532999630000047
与
Figure BDA0002532999630000048
的内积运算结果。in:
Figure BDA0002532999630000046
for
Figure BDA0002532999630000047
and
Figure BDA0002532999630000048
The result of the inner product operation.

进一步地，所述步骤(3)中对于稀疏视角情况下的CT图像重建方程，首先引入非奇异矩阵Pcone，将式(2)改写为如下形式：Further, in the step (3), for the CT image reconstruction equation in the case of sparse viewing angles, a non-singular matrix P cone is first introduced, and the formula (2) is rewritten into the following form:

Figure BDA0002532999630000049
Figure BDA0002532999630000049

式(8)对应的拉格朗日方程

Figure BDA00025329996300000410
定义如下：The Lagrange equation corresponding to formula (8)
Figure BDA00025329996300000410
Defined as follows:

Figure BDA00025329996300000411
Figure BDA00025329996300000411

进而将式(9)转换成鞍点问题，其对应的目标函数如下：Then, formula (9) is transformed into a saddle point problem, and the corresponding objective function is as follows:

Figure BDA00025329996300000412
Figure BDA00025329996300000412

其中：

Figure BDA00025329996300000413
为拉格朗日向量。in:
Figure BDA00025329996300000413
is a Lagrangian vector.

进一步地，所述非奇异矩阵Pcone的表达式如下：Further, the expression of the non-singular matrix P cone is as follows:

Figure BDA00025329996300000414
Figure BDA00025329996300000414

P＝F-1R(ω)FP=F -1 R(ω)F

其中：在低剂量情况下

Figure BDA00025329996300000415
在稀疏视角情况下
Figure BDA00025329996300000416
Gc为对角矩阵且维度为n，n为圆周锥束CT系统中水平方向每行探测器单元的个数，Gc中的每个对角元素为对应角度方向对应行中每个探测器单元测得投影数据的修正项值，F和F-1分别为一维傅里叶变换算子和傅里叶逆变换算子，ω为对应角度方向下所采集得到的投影数据向量经傅里叶变换后的频域变量，τ为给定的参数。Of which: at low doses
Figure BDA00025329996300000415
In the case of sparse viewing angles
Figure BDA00025329996300000416
G c is a diagonal matrix with dimension n, n is the number of detector units in each horizontal row in the circular cone beam CT system, and each diagonal element in G c is each detector in the corresponding row in the corresponding angular direction The correction term value of the projection data measured by the unit, F and F -1 are the one-dimensional Fourier transform operator and the Fourier inverse operator respectively, ω is the projection data vector collected in the corresponding angle direction after Fourier The frequency domain variable after leaf transformation, τ is a given parameter.

进一步地，若圆周锥束CT系统采用的是平面探测器，则对角矩阵Gc中的对角元素为

Figure BDA0002532999630000051
若圆周锥束CT系统采用的是柱面探测器，则对角矩阵Gc中的对角元素为
Figure BDA0002532999630000052
其中：s为光线与对应探测器单元在水平方向上的交点位置值，h为光线与对应探测器单元在竖直方向上的交点位置值，D为光源到旋转中心的距离，γ为光线在锥束的水平中心平面上与中心光线之间的夹角。Further, if the circular cone beam CT system adopts a plane detector, the diagonal elements in the diagonal matrix G c are:
Figure BDA0002532999630000051
If the circular cone beam CT system uses a cylindrical detector, the diagonal elements in the diagonal matrix G c are
Figure BDA0002532999630000052
Where: s is the position of the intersection of the light and the corresponding detector unit in the horizontal direction, h is the position of the intersection of the light and the corresponding detector unit in the vertical direction, D is the distance from the light source to the rotation center, γ is the light in the The angle between the horizontal center plane of the cone beam and the center ray.

进一步地，所述步骤(4)中采用交替投影接近算法对目标函数进行优化求解。Further, in the step (4), an alternate projection approach algorithm is used to optimize and solve the objective function.

进一步地，所述惩罚项

Figure BDA0002532999630000053
采用全变分算子。Further, the penalty item
Figure BDA0002532999630000053
A total variation operator is used.

进一步地，对于圆周锥束CT系统，所述非奇异矩阵Pcone不是直接作用于所有投影数据，而是对每个角度每行探测器单元测得的投影数据分别进行处理。Further, for a circular cone beam CT system, the non-singular matrix P cone does not directly act on all projection data, but separately processes the projection data measured by each row of detector units at each angle.

本发明的圆周锥束CT图像重建算法通过利用低剂量和稀疏视角情况下的CT图像重建模型，结合了解析重建算法速度快和迭代重建算法效果好的优势，有效的提升了图像重建的质量和速度，在对该模型进行求解的过程中引入基于FDK算法的预处理矩阵来对三维的迭代重建过程进行加速，并使用交替投影接近算法来对其进行求解，反复迭代直至终止条件被满足；与现有重建算法的实验对比表明，本发明能取得较好的重建效果。The circular cone beam CT image reconstruction algorithm of the present invention combines the advantages of fast analytical reconstruction algorithm and good effect of iterative reconstruction algorithm by using the CT image reconstruction model under the condition of low dose and sparse viewing angle, and effectively improves the quality and efficiency of image reconstruction. In the process of solving the model, a preprocessing matrix based on the FDK algorithm is introduced to accelerate the three-dimensional iterative reconstruction process, and the alternating projection approach algorithm is used to solve it, and iteratively iterates until the termination condition is satisfied; and The experimental comparison of the existing reconstruction algorithms shows that the present invention can achieve better reconstruction effect.

附图说明Description of drawings

图1为本发明圆周锥束CT图像重建算法的流程示意图。FIG. 1 is a schematic flowchart of the reconstruction algorithm of the circular cone beam CT image according to the present invention.

图2为三维Shepp-Logan头部模型的截面图，显示灰度范围为[1.00，1.05]，其中(a)和(b)分别为头部模型两个垂直方向的竖直中轴横截面图，(c)和(d)分别为(a)中从上往下两条虚线位置L1和L2处的水平方向横截面。Figure 2 is a cross-sectional view of the 3D Shepp-Logan head model, showing a grayscale range of [1.00, 1.05], where (a) and (b) are the vertical mid-axis cross-sectional views of the head model in two vertical directions, respectively , (c) and (d) are the horizontal cross-sections at the positions L1 and L2 of the two dashed lines from top to bottom in (a), respectively.

图3为重建过程中三种方法得到的重建图像相比于真值图像的均方根误差随迭代次数的变化曲线图，投影数据的角度范围为360°，角度数为120。Figure 3 is a graph showing the variation of the root mean square error of the reconstructed image obtained by the three methods compared with the true value image with the number of iterations in the reconstruction process. The angle range of the projection data is 360°, and the number of angles is 120.

图4为三种方法得到的重建结果的竖直中轴横截面图，其中(a)和(b)分别为两个不同方向的横截面，图中的重建图像按行从上到下对应为采用本发明方法、ART+TV方法和FISTA方法重建得到的结果，按列从左到右对应为第5、10、15和20次迭代得到的重建结果。Figure 4 is a vertical mid-axis cross-sectional view of the reconstruction results obtained by the three methods, in which (a) and (b) are cross-sections in two different directions, respectively, and the reconstructed images in the figure correspond from top to bottom in rows as The reconstruction results obtained by the method of the present invention, the ART+TV method and the FISTA method correspond to the reconstruction results obtained by the 5th, 10th, 15th and 20th iterations from left to right in columns.

图5为三种方法得到的重建结果的距离中轴面较近的水平横截面图，图中的重建图像按行从上到下对应为采用本发明方法、ART+TV方法和FISTA方法重建得到的结果，按列从左到右对应为第5、10、15和20次迭代得到的重建结果。Figure 5 is a horizontal cross-sectional view of the reconstruction results obtained by the three methods, which are closer to the mid-axis plane. The reconstructed images in the figure correspond to the reconstruction results obtained by the method of the present invention, the ART+TV method and the FISTA method from top to bottom. The results of the column correspond to the reconstruction results obtained at the 5th, 10th, 15th and 20th iterations from left to right.

图6为三种方法得到的重建结果的距离中轴面较远的水平横截面图，图中的重建图像按行从上到下对应为采用本发明方法、ART+TV方法和FISTA方法重建得到的结果，按列从左到右对应为第5、10、15和20次迭代得到的重建结果。Figure 6 is a horizontal cross-sectional view of the reconstruction results obtained by the three methods farther from the mid-axis plane. The reconstructed images in the figure correspond from top to bottom by the method of the present invention, the ART+TV method and the FISTA method. The results of the column correspond to the reconstruction results obtained at the 5th, 10th, 15th and 20th iterations from left to right.

具体实施方式Detailed ways

为了更为具体地描述本发明，下面结合附图及具体实施方式对本发明的技术方案进行详细说明。In order to describe the present invention more specifically, the technical solutions of the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

如图1所示，本发明基于FDK型预处理矩阵的圆周锥束CT快速迭代重建算法，包括如下步骤：As shown in Figure 1, the present invention based on the FDK type preprocessing matrix fast iterative reconstruction algorithm of circular cone beam CT, including the following steps:

(1)采集探测器在不同角度方向测量得到的CT图像的投影数据，组成投影数据集

Figure BDA0002532999630000061
投影数据集
Figure BDA0002532999630000062
由对应各个角度方向下所采集得到的投影数据向量组成，投影向量维度为Ndet且向量中每一元素值为对应探测器测得的投影数据，Ndet为探测器个数，若投影角度数为Ndeg，则投影数据的总数目为Ndeg×Ndet。(1) Collect the projection data of CT images measured by the detector in different angular directions to form a projection data set
Figure BDA0002532999630000061
Projection dataset
Figure BDA0002532999630000062
It is composed of projection data vectors collected from corresponding angles. The dimension of the projection vector is N det and each element in the vector is the projection data measured by the corresponding detector. N det is the number of detectors. If the number of projection angles is N deg , then the total number of projection data is N deg ×N det .

(2)分别建立低剂量和稀疏视角情况下的CT图像重建模型：(2) Establish CT image reconstruction models under low-dose and sparse viewing angles, respectively:

低剂量情况下的CT图像重建模型可以表达如下：The CT image reconstruction model in the case of low dose can be expressed as follows:

Figure BDA0002532999630000063
Figure BDA0002532999630000063

稀疏视角情况下的CT图像重建模型可以表达如下：The CT image reconstruction model in the case of sparse viewing angles can be expressed as follows:

Figure BDA0002532999630000064
Figure BDA0002532999630000064

其中：向量

Figure BDA0002532999630000071
为离散表达的图像数组，其中的每一个元素值为待重建CT图像中对应像素点处的X光吸收系数；向量
Figure BDA0002532999630000072
为对应的投影数据；A为系统矩阵；
Figure BDA0002532999630000073
为用来对目标方程进行约束的惩罚项；β为给定的权重系数。where: vector
Figure BDA0002532999630000071
is a discretely expressed image array, where each element value is the X-ray absorption coefficient at the corresponding pixel in the CT image to be reconstructed; vector
Figure BDA0002532999630000072
is the corresponding projection data; A is the system matrix;
Figure BDA0002532999630000073
is the penalty term used to constrain the objective equation; β is the given weight coefficient.

(3)对两种模式下的CT图像重建方程进行预处理得到对应的目标函数，然后利用拉格朗日对偶将重建方程转换成鞍点问题：(3) Preprocess the CT image reconstruction equations in the two modes to obtain the corresponding objective functions, and then use Lagrangian duality to convert the reconstruction equations into a saddle point problem:

对于低剂量情况下的CT图像重建模型，首先引入一个额外的变量

Figure BDA00025329996300000717
将公式(1)改写成如下形式：For the CT image reconstruction model in the low-dose case, an additional variable is first introduced
Figure BDA00025329996300000717
Rewrite formula (1) into the following form:

Figure BDA0002532999630000074
Figure BDA0002532999630000074

其中：

Figure BDA0002532999630000075
可以理解成对
Figure BDA0002532999630000076
进行正投影计算得到的投影数据。in:
Figure BDA0002532999630000075
understandable in pairs
Figure BDA0002532999630000076
Projection data obtained by performing an orthographic projection calculation.

接着引入非奇异矩阵Pcone，将公式(3)进一步改写为如下形式：Next, a non-singular matrix P cone is introduced, and formula (3) is further rewritten into the following form:

Figure BDA0002532999630000077
Figure BDA0002532999630000077

其对应的拉格朗日方程

Figure BDA0002532999630000078
定义如下：Its corresponding Lagrange equation
Figure BDA0002532999630000078
Defined as follows:

Figure BDA0002532999630000079
Figure BDA0002532999630000079

其中：

Figure BDA00025329996300000710
为拉格朗日向量，T表示转置。in:
Figure BDA00025329996300000710
is a Lagrangian vector, and T represents the transpose.

则公式(4)可以转换成如下鞍点问题：Then formula (4) can be transformed into the following saddle point problem:

Figure BDA00025329996300000711
Figure BDA00025329996300000711

最后通过关于变量

Figure BDA00025329996300000712
进行极小化计算可以将其从鞍点问题中消去，得到如下目标函数：Finally pass about the variable
Figure BDA00025329996300000712
It can be eliminated from the saddle point problem by performing a minimization calculation, and the following objective function is obtained:

Figure BDA00025329996300000713
Figure BDA00025329996300000713

Figure BDA00025329996300000714
Figure BDA00025329996300000714

P＝F-1R(ω)FP=F -1 R(ω)F

Figure BDA00025329996300000715
Figure BDA00025329996300000715

其中：

Figure BDA00025329996300000716
为拉格朗日向量；T表示转置；(·，·)为内积运算；Gc为对角矩阵，其对角元素为每一个投影数据对应的修正项的值；Gc 1/2为Gc的均方根矩阵，由于Gc为对角矩阵，因此对其所有对角元素进行开方即可得到Gc 1/2；F和F-1分别为一维傅里叶变换算子和傅里叶逆变换算子；ω为对应角度方向下所采集得到的投影数据向量经傅里叶变换后的频域变量，m为角度方向数量，τ为给定的参数；对于圆周锥束CT，矩阵Pcone不是直接作用于每个角度的所有投影数据，而是对每个角度的投影数据按探测器的行分别对其进行处理。in:
Figure BDA00025329996300000716
is a Lagrangian vector; T means transposition; (·,·) is an inner product operation; G c is a diagonal matrix, and its diagonal elements are the value of the correction term corresponding to each projection data; G c 1/2 is the root mean square matrix of G c . Since G c is a diagonal matrix, G c 1/2 can be obtained by taking the square root of all its diagonal elements; F and F -1 are respectively the one-dimensional Fourier transform calculation and the inverse Fourier operator; ω is the frequency domain variable after the Fourier transform of the projection data vector collected in the corresponding angular direction, m is the number of angular directions, and τ is the given parameter; for the circular cone Beam CT, matrix Pcone does not directly act on all projection data of each angle, but the projection data of each angle is processed separately by the row of the detector.

对角矩阵Gc中每个对角元素的具体计算方式如下：The specific calculation method of each diagonal element in the diagonal matrix G c is as follows:

当圆周锥束CT使用的探测器为平面探测器时，假设投影数据为Rβ(s，h)，β为光源与旋转中心的连线与坐标轴之间的夹角，s为光线与探测器在水平方向上的交点位置，h为光线与探测器在竖直方向上的交点位置，则对角矩阵Gc中每个对角元素的具体计算方式为

Figure BDA0002532999630000081
D为光源到旋转中心的距离。When the detector used in circular cone beam CT is a plane detector, the projection data is assumed to be R β (s, h), β is the angle between the line connecting the light source and the rotation center and the coordinate axis, s is the light and the detection The intersection position of the detector in the horizontal direction, h is the intersection position of the light and the detector in the vertical direction, then the specific calculation method of each diagonal element in the diagonal matrix G c is as follows
Figure BDA0002532999630000081
D is the distance from the light source to the center of rotation.

当圆周锥束CT使用的探测器为柱面探测器时，假设投影数据为Rβ(γ，h)，为光源与旋转中心的连线与坐标轴之间的夹角，γ为光线在锥束的水平中心平面上与中心光线之间的夹角，h为光线与探测器在竖直方向上的交点位置，则对角矩阵Gc中每个对角元素的具体计算方式为

Figure BDA0002532999630000082
When the detector used in the circular cone beam CT is a cylindrical detector, it is assumed that the projection data is R β (γ, h), which is the angle between the line connecting the light source and the rotation center and the coordinate axis, and γ is the light beam in the cone The angle between the horizontal central plane of the beam and the central ray, h is the position of the intersection of the ray and the detector in the vertical direction, then the specific calculation method of each diagonal element in the diagonal matrix G c is:
Figure BDA0002532999630000082

对于稀疏视角情况下的CT图像重建模型，同样引入非奇异矩阵Pcone，则公式(2)可以改写成如下形式：For the CT image reconstruction model in the case of sparse viewing angles, the non-singular matrix P cone is also introduced, and the formula (2) can be rewritten as the following form:

Figure BDA0002532999630000083
Figure BDA0002532999630000083

其对应的拉格朗日方程

Figure BDA0002532999630000084
定义如下：Its corresponding Lagrange equation
Figure BDA0002532999630000084
Defined as follows:

Figure BDA0002532999630000085
Figure BDA0002532999630000085

则将公式(8)转换成了鞍点问题，其对应的目标函数如下：Then formula (8) is converted into a saddle point problem, and its corresponding objective function is as follows:

Figure BDA0002532999630000086
Figure BDA0002532999630000086

Figure BDA0002532999630000087
Figure BDA0002532999630000087

P＝F-1R(ω)FP=F -1 R(ω)F

Figure BDA0002532999630000088
Figure BDA0002532999630000088

(4)使用交替投影接近算法来对目标函数(7)和(10)进行求解。(4) The objective functions (7) and (10) are solved using an alternate projection approximation algorithm.

4.1初始化

Figure BDA0002532999630000089
并设置各项参数的值：总迭代次数Niter、迭代终止阈值δ、惩罚项权重系数β、参数τ、参数σ；其中，总迭代次数Niter的取值范围为1～50000，迭代终止阈值δ的取值范围为10-10～1，惩罚项权重参数β的取值范围为10-6～1，参数τ的取值范围为10-3～103，参数σ的取值范围为10-3～103。4.1 Initialization
Figure BDA0002532999630000089
And set the value of each parameter: the total number of iterations Niter , the iteration termination threshold δ, the penalty item weight coefficient β, the parameter τ, the parameter σ; among them, the value range of the total iteration number Niter is 1-50000, and the iteration termination threshold The value range of δ is 10 -10 to 1, the value range of penalty item weight parameter β is 10 -6 to 1, the value range of parameter τ is 10 -3 to 10 3 , and the value range of parameter σ is 10 -3 to 10 3 .

4.2初始令迭代计数k＝1；4.2 Initially let the iteration count k=1;

4.3计算

Figure BDA0002532999630000091
的值，当k为1时：4.3 Calculation
Figure BDA0002532999630000091
The value of , when k is 1:

Figure BDA0002532999630000092
Figure BDA0002532999630000092

当k不为1时：When k is not 1:

低剂量CT：

Figure BDA0002532999630000093
Low-dose CT:
Figure BDA0002532999630000093

稀疏视角CT：

Figure BDA0002532999630000094
Sparse view CT:
Figure BDA0002532999630000094

4.4对

Figure BDA0002532999630000095
进行更新：4.4 pair
Figure BDA0002532999630000095
To update:

Figure BDA0002532999630000096
Figure BDA0002532999630000096

若选择全变分作为惩罚项，则可使用Chambolle的TV去噪算法来对上式进行求解，需迭代多次直至迭代终止条件被满足。If the total variation is selected as the penalty term, Chambolle's TV denoising algorithm can be used to solve the above equation, and it needs to iterate many times until the iteration termination condition is satisfied.

4.5对

Figure BDA0002532999630000097
进行更新：4.5 pairs
Figure BDA0002532999630000097
To update:

低剂量CT：

Figure BDA0002532999630000098
Low-dose CT:
Figure BDA0002532999630000098

稀疏视角CT：

Figure BDA0002532999630000099
Sparse view CT:
Figure BDA0002532999630000099

4.6判断重建图像

Figure BDA00025329996300000910
与重建图像
Figure BDA00025329996300000911
的差值图像的所有像素值的平方和是否小于迭代终止阈值δ或k是否大于Niter：若是，执行步骤4.7；若否，令k＝k+1，执行步骤4.3～4.5。4.6 Judging the reconstructed image
Figure BDA00025329996300000910
with the reconstructed image
Figure BDA00025329996300000911
Whether the square sum of all pixel values of the difference image is less than the iteration termination threshold δ or whether k is greater than Niter : if yes, go to step 4.7; if not, set k=k+1, go to steps 4.3 to 4.5.

4.7终止迭代，得到最终重建图像

Figure BDA00025329996300000912
4.7 Terminate the iteration to get the final reconstructed image
Figure BDA00025329996300000912

以下我们通过对三维Shepp-Logan头部模型的正弦图进行重建来验证本发明方法的实用性和可靠性，三维Shepp-Logan头部模型图的截面图如图2所示，由于其大部分结构的灰度值十分接近，为了清楚的显示其结构，这里选择灰度范围[1.00，1.05]对图像进行显示。In the following, we verify the practicability and reliability of the method of the present invention by reconstructing the sine diagram of the three-dimensional Shepp-Logan head model. The cross-sectional view of the three-dimensional Shepp-Logan head model is shown in Figure 2. The gray value of is very close. In order to clearly display its structure, the gray scale range [1.00, 1.05] is selected to display the image.

采用本发明方法、ART+TV方法和FISTA方法对投影数据(采用的投影角度范围为360°，角度数为120)进行重建，图3给出了重建过程中三种方法得到的重建图像相比于真值图的均方根误差随迭代次数的变化，可以看到三种方法都随着迭代次数的增加而逐渐收敛，本发明方法的重建结果均方根误差更小，且其收敛速度较其它两种方法快。The method of the present invention, the ART+TV method and the FISTA method are used to reconstruct the projection data (the range of the projection angle used is 360° and the number of angles is 120). Figure 3 shows the comparison of the reconstructed images obtained by the three methods in the reconstruction process As the root mean square error of the ground truth map changes with the number of iterations, it can be seen that all three methods gradually converge with the increase of the number of iterations. The reconstruction result of the method of the present invention has a smaller root mean square error and a faster convergence speed The other two methods are faster.

我们选择最大迭代次数(即20次)时的重建图像加以显示，图4的(a)和(b)分别为重建结果两个不同方向的竖直中轴横截面图，图5为重建结果的距离水平中轴面较近的水平横截面，图6为重建结果的距离水平中轴面较远的水平横截面。在图4的(a)和(b)、图5、图6中，重建图像按行从上到下分别由本发明的方法、ART+TV方法和FISTA方法重建得到，按列从左到右分别为第5、10、15和20次迭代得到的重建结果；从这些结果中可以看到本发明方法重建效果较好，不管是距离水平中轴较近的水平横截面还是较远的水平横截面重建结果都很好，并且图像的边界和内部细节都恢复的较好，且无明显的伪影，另外从重建结果和均方根误差图中很容易看出本文的方法收敛速度快，能够以更少的迭代次数得到更好的重建结果。We select the reconstructed images when the maximum number of iterations (ie, 20 times) is selected for display. Figure 4 (a) and (b) are the vertical mid-axis cross-sectional views of the reconstruction results in two different directions, and Figure 5 is the reconstruction results. The horizontal cross section that is closer to the horizontal mid-axis plane, Figure 6 shows the horizontal cross-section farther from the horizontal mid-axis plane of the reconstruction result. In (a) and (b) of Fig. 4, Fig. 5, Fig. 6, the reconstructed images are reconstructed by the method of the present invention, the ART+TV method and the FISTA method respectively from top to bottom in rows, and from left to right in columns The reconstruction results obtained for the 5th, 10th, 15th and 20th iterations; it can be seen from these results that the reconstruction effect of the method of the present invention is better, no matter it is the horizontal cross-section that is closer to the horizontal axis or the horizontal cross-section that is farther away The reconstruction results are very good, and the boundaries and internal details of the image are recovered well, and there are no obvious artifacts. In addition, it is easy to see from the reconstruction results and the root mean square error graph that the method in this paper has a fast convergence speed and can be Fewer iterations yield better reconstruction results.

上述对实施例的描述是为便于本技术领域的普通技术人员能理解和应用本发明。熟悉本领域技术的人员显然可以容易地对上述实施例做出各种修改，并把在此说明的一般原理应用到其他实施例中而不必经过创造性的劳动。因此，本发明不限于上述实施例，本领域技术人员根据本发明的揭示，对于本发明做出的改进和修改都应该在本发明的保护范围之内。The above description of the embodiments is for the convenience of those of ordinary skill in the art to understand and apply the present invention. It will be apparent to those skilled in the art that various modifications to the above-described embodiments can be readily made, and the general principles described herein can be applied to other embodiments without inventive effort. Therefore, the present invention is not limited to the above-mentioned embodiments, and improvements and modifications made by those skilled in the art according to the disclosure of the present invention should all fall within the protection scope of the present invention.

Claims (9)

1. A fast iterative reconstruction algorithm of a circular cone beam CT based on an FDK type preprocessing matrix comprises the following steps:
(1) acquiring projection data of CT images in different angle directions by using a circumferential cone beam CT system to form a projection data set
Figure FDA0002532999620000011
If projection data of m angular directions are acquired during the measurement, a projection data set is formed
Figure FDA0002532999620000012
Is m × N, N being the number of detector units in a circumferential cone-beam CT system, the projection data set
Figure FDA0002532999620000013
The projection data vector is formed by projection data vectors acquired under the corresponding m angle directions, the dimensionality of the projection data vector is N, and each element value in the vector is projection data measured by a corresponding detector unit;
(2) respectively establishing CT image reconstruction models under the conditions of low dose and sparse view angle;
(3) preprocessing the CT image reconstruction equation under the two conditions, and converting the preprocessed CT image reconstruction equation into a saddle point problem by using a Lagrangian dual method to obtain a corresponding target function;
(4) and selecting a corresponding objective function according to the actual situation, and carrying out optimization solution on the objective function to reconstruct to obtain the CT image.
2. The circular cone beam CT fast iterative reconstruction algorithm according to claim 1, characterized in that: the CT image reconstruction model under the condition of low dose in the step (2) is expressed as follows:
Figure FDA0002532999620000014
the CT image reconstruction model under sparse view angle condition is expressed as follows:
Figure FDA0002532999620000015
wherein:
Figure FDA0002532999620000016
is a CT image data set and
Figure FDA0002532999620000017
wherein each element value is the X-ray absorption coefficient of the corresponding pixel point in the CT image to be reconstructed, J is the number of pixel points of the CT image to be reconstructed, T represents transposition, A is a system matrix,
Figure FDA0002532999620000018
for the penalty term used to constrain the objective function, β is given a weighting factor, | | | | represents a 2-norm.
3. The circular cone beam CT fast iterative reconstruction algorithm according to claim 2, characterized in that: for the CT image reconstruction equation under the condition of low dose in the step (3), firstly, an additional variable is introduced
Figure FDA0002532999620000019
Rewriting formula (1) to the following form:
Figure FDA0002532999620000021
wherein: variables of
Figure FDA0002532999620000022
Expressed as a vector of
Figure FDA0002532999620000023
Projection data obtained by performing orthographic projection calculation;
then, introduce the non-singular matrix PcomeFormula (3) is further rewritten as follows:
Figure FDA0002532999620000024
lagrange equation corresponding to equation (4)
Figure FDA0002532999620000025
The definition is as follows:
Figure FDA0002532999620000026
wherein:
Figure FDA0002532999620000027
is a Lagrangian vector;
further, equation (4) is converted into the following saddle point problem:
Figure FDA0002532999620000028
finally, through the pair of variables
Figure FDA0002532999620000029
Minimization calculations can be performed to eliminate it from the saddle point problem, resulting in the following objective function:
Figure FDA00025329996200000210
wherein:
Figure FDA00025329996200000211
is composed of
Figure FDA00025329996200000212
And
Figure FDA00025329996200000213
the inner product operation result of (2).
4. The circular cone beam CT fast iterative reconstruction algorithm according to claim 2, characterized in that: for the CT image reconstruction equation under the condition of sparse view angle in the step (3), firstly, a non-singular matrix P is introducedcomeThe formula (2) is rewritten as follows:
Figure FDA00025329996200000214
lagrange equation corresponding to equation (8)
Figure FDA00025329996200000215
The definition is as follows:
Figure FDA00025329996200000216
further, equation (9) is converted into the saddle point problem, and the corresponding objective function is as follows:
Figure FDA00025329996200000217
wherein:
Figure FDA0002532999620000031
is the lagrange vector.
5. The circumferential cone beam CT fast iterative reconstruction algorithm according to claim 3 or 4, characterized in that: the nonsingular matrix PcomeThe expression of (a) is as follows:
Figure FDA0002532999620000032
P=F-1R(ω)F
wherein: at low dose
Figure FDA0002532999620000033
In sparse view conditions
Figure FDA0002532999620000034
GcIs a diagonal matrix with dimension n, n is the number of detector units in each horizontal row in the circumferential cone-beam CT system, GcEach diagonal element in (a) is a correction term value, F and F, of the projection data measured by each detector unit in the corresponding line in the corresponding angular direction-1Respectively, a one-dimensional Fourier transform operator andand omega is a frequency domain variable obtained by Fourier transformation of the projection data vector acquired in the corresponding angle direction, and tau is a given parameter.
6. The circular cone beam CT fast iterative reconstruction algorithm according to claim 5, characterized in that: if the circular cone-beam CT system uses a planar detector, the diagonal matrix GcThe diagonal element in (A) is
Figure FDA0002532999620000035
Figure FDA0002532999620000036
If the circular cone beam CT system adopts a cylindrical surface detector, the diagonal matrix GcThe diagonal element in (A) is
Figure FDA0002532999620000037
Wherein: s is the intersection point position value of the light and the corresponding detector unit in the horizontal direction, h is the intersection point position value of the light and the corresponding detector unit in the vertical direction, D is the distance from the light source to the rotation center, and gamma is the included angle between the light and the central light on the horizontal central plane of the cone beam.
7. The circular cone beam CT fast iterative reconstruction algorithm according to claim 1, characterized in that: and (4) carrying out optimization solution on the objective function by adopting an alternative projection approximation algorithm.
8. The circular cone beam CT fast iterative reconstruction algorithm according to claim 2, characterized in that: the penalty term
Figure FDA0002532999620000038
And adopting a total variation operator.
9. The circumferential cone beam CT fast iterative reconstruction algorithm according to claim 3 or 4, characterized in that: for a circular cone beam CT systemSystem, said non-singular matrix PcomeInstead of acting directly on all projection data, the projection data measured by each row of detector units for each angle is processed separately.
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