有限角度CT图像重建：基于l0梯度最小化算法

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"基于l（0）梯度最小化的有限角度CT图像重建" 本文是一篇研究论文，关注的是在计算机断层摄影（CT）成像领域中如何从有限的投影数据中重建高质量的图像，特别是在有限的角度扫描下。有限角度CT成像能够降低对患者的辐射剂量，但同时也带来了一个挑战：由于可用的投影数据不完整，导致重建图像时会出现明显的伪影，如边缘附近的条纹和渐变伪影。为解决这一问题，作者提出了一种基于l（0）梯度最小化的图像重建算法。l（0）范数通常用于表示非零元素的数量，它被用来作为正则化函数，旨在保留图像的边缘信息并减少噪声。在提出的重建模型框架中，图像的梯度的l（0）-范数被用作正则项，以控制图像的突变，从而有助于消除伪影。文章详细描述了如何将优化问题分解为多个子问题，并采用交替迭代的方法来求解这些子问题。通过数值实验，该方法的有效性和可行性得到了验证。实验结果通过比较不同重建算法的峰值信噪比（PSNR）和归一化均方根距离（NRMSD）进行了评估。结果显示，在不同的扫描角度范围内，提出的算法与其他传统重建算法相比具有显著的统计差异（p < 0.0001），并且在抑制条纹伪影和渐变伪影方面表现出色。这篇论文为有限角度CT图像重建提供了一个创新的解决方案，利用l（0）梯度最小化技术有效地改善了图像质量，降低了伪影的影响，这对于医疗和工业CT成像领域具有重要的实践意义，特别是在减少辐射剂量的同时提高图像诊断准确性方面。

coefficient sparsity in wavelet domains for limited-angle CT image reconstruction can be

found in [21–23]. In our work, we focus the regularization in the image domain just as TV reg-

ularization done. Thus, the TVM based reconstruction results are compared with our results.

In recent years, a novel regularization method based on the ℓ

-norm of image gradient has

been applied in the image smoothing [24], image segmentation [25], image super-resolution

and blur deconvolution [26], visual enhancement [27], disparity and optical flow partitioning

[28]. Different from the ℓ

-norm of image u,||u||

, which is the number of its non-zero coeffi-

cients, the ℓ

-norm of image gradient is denoted as

jjrujj

#fpjj@

jþj@

j 6¼ 0g; ð2Þ

where the gradient of 2D image at the pixel point p is denoted as ru

=(@

)

, @

and

represent the differences in x direction and in y direction respectively. #{} is counting

operator, counting the number of p that satisﬁes |@

|+|@

|6¼0. As the ℓ

-norm of image gra-

dient does not count on gradient magnitude, the large gradient magnitudes will not be penal-

ized, thus the edge can be effectively retained [24].

To better preserve the edges and suppress the artifacts to limited-angle CT image recon-

struction, we developed an alternating iterative reconstruction algorithm for limited-angle CT

based on ℓ

gradient minimization. In this paper, different from the ℓ

-norm of the image gra-

dient magnitude mentioned above, the ℓ

-norm of image gradient was taken as the regulariza-

tion function of the new optimization problem. We converted the optimization problem into a

few sub-problems, and solved these problems alternately. From the experimental results, it is

shown that by the developed algorithm the streak artifacts and gradual changed artifacts

nearby edges can be effectively reduced.

The rest of the paper is organized as follows. In section Method, our reconstruction algorithm

for limited-angle tomography is described, together with an efficient numerical scheme. More-

over, the performance evaluations are also outlined in this section. In the following section,

experimental results and discussion are presented and conclusions are given in final section.

Method

The fan-beam X-ray CT has been widely used in medical diagnosis, which will be the scanning

geometry that we focus in this paper. Fig 1 shows the scanning geometry configuration for cir-

cular and limited-angle fan-beam CT. For limited-angle tomography, in this paper, the scan-

ning angular range is limited within [0,θ], where θ is the maximum rotation angle of the X-ray

source, usually less than 180°.

As described in detail in S1 Appendix, we approximate the CT imaging model as following

discrete linear system [11]:

Au ¼ g ð3Þ

where u is the unknown object to be reconstructed, g is the measured projection data, A is the

system matrix which represents the forward projection.

In some practical CT imaging, when projection data are incomplete, the system Eq (3) is

underdetermined. To find the solution to this problem, we usually need to acquire the optimal

solution u



satisfying the optimization problem in the following form [29]:



¼ min

FðuÞ :¼ DðuÞþl  C ðuÞ: ð4Þ

where D(u) is data ﬁdelity term; C(u) represents the regularization term; λ is the penalty

parameter.

ℓ

Gradient Based Image Reconstruction for Limited-Angle Tomography

PLOS ONE | DOI:10.1371/journal.pone.0130793 July 9, 2015 3/15

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