基于l0梯度最小化的有限角CT图像重建算法：提升边缘伪影抑制

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本文主要探讨了基于l（0）梯度最小化的有限角度计算机断层扫描（Computed Tomography, CT）图像重建方法。在现代医学和工业CT成像中，由于扫描环境的限制以及对患者辐射风险的关注，如何从有限的投影数据中高效地生成高质量图像是一个亟待解决的问题。传统的有限角度CT图像重建，如滤波反投影（Filtered Back Projection, FBP）算法，常导致边缘区域出现明显的条纹伪影和逐渐变化的伪影，这在图像质量上产生了负面影响。为改善这一问题，研究者提出了一种新的重建策略，即利用l（0）梯度最小化。l（0）范数作为一种稀疏性推广，旨在鼓励图像中的像素分布尽可能接近真实场景中的稀疏特性。通过将图像梯度的l（0）-范数引入到重建模型中，算法能够在保持图像细节的同时，抑制边缘伪影和伪影的逐渐变化。这种方法将原问题分解为一系列可优化的子问题，并通过交替迭代方式进行求解，这种方法有助于提升重建的稳定性与精度。作者们进行了大量的数值实验，通过比较峰峰值信噪比（Peak Signal-to-Noise Ratio, PSNR）和归一化均方根距离（Normalized Root Mean Square Error, NRMSD）等指标，定量评估了新算法与传统方法在不同扫描角度下的性能差异。实验结果显示，基于l（0）梯度最小化的有限角度CT图像重建算法在抑制伪影方面表现出显著优势，特别是在边缘伪影的控制和整体图像质量的提升上，其效果优于常规的FBP算法。这些发现对于优化有限角度CT成像技术，降低辐射暴露并提高图像质量具有重要的理论价值和实际应用意义。总结来说，这项研究提供了一种创新的图像重建策略，它在有限角度CT成像领域具有潜在的应用前景，对于提高CT成像的精确性和辐射安全性具有重要意义。通过l（0）梯度最小化方法，研究人员有望在实际应用中看到更加清晰、无伪影的CT图像，从而推动医学诊断和工业检测的进一步发展。

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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