基于l（0）梯度最小化的有限角度CT图像重建资源-CSDN文库

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在计算机断层摄影（CT）成像的医学和工业应用中，受扫描环境和对患者施加过多X射线辐射的风险所限制，从有限的投影数据重建高质量的CT图像已成为热门话题。在有限的扫描角度范围内进行X射线成像是一种有效的成像方式，可以减少对患者的辐射剂量。由于此模式中可用的投影数据不完整，因此有限角度CT图像重建实际上是一个不适定的逆问题。为了解决该问题，通过常规的滤波反投影（FBP）算法重建的图像经常在边缘附近产生明显的条纹伪影和逐渐变化的伪影。基于总变化量最小化（TVM）的图像重建可以显着减少少数视图CT中的条纹伪影，但受限于有限角度CT中边缘附近的伪影逐渐变化。为了抑制这种伪像，我们针对有限角度CT开发了一种基于l（0）梯度最小化的图像重建算法。在已开发的重建模型框架内，将图像梯度的l（0）-范数作为正则化函数。我们将优化问题转化为几个优化子问题，然后以交替迭代的方式解决了这些子问题。进行了数值实验，以验证所开发算法的效率和可行性。根据性能评估的峰峰值信噪比（PSNR）和归一化均方根距离（NRMSD）的统计分析结果，表明在不同扫描角度范围内，不同算法之间存在显着的统计学差异（p <0.0001）。从实验结果还可以看出，该算法在同时抑制边缘附近的条纹伪影和渐进伪影方面优于传统的重建算法。随着医疗技术的快速发展，计算机断层扫描（CT）已成为临床上不可或缺的成像工具。其主要优势在于能够非侵入性地提供体内结构的详细横截面图像。然而，传统CT扫描过程中，患者需要接受一定量的X射线辐射，这对健康可能造成不利影响。因此，如何在降低辐射剂量的同时，从有限的角度采集数据并重建出高质量的CT图像，是该领域研究的热点问题。为解决上述挑战，提出了一种基于l（0）梯度最小化的有限角度CT图像重建算法。这一研究的核心在于利用图像梯度的l（0）-范数作为正则化函数。此处的l（0）-范数，与通常应用的l（2）-范数不同，它更倾向于得到一个稀疏解，即图像中的大部分像素值为零，仅保留关键特征。这种特性有助于在重建图像过程中，保留边缘等重要视觉信息，同时滤除噪声和伪影。研究中的算法优化问题被分解成若干子问题，并通过交替迭代的方法逐一解决。这种方法不仅简化了整个优化过程，还确保了在去除伪影和保留图像细节之间取得良好的平衡。数值实验验证了所开发算法的效率和可行性，从统计分析的角度证明了其在不同扫描角度和不同条件下，相较于其他传统算法的显著优势。具体来说，通过对比峰峰值信噪比（PSNR）和归一化均方根距离（NRMSD）等性能评估指标，本研究提出的算法在降低条纹伪影和边缘渐变伪影方面表现尤为出色。PSNR的提高意味着重建图像的细节保留更加完整，而NRMSD的降低则表明图像差异的缩小，这些都是提高图像质量的关键因素。此外，该算法的贡献不仅仅在于技术实现上的突破，更在于为处理有限角度CT图像重建的不适定问题提供了新的思路。通过有效减少条纹伪影和边缘渐变伪影，该算法能大幅提高图像的诊断质量和准确性。未来的研究工作可能会围绕进一步改进算法性能，例如通过算法迭代策略的优化，或结合其他图像处理正则化技术，使图像重建更加高效和精确。基于l（0）梯度最小化的有限角度CT图像重建方法，展示了其在减少辐射剂量的同时提高图像质量的巨大潜力。该技术在临床诊断和工业检测中，都具有广泛的应用前景。随着研究的深入和技术的成熟，这项研究将为患者和行业带来真正的益处，实现高精度、低风险的医学成像。

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RESEARCH ARTICLE

ℓ

Gradient Minimization Based Image

Reconstruction for Limited-Angle Computed

Tomography

Wei Yu

1,3

, Li Zeng

2,3

1 School of Biomedical Engineering, Hubei University of Science and Technology, Xianning, China,

2 College of Mathematics and Statistics, Chongqing University, Chongqing, China, 3 Engineering Research

Center of Industrial Computed Tomography Nondestructive Testing of the Education Ministry of China,

Chongqing University, Chongqing, China

* drlizeng@cqu.edu.cn

Abstract

In medical and industrial applications of computed tomography (CT) imaging, limited by the

scanning environment and the risk of excessive X-ray radiation exposure imposed to the

patients, reco nstructing high quality CT images from limited projection data has become a

hot topic. X-ray imaging in limited scanning angular range is an effective imaging modality

to reduce the radiation dose to the patients. As the projection data available in this modality

are incomplete, limited-angle CT image reconstruction is actually an ill-posed inverse prob-

lem. To solve the problem, image reconstructed by conventional filtered back projection

(FBP) algori thm frequently results in conspicuous streak artifacts and gradual changed arti-

facts nearby edges. Image reconstruction based on total variation minimization (TVM) can

significantly reduce streak artifacts in few-view CT, but it suffers from the gradual changed

artifacts nearby edges in limited-angle CT. To suppress this kind of artifacts, we develop an

image reconstruction algorithm based on ℓ

gradient minimization for limited-angle CT in

this paper. The ℓ

-norm of the image gradient is taken as the regularization function in the

framework of developed reconstruction model. We transformed the optimization problem

into a few optimization sub-problems and then, solved these sub-problems in the man ner of

alternating iteration. Numerical experiments are performed to validate the ef ficiency and the

feasibility of the developed algorithm. From the statistical analy sis results of the perfor-

mance evaluations peak signal-to-noise ratio (PSNR) and normalized root mean square

distance (NRMSD), it shows that there are significant statistical differences between differ-

ent algorithms from different scanning angular ranges (p<0.0001). From the experimental

results, it also indicates that the developed algorithm outperforms classical reconstruction

algorithms in suppressing the streak artifacts and the gradua l changed artifacts nearby

edges simultaneously.

PLOS ONE | DOI:10.1371/journal.pone.0130793 July 9, 2015 1/15

OPEN ACCESS

Citation: Yu W, Zeng L (2015) ℓ

Gradient

Minimization Based Image Reconstruction for

Limited-Angle Computed Tomography. PLoS ONE

10(7): e0130793. doi:10.1371/journal.pone.0130793

Editor: Huafeng Liu, Zhejiang Univ, CHINA

Received: February 5, 2015

Accepted: May 26, 2015

Published: July 9, 2015

article distributed under the terms of the Creative

Commons Attribution License, which permits

unrestricted use, distribution, and reproduction in any

medium, provided the original author and source are

credited.

Data Availability Statement: All relevant data are

within the paper and its Supporting Information files.

Funding: This work is supported by the National

Natural Science Foundation of China under grant

(61271313) (http://www.nsfc.gov.cn), National

Instrumentation Program of China (2013YQ030629)

(http://www.most.gov.cn), and Chongqing science and

technology research plan project (cstc2012gg-

yyjs70016). The funders had no role in study design,

data collection and analysis, decision to publish, or

preparation of the manuscript.

Competing Interests: The authors have declared

that no competing interests exist.

Introduction

As an important nondestructive testing method, CT shows large-scale applications in many

fields such as medical diagnosis, industrial nondestructive testing, etc. In practical applications

of CT imaging, when projection data obtained are adequate and complete, the FBP algorithm,

which has been commonly utilized in commercial CT [1], can reconstruct images accurately.

However, limited by the scanning environment and the excessive radiation dose imposed to the

patients, it is desired that high quality CT images can be reconstructed from low-dose projec-

tion data [2,3,4]. To reduce the radiation dose to the patients, an effective imaging modality is

X-ray imaging in limited scanning angular range. It is possible that the effective scanning angu-

lar range doesn’t satisfy the condition of short scan [5], i.e., the effective scanning angular range

is less than 180° plus fan angle. In this case, significant streak artifacts and gradual changed arti-

facts nearby edges are present in reconstructed images by conventional FBP algorithm and con-

sequently, images are distorted [6]. In the medical domain, especially for dental CT [7,8], C-

arm tomosynthesis [9], imaging in the chest and the breast [10] etc., as X-ray ionizing radiation

is harmful to human bodies, it is in urgent need to use shorter time of exposure and fewer pro-

jection data to reconstruct approximately accurate images. Therefore, to reconstruct high-qual-

ity images using limited-angle projection data has been a research focus all along.

Recently, the iterative reconstruction algorithm shows more advantages than conventional

FBP algorithm in dealing with the reconstruction problem with incomplete projection data. As

early as 1980s, the algebraic reconstruction technique (ART) and simultaneous algebraic

reconstruction technique (SART) were utilized by some researchers to investigate CT image

reconstruction [11, 12]. While for incomplete projection data, obvious artifacts and noise are

present in reconstructed images obtained by the two algorithms.

In 1992, Rudin et al. proposed an image denoising method based on total variation (TV) of

image [13], and they showed that this method can well protect the edge during the denoising

process. Assuming that the pixel value of image at positio n (x,y) is labeled by u

x,y

, the TV of

image can be expressed as

jjujj

x;y

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðu

x;y

 u

x1;y

þðu

x;y

 u

x;y1

: ð1Þ

The TV is essentially the ℓ

-norm of the image gradient magnitude. In image domains,

images consisting of image gradient magnitudes are approximately sparse. To utilize the spar-

sity of gradient image, the TV norm can be taken as a regularization function. Furthermore, in

2006, Sidky et al.[14] adapted the TV minimization to consider the sparsity of the image gradi-

ent magnitudes, and then proposed an accurate algorithm for CT image reconstruction from

few-view and limited-angle projection data. This algorithm is called TVM based algorithm

hereafter. The TVM based algorithm can obtain accurate images from incomplete projections

especially in the sparse angular sampling over 360°. While the scanning angula r range is limited

and less than 180°(such as 90° and 120°), the reconst ruction results suffer from gradual

changed artifacts nearby the edges of the objects [14, 15], although it shows superiority to sup-

press streak artifacts. To further improve the quality of CT images for limited-angle tomogra-

phy, some scholars have advanced the conventional TV based image reconstruction algorithm

[16–18]. Although these methods improve the performance on reducing gradual changed arti-

facts nearby edges, however, the edge information of the objects may have a certain degree of

distortion. With the aim to make the most of previously reconstructed CT images, by means of

the constraint of TV, the reconstruction algorithms can generate better images [19,20], while

their applications are limited to some extent as the image database is often needed before

image reconstruction. Other reconstruction algorithms based on the prior knowledge of image

ℓ

Gradient Based Image Reconstruction for Limited-Angle Tomography

PLOS ONE | DOI:10.1371/journal.pone.0130793 July 9, 2015 2/15

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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