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A Novel Weighted Total Difference Based Image

Reconstruction Algorithm for Few-View Computed

Tomography

Wei Yu

1,3

, Li Zeng

1,2

1 Key Laboratory of Optoelectronic Technolo g y and System of the Education Ministry of China, Chongqing University, Chongqing, 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

Abstract

In practical applications of computed tomography (CT) imaging, due to the risk of high radiation dose imposed on the

patients, it is desired that high quality CT images can be accurately reconstructed from limited projection data. While with

limited projections, the images reconstructed often suffer severe artifacts and the edges of the objects are blurred. In recent

years, the compressed sensing based reconstruction algorithm has attracted major attention for CT reconstruction from a

limited number of projections. In this paper, to eliminate the streak artifacts and preserve the edge structure information of

the object, we present a novel iterative reconstruction algorithm based on weighted total difference (WTD) minimization,

and demonstrate the superior performance of this algorithm. The WTD measure enforces both the sparsity and the

directional continuity in the gradient domain, while the conventional total difference (TD) measure simply enforces the

gradient sparsity horizontally and vertically. To solve our WTD-based few-view CT reconstruction model, we use the soft-

threshold filtering approach. Numerical experiments are performed to validate the efficiency and the feasibility of our

algorithm. For a typical slice of FORBILD head phantom, using 40 projections in the experiments, our algorithm outperforms

the TD-based algorithm with more than 60% gains in terms of the root-mean-square error (RMSE), normalized root mean

square distance (NRMSD) and normalized mean absolute distance (NMAD) measures and with more than 10% gains in terms

of the peak signal-to-noise ratio (PSNR) measure. While for the experiments of noisy projections, our algorithm outperforms

the TD-based algorithm with more than 15% gains in terms of the RMSE, NRMSD and NMAD measures and with more than

4% gains in terms of the PSNR measure. The experimental results indicate that our algorithm achieves better performance in

terms of suppressing streak artifacts and preserving the edge structure information of the object.

Citation: Yu W, Zeng L (2014) A Novel Weighted Total Difference Based Image Reconstruction Algorithm for Few-View Computed Tomography. PLoS ONE 9(10):

e109345. doi:10.1371/journal.pone.0109345

Editor: Christof Markus Aegerter, University of Zurich, Switzerland

Received April 21, 2014; Accepted September 9, 2014; Published October 2, 2014

Copyright: ß 2014 Yu, Zeng. This is an open-access 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: The authors confirm that all data underlying the findings are fully available without restriction. All relevant data are within the paper.

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) (http://www.ctin.ac.cn/). 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.

* Email: drlizeng@cqu.edu.cn

Introduction

As an extremely valuable diagnostic tool, computed tomogra-

phy (CT) has been widely used in medical area. With this powerful

tool, many valuable internal features can be extract without

cutting the object [1,2]. However, during clinical exams, excessive

X-ray radiation exposure may increase the lifetime cancer risk

[3,4]. Thus, it has great significance to use shorter time of

radiation exposure and lower patient radiation dose to reconstruct

numerically accurate tomographic images. To reduce radiation

dose, few-view CT has been an important CT imaging modality.

In this scanning data situation, tomographic image is reconstruct-

ed from the projection data collected by sparse angular sampling

[5–9]. For few-view CT, due to the projection data obtained is not

theoretically sufficient for exact reconstruction of tomographic

images, conspicuous streak artifacts are present in reconstructed

images by conventional analytic algorithms such as filtered back-

projection [5,10–12]. In this paper, we mainly focus the iterative

reconstruction algorithm for few-view CT.

Since the development of the large computational capacities in

graphical processing unit and the ongoing efforts towards lower

doses have made in CT, iterative reconstruction has become a hot

topic for all major vendors of clinical CT systems in the past years

[13–17]. The algebraic reconstruction technique and simultaneous

algebraic reconstruction technique (SART) are two classical

reconstruction algorithms for CT image reconstruction [18,19].

Since the projection data are incomplete, using the two algorithms,

obvious artifacts and noise are present in reconstructed images.

With the development of compressed sensing theory [20–22],

compressed sensing based iterative reconstruction algorithm has

drawn much attention in the medical imaging and other

tomographic imaging modalities. By adopting the compressed

sensing based iterative reconstruction algorithm, the image can be

reconstructed from rather limited projection data [23]. In

mathematics, actually, CT image reconstruction with few-view

PLOS ONE | www.plosone.org 1 October 2014 | Volume 9 | Issue 10 | e109345

projection data is taken as an ill-posed inverse problem. To solve

this problem, regularization method is usually adopted, and

corresponding unconstrained optimization problem can be

formulated [24]. In the unconstrained optimization problem, the

objective function usually contains two terms. The first term is

data fidelity term which constraints the data consistency between

measured projection data and model data. The second term is

regularization term which is designed according to the priori

information of the image.

In the CT reconstruction field, it is likely that images are not

sparse themselves, but image coefficients in some transform

domains show sparsity. In image gradient transform, the L1 norm

of the image gradient magnitudes (also known as total variation

(TV) of image) are approximately sparse. If D

and D

represent

the horizontal and vertical gradient operators respectively, then

TV regularization term can be expressed as

TV(u)~

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

z(D

. This regularization term which

was originally proposed for image denoising [25], has been

extended in the field of few-view CT image reconstruction [5].

Subsequently, many other related reconstruction algorithms have

been developed [6–8,23,26].

In 1996, Li and Santosa suggested that total difference (TD)

which was defined as TD(u)~DDD

uDD

zDDD

uDD

, was a reliable

and computationally efficient approximation to the TV in image

restoration [27]. In 2010, by constructing a pseudo-inverse of the

discrete TD, a TD minimization algorithm with soft-threshold

filtering (TDM-STF) was developed for few-view CT. It can

improve the convergence and efficiency of TV based minimization

methods [28]. In the TDM-STF algorithm, TD is taken as the

regularization term. Later, the TDM-STF algorithm was applied

to a multisource x-ray interior imaging system [29]. In their work,

they accelerated the convergence speed of TDM-STF by

incorporating a fast iterative shrinkage thresholding algorithm

[30]. It shows that when obtaining the same image quality, the

TDM-STF may need less iterations and total computational cost

for practical application. However, the TD seeks the gradient

sparsity horizontally and vertically, but fails to enforce the gradient

continuity. Thus, the TD is prone to recovering an image of sharp

horizontal and vertical edges. To overcome this shortcoming of

TD, a new measure (called weighted total difference (WTD)

measure hereafter) was utilized by Shu and Ahuja for compressive

sampling [31]. In their work, they proposed a hybrid compressive

sampling method for recovering a piecewise smooth image from

limited measurements. Since the WTD measure exploits the

continuity and sparsity simultaneously in the partial gradient

domain, all possible sharper edges of the image can be recovered

from limited measurements. In [31], WTD measure was taken as

the regularization term. Note that model investigated in [31] is

different from the model for CT reconstruction since WTD

combines two complementary sampling systems for image

recovery. In this work, we consider incorporating WTD measure

into the model for CT reconstruction due to the good property of

WTD.

With the aim to eliminate the undesired streak artifacts and

preserve the edge structure information of the object, in this paper,

we propose a novel reconstruction algorithm based on WTD

minimization for few-view CT. The proposed reconstruction

model combines the CT imaging model and the WTD measure.

In the proposed reconstruction model, the WTD measure is taken

as the regularization term. The differences between the WTD in

current study and in the study [31] lie in the application fields and

the corresponding problems need to solve. To solve our model

effectively, the soft-threshold filtering (STF) method and a fast

iterative shrinkage thresholding algorithm are employed to

accelerate the converging speed of our algorithm. For simplicity,

in the following sections, we referred to our algorithm as WTDM-

STF.

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

we illustrate the CT imaging model and describe the WTDM-STF

reconstruction algorithm for few-view CT, together with an

efficient iterative scheme. Moreover, the data acquisitions and

performance evaluations are also outlined in this section. In the

following section, numerical results and discussion are presented

and conclusions are given in final section.

Methods

CT imaging model

The model of fan beam CT imaging can be approximated as

following discrete linear system [19]:

Au~g, ð1Þ

where g~½g

,:::,g



is the measured projection data

which can be represented by a vector of size M , M is the number

of the transmission rays, g

is the ray-sum measured with the ith

ray, u~½u

,:::,u



denotes the image to be reconstructed

which can be represented by a vector of size N, N is the number of

image pixels. A~(a

i,j

) [R

is the system matrix which

represents forward projection. The system matrix weight a

i,j

represents the contribution of the jth pixel to the ith ray-sum. In

our experiments, the system matrix weights a

i,j

are computed by

calculating the intersection length of the ith ray through the jth

pixel. Given the projection data acquired from the detector, the

aim for image reconstruction is to solve the Eq. (1) for u from the

measured data g. As for few-view CT problem, the number of the

measured projection data M is much smaller than the number of

image pixels N, then, the Eq. (1) is underdetermined. Therefore,

Figure 1. A typical slice of the FORBILD head phantom.

doi:10.1371/journal.pone.0109345.g001

Image Reconstruction for Few-View Computed Tomography

PLOS ONE | www.plosone.org 2 October 2014 | Volume 9 | Issue 10 | e109345

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