A study on projection distribution of few-view reconstructionwith total variation constraint

Li Zhang, Xinhui Duan, Yuxiang Xing, Zhiqiang Chen and Jianping Chenga

aDepartment of Engineering Physics, Tsinghua University, Beijing, China

ABSTRACT

In today’s tomographic imaging, there are more incomplete data systems, such as few-view system. The advan-tage of few-view tomography is less x-ray dose and reduced scanning time. In this work, we study the projectiondistribution in few-view fan-beam imaging. It is one of the fundamental problems in few-view imaging becauseof its severe lack of projection data. The aim is to reduce data redundancy and to improve the quality of re-constructed images by research on projection distribution schemes. The reconstruction algorithm for few-viewimaging is based on algebraic reconstruction techniques (ART) and total variation (TV) constraint approachedby E. Sidky .et al in 2006. Study of few-view fan-beam projection distribution is performed mainly throughcomparison of several distribution types in projection space and reconstructed images. Results show that thedistribution called short-scan type obtains the best image in five typical distributions.

Keywords: few-view imaging, projection distribution, computed tomography, total variation

1. INTRODUCTION

In applications of CT imaging nowadays, insufficient data problems occur more frequently. This is caused by theconstraints of the imaging hardware, scanning geometry or radiation dose. The insufficient data problem cantake various forms, but in this work we only consider the few-view projection problem in fan-beam tomography;namely, CT projections are fan-beam taken at few views. Because the projection data is severely undersampledin the angular direction and the inherent symmetry of CT projections may cause redundancy, it is importantto resort to an efficient sampling scheme if the geometry can be chosen freely. More efficient sampling schemesof common fan-beam CT were studied (see e.g. Section 4.3 in Ref. 1), but they may be inconvenient to realizein practice. Changing projection positions on the trajectory is an easier method to adjust sampling schemesin few-view imaging. It also greatly affects the quality of the reconstructed image. Thus different projectionarrangement schemes in few-view imaging are studied in this paper and we will give our suggestion. We willmainly compare four typical projection distributions of few-view imaging. The projections in these schemes arealmost uniformly distributed on a circle trajectory. This may help to avoid obvious anisotropic in reconstructedimages.

There are mainly two types of approaches to overcome data insufficiency in incomplete data tomographicimaging. One estimates the missing data regions from the measured data set, and then reconstructs the imagewith analytic algorithms. Such approaches may be useful for specific situations, but it is difficult to make generalconclusions on the implementation of such an approach. The other approach solves the image model from theavailable measurements in an iterative form. These algorithms differ in the mathematical model for the imagefunction and the numeric process of the iterative scheme. This approach has achieved varying degree successfor tomographic reconstruction from incomplete data. The reconstruction algorithm in our study bases on thesecond approach.

It is often true that tomographic images are relatively constant or smooth in the internal region of objects andrapid variation only occurs at the boundary of structures. The sparse nature will show in its gradient image. Totake advantage of this sparseness, total variation (TV) is minimized as the objective function, which is definedas follows:

Further author information: (Send correspondence to Li Zhang)ZhangLi: E-mail: zli@mail.tsinghua.edu.cn, Telephone: +86 010 62780909 86201Xinhui Duan: E-mail: dxhcom@gmail.com, Telephone: +86 010 62780909 86209

Medical Imaging 2008: Physics of Medical Imaging, edited by Jiang Hsieh, Ehsan Samei,Proc. of SPIE Vol. 6913, 69132W, (2008) · 1605-7422/08/$18 · doi: 10.1117/12.769779

Proc. of SPIE Vol. 6913 69132W-12008 SPIE Digital Library -- Subscriber Archive Copy

TV(u) =∫

Ω

|∇u| =∫

Ω

√u2

x + u2y (1)

Here, u refers to a 2D image function and ux, uy are the derivatives of u.

TV minimization is a popular image restoration method with edge preserving.2, 3 Many researches have beendone on its faster numerical schemes4 and the property of its solutions;5, 6 Also, TV is used as a regularizationfunction for tomographic reconstruction.7 The reconstruction algorithm used in our work is closer to the secondone.

The few-view reconstruction algorithm in our study is derived from a recently developed TV-based algorithmfor recovering an image from its sparse linear measurements, such as Fourier transform (FT) samples by Candeset al.8 In that work, they reconstructed a Shepp-Logan phantom image using only 22 parallel-beam projectionsaccurately with an FT-TV form. In 2006, Sidky .et al extended this algorithm to the divergent-beam CT with analgebraic reconstruction techniques (ART)-TV form.9 The reconstruction part of our work follows the frameworkof ART-TV form algorithm for few-view imaging. It is very flexible to implement different projection schemesand comparable results can be obtained.

2. PROJECTION DISTRIBUTION PROBLEM IN FEW-VIEW PROJECTIONS

For the sake of simplicity, we restrict our attention to the case satisfying the following conditions:

1. We only consider the fan-beam sampling problem in few-view imaging and different sampling lattices areobtained only from different projection distributions. Much research has been done on more efficientsampling schemes for fan-beam projection as referred in Sec. 1 , but we just focus a specific situation:projection distribution.

2. We suppose the prior information on the image model piece-wise smooth. This is because TV is used asa constraint in the reconstruction algorithm for few-view projections. It is obvious that the reconstructedimage quality varies with the objects scanned. However, we don’t consider this factor in this paper. Wefocus on the sampling lattice itself and we suppose to know little information about the scanning object.Thus additionally we only suppose the image band-limited and with compact support just as common CTimaging.

3. We apply ART-TV reconstruction method for few-view imaging. This algorithm is effective in few-viewreconstruction and convenient to implement in our study. We compare the reconstruction results of differentprojection distributions by this algorithm. Although the result concluded from other methods can be ofslight difference, the tendency will be the similar.

In fan-beam CT, only 180 + 2γm angle projections are needed to reconstruct an exact image (2γm the fanangle). This case is referred as the short-scan projection.10 The inherent symmetry of fan-beam projectionleads to data redundancy. The same case occurs in few-view fan-beam projections. Further more, the few-viewproblem is more sensitive to data redundancy because of the sever lack of projection data. Thus in our study,we try to reduce the redundancy by choosing a better projection distribution.

Half of the projection data is redundant in full-scan fan-beam CT. In short-scan projection, the redundantdata in projection space is represented as the shaded regions in Fig 1. When the projection angle is from 0to βN ∈ (0, 180 + 2γm), the projection data become limited-angle. The few-view fan-beam projection has thesimilar problem. In our study, projection sampling lattices are mainly obtained by altering βN ∈ (0, 360] witha fixed number of projections uniformly distributed in [0, βN ] and βN = 180 + 2γm is still a turning point inthis scheme. Other distribution schemes could be such as suggested by Ref. 9, 20 views specified by:

θi =

18 ∗ (i − 1) 1 ≤ i ≤ 1018 ∗ (i − 0.5) 10 < i ≤ 20 (2)

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4 m

0

180 2 m

S1

S2

180

2 m

N

Figure 1. Fan-beam projection data in γ−β coordinate system. If the fan-beam projections are measured in [0, βN ], βN ∈(180 − 2γm, 180 + 2γm) (2γm the fan angle), the data within the area S1 are missing, and S2 redundant. The ratio ofS1 to S2 will determine the quality of reconstructions.

Here θi refers to the source position angle, and i = 1, 2, ..., 20 is the sequence number of the projection. Thisscheme shifts the second half of the angular measurements to reduce the redundancy and will be referred asScheme 2 in the following.

The reconstruction process follows the framework of ART-TV and the implement details can be found inRef. 9. In this framework, the reconstruction is mainly divided into two relatively independent steps comparedto the compact form of FT-TV in Ref. 8:

1. ART reconstruction to enforce consistency with the projection data.

fk+1 = fk +gi − Hifk

‖Hi‖2Hi, i = 1, ..., M (3)

2. TV minimization with gradient descent.

fk+1,j+1 = fk+1,j + λ∂TV(fk+1,j)

∂fk+1,j, j = 1, ..., N (4)

The separate steps of reconstruction and constraint on images make this algorithm applicable for many typesof incomplete data problems. However, the discrete error is also larger than FT-TV form. This may result inmore quantity demand of projection data than FT-TV form for the same quality of reconstructed images. It isanother reason why we study the projection distribution in few-view imaging.

In ART step, image f is reconstructed from M projections of g. The most advantage of ART in our work isstable and convenient for various consistent projection sampling data even for nonuniform cases.

TV minimization is more like a common image restoration process in ART-TV form. In TV step, N iterationof TV minimization are accomplished by use of gradient descent method. λ is a parameter controlling thegradient descent procedure of TV minimization. Taking the derivative of TV(u) with respect to each pixel valueresults in a singular form. Thus we use the following approximate derivative. Here ε is a small positive numberto avoid singularity in uniform areas.

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∂TV∂ux,y

=2ux,y − ux−1,y − ux,y−1√

ε + (ux,y − ux−1,y)2 + (ux,y − ux,y−1)2

− ux+1,y − ux,y√ε + (ux+1,y − ux,y)2 + (ux+1,y − ux+1,y−1)2

− ux,y+1 − ux,y√ε + (ux,y+1 − ux,y)2 + (ux,y+1 − ux−1,y+1)2

(5)

3. NUMERICAL RESULTS

In this section, we show the results on projection distribution study by simulation experiment. Five types ofprojection data are generated. Four of them are uniformly distributed within [0, βN ], and the other is Scheme 2.βN is set to 360 (one circle), 180 + 2γm (short-scan type), 180 (half circle) and 120 (less than half circle).Scheme βN = 180 + 2γm is also called short-scan type as referred in Sec. 2. We think the five types representthe typical situations of the uniform projection distribution of few-view imaging. Others will obtain the similarresult or the quality of reconstructed images between two of the five cases.

To analysis the redundancy of the sampling schemes, we plot five types of projection data in parallel-beam(r, θ) space in Fig. 2, where r is the distance from the rotation center to rays and θ is the angle from x axisto the distance vector. The projection number is 20 and the fan angle is 40. One projection consists of 300equiangular rays. This figure helps to compare these schemes in the same coordinate system. When βN becomessmaller, the sampling lattice are denser, but regions without sampling may occur. If βN < 220, a blank areaemerges in the projection space as shown in Fig. 1. This will turn the few-view problem into a few-view limitedangle one with greater singularity, which causes the reconstruction more difficult. Thus we focus our attentionon comparing the three sampling schemes of full space coverage: βN = 360, βN = 220 and Scheme 2. Weexpect to estimate the redundancy of the schemes. As shown in Fig. 2, one projection data forms a curve in(r, θ) space. All the curves cross each other in projection space. The crossings present the redundant data inprojections. In a sense, the number of crossings approximately shows the degree of redundancy. The crossingsin the three sampling schemes are: βN = 360 50, βN = 220 28, and Scheme 2 42. Fewer crossings imply lowerredundancy and better quality of the reconstructed image with higher possibility, so the scheme of βN = 220

may achieve the best result. The conclusion is reasonable in this case and is needed to be demonstrated in imagereconstruction. In other ways, the number of crossings is strongly depend on the fan angle. When the fan anglebecomes smaller, the number of crossings is also reduced, and the three schemes become similar. For example,when the fan angle is 20, the crossing number are: βN = 360 30, βN = 220 10, and Scheme 2 21. It is obviousthat in the parallel-beam projection the three are all the same. Thus reducing the fan angle is an approach todepress the redundancy if possible.

To test the different sampling schemes, reconstructions are carried out. Results are shown in Fig. 3. In ournumerical demonstrations, a high contrast Shepp-Logan phantom is used. The geometry parameters are thesame as Fig. 2. The number of iteration of each scheme is 100 and every loop contains 20 iterations of TVminimization. For a quantitative comparison, the image horizontal profiles are also plotted. To compare theimage quality more directly, the distances from the phantom are computed as follows:

D =

∑x,y |u − u0|∑

x,y |u0| (6)

where u0 is the phantom image and u is the reconstructed image.

The results in Fig. 3 indicate that the projection scheme of βN = 180+2γm obtains the most accurate imagein five types. The profiles in Fig. 3 (b) show that it can almost recover the attenuation coefficients of the objectcompared with the loss of contrast in the case of distributing view samples in full 360 degree. It is also moreaccurate than the result from Scheme 2. The relative distance shown in Fig. 3 (c) gives the same results. It isalso demonstrated that the few-view problem is very sensitive to the limited coverage of view angles. With thecoverage of view angles getting smaller, severe artifacts appear. The using of TV in our reconstruction algorithm

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−4 −2 0 2 40

20

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160

18020(360)

−4 −2 0 2 40

20

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60

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18020(360+)

−4 −2 0 2 40

20

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−4 −2 0 2 40

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140

160

18020(180)

−4 −3 −2 −1 0 1 2 3 40

20

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18020(120)

Figure 2. Five types of few-view fan-beam projection data in parallel-beam (r, θ) space. Top row: βN = 360, Scheme 2.Middle row: βN = 220 and βN = 180. Bottom row: βN = 120.

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50 100 150 2000.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Pixel Number

Horizontal Profiles

20(360)20(360+)20(220)20(180)20(120)phantom

360 360+ 220 180 1200

0.05

0.1

0.15

0.2

0.25

Dis

tanc

e,|x

−p|

/|x|

βN

Distance from phantom

(a)(b) (c)

Figure 3. (a)(upper row 6 images): The phantom and images reconstructed from 20 projections with different βN , inorder, βN = 360, 360+, 220, 180, 120, and the phantom, where ”360+” refers to Scheme 2. The display gray scaleis [0, 2]. (b): horizontal profiles of the images; (c): the distance of the five reconstruction images from the phantom.Notation: ”20(360)” in top left of the images means 20 uniform projections in 360, and others follow similarly.

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may reduce the difference in the images reconstructed from the different sampling schemes. Other algorithmsensitive to sampling lattices may enlarge the reconstruction difference, but the relative sequence may be thesame.

4. DISCUSSION AND CONCLUSION

In this article, we evaluate different projection distribution schemes in few-view imaging. The reconstructionalgorithm with TV constraint is applied in the form of ART-TV iteration. The numerical experiments show thatthe best one in the five schemes is uniformly distributed in the angle range of [0, 180 + 2γm], or the so-calledshort-scan type. This scheme reduces data redundancy and avoids more data singularity compared to the others.This is the sampling scheme we suggest for the few-view imaging.

In our study, five typical sampling schemes of uniform distribution are chosen. We also compare them withthe short-scan projection in fan beam and the limited-angle projection. The aim is to reduce redundant datawithout increasing singularity of the problem. However, measurement of data redundancy is difficult for few-viewsampling schemes. In other CT imaging modalities such as full angle fan-beam CT, redundant data often refersto the overlapped or double-sampled region of projections, but there is no ”region” in the projection space offew-view imaging because of the sparseness of the data. As a result in our special case, a simple trick is use toapproximately estimate the redundancy: enumerate the crossings in projection space. This means works in ourstudy and there may be other methods having the same effect.

In our work, the reason why we choose uniformly distributed projection schemes is that little priori informa-tion is supposed to be known about the scanning object. But in actual applications, we often know about theobject to be scanned. In this sense, nonuniform projection or other distributions may be more suitable for thespecial case. In the other hand, the priori always plays a central role in the iterative reconstruction algorithmsuch as TV in ours. It may reduce the impact of different sampling schemes. In this case, the criterion forchoosing the scheme may not only be the least redundancy of projection data, but also most convenient forspecial priori.

The future work will be on the study of the sampling schemes for various few-view reconstruction algorithmsbecause the reconstruction is a major part of few-view imaging study. Other research work may include optimizingsampling schemes for practical usage and nonuniform sampling lattice for special applications.

ACKNOWLEDGMENTS

This work is supported by National Natural Science Foundation of China (No. 10575059), New Century ExcellentTalents in University (NCET), National Natural Science Foundation of China for Young Scholars (No. 10605015),and National Natural Science Foundation of China (No. 60772051).

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