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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 23 — Nov. 18, 2013
  • pp: 27946–27963
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An approach to increasing the resolution of industrial CT images based on an aperture collimator

Yining Zhu, Defeng Chen, Yunsong Zhao, Hongwei Li, and Peng Zhang  »View Author Affiliations


Optics Express, Vol. 21, Issue 23, pp. 27946-27963 (2013)
http://dx.doi.org/10.1364/OE.21.027946


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Abstract

The spatial resolution of CT images is dominated by the focal spot size when it is large relative to the detector cells. We propose an approach to increase the spatial resolution by utilizing an aperture collimator. The aperture collimator is specially designed and placed in front of the X-ray source so that the rays penetrating the collimator form a set of narrow fan beams. Then an iterative algorithm is introduced to reconstruct CT images from the data obtained by scanning the narrow fan beams. Numerical experiments show that the proposed approach could significantly increase the resolution of the CT images. Furthermore, this approach is also robust against some challenging cases, such as the examination of low contrast object, reconstruction based on multi-energy data and perturbation of geometric errors in CT systems.

© 2013 OSA

1. Introduction

Spatial resolution is an important performance index of Computerized Tomography (CT) systems, that reflects the capability of a CT system to distinguish details. Generally speaking, the spatial resolution of a CT system is determined by many factors, including the X-ray focal spot size, detector cell size, ratio of the source-to-detector distance (SDD) to the source-to-origin (rotation center) distance (SOD), field of view, mechanical precision, and reconstruction algorithms.

To increase the spatial resolution of CT images, many methods have been proposed in the literature such as quarter-detector-offset [1

1. H. Hu, “Multi-slice helical CT: scan and reconstruction,” Medical Physics 26, 5–18 (1999). [CrossRef] [PubMed]

], focal-spot-wobbling [2

2. J. Hsieh, Computed Tomography: Principles, Design, Artifacts, and Recent Advances (SPIE Press, 2003, vol. PM188).

] and the virtual-detector based algorithms [3

3. H. Zhang, J. Tian, M. Chen, and P. Zhang, “A novel scanning mode and image reconstruction method on super-resolution CT,” Chin. J. Stereol. Image Analysis 9, 154–157 (2005).

6

6. J. Hsieh, M.F. Gard, and S. Gravelle, “Reconstruction technique for focal spot wobbling,”Proc. of SPIE Medical Imaging VI 1652, 175–182 (1992). [CrossRef]

]. However, when the ratio of the focal spot size to the detector cell size is relatively large, the spatial resolution of CT images will be dominated by the focal spot size, and the methods mentioned above become almost invalid. For instance, the focal spot size of the X-ray tube for 450 keV industrial CT is typically about 3.0 to 6.5 mm following the European standard (EN 12543), but the detector cell size is about 0.083 to 0.4 mm. In this situation, the projection data will be blurred and result in low resolution CT images.

We have designed a collimator with a set of apertures to tackle the above-mentioned problem. The collimator is placed in front of the X-ray source, such that the rays penetrating the collimator form a set of narrow fan beams, and each of the fan beams is emitted from a segment of the focal spot. In this case, each detector cell is limited to receiving photons from a few narrow fan beams (generally 2 to 3 fan beams). Such a projection image can be taken as a combination of a set of projection images under a set of narrow fan beams. We find that the projection image scanned with an aperture collimator has a higher resolution than that scanned without an aperture collimator. Conventional algorithms were not able to be utilized directly to reconstruct CT images from such scanned projection images. Hence, we will propose an iterative reconstruction algorithm to deal with such projection data.

The basic idea of our method is somewhat similar to coded aperture (CA) imaging which was originally developed in X-ray astronomy [7

7. E. Caroli, J. B. Stephen, G. Cocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45, 349–403 (1987). [CrossRef]

10

10. G. Skinner, “Imaging with coded-aperture masks,” Nucl. Instr. Meth. Phys. Res. 221, 33–40 (1984). [CrossRef]

]. In CA imaging, a mask with numerous apertures drilled in a certain pattern is placed parallel to the X-ray detector surface. The recorded count (measurement data) in each detector cell is the summation of the X-ray flux emitted from all stars modulated by the mask. The X-ray flux (measurement data) could then be decoded by some reconstruction method, so that the distribution of X-ray flux intensities emitted from the stars could be identified, i.e., the stars can be located.

The collimator with a CA was then applied widely to emission tomography. In [11

11. S. Webb, The Physics of Medical Imaging (Taylor & Francis, 2010).

], Webb described several types of limited-angle emission tomography systems. Due to the high degree of focus of the collimator, the camera was able to selectively receive the only gamma photons just within a small range of incidence angle. Therefore, the radioactive distribution of the plane “in focus” at a certain depth is obtained. By moving the detector device, the images (radioactive distribution) of planes at different depths are obtained. However, the images from the emission tomography contain blurred information from above and below the non-focal plane. Hence, the spatial resolution of the depth (thickness of the focal plane) is lower than the in-slice resolution. On the other hand, the resolution of the focal plane (in-slice) decreases rapidly with increasing depth. More recently, the fixed collimator developed into two categories: coded apertures and segmented apertures. In [12

12. G. Knoll, W. Rogers, K. Koral, J. Stamos, and N. Clinthorne, “Application of coded apertures in tomographic head scanning,” Nucl. Instr. Meth. Phys. Res. 221, 226–232 (1984). [CrossRef]

], Knoll proposed a head scanning apparatus that was based on ring detectors and coded apertures. This device could obtain scanning data that are coded by the rotation of the apertures. The coded scanning data were then decoded and high resolution images were reconstructed. In recent years, the coded aperture imaging technique has been extended to other nuclear medicine imaging fields [13

13. W. E. Smith, R. G. Paxman, and H. H. Barrett, “Image reconstruction from coded data: I. reconstruction algorithms and experimental results.” J. Opt. Soc. Am. A 2, 491–500 (1985). [CrossRef] [PubMed]

20

20. E. E. Fenimore and T. M. Cannon, “Uniformly redundant arrays: digital reconstruction methods.” Appl. Opt. 20, 1858–1864 (1981). [CrossRef] [PubMed]

] and X-ray phase contrast imaging [21

21. A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91, 074106 (2007). [CrossRef]

]. Reference structure tomography (RST) is one of progresses in coded-aperture imaging in recent years. In [22

22. D. J. Brady, N. P. Pitsianis, and X. Sun, “Reference structure tomography,” JOSA. A. 21, 1140–1147 (2004). [CrossRef] [PubMed]

], Brady introduced RST that utilizes multidimensional modulations to encode mappings between radiating objects and measurements. In [23

23. K. Choi and D. J. Brady, “Coded aperture computed tomography,” Proc. of SPIE 74680B, 1–4 (2009).

], Brady combined compressed sensing (CS) into coded imaging, by which could reconstruct sparse object with fewer samplings in tomography.

The coded aperture imaging problem for X-ray astronomy can be modeled as a convolution process, e.g., the convolution of the aperture function and the flux intensity distribution from stars or radioactive sources. Hence, to recover the flux intensity distribution, it is only necessary to perform some deconvolution process. The coded aperture (with different patterns or shifting) will help to modulate the flux intensity distributions, and leads to sparse equations in discrete situation, which are easier to solve. In [24

24. J. Fleming and B. Goddard, “An evaluation of techniques for stationary coded aperture three-dimensional imaging in nuclear medicine,” Nucl. Instr. Meth. Phys. Res. 221, 242–246 (1984). [CrossRef]

], Fleming transformed the CA data using deconvolution and algebraic iterative algorithm (ART) respectively and evaluated the results reconstructed from these two methods. It should be pointed out that the mathematical models used by these methods are linear.

However, the task for the X-ray CT imaging is to recover the linear attenuation coefficient distribution of the object under examination, where the flux intensity of the X-ray source is known. In an ideal CT model, the X-rays are assumed to be monochrome and the size of the source focal spot is ignored. Therefore, the reconstructed images represent the linear attenuation coefficient distribution of the object for the monochrome X-rays. In this situation, the projection data, which is the negative logarithm of the scanned data (with object/without object), has a linear relationship with the attenuation coefficient distribution of the object. However, if the focal spot size is taken into account, this linear relation would dissolve and turn into a nonlinear one. This is the fundamental difference from the X-ray astronomy problem. In spite of this, the aperture collimator might reduce the coupling between different parts of the attenuation coefficient distribution in the projection data, which will make the nonlinear equations easier to solve (see Section 2 for details). At this point, the principle of our method is the same as that of coded aperture imaging.

2. Imaging model with the aperture collimator

As mentioned above, the effect of the X-ray focal spot size in an ideal CT model is ignored. Therefore, the rays emitted from the focal spot to the line detector form a fan-beam. However, when the ratio of the focal spot size to the detector cell size is relatively large, the focal spot size can no longer be ignored. In two dimensional CT, the X-ray focal spot can be seen as a bar-shaped spot (i.e. short line segment) because the variance of the focal spot is very small along the direction vertical to the detector, compared to the source-to-detector distance. Under this assumption, the X-rays emitted from the bar-shaped spot could be seen as a combination of a series of fan beams as shown in Fig. 1. If the scattering effect is ignored, the data scanned with an aperture collimator for a bar focal spot can be modeled as
I(β,u)=s0s1I0(s)exp{yL(s,u)[f(R(β)y)+λg(y)]dL}ds,
(1)
where f (x) is the linear attenuation coefficient distribution of the scanned object at point x, β denotes the rotation angle, R(β) is the relative rotation matrix, u is the detector coordinate, s ∈ [s0, s1] is the position on the bar-shaped spot, and L(s, u) represents the X-rays emitted from the source position s to the detector coordinate u. I0(s) denotes the photon intensity distribution with energy E0 emitted from the position s, and I(β, u) denotes the photos remaining after penetrating the scanned object (the photons that were not eliminated due to attenuation while propagating from s to u). λ is the linear attenuation coefficient of the collimator material, and g(y) is the characteristic function of the aperture collimator distribution. For convenience, all these symbols mentioned above are listed in Table 1.

Fig. 1 Schematic diagram of the scanning configuration with a bar-shaped focal spot.

Table 1. Explanations of the symbols in Eq. (1)

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Fig. 2 Structure of the aperture collimator.

Table 2. The explanations of symbols in formula (2)

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Setting Î0,u,s = I0,u,s exp(−λgu,s), which represents the intensity distribution of the X-rays from each discrete focal spot after penetrating the aperture collimator, then Eq. (2) can be rewritten as
In=s=1SI^0,u,sexp(j=1Jrn,s,jfj),(1nN).
(3)

The CT reconstruction problem in this situation raises two subproblems: (1) estimate the intensity distribution Î0,u,s of the flux after penetrating the collimator from each segmented focal spot; (2) reconstruct the CT images under the assumption that Î0,u,s are known.

For a real CT system, Î0,u,s is independent of the scanned object, but difficult to measure directly. The estimation of Î0,u,s is somewhat similar to the coded aperture imaging in X-ray astronomy, e.g., Î0,u,s may be estimated by utilizing of specially designed masks or phantoms. For the first subproblem, we just estimate Î0,u,s by forward-projection simulation in an ideal condition. Hence, the aim of this paper is to solve the second subproblem.

Due to the obstruction of the aperture collimator, the sinogram scanned with the designed aperture collimator can be treated as a combination of data scanned under a set of discrete narrow fan beams. By changing the widths of the apertures, the thickness of the collimator, and the ratio of the SDD to the SOD, we can modulate the number of the virtual focal spots from which a detector cell could receive photons.

To design a mask for the CA imaging, several factors should be taken into account and balanced, such as spatial resolution, FOV (Filed of view) and image noise level. Similarly, we design our collimator with an arc-shaped distribution of apertures based on the following considerations:
  1. Data completeness: the combination of narrow fan beams passing through the collimator should be able to cover the whole object;
  2. High signal-to-noise ratio (SNR): the apertures should be made and distributed such that there are as many X-rays as possible which penetrate the collimator, since there is a positive correlation between the noise level and the dose;
  3. High spatial resolution: the aperture collimator should be placed in a suitable position to modulate the fan beams such that each detector unit receives photons from only a tiny region of the bar-shaped focal spot.

These considerations are in competition with one another. For example, a collimator with one or a few tiny apertures will increase the resolution of the scanning data, while the filed of view is limited because the fan beam from the tiny hole is too narrow to cover the whole object. Meanwhile, the SNR will also decrease.

3. The reconstruction algorithm

In this section, we propose an algorithm to reconstruct the image fj from the measured data In under the assumption that Î0,u,s are known.

Let
I^n,s=I^0,u,sexp(j=1Jrn,s,jfj),
pn,s=lnI^n,sI^0,u,s
=j=1Jrn,s,jfj.
Let F be the image vector; R, the projection matrix; and P, the projection data vector, where
F=(f1fjfJ);P=(P1PsPS),Ps=(p1,spn,spN,s);R=(R1RsRS),Rs=(r1,s,1r1,s,jr1,s,Jrn,s,1rn,s,jrn,s,JrN,s,1rN,s,JrN,s,J).
Rs indicates the projection matrix of s-th virtual focal spot, which is the sub matrix of R. Then we obtain the linear system
P=RF.
(4)

If pn,s is known for any n, s, then we can solve (4) using simultaneous algebraic reconstruction technique (SART) as follows,
fj(k+1)=fj(k)+1s=1Sn=1Nrn,s,js=1Sn=1Nrn,s,jεn,s(k)l=1Jrn,s,l.
(5)
where εn,s(k)=pn,srn,sF(k) is the residual error for the k-th iteration and rn,s is the n-th row of matrix Rs. However, pn,s cannot be measured directly, which means that we cannot calculate εn,sk directly. Therefore, Eq. (5) cannot be applied directly to recover F. In the following, we will propose a method to estimate εn,s(k).

Let I^0,u=s=1SI^0,u,s. According to the design of the collimator, there are only several indices u for a given s, such that Î0,u,s is nonzero. Denoting In(k) as the summation of the photons integrated by forward-projection in the k-th iteration, then we can calculate the sum of the residual errors of the n-th data εn(k) from Î0,u and In(k)
εn(k)=ln(In/I^0,u)(ln(In(k)/I^0,u))=ln(In(k)/In).
(6)
On the other hand,
εn(k)=s=1Sεn,s(k).
ωu,s=I^0,u,s/I^0,u.
Since ωu,s is independent of the object, we have
ωn,s=ωmod(n,U),s=ωu,s.
(7)
It is easy to see that
s=1Sωu,s=1.
(8)
Now substituting ωu,sεn(k) for εn,s(k) into (5), we obtain a new iterative formula for reconstruction,
fj(k+1)=fj(k)+1s=1Sn=1Nrn,s,js=1Sn=1Nrn,s,jωu,sεn(k)l=1Jrn,s,l.
(9)
We will name our method Aperture Collimator-SART (AC-SART).

Furthermore, we should discuss the reasonableness of the substitution for εn,s(k) by ωu,sεn(k). According to Eq. (2), the relationship between the source intensity, the projection data and the object can be written as:
pn=log(s=1S[I^0,u,ss=1SI^0,u,sexp(l=1Jrn,j,sfj)]).
(10)
In real CT imaging, l=1Jrn,j,sfj is usually smaller than 1 as well as s=1SI^0,u,ss=1SI^0,u,sl=1Jrn,j,sfj. Hence, pn can be linearized by Taylor expansions as follows,
pn=log(s=1S[I^0,u,ss=1SI^0,u,sexp(l=1Jrn,j,sfj)])log(s=1SI^0,u,ss=1SI^0,u,s(1l=1Jrn,j,sfj))=log(1s=1SI^0,u,ss=1SI^0,u,sl=1Jrn,j,sfj)s=1SI^0,u,ss=1SI^0,u,sl=1Jrn,j,sfj.
(11)
We can find that the linear coefficient I^0,u,ss=1SI^0,u,s is exactly the same as ωu,s. This means that the proposed algorithm is a reasonable linear method to solve the high resolution CT imaging problem based on the collimator and virtual spots.

4. Numerical experiments

A series of numerical experiments are carried out to verify the effectiveness of the proposed approach. To make the experiments more realistic, we simulate a CT system which has exactly the same parameters as a real CT in our laboratory, e.g. the geometrical parameters are: SDD is 1000 mm, SOD is 750 mm, number of detector cells is 1800, width of each cell is 0.127 mm, number of sampling angles is 720 (uniformly distributed), and the size of the focal spot is 6 mm following the European standard (EN 12543). In our simulations, we set I0,n = 106, and we set the energy of X-ray to 300 keV. The focal spot is divided into 21 virtual focal spots and the flux intensity emitted from each virtual focal spot obeys a Gaussian distribution,
I(s)=12πexp(s2),
where s denotes the distance from the focal spots to the center of the whole focus.

Tungsten of density 19.35 g/cm3 was selected as the material for the aperture collimator. According to data from National Institute of Standards and Technology (NIST) [25

25. J. H. Hubbell and S. M. Seltzer, “Tables of x-ray mass attenuation coefficients and mass energy-absorption coefficients from 1 kev to 20 mev for elements z = 1 to 92 and 48 additional substances of dosimetric interest,” http://www.nist.gov/pml/data/xraycoef/.

], the mass attenuation coefficient of tungsten is 0.3238 cm2/g for 300 keV. The parameters of the aperture collimator are listed in Table 3, as mentioned in Section 2. All of the raw simulation data are polluted with Poisson noise.

Table 3. Parameters of the collimator

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4.1. Spatial-resolution test experiments

The spatial resolution is theoretically described by a MTF (modulation transfer function). However, in practice, it is often specified as line pairs per millimeter (lp/mm) and directly measured by some typical models such as bar or hole patterns of various spatial frequencies. Hence, we use the phantom with the pattern shown in Fig. 3 to test the spatial resolution. The phantom consists of: 4 bar patterns with respective resolutions 1.9179 lp/mm (left), 1.4049 lp/mm (two in the middle) and 1.1507 lp/mm(right); 5 concentric ring patterns with thickness 0.2607 mm and distances of 0.2607 mm between each other; and some tiny squares with width 0.3476 mm and distance 0.3476 mm between each other. Aluminum was selected as the material for the phantom.

Fig. 3 Phantom for the spatial resolution test.

First, we simulate the scanning process of the phantom with and without the aperture collimator, respectively. The image of the scanned data without the collimator looks blurred [Fig. 4(a)], while the image of the scanned data with the collimator looks clear but with some dark and light bands [Fig. 4(b)]. As mentioned above, the count of the data scanned with collimator is lower than the collimator-free one due to the shielding of the aperture collimator. The total dose is reduced to about 35% after applying the collimator. Hence, the current of X-ray tube should be increased and/or the scanning time should be extended in practical experiments.

Fig. 4 Image of the scanned data: (a) is scanned without the collimator; (b) is scanned with the collimator.

Figure 5 shows the image reconstructed from the data scanned without the collimator by SART algorithm. Because of the limit of space, the image reconstructed by Filtered Back-Projection (FBP) algorithm [26

26. A. C. Kak and M. Slaney, Principles of Tomographic Imaging (IEEE Engineering in Medicine and Biology Society, 2001). [CrossRef]

] without the collimator is omitted, which is almost the same as Fig. 5. We magnified the image and found that the patterns with high frequencies, such as the bar or ring patterns, were blurred and undistinguishable. Figure 6 shows the image reconstructed from the data scanned with the collimator by using the proposed AC-SART algorithm. It is easy to see that higher spatial resolution is obtained, and all the details can be discerned clearly in the magnified image.

Fig. 5 CT image reconstructed by SART from the data scanned without the collimator.
Fig. 6 CT image reconstructed by AC-SART from the data scanned with the collimator.

4.2. Density-contrast test experiments

In this experiment, we test the density resolution of our method. Therefore, we used a phantom as shown in Fig. 7, which was extracted and then simplified from the middle slice of the FORBILD HEAD [27

27. G. Lauritsch and H. Bruder, “Head phantom,” http://www.imp.uni-erlangen.de/phantoms/head (2012).

]. The phantom has two circles that have very low contrast with the background (about 0.5%). Figure 8(a) shows the image reconstructed by SART algorithm from the the data scanned without the collimator, while figure 8(b) shows the image reconstructed by our AC-SART from the data scanned with the collimator. Comparing the two CT images with the display window [1.0, 1.1], one could see that their density resolutions are almost the same.

Fig. 7 Pattern for testing the density resolution.
Fig. 8 Results of the density resolution test: (a) is reconstructed by SART from the data scanned without the collimator; (b) is reconstructed by AC-SART from the data scanned with the collimator.

4.3. Experiments with multi-energy scanned data

Fig. 9 Energy spectrum of the X-ray source.
Fig. 10 Mass attenuation coefficients of Tungsten and Aluminum.

In this experiment, we test our method for its ability of keeping the spatial resolution under multi-energy X-rays. To do so, we simulated the scanning process that was performed in the experiment described in Section 4.1 under the multi-energy X-rays specified above. Figure 11 shows the CT image reconstructed by our approach from the multi-energetic scanned data. We see that the high resolution is preserved.

Fig. 11 Reconstruction result for multi-energy X-rays.

4.4. Experiments on the perturbation of the geometrical settings

As described in Section 2, the parameters Î0,u,s, which may be estimated by means of masks or phantoms with special constructions, are very important in the AC-SART. We carry out some experiments to test the effect of the estimation error on Î0,u,s.

In our method, the Î0,u,s is calculated by gn,s, which indicates the intersection-length between the rays from s to u and the collimator. We add random noise to the intersection-length as follows
{gn,s*=gn,s(1+t),gn,s*[0,gmax]t~Gaussian(0,σ2),
(12)(13)
where gn,s* indicates the intersection-length with errors, gmax denotes the maximum intersection-length, and t obeys a Gaussian distribution. Figure 12 shows the simulation results for the 11-th virtual focus, to compare the intersection-length under ideal and noisy conditions (σ = 0.1 and σ = 0.2).

Fig. 12 Disturbance errors with different σ values: (a) and (b) are the intersect-lengths between rays from the 11-th virtual focus to detector bins and collimator, and σ is 0.1 in (a) and 0.2 in (b); (c) and (d) are the intersect-lengths between rays from the virtual focus to the 900-th bin and the collimator, and σ is 0.1 in (c) and 0.2 in (d).

We use the ideal intersection-length to estimate I0,u,s and then generate the raw data by adding noise as described above. Figure 13 shows the reconstructed images. When σ = 0.1, t will be distributed within [−0.3, 0.3]. The reconstructed image [Fig. 13(a)] still has high quality and high spatial resolution. When σ = 0.2, the range of t is from −0.6 to 0.6, and the estimation error grows rapidly which leads to large residual errors in the back-projection process. Therefore, the reconstructed CT image suffers from ring artifacts, as shown in Fig. 13(b).

Fig. 13 CT images with different perturbations: (a) is reconstructed with σ = 0.1, and (b) is reconstructed with σ = 0.2.

5. Conclusion

In this paper, we have studied an approach to increase the resolution of industrial CT images by utilizing an aperture collimator. The aperture collimator is designed such that the CT problem can be treated as an image reconstruction problem under a series of narrow fan beams emitted from a set of segmented (tiny) focal spots. The CT reconstruction problem in this situation is then split into two subproblems: (1) estimate the intensity distribution Î0,u,s of the flux after penetrating the collimator from each segmented focal spot; (2) reconstruct the CT images under the assumption that Î0,u,s are known. While the first subproblem is solved approximately by forward-projection simulation, an algorithm named AC-SART has been proposed to solve the second subproblem robustly. The numerical experiments show that the algorithm is able to reconstruct CT images of higher resolution than conventional algorithms. Meanwhile, it could maintain a favorable low contrast and is robust for multi-energy data as well as geometrical perturbations of CT systems.

The parameters for the scanning configuration and the parameters of aperture collimator are chosen based on trial and error and our experience with CT systems. In addition, the computational cost of the AC-SART is much higher than conventional SART for data scanned without a collimator. However, new computational techniques, such as GPUs, make it possible to utilize our proposed method.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under the grants 61127003, 60971131, and 61371195, and by the Beijing Education Committee under the grants PHR20110509 and KZ201110028034.

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S. D. Metzler, J. E. Bowsher, M. F. Smith, and R. J. Jaszczak, “Analytic determination of pinhole collimator sensitivity with penetration,” IEEE Trans. Medical Imaging 20, 730–741 (2001). [CrossRef]

20.

E. E. Fenimore and T. M. Cannon, “Uniformly redundant arrays: digital reconstruction methods.” Appl. Opt. 20, 1858–1864 (1981). [CrossRef] [PubMed]

21.

A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91, 074106 (2007). [CrossRef]

22.

D. J. Brady, N. P. Pitsianis, and X. Sun, “Reference structure tomography,” JOSA. A. 21, 1140–1147 (2004). [CrossRef] [PubMed]

23.

K. Choi and D. J. Brady, “Coded aperture computed tomography,” Proc. of SPIE 74680B, 1–4 (2009).

24.

J. Fleming and B. Goddard, “An evaluation of techniques for stationary coded aperture three-dimensional imaging in nuclear medicine,” Nucl. Instr. Meth. Phys. Res. 221, 242–246 (1984). [CrossRef]

25.

J. H. Hubbell and S. M. Seltzer, “Tables of x-ray mass attenuation coefficients and mass energy-absorption coefficients from 1 kev to 20 mev for elements z = 1 to 92 and 48 additional substances of dosimetric interest,” http://www.nist.gov/pml/data/xraycoef/.

26.

A. C. Kak and M. Slaney, Principles of Tomographic Imaging (IEEE Engineering in Medicine and Biology Society, 2001). [CrossRef]

27.

G. Lauritsch and H. Bruder, “Head phantom,” http://www.imp.uni-erlangen.de/phantoms/head (2012).

OCIS Codes
(170.7440) Medical optics and biotechnology : X-ray imaging
(340.7430) X-ray optics : X-ray coded apertures

ToC Category:
Medical Optics and Biotechnology

History
Original Manuscript: April 16, 2013
Revised Manuscript: September 20, 2013
Manuscript Accepted: October 23, 2013
Published: November 7, 2013

Virtual Issues
Vol. 9, Iss. 1 Virtual Journal for Biomedical Optics

Citation
Yining Zhu, Defeng Chen, Yunsong Zhao, Hongwei Li, and Peng Zhang, "An approach to increasing the resolution of industrial CT images based on an aperture collimator," Opt. Express 21, 27946-27963 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-27946


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References

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  16. R. J.  Jaszczak, J.  Li, H.  Wang, M. R.  Zalutsky, R. E.  Coleman, “Pinhole collimation for ultra-high-resolution, small-field-of-view SPECT.” Phys. Med. Biol. 39, 425–437 (1994). [CrossRef] [PubMed]
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  19. S. D.  Metzler, J. E.  Bowsher, M. F.  Smith, R. J.  Jaszczak, “Analytic determination of pinhole collimator sensitivity with penetration,” IEEE Trans. Medical Imaging 20, 730–741 (2001). [CrossRef]
  20. E. E.  Fenimore, T. M.  Cannon, “Uniformly redundant arrays: digital reconstruction methods.” Appl. Opt. 20, 1858–1864 (1981). [CrossRef] [PubMed]
  21. A.  Olivo, R.  Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91, 074106 (2007). [CrossRef]
  22. D. J.  Brady, N. P.  Pitsianis, X.  Sun, “Reference structure tomography,” JOSA. A. 21, 1140–1147 (2004). [CrossRef] [PubMed]
  23. K.  Choi, D. J.  Brady, “Coded aperture computed tomography,” Proc. of SPIE 74680B, 1–4 (2009).
  24. J.  Fleming, B.  Goddard, “An evaluation of techniques for stationary coded aperture three-dimensional imaging in nuclear medicine,” Nucl. Instr. Meth. Phys. Res. 221, 242–246 (1984). [CrossRef]
  25. J. H.  Hubbell, S. M.  Seltzer, “Tables of x-ray mass attenuation coefficients and mass energy-absorption coefficients from 1 kev to 20 mev for elements z = 1 to 92 and 48 additional substances of dosimetric interest,” http://www.nist.gov/pml/data/xraycoef/ .
  26. A. C.  Kak, M.  Slaney, Principles of Tomographic Imaging (IEEE Engineering in Medicine and Biology Society, 2001). [CrossRef]
  27. G.  Lauritsch, H.  Bruder, “Head phantom,” http://www.imp.uni-erlangen.de/phantoms/head (2012).

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