Grating-based X-ray Tomosynthesis on Biological Sample 1,31,31,31,32Ran Zhang, Zhiqiang Chen, Li Zhang, Xiaolei Jiang, Hongxia Yin, 2Zhenchang Wang
5 (1. Department of Engineering Physics, Tsinghua Universiy, 100084;
2. Medical Imaging Center, Beijing TongRen Hospital, 100730;
3. Key Laboratory of Particle & Radiation Imaging (Tsinghua University), Ministry of Education,
Abstract: X-ray grating-based imaging can provide additional contrast mechanisms other than
10 absorption, namely (differential) phase shift and small-angle scattering. For many biological samples which have poor absorption contrast, the (differential) phase shift and small-angle scattering contrast may provide useful information. Tomosynthesis is a three-dimensional reconstruction technique that aims to separate the overlapping structure of the object with a few projections from a limited angle. In this work, grating-based tomosynthesis with synchrotron radiation source has been performed.
15 Tomosynthetic results using both analytical and iterative reconstruction methods have been compared. It is seen from the results that images reconstructed using iterative method with total variation constraint have better visual effect.
Key words: grating-based imaging; reconstruction; tomosynthesis
20 Since its discovery over a century ago, X-ray has found its important application in many fields such as biological research, clinical diagnosis and material science. Conventional X-ray imaging is based on the absorption property of the object, which is related to the imaginary part of the object’s complex refractive index. However, the phase shift of the X-ray beam, which is related to the real part of the object’s complex refractive index, can’t be measured using
25 conventional X-ray imaging method. This problem was successfully resolved by the so called grating interferometry [1-4]. Moreover, small-angle scattering information caused by sub-micron scale inhomogeneities in the object can also be retrieved in grating-based X-ray imaging [5-8]. The retrieved image is referred to as dark field image. Therefore, three images, namely absorption image, phase contrast image and dark-field image can be retrieved simultaneously in a single
30 procedure using grating-based X-ray imaging. For many biological samples consist of low-Z elements, the absorption image provides little information due to the weak absorption. Grating-based method, with three complementary modalities of contrast, offers more information and therefore can be used in various fields such as material science and medical imaging.
Tomosynthesis is a section reconstruction technique that aims to separate the overlapping
35 structures in radiography at lower dose than computed tomography (CT) . The projections are typically only a few tens acquired within a limited angle. The quality of the reconstructed image is determined by the projection data as well as the reconstruction algorithm used. In this work, we performed grating-based X-ray tomosynthesis using analytical and iterative method, the results are compared and discussed.
Foundations: National Natural Science Foundation of China (No. 10905031) , Specialized Research Fund for the Doctoral Program of Higher Education (No. 20090002120016)
Brief author introduction:Ran Zhang(1986-),Male,PhD student,X-ray grating-based imaging
Correspondance author: Zhiqiang Chen(1971-),Male,Professor,Research interests: X-ray imaging and its application. E-mail: email@example.com
- 1 -
40 1 Methods
1.1 Grating-based X-ray Imaging Using Synchrotron Source
Fig. 1 shows a grating-based X-ray imaging system using synchrotron source.
45 Fig. 1 Schematic diagram of grating-based X-ray imaging system using synchrotron source
The first grating G1, also known as the phase grating, introduces a periodic phase shift in the
incident X-ray beam and generates an interference pattern at a distance d behind G1, where the
second grating G2 is located. This is the so-called Talbot effect and the distance d corresponds to 50 the Talbot distance [10, 11]. G2 is an absorption grating with the same period as the interference
pattern, acting as a transmission mask. The detector is immediately placed behind the second
grating. A phase-stepping method is used in data acquisition , where one of the gratings is
scanned along the transverse direction x within one of the grating period. For every step of the
scan, the image on the detector is recorded. After phase stepping, the oscillation curve for every 55 detector pixel is acquired, which is the intensity (counts) variation as a function of the grating
position. Absorption, (differential) phase contrast and dark-field information can be retrieved from
the oscillation curve.
1.2 Information Retrieval
The simulated oscillation curve for one detector pixel is shown in Fig. 2.
Fig. 2 Simulated intensity oscillation curves. Red: with the sample. Blue: without the sample
- 2 -
Here, the red curve and blue curve are acquired with or without the sample, respectively. As
can be noticed from Fig. 2, the mean value ( I , I), the phase ( φ, φ) and the amplitude s b s b
( ?I , ?I) of the curve are different with or without the sample. The intensity oscillation curve is s b
65 approximately sinusoidal, which can be expressed by
(1) I (k ) ? I + ?Icos(2π k χ / p+ s s s 2
φ) s (2)
I(k ) ? I+ ?Icos(2π k χ / p+ b b b 2
where k is the index of the steps, χ is the step length and pis the period of G2. Absorption, 2
(differential) phase contrast and dark-field image are related to the oscillation curves by
? ln I =µ t dt70 ( ) () (3) / I s b ? l
φ ? φ / λ d = ?Φ / ?x(4) p () () 2 s b 2 2 2 ( ? p ln ?I ) (5) s (t )dt/ ?I / 2π d = ( ) ? 2 s b l
where µ is the linear absorption coefficient, λ is the X-ray wavelength, Φ is the phase shift of
the wave-front, s is the scattering parameter . The phase shift Φ is further related to the real
75 part of the sample’s complex refractive index by δ
Φ = 2π / λ δ t dt( ) () (6) ? l
1.3 Multiple Information Tomosynthesis
The geometry of tomosynthesis is shown in Fig.1, where the sample is rotated along the y
axis within a limited angular range and the tomosynthetic plane is parallel to the x-y plane. When 80 synchrotron source is used, the X-ray beam is approximately parallel. In this case, the
tomosynthesis reconstruction is equivalent to parallel beam CT problem with limited angle and
For the reconstruction of µ in (3) and s in (4), conventional methods, such as filtered
back-projection (FBP) and algebraic reconstruction technique (ART) can be used because of the 85 simple line integral relation between the projection and the reconstructed parameter.
For the reconstruction of δ , however, FBP and ART can’t be used directly as the line integral
of δ is related to the phase shift Φ , instead of ?Φ / ?x . Intuitively, we can do the integral along
x direction to get Φ first, but this operation will introduce severe stripe artifacts. In order to
reconstruct δ directly from the projection, i.e. ?Φ / ?x , two approaches can be applied.
90 The first way is an analytical method similar to FBP. In fact, the only modification is the
ramp filter ω is replaced with a Hilbert filter
(7) 1/ i 2π sgn ω () ()
given the fact that a ramp filtering process can be viewed as the combination of a Hilbert filtering and a differentiation operation. In the paragraph below, this method is referred to as Hilbert
filtered back-projection (HFBP). 95
Another way is an iterative method similar to ART. The system matrix in ART is replaced with a linear partial derivative matrix , where the differentiation process is discretized and
interpolation is adopted. In the paragraph below, this method is referred to as differential algebraic reconstruction technique (DART). This method, like all other iterative algorithms, is able to work
with various constraints to improve the reconstruction results, especially when the projection data 100
- 3 -
- 4 -
constraint, where the total variation (TV) of the reconstructed image is minimized during the iteration . 2 Experiments
105 The experiments were carried out at TOMCAT beamline  at the Swiss Light Source, Paul
Scherrer Institut in Switzerland. The beam energy is 25 keV and the detector is a scintillator-CCD
detector with a pixel size of 1.75µ m × 1.75µ m . The period of the two gratings are 4µ m and 2µ m , D respectively. The sample is rotated by the sample manipulator over a range of . The ?30 D 3with 8 phase steps in each angle. The sample used projections were taken with an interval of
was a genuine pig cochlea, which is a very complex organ and is often used in hearing research. 110 D Fig. 3 shows the projections at 0. As can be noticed, structures at different ‘depth’ are
overlapped which makes the interpretation of the results very difficult.
Fig. 3 Projections at 0 degree. (a), (b), (c): absorption, differential phase contrast and dark-field projection, 115 respectively. (d) is the phase contrast projection acquired by integrating (c) along the x direction. Fig. 3(d) is the phase contrast projection acquired by integrating the differential phase contrast projection along the x direction, where severe stripe artifacts can be found. Fig. 4 shows the tomosynthetic results using analytical methods, for absorption and dark-field projection, FBP is used to reconstruct µ and s , while for the differential phase shift projection,
120 HFBP is used to reconstruct δ . Reconstruction results for three different tomosynthetic planes
are compared. It can be seen from Fig. 4 that structures at different tomosynthetic planes are
D separated from the radiographs at 0. However, some stripe artifacts can be found in the images.
- 5 -
Fig. 4 Tomosynthetic planes reconstructed using analytical methods.From left to right: µ , δ and s . (a)-(c):
tomosynthetic plane?. (d)-(f): tomosynthetic plane?.(g)-(h): tomosynthetic plane ?. 125
Fig. 5 shows the tomosynthetic results using iterative methods with TV constraint. For s , while for the absorption and dark-field projection, ART is used to reconstruct µ and differential phase shift projection, DART is used to reconstruct δ . Reconstruction results for
130 three different tomosynthetic planes are compared. The results show that, compared with the
analytical approach, the iterative methods with TV constraint have better performance, as less
stripe artifacts are found in the image.
- 6 -
Fig. 5 Tomosynthetic planes reconstructed using iterative methods. From left to right: µ , δ and s . (a)-(c):
tomosynthetic plane?. (d)-(f): tomosynthetic plane?.(g)-(h): tomosynthetic plane ?. 135
To investigate the effect of limited-angle and few-view on the tomosynthesis result, more experiments were carried out. Besides the original condition, where 21 projections are evenly D 60, two more conditions are compared. In the first one, 21 projections are distributed over D 140 evenly distributed over 120, i.e. both the sampling interval and the angular range are increased. D In the second condition, 21 projections are evenly distributed over 20, i.e. both the sampling
interval and the angular range are decreased. The results are shown in Fig. 6. For comparison, the reconstruction results using the complete projection data (541 projections evenly distributed over
D 180) are also shown in Fig. 6. It can be seen from the results that the few-view condition (large
sampling interval) is responsible for the stripe artifacts while the limited-angle condition 145
introduces more overlapping structures.
- 7 -
Fig.6 The effect of limited angle and few views on the results. From left to right: µ , δ and s . (a)-(c):21
projections covering 20 degrees. (d)-(f): 21 projections covering 60 degrees.(g)-(i): 21 projections covering 120
degrees.(j)-(l): 541 projections covering180 degrees. 150 3 Conclusion In this paper, grating-based X-ray tomosynthesis on a genuine pig cochlea was performed using synchrotron source. Two types of reconstruction algorithms were used and the results were
compared. From the results it is seen that images reconstructed using iterative methods with TV 155
constraint have less stripe artifacts. These results demonstrate that grating-based X-ray
tomosynthesis can be used to reconstruct slices of the sample, thus remove the overlapping
- 8 -
structures in conventional radiography. The phase contrast information and dark-field information, together with the absorption information can provide comprehensive information about the sample
and may be used in medical research and diagnosis. 160 Acknowledgements The actual experiments were performed on the TOMCAT beamline at the Swiss Light Source,
Paul Scherrer Institut, Villigen, Switzerland. We are grateful to Samuel McDonald at Swiss Light Source whose outstanding efforts have made these experiments possible. 165 References  C. David, B. Nohammer, H. H. Solak, and E. Ziegler. Differential x-ray phase contrast imaging using a shearing interferometer[J].Appl. Phys. Lett.,2002, 81:3287-3289.  Momose, A. Phase-sensitive imaging and phase tomography using X-ray interferometers[J]. Optics 170 Express,2003,11(19):2303-2314.  Weitkamp, T., Diaz, A., David, C., Pfeiffer, F., Stampanoni, M., Cloetens, P. & Ziegler, E.X-ray phase imaging with a grating interferometer[J].Optics Express,2005, 13(16):6296-6304.  Pfeiffer, F., Weitkamp, T., Bunk, O. & David, C.Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources[J].Nature Physics,2006,2(4),:258-261.
175  Pfeiffer, F., Bech, M., Bunk, O., Kraft, P., Eikenberry, E.F., Bronnimann, C., Grünzweig, C. & David, C.
Hard-X-ray dark-field imaging using a grating interferometer[J]. Nature Materials,2008,7(2):134-137.  Wang, Z., Kang, K., Huang, Z. & Chen, Z. Quantitative grating-based x-ray dark-field computed tomography[J]. Applied Physics Letters,2009, 95(9):094105.  Yashiro, W., Terui, Y., Kawabata, K. & Momose, A. On the origin of visibility contrast in x-ray Talbot 180 interferometry[J].Optics Express,2010, 18(16):16890-16901.
 Modregger, P., Scattarella, F., Pinzer, B.R., David, C., Bellotti, R. & Stampanoni, M. Imaging the Ultrasmall-Angle X-Ray Scattering Distribution with Grating Interferometry[J]. Physical Review Letters,2012, 108:048101.  J. T. Dobbins. Tomosynthesis imaging: At a translational crossroads[J].Medical Physics, 2009,36(6):1956.
 M. V. Berry, S. Klein. Integer, fractional and fractal Talbot effects[J]. Journal of modern optics,1996, 43(10): 185
 P. Cloetens, J. P. Guigay, C. De Martino, J. Baruchel, and M. Schlenker. Fractional Talbot imaging of phase gratings with hard x rays[J].Optics Letters,1997, 22(14): 1059.  Z. Wang, L. Zhang, and Z. Huang. Linear partial derivative matrix for iterative algorithm to reconstruct 190 refractive index from refraction angle data[C].Nuclear Science Symposium Conference Record (NSS/MIC), pp.
3786-3788, 2009 IEEE. 13] E. Sidky, C. Kao and X. Pan. Accurate image reconstruction from few-views and limited-angle data in [ divergent-beam CT[J]. Journal of X-Ray Science and Technology, 2006,14:119-139.  M. Stampanoni et al.Trends in synchrotron-based tomographic imaging: the SLS experience[C]. Proceedings
of SPIE,2006,6318: 63180M. 195
基于同步辐射的生物样品 X 射线光栅层析
实验 1,31,31,31,322 张冉(陈志强(张丽(姜晓磊(尹红霞(王振常
！1. 清华大学工程物理系(北京 100084，200 2. 北京同仁医院医学影像中心(北京 100730， 3. 教育部粒子信息获取与处理实验室(北京 100084， 摘要；X 射线光栅成像方法可以同时得到吸收、相衬和暗场三种信息。对于生物样品的成像( 传统的吸收信息往往不能提供足够的信息和图像对比度(而光栅成像提供的相衬和暗场信息
205 能够作为很好的补充(提高图像对比度。层析摄影是一种三维重建的技术(能够解决传统 X 射线照相中不同平面的物体相互重叠的问题(而需要的投影数又远远少于计算机断层成像。
本文实现了基于同步辐射的生物样品光栅层析实验(分别采用解析算法和迭代算法进行重 建(并对结果进行了分析和比较。 关键词；光栅成像，图像重建，层析摄影
- 9 -