Bio-Medical Materials and Engineering 24 (2014) 45–51 DOI 10.3233/BME-130782 IOS Press
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Single-slice reconstruction method for helical cone-beam differential phase-contrast CT Jian Fu ∗ and Liyuan Chen Research Center of Digital Radiation Imaging and Biomedical Imaging, Beijing University of Aeronautics and Astronautics, 100191 Beiijng, People’s Republic of China E-mail: [email protected], [email protected]
Abstract. X-ray phase-contrast computed tomography (PC-CT) can provide the internal structure information of biomedical specimens with high-quality cross-section images and has become an invaluable analysis tool. Here a simple and fast reconstruction algorithm is reported for helical cone-beam differential PC-CT (DPC-CT), which is called the DPC-CB-SSRB algorithm. It combines the existing CB-SSRB method of helical cone-beam absorption-contrast CT with the differential nature of DPC imaging. The reconstruction can be performed using 2D fan-beam filtered back projection algorithm with the Hilbert imaginary filter. The quality of the results for large helical pitches is surprisingly good. In particular, with this algorithm comparable quality is obtained using helical cone-beam DPC-CT data with a normalized pitch of 10 to that obtained using the traditional inter-row interpolation reconstruction with a normalized pitch of 2. This method will push the future medical helical cone-beam DPC-CT imaging applications. Keywords: X-ray differential phase-contrast imaging, computed tomography, helical scanning, reconstruction algorithm
1. Introduction X-ray phase-contrast computed tomography (PC-CT) uses the phase shift that the X-rays undergo when passing through matter, rather than their attenuation, as the imaging signal and may provide better image quality in soft-tissue and low atomic number samples. Over the last years, several PC-CT methods have been developed [1–19]. One of the recent developments is differential PC-CT (DPC-CT), which is based on a grating interferometer [8, 10, 11]. DPC-CT has first been implemented at X-ray synchrotron radiation sources [8, 12] and recently transferred to lab-based X-ray tube sources [13, 15]. Most existing DPC-CT approaches include essentially two steps: (i) retrieval of the DPC projections and (ii) phase reconstruction. The first step can be accomplished by using a phase-stepping procedure [7, 8, 11, 12], a reverse projection method [20], or a single-shot Fourier-based phase-extraction method [21]. The second step has so far been solved by using the filtered back-projection (FBP) algorithm with an imaginary Hilbert filter [13,22]. Several experimental case studies reported in the literatures demonstrate that DPCCT offers improved soft-tissue contrast and much more internal structure details than absorption-contrast CT. *
Corresponding author. E-mail: [email protected].
0959-2989/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved
46
J. Fu and L. Chen / Single-slice reconstruction method for helical cone-beam differential phase-contrast CT
In the absorption-contrast CT, extended long objects are impossible to be covered in one single axial scan due to the limited axial extension of the detector [23–25], which is the famous long object problem. For an imaging system with sufficient detector coverage in the axial direction and negligible cone beam angle, a single axial scan may be sufficient for artifact-free imaging. However, for extended long objects, a single axial scan would result in artifacts as cone angle becomes large. Since the 1990s, many attempts have been made to address this issue. The helical or spiral CT undoubtedly is the most successful one [23–26], which has been historically one of the major enabling factors for absorption-contrast CT to be widely used in both clinical and industrial applications [27]. In the helical CT, the scanned sample is continuously translated through the gantry while data are acquired and a large axial coverage and volumetric scan with sufficient reconstruction accuracy could be achieved. Recently, the concept of helical cone-beam CT was extended to DPC-CT and two algorithms were proposed and investigated, which demonstrated the feasibility of the helical acquisition and reconstruction for DPC-CT. The first one [27] combines the well-known inter-row interpolation algorithms of helical absorption-contrast CT with DPC imaging. It approximates the cone-beam geometry by multiple parallel fan-beams such that the 2D sinogram of a given slice can be synthesized from the neighboring fan-beams by interpolation methods. Then the classical FBP algorithm with an imaginary Hilbert filter [13] is applied to the sinograms to reconstruct the target slices. In the case with a small cone-beam angle and a small pitch, this algorithm can yield excellent results. The second one [28] is based on the back-projection filtered (BPF) algorithm of fan-beam DPC-CT and the authors claim that it can provide better results than the first one. However, this approach adopts a helical cone-beam geometry consisting of a curved detector and three cylindrical gratings. The new aspect of the work presented here is to develop an alternative reconstruction algorithm for the helical cone-beam DPC-CT. Different from the above algorithms, it adopts an imaging geometry consisting of much more accessible flat gratings and detectors and can provide high quality reconstruction even when the pitch of the helix is large. Combining the existing CB-SSRB method of helical cone-beam absorption-contrast CT [25] with the differential nature of DPC imaging, this algorithm, which is called the DPC-CB-SSRB algorithm, is devised in section 2. Results from simulated data are given in section 3 for helical pitches of 25mm and 100mm and a detector row thickness of 5mm. Simulations for an inter-row interpolation helical DPC-CT algorithm are presented for comparison. Some technical aspects and limitations of the DPC-CB-SSRB method are discussed with conclusions in section 4.
2. The DPC-CB-SSRB algorithm 2.1. Scanner geometry The adopted helical cone-beam DPC-CT acquisition scheme is depicted in Fig.1. The grating G0 is placed close to the X-ray tube anode and creates an array of individually coherent but mutually incoherent sources. The phase grating G1 and the absorption grating G2 are positioned to form the Talbot interferometer. The gantry holding the object can rotate and translate vertically. When the scanning is executed, the gantry rotates and translates vertically at the same time, which leads to a relative helical orbit of the source-detector assembly. Each position in the helical orbit represents a view angle, under which a phase-stepping procedure is executed and the DPC projection is obtained through Fourier analysis to the interference pattern intensity images detected by the flat panel detector [11]. Finally, the DPC-CT slices are reconstructed by the algorithm presented in the following.
J. Fu and L. Chen / Single-slice reconstruction method for helical cone-beam differential phase-contrast CT
47
Fig. 1. (a) Scanner geometry. The rotation and the vertical translation of the object lead to a helix of pitch h and radius xc for the source-detector assembly. The field-of-view is a cylinder of radius r, centred on the rotation axis of the scanner (denoted z-axis). (b) The DPC projection acquisition scheme with three flat gratings and a flat detector at each view angle. (x, y, z) and (x , y , z ) are two sets of coordinates describing the inspected object and the detector respectively. y and z are the indexes of detector channels. xc is the distance from the X-ray tube to the rotation center and xc is the distance from the X-ray tube to the detector.
2.2. Reconstruction algorithm A three dimensional object can be described by a complex refractive index distribution n(x, y, z) = 1 − δ(x, y, z) + iβ(x, y, z). δ is the real part of the complex refractive index distribution of an object and proportional to the phase shift cross-section. In the DPC imaging, the effect of variations of δ of the object is measured by evaluating the tiny refraction angle of X-rays α induced by the specimen. Correspondingly, the retrieved DPC projection in Fig.1 can be expressed by Eq.(1). In this equation, the line integral P is nothing but a Radon transform [29] of δ along the incident ray direction l. θ represents the view angle and its value is bigger than 2π. The helical cone-beam DPC-CT algorithm aims to reconstruct δ(x, y, z) from the measured α. α(y , z , θ) = ∂(
l
δ(x, y, z)dl)/∂y = ∂P/∂y
(1)
For the helical cone-beam DPC-CT geometry in Fig.1, a reconstruction algorithm is developed combining the one in the reference [25] with the differential nature of DPC imaging. This algorithm conceptually converts the projection α(y , z , θ) to a stack of fan-beam sinograms p(y , γ), each associated with one axial z-slice. Here, γ represents the fan-beam scanning view angle and its value belongs to [0, 2π]. Once the fan-beam sinograms are built, reconstruction is performed for each z-slice using the classical fan-beam filter back-projection algorithm with an imaginary filter [13].
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J. Fu and L. Chen / Single-slice reconstruction method for helical cone-beam differential phase-contrast CT
Mathematically the conversion equation between α(y , z , θ) and p(y , γ) is
p(y , γ) ≈
y 2 + x2 c
y 2 + z 2 + x2 c
α(y , z , θ),
(2)
where z ≈
y 2 + x2 c Δz. xc xc
(3)
Δz is the vertical distance between the converted fan-beam focus spot and the actual helical one. The scaling factor in Eq.(2) accounts for the difference of the length of the X-ray beam in fan-beam circular orbit and helical orbit. Eq.(3) describes a parabola on the flat panel detector for the helical projection at the view angle θ. The measurements on this parabola provide the estimates of all the DPC data in the fan-beam projection at view angle γ and a given axial position. So, in this DPC-CB-SSRB algorithm, each of the fan-beam data is fully estimated from one of the single helical data. To obtain a full fan-beam sinogram for a given z-slice, this conversion step requires only the data for the 2π segment of the helix that is centred on the z-slice. The maximum distance Δz between a helical focus spot and a converted fan-bean focus spot is then equal to 0.5h. In pseudo code, this algorithm consists of three steps: (i) Fix the axial position of the reconstructed slice z, and determine the angular range of θ under which the helical data α(y , z , θ) will be involved into the conversion to estimate the fan-beam sinogram pz (y , γ). (ii) Calculate one complete fan-beam sinogram using Eqs. (2) and (3). (iii) Apply the short-scan fan-beam FBP algorithm with an imaginary filter to the reconstruction of the slice image.
3. Simulations and results Numerical simulations are presented in this section to demonstrate our algorithm. All the simulations are conducted on our workstation with an Intel Pentium G630 CPU and 4GB RAM. The VC++6.0 and Matlab R2008 are chosen to implement the algorithms. Reconstructions are performed for two phantoms: the 3D Shepp phantom and the discs phantom [25]. The Shepp phantom is a low-contrast object, which fits to a cylinder of radius r = 90mm and height 184mm and consists of ellipsoids. The values of this phantom are linearly mapped into 0 − 2.0 × 10−6 to represent the distribution of δ. The discs phantom is a stack of six discs of value 1.0 × 10−6 , of radius 70mm and thickness 8mm. These discs are centred on the rotation axis and are separated by 10mm, for a total axial extent of 98mm. The DPC projections α are analytically calculated according to Eq.(1) and the partial derivatives in Eq.(1) are directly calculated by the three-point difference method. Three datasets are created for each phantom. All of them are generated by using the helical path of radius xc = 300mm, with 300 projections per 2π rotation. The detector is always a distance xc = 600mm from the source. The next simulation and analysis procedure is similar to the one in the reference [25]. The value of the pitch is modified from one set to the next, beginning with h = 5mm, and then moving to the larger values h = 25mm and h = 100mm. There are 216 pixels on each detector row, with a spacing Δy = 2mm. Each detector row has a thickness Δz = 5mm. So the normalized pitch (xc h)/(xc Δz ) [23, 26] is respectively equal to 2, 10 and 40. To account for the limited scanner
J. Fu and L. Chen / Single-slice reconstruction method for helical cone-beam differential phase-contrast CT
49
resolution, an approach similar to the one in the reference [25] is followed. The simulated X-ray source as a focal spot of radius 0.5mm is modelled using a 2 × 2 array of point sources. The detector elements are simulated by breaking up the detector pixels into 3 × 5 subpixels (in z and y ). The DPC projections from the detector subpixels are calculated analytically and then summed to generate the simulated DPC data. Physical effects such as beam hardening, scattering and finite photon counts are not included. Reconstructions are carried out on a grid of 2003 cubic voxels of side 1mm. The results of the Shepp phantom and the discs phantom reconstructed by the DPC-CB-SSRB algorithm and the inter-row interpolation algorithm [27] are displayed in Figs.2 and 3. The reconstructions of the first dataset with a pitch of 5mm are used to investigate the performance of the DPC-CB-SSRB algorithm at small pitch values, as shown in Figs.2(a), 2(b) and 3(a), 3(b). These images show that both of the algorithms can provide the high quality results in a small pitch case. The reconstructions of the second dataset with a pitch of 25mm are used to make a performance comparison between the DPC-CB-SSRB algorithm and the inter-row interpolation algorithm, as shown in Figs.2(c), 2(d) and 3(c), 3(d). Obviously, in this case the reconstruction quality of the inter-row interpolation algorithm is limited since the small cone-beam angle assumption becomes invalid. Figs.2(b), 2(c) and 3(b), 3(c) also show that even though the pitch of the helix is five times bigger, the DPC-CB-SSRB algorithm provides reconstructions of comparable quality to the inter-row interpolation algorithm with a pitch of 5mm. The reconstructions of the third dataset with a pitch of 100mm are used to study the behaviour of the DPC-CB-SSRB algorithm at large pitch values. The reconstructions for this dataset are shown in Figs.2(e) and 3(e). For comparison, Figs.2(f) and 3(f) illustrate the results obtained by the inter-row interpolation algorithm. Despite the large pitch value, the DPC-CB-SSRB algorithm provides surprisingly good results in comparison with the inter-row interpolation algorithm. The reconstruction artifacts are mainly limited to moderate axial distortions of objects. 4. Discussion and conclusions In this paper, an alternative reconstruction algorithm is represented for the helical cone-beam DPC-CT, which is validated by numerical simulation. Different from the existing algorithms, it adopts a cone-beam imaging geometry with flat gratings and detectors, which can provide better reconstruction when the helical pitch is large. Its fundamentals were first disclosed in the reference [25]. This algorithm is not exact but can be performed using 2D fan-beam filtered back projection algorithm with the Hilbert imaginary filter. The data conversion operation from cone-beam projections to fan-beam sinograms makes a decisive contribution to the excellent performance of this algorithm under large pitches. As can be expected for medical applications, particularly where DPC-CT has been proven to be a uniquely powerful method, this reconstruction method will provide a possible resolution for some imaging requirements and push the future medical and industrial helical cone-beam DPC-CT imaging applications. Acknowledgements The authors are grateful to the group of biophysics at the Department of Physics of Technische Universität München (TUM), Germany, for introducing us to the field of X-ray phase contrast imaging and for the many shared valuable discussions. This work is partly supported by the National Natural Science Foundation of China (grant no. 11179009 and 50875013), the China Beijing municipal natural science foundation (grant no. 4102036) and the Beijing NOVA program (grant no. 2009A09).
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J. Fu and L. Chen / Single-slice reconstruction method for helical cone-beam differential phase-contrast CT
Fig. 2. Central vertical slice through the Shepp phantom. Greyscale [0.2 × 10−6 1.8 × 10−6 ]. The graphs below and to the right of the pictures are the profiles along the horizontal and vertical lines. (a), (c) and (e) are the results of the DPC-CB-SSRB algorithm for normalized pitches equal to 2, 10 and 40 respectively. (b), (d) and (e) are the results of the inter-row interpolation algorithm for normalized pitches equal to 2, 10 and 40 respectively.
Fig. 3. Central vertical slice through the discs phantom. Greyscale [−0.1 × 10−6 0.5 × 10−6 ]. The graphs below and to the right of the pictures are the profiles along the horizontal and vertical lines. (a), (c) and (e) are the results of the DPC-CB-SSRB algorithm for normalized pitches equal to 2, 10 and 40 respectively. (b), (d) and (e) are the results of the inter-row interpolation algorithm for normalized pitches equal to 2, 10 and 40 respectively.
J. Fu and L. Chen / Single-slice reconstruction method for helical cone-beam differential phase-contrast CT
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References [1] C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov and V. Kohn, Phase-contrast microtomography with coherent high-energy synchrotron x rays, Appl. Phys. Lett. 69 (1996),1826–1828. [2] F. Beckmann, K. Heise, B. Kolsch, U. Bonse, M. Rajewsky and T. Biermann, Three-dimensional imaging of nerve tissue by x-ray phase-contrast microtomography, Biophys. J. 76 (1999),98–102. [3] P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay and M. Schlenker, Holotomography:quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays, Appl. Phys. Lett. 75 (1999),2912-2924. [4] F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion and W. C. Thomlinson, Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method, Phys. Med.Biol. 45 (2000), 933–946. [5] S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson and S. W. Wilkins, X-ray phase-contrast microscopy and microtomography, Opt. Express 11 (2003),2289–2302. [6] P. J. McMahon, A. G. Peele, D. Paterson, J. J. A. Lin, T. H. K. Irving, I. McNulty and K. A. Nugent, Quantitative x-ray phase tomography with sub-micron resolution, Opt. Commun. 217 (2003),53-57. [7] A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai and Y. Suzuki, Demonstration of x-ray Talbot interferometry, Jpn. J. Appl. Phys. 42 (2003),L866-L868. [8] T. Weitkamp and A. Diaz and C. David and F. Pfeiffer and M. Stampanoni and P. Cloetens and E. Ziegler, X-ray phase imaging with a grating interferometer, Opt. Express 13 (2005),6296-6304. [9] A. Groso, M. Stampanoni, R. Abela, P. Schneider, S. Linga and R. Müller, Phase contrast tomography: an alternative approach, Appl. Phys. Lett. 88 (2006), 214104. [10] A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki and T. Hattori, Phase tomography by x-ray Talbot interferometry for biological imaging, Jpn. J. Appl. Phys. 45 (2006), 5254-5262. [11] F. Pfeiffer, T. Weitkamp, O. Bunk and C. David, Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources, Nature Phys. 2 (2006), 258-261. [12] A. Momose, Recent advances in x-ray phase imaging, Jpn. J. Appl. Phys. 44 (2005),6355–6359. [13] F. Pfeiffer, C. Kottler, O. Bunk and C. David, Hard x-ray phase tomography with low-brilliance sources, Phys. Rev. Lett. 98 (2007), 108105. [14] A. Bravin, J. Keyrilainen, M. Fernandez, S. Fiedler, C. Nemoz, M. L. Karjalainen-Lindsberg, M. Tenhunen, P. Virkkunen, M. Leidenius, K. Von Smitten, P. Sipila and P. Suortti, High-resolution CT by diffraction-enhanced x-ray imaging: mapping of breast tissue samples and comparison with their histo-pathology, Phys. Med. Biol. 52 (2007), 2197–2211. [15] M. Bech, T. H. Jensen, R. Feidenhans’l, O. Bunk, C. David and F. Pfeiffer, Soft-tissue phase-contrast tomography with an x-ray tube source, Phys. Med. Biol. 54 (2009), 2747-2753. [16] J. Fu, P. Li, Q. L. Wang, S. Y. Wang, M. Bech, A. Tapfer, D. Hahn and F. Pfeiffer, A reconstruction method for equidistant fan beam differential phase contrast computed tomography, Phys. Med. Biol. 56, (2011), 4529-4538. [17] J. Fu, A. Velroyen, R. B. Tan, J. W. Zhang, L. Y. Chen, A. Tapfer, M. Bech and F. Pfeiffer, A reconstruction method for cone-beam differential x-ray phase-contrast computed tomography, Optics Express 20, (2012), 21512-21519. [18] J. Fu, S. Schleede, R. Tan, L. Chen, M. Bech, K. Achterhold, M. Gifford, R. Loewen, R. Ruth and F. Pfeiffer, An algebraic iterative reconstruction technique for differential x-ray phase-contrast computed tomography, Z. Med. Phys. in press, (2012), http://dx.doi.org/10.1016/j.zemedi.2012.11.002. [19] J. Fu, Phase Contrast Computed Tomography, Computed Tomography - Clinical Applications, Dr. Luca Saba (Ed.), ISBN: 978-953-307-378-1, InTech, DOI: 10.5772/22749. [20] P. P. Zhu, K. Zhang, Z. L. Wang, Y. J. Liu, X. S. Liu, Z. Y. Wu, S. A. McDonald, F. Marone and M. Stampanoni, Lowdose,simple, and fast grating-based X-ray phase´lCcontrast imaging, Proc. Natl Acad. Sci. USA 107 (2010),13576-13581. [21] M. Takeda, H. Ina and S. Kobayashi, Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry, J. Opt. Soc. Am. 72 (1982), 156-160. [22] I. Jerjen, V. Revol, C. Kottler, Th. Luethi, U. Sennhauser, R. Kaufmann and C. Urban, Phase contrast cone beam tomography with an x-ray grating interferometer, AIP Conf. Proc. CP1236 (2010), 227-231. [23] K. Taguchi and H. Aradeate, Algorithm for image reconstruction in multi-slice helical CT, Med. Phys. 25 (1998),550-561. [24] H. Hu, Multi-slice helical CT: Scan and reconstruction, Med. Phys. 26 (1999), 5–18. [25] F. Noo, M. Defrise and R. Clackdoyle, Single-slice rebinning method for helical cone-beam CT, Phys. Med. Biol. 44 (1999), 561-570. [26] W. A. Kalender, Technical foundations of spiral CT, Semin. Ultrasound CT MRI 15 (1994), 81-89. [27] Z. H. Qi, P. Theriault-Lauzier, N. Bevins, J. Zambelli, K. Li and G. H. Chen, Helical x-ray differential phase contrast computed tomography, Proc. SPIE 7691 (2011), 79611Q. [28] J. Li, P. P. Zhu and Y. Sun, An approximate reconstruction method for helical cone-beam differential phase-contrast computed tomography images, Phys. Med. Biol. 57 (2012), 2347-2356. [29] A. C. Kak and M. Slaney, Principles of Computerized Tomography, IEEE Press, 1987.
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Single-slice reconstruction method for helical cone-beam differential phase-contrast CT Jian Fu ∗ and Liyuan Chen Research Center of Digital Radiation Imaging and Biomedical Imaging, Beijing University of Aeronautics and Astronautics, 100191 Beiijng, People’s Republic of China E-mail: [email protected], [email protected]
Abstract. X-ray phase-contrast computed tomography (PC-CT) can provide the internal structure information of biomedical specimens with high-quality cross-section images and has become an invaluable analysis tool. Here a simple and fast reconstruction algorithm is reported for helical cone-beam differential PC-CT (DPC-CT), which is called the DPC-CB-SSRB algorithm. It combines the existing CB-SSRB method of helical cone-beam absorption-contrast CT with the differential nature of DPC imaging. The reconstruction can be performed using 2D fan-beam filtered back projection algorithm with the Hilbert imaginary filter. The quality of the results for large helical pitches is surprisingly good. In particular, with this algorithm comparable quality is obtained using helical cone-beam DPC-CT data with a normalized pitch of 10 to that obtained using the traditional inter-row interpolation reconstruction with a normalized pitch of 2. This method will push the future medical helical cone-beam DPC-CT imaging applications. Keywords: X-ray differential phase-contrast imaging, computed tomography, helical scanning, reconstruction algorithm
1. Introduction X-ray phase-contrast computed tomography (PC-CT) uses the phase shift that the X-rays undergo when passing through matter, rather than their attenuation, as the imaging signal and may provide better image quality in soft-tissue and low atomic number samples. Over the last years, several PC-CT methods have been developed [1–19]. One of the recent developments is differential PC-CT (DPC-CT), which is based on a grating interferometer [8, 10, 11]. DPC-CT has first been implemented at X-ray synchrotron radiation sources [8, 12] and recently transferred to lab-based X-ray tube sources [13, 15]. Most existing DPC-CT approaches include essentially two steps: (i) retrieval of the DPC projections and (ii) phase reconstruction. The first step can be accomplished by using a phase-stepping procedure [7, 8, 11, 12], a reverse projection method [20], or a single-shot Fourier-based phase-extraction method [21]. The second step has so far been solved by using the filtered back-projection (FBP) algorithm with an imaginary Hilbert filter [13,22]. Several experimental case studies reported in the literatures demonstrate that DPCCT offers improved soft-tissue contrast and much more internal structure details than absorption-contrast CT. *
Corresponding author. E-mail: [email protected].
0959-2989/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved
46
J. Fu and L. Chen / Single-slice reconstruction method for helical cone-beam differential phase-contrast CT
In the absorption-contrast CT, extended long objects are impossible to be covered in one single axial scan due to the limited axial extension of the detector [23–25], which is the famous long object problem. For an imaging system with sufficient detector coverage in the axial direction and negligible cone beam angle, a single axial scan may be sufficient for artifact-free imaging. However, for extended long objects, a single axial scan would result in artifacts as cone angle becomes large. Since the 1990s, many attempts have been made to address this issue. The helical or spiral CT undoubtedly is the most successful one [23–26], which has been historically one of the major enabling factors for absorption-contrast CT to be widely used in both clinical and industrial applications [27]. In the helical CT, the scanned sample is continuously translated through the gantry while data are acquired and a large axial coverage and volumetric scan with sufficient reconstruction accuracy could be achieved. Recently, the concept of helical cone-beam CT was extended to DPC-CT and two algorithms were proposed and investigated, which demonstrated the feasibility of the helical acquisition and reconstruction for DPC-CT. The first one [27] combines the well-known inter-row interpolation algorithms of helical absorption-contrast CT with DPC imaging. It approximates the cone-beam geometry by multiple parallel fan-beams such that the 2D sinogram of a given slice can be synthesized from the neighboring fan-beams by interpolation methods. Then the classical FBP algorithm with an imaginary Hilbert filter [13] is applied to the sinograms to reconstruct the target slices. In the case with a small cone-beam angle and a small pitch, this algorithm can yield excellent results. The second one [28] is based on the back-projection filtered (BPF) algorithm of fan-beam DPC-CT and the authors claim that it can provide better results than the first one. However, this approach adopts a helical cone-beam geometry consisting of a curved detector and three cylindrical gratings. The new aspect of the work presented here is to develop an alternative reconstruction algorithm for the helical cone-beam DPC-CT. Different from the above algorithms, it adopts an imaging geometry consisting of much more accessible flat gratings and detectors and can provide high quality reconstruction even when the pitch of the helix is large. Combining the existing CB-SSRB method of helical cone-beam absorption-contrast CT [25] with the differential nature of DPC imaging, this algorithm, which is called the DPC-CB-SSRB algorithm, is devised in section 2. Results from simulated data are given in section 3 for helical pitches of 25mm and 100mm and a detector row thickness of 5mm. Simulations for an inter-row interpolation helical DPC-CT algorithm are presented for comparison. Some technical aspects and limitations of the DPC-CB-SSRB method are discussed with conclusions in section 4.
2. The DPC-CB-SSRB algorithm 2.1. Scanner geometry The adopted helical cone-beam DPC-CT acquisition scheme is depicted in Fig.1. The grating G0 is placed close to the X-ray tube anode and creates an array of individually coherent but mutually incoherent sources. The phase grating G1 and the absorption grating G2 are positioned to form the Talbot interferometer. The gantry holding the object can rotate and translate vertically. When the scanning is executed, the gantry rotates and translates vertically at the same time, which leads to a relative helical orbit of the source-detector assembly. Each position in the helical orbit represents a view angle, under which a phase-stepping procedure is executed and the DPC projection is obtained through Fourier analysis to the interference pattern intensity images detected by the flat panel detector [11]. Finally, the DPC-CT slices are reconstructed by the algorithm presented in the following.
J. Fu and L. Chen / Single-slice reconstruction method for helical cone-beam differential phase-contrast CT
47
Fig. 1. (a) Scanner geometry. The rotation and the vertical translation of the object lead to a helix of pitch h and radius xc for the source-detector assembly. The field-of-view is a cylinder of radius r, centred on the rotation axis of the scanner (denoted z-axis). (b) The DPC projection acquisition scheme with three flat gratings and a flat detector at each view angle. (x, y, z) and (x , y , z ) are two sets of coordinates describing the inspected object and the detector respectively. y and z are the indexes of detector channels. xc is the distance from the X-ray tube to the rotation center and xc is the distance from the X-ray tube to the detector.
2.2. Reconstruction algorithm A three dimensional object can be described by a complex refractive index distribution n(x, y, z) = 1 − δ(x, y, z) + iβ(x, y, z). δ is the real part of the complex refractive index distribution of an object and proportional to the phase shift cross-section. In the DPC imaging, the effect of variations of δ of the object is measured by evaluating the tiny refraction angle of X-rays α induced by the specimen. Correspondingly, the retrieved DPC projection in Fig.1 can be expressed by Eq.(1). In this equation, the line integral P is nothing but a Radon transform [29] of δ along the incident ray direction l. θ represents the view angle and its value is bigger than 2π. The helical cone-beam DPC-CT algorithm aims to reconstruct δ(x, y, z) from the measured α. α(y , z , θ) = ∂(
l
δ(x, y, z)dl)/∂y = ∂P/∂y
(1)
For the helical cone-beam DPC-CT geometry in Fig.1, a reconstruction algorithm is developed combining the one in the reference [25] with the differential nature of DPC imaging. This algorithm conceptually converts the projection α(y , z , θ) to a stack of fan-beam sinograms p(y , γ), each associated with one axial z-slice. Here, γ represents the fan-beam scanning view angle and its value belongs to [0, 2π]. Once the fan-beam sinograms are built, reconstruction is performed for each z-slice using the classical fan-beam filter back-projection algorithm with an imaginary filter [13].
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J. Fu and L. Chen / Single-slice reconstruction method for helical cone-beam differential phase-contrast CT
Mathematically the conversion equation between α(y , z , θ) and p(y , γ) is
p(y , γ) ≈
y 2 + x2 c
y 2 + z 2 + x2 c
α(y , z , θ),
(2)
where z ≈
y 2 + x2 c Δz. xc xc
(3)
Δz is the vertical distance between the converted fan-beam focus spot and the actual helical one. The scaling factor in Eq.(2) accounts for the difference of the length of the X-ray beam in fan-beam circular orbit and helical orbit. Eq.(3) describes a parabola on the flat panel detector for the helical projection at the view angle θ. The measurements on this parabola provide the estimates of all the DPC data in the fan-beam projection at view angle γ and a given axial position. So, in this DPC-CB-SSRB algorithm, each of the fan-beam data is fully estimated from one of the single helical data. To obtain a full fan-beam sinogram for a given z-slice, this conversion step requires only the data for the 2π segment of the helix that is centred on the z-slice. The maximum distance Δz between a helical focus spot and a converted fan-bean focus spot is then equal to 0.5h. In pseudo code, this algorithm consists of three steps: (i) Fix the axial position of the reconstructed slice z, and determine the angular range of θ under which the helical data α(y , z , θ) will be involved into the conversion to estimate the fan-beam sinogram pz (y , γ). (ii) Calculate one complete fan-beam sinogram using Eqs. (2) and (3). (iii) Apply the short-scan fan-beam FBP algorithm with an imaginary filter to the reconstruction of the slice image.
3. Simulations and results Numerical simulations are presented in this section to demonstrate our algorithm. All the simulations are conducted on our workstation with an Intel Pentium G630 CPU and 4GB RAM. The VC++6.0 and Matlab R2008 are chosen to implement the algorithms. Reconstructions are performed for two phantoms: the 3D Shepp phantom and the discs phantom [25]. The Shepp phantom is a low-contrast object, which fits to a cylinder of radius r = 90mm and height 184mm and consists of ellipsoids. The values of this phantom are linearly mapped into 0 − 2.0 × 10−6 to represent the distribution of δ. The discs phantom is a stack of six discs of value 1.0 × 10−6 , of radius 70mm and thickness 8mm. These discs are centred on the rotation axis and are separated by 10mm, for a total axial extent of 98mm. The DPC projections α are analytically calculated according to Eq.(1) and the partial derivatives in Eq.(1) are directly calculated by the three-point difference method. Three datasets are created for each phantom. All of them are generated by using the helical path of radius xc = 300mm, with 300 projections per 2π rotation. The detector is always a distance xc = 600mm from the source. The next simulation and analysis procedure is similar to the one in the reference [25]. The value of the pitch is modified from one set to the next, beginning with h = 5mm, and then moving to the larger values h = 25mm and h = 100mm. There are 216 pixels on each detector row, with a spacing Δy = 2mm. Each detector row has a thickness Δz = 5mm. So the normalized pitch (xc h)/(xc Δz ) [23, 26] is respectively equal to 2, 10 and 40. To account for the limited scanner
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resolution, an approach similar to the one in the reference [25] is followed. The simulated X-ray source as a focal spot of radius 0.5mm is modelled using a 2 × 2 array of point sources. The detector elements are simulated by breaking up the detector pixels into 3 × 5 subpixels (in z and y ). The DPC projections from the detector subpixels are calculated analytically and then summed to generate the simulated DPC data. Physical effects such as beam hardening, scattering and finite photon counts are not included. Reconstructions are carried out on a grid of 2003 cubic voxels of side 1mm. The results of the Shepp phantom and the discs phantom reconstructed by the DPC-CB-SSRB algorithm and the inter-row interpolation algorithm [27] are displayed in Figs.2 and 3. The reconstructions of the first dataset with a pitch of 5mm are used to investigate the performance of the DPC-CB-SSRB algorithm at small pitch values, as shown in Figs.2(a), 2(b) and 3(a), 3(b). These images show that both of the algorithms can provide the high quality results in a small pitch case. The reconstructions of the second dataset with a pitch of 25mm are used to make a performance comparison between the DPC-CB-SSRB algorithm and the inter-row interpolation algorithm, as shown in Figs.2(c), 2(d) and 3(c), 3(d). Obviously, in this case the reconstruction quality of the inter-row interpolation algorithm is limited since the small cone-beam angle assumption becomes invalid. Figs.2(b), 2(c) and 3(b), 3(c) also show that even though the pitch of the helix is five times bigger, the DPC-CB-SSRB algorithm provides reconstructions of comparable quality to the inter-row interpolation algorithm with a pitch of 5mm. The reconstructions of the third dataset with a pitch of 100mm are used to study the behaviour of the DPC-CB-SSRB algorithm at large pitch values. The reconstructions for this dataset are shown in Figs.2(e) and 3(e). For comparison, Figs.2(f) and 3(f) illustrate the results obtained by the inter-row interpolation algorithm. Despite the large pitch value, the DPC-CB-SSRB algorithm provides surprisingly good results in comparison with the inter-row interpolation algorithm. The reconstruction artifacts are mainly limited to moderate axial distortions of objects. 4. Discussion and conclusions In this paper, an alternative reconstruction algorithm is represented for the helical cone-beam DPC-CT, which is validated by numerical simulation. Different from the existing algorithms, it adopts a cone-beam imaging geometry with flat gratings and detectors, which can provide better reconstruction when the helical pitch is large. Its fundamentals were first disclosed in the reference [25]. This algorithm is not exact but can be performed using 2D fan-beam filtered back projection algorithm with the Hilbert imaginary filter. The data conversion operation from cone-beam projections to fan-beam sinograms makes a decisive contribution to the excellent performance of this algorithm under large pitches. As can be expected for medical applications, particularly where DPC-CT has been proven to be a uniquely powerful method, this reconstruction method will provide a possible resolution for some imaging requirements and push the future medical and industrial helical cone-beam DPC-CT imaging applications. Acknowledgements The authors are grateful to the group of biophysics at the Department of Physics of Technische Universität München (TUM), Germany, for introducing us to the field of X-ray phase contrast imaging and for the many shared valuable discussions. This work is partly supported by the National Natural Science Foundation of China (grant no. 11179009 and 50875013), the China Beijing municipal natural science foundation (grant no. 4102036) and the Beijing NOVA program (grant no. 2009A09).
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J. Fu and L. Chen / Single-slice reconstruction method for helical cone-beam differential phase-contrast CT
Fig. 2. Central vertical slice through the Shepp phantom. Greyscale [0.2 × 10−6 1.8 × 10−6 ]. The graphs below and to the right of the pictures are the profiles along the horizontal and vertical lines. (a), (c) and (e) are the results of the DPC-CB-SSRB algorithm for normalized pitches equal to 2, 10 and 40 respectively. (b), (d) and (e) are the results of the inter-row interpolation algorithm for normalized pitches equal to 2, 10 and 40 respectively.
Fig. 3. Central vertical slice through the discs phantom. Greyscale [−0.1 × 10−6 0.5 × 10−6 ]. The graphs below and to the right of the pictures are the profiles along the horizontal and vertical lines. (a), (c) and (e) are the results of the DPC-CB-SSRB algorithm for normalized pitches equal to 2, 10 and 40 respectively. (b), (d) and (e) are the results of the inter-row interpolation algorithm for normalized pitches equal to 2, 10 and 40 respectively.
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