Research Article Direct Fan-Beam Reconstruction Algorithm via Filtered Backprojection for Differential Phase-Contrast Computed Tomography

Transcription

1 Hindawi Publishing Corporation X-Ra Optics and Instrumentation Volume, Article ID 57, pages doi:.55//57 Research Article Direct Fan-Beam Reconstruction Algorithm via Filtered Backprojection for Differential Phase-Contrast Computed Tomograph Zhihua Qi and Guang-Hong Chen Department of Medical Phsics, Universit of Wisconsin-Madison, Highland Avenue, Madison, WI , USA Correspondence should be addressed to Guang-Hong Chen, Received December 7; Revised April ; Accepted Ma Recommended b Christian David Recentl, a novel data acquisition method has been proposed and eperimentall implemented for differential phase-contrast computed tomograph (DPC-CT), in which a conventional X-ra tube and a Talbot-Lau-tpe interferometer were utilized in data acquisition. The divergent nature of the data acquisition sstem requires a divergent-beam image reconstruction algorithm for DPC-CT. This paper focuses on addressing this image reconstruction issue. We have developed a filtered backprojection algorithm to directl reconstruct the DPC-CTimages from acquiredfan-beam data. The developed algorithm allows one to directl reconstruct the decrement of the real part of the refractive inde from the measured data. In order to accuratel reconstruct an image, the data need to be acquired over an angular range of at least plus the fan angle. As opposed to the parallel beam data acquisition and reconstruction methods, a -rotation angle for the data acquisition sstem does not provide sufficient data for an accurate reconstruction of the entire field of view. Numerical simulations have been conducted to validate the image reconstruction algorithm. Copright Z. Qi and G.-H. Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in an medium, provided the original work is properl cited.. Introduction In recent ears, X-ra phase-contrast imaging methods have attracted significant interest. Based upon the phase retrieval method, phase-contrast imaging can be classified into three categories: interferometric methods [, ], analzer crstal methods [, ], and free-space propagation methods [5 7]. These methods generall require ecellent spatial coherence and/or ecellent spatial resolution of the detectors. Phasecontrast imaging is thus normall conducted using highbrilliance snchrotron beam lines or microfocus X-ra tubes. It is difficult to adapt these phase retrieval methods to a clinical setting for medical imaging. The shearing interferometr method has been introduced to measure the gradient of the local phase shift distribution using snchrotron beam lines [ ]. The ke theoretical foundation is the Talbot effect [ ] and Moiré interferometr [5]. Computed tomograph imaging methods have been developed [, ] based on the phasecontrast projection images. The shearing interferometr method onl requires a mild spatial coherence length, which opens up a new opportunit to implement phase-contrast imaging using a conventional X-ra tube. A method which takes advantage of these characteristics was proposed and eperimentall implement in X- ra differential phase-contrast imaging [7] and differential phase-contrast computed tomograph (DPC-CT) [, 9]. In this scheme, Pfeiffer et al. [7 9] proposed to mount an absorption grating in front of a low-brilliance X-ra source to generate an arra of equidistant secondar line sources. The coherence length of the beam is determined not b the focal spot size of the X-ra tube, but rather b the width of the opening of the absorption grating, which is about 5 μm. This allows the use of a conventional X-ra tube, where the high X-ra output can be turned into man mutuall incoherent line sources. In this paper, we refer to

2 X-Ra Optics and Instrumentation this grating as a beam slicer. Using this method, the spatial coherence length of each line source has been improved b a factor of ten or more over that of the X-ra tube without the grating, thus enabling significant reduction of the distance between the X-ra source and the image object. As a consequence, this method enables construction of an X-ra phase-contrast imaging sstem with compact size, which is often a requirement in medical and nondestructive inspection applications. For a given X-ra detector with rectangular geometr, the fan angle is determined b the angle given b the following formula: ( ) H γ m = tan, () D where H is the width of the detector and D is the distance from the X-ra focal spot to the detector. Therefore, when the source to detector distance D is ver long, the divergence of the X-ra beam is negligible, which is often the case for snchrotron beam line eperiments and inline holograph phase-contrast imaging. In this case, the beams are well approimated as plane wave, and a parallel-beam image reconstruction method is sufficient for image reconstruction in DPC-CT. However, using the new phase retrieval method b Pfeiffer et al. [7 9], the distance D is significantl reduced (D cm in [9]). Note that the effective detector size is limited b the size of the third grating in front of the detector, about cm in [9]. Using these parameters, the fan angle γ m in [9] canbeestimatedas γ m = tan ( H D ). () In this case, the parallel-beam approimation is still acceptable (see Section.). Thus, the well-known parallel-beam image reconstruction method [] can be adapted to reconstruct DPC-CT images [,, ]. When a larger image object is scanned with the same detector-to-source distance D, the fan angle γ m is increased and the parallel-beam approimation becomes less accurate. When the fan angle is increases beond 5, the divergent nature of the data acquisition sstem is no longer negligible (see Section.). For eample, if the size of the image object is doubled in the above estimation, the fan angle is nearl 5. For larger objects, a divergent beam image reconstruction algorithm would be desirable for this new DPC-CT data acquisition method. In this paper, we present an image reconstruction formula for fan-beam DPC-CT which can be utilized to accuratel reconstruct DPC-CT images at an fan angle provided that the data is acquired within the angular range of plus fan angle.. Derivation of the Fan-Beam Image Reconstruction algorithm In general, the eperimental results should be eplained b wave optics. However, for convenience, an effective geometrical model is utilized in this paper to develop the image reconstruction algorithm. Θ(ρ, θ) Figure : A geometric model of the parallel-beam data acquisition model. u Θ(t, u) o ρ θ o R t S(R cos t, R sin t) Figure : Geometrical model for fan-beam DPC-CT data acquisition. R is the radius of the source trajector, and D is source-todetector distance... Brief Review of the Parallel-Beam Reconstruction of the Differential Phase-Contrast X-Ra CT As eplained b Pfeiffer et al. [], the fundamental idea in differential phase retrieval is to locall measure the angular change Θ(ρ, θ) of the incident X-ra beam (Figure ). Here, the measured data is labeled b the distance ρ, whichis measured from the origin of the coordinate sstem to the incident ra direction and the angle θ, which specifies the direction of the incident ra. The quantit Θ(ρ, θ) is related to the local gradient of the object s phase shift can be written as follows [5, ]: D Θ(ρ, θ) = dl(, ), () ρ l where (, ) is the decrement of the real part of the object s refractive inde, n = +iβ. The letter l labels the incident ra; therefore, the above line integral is performed along the incident ra direction. Note that the integral in () is simpl

3 X-Ra Optics and Instrumentation (a) (b) (c) (d) Figure : Reconstruction results for a fan-beam full scan. (a): Reconstructed image. (b), (c), and (d) Image intensit plots corresponding to lines,, and, respectivel. The solid line represents the reconstructed image, and the dashed line represents the ground truth. The abscissas of (b), (c), and (d) are normalized to.5 m. the Radon transform R (ρ, θ) of the target function (, ). Thus, ()canberewrittenas Θ(ρ, θ) = ρ R (ρ, θ). () After a Fourier transform with respect to the variable ρ, () is recast into the following form: Θ(ω, θ) = iπω R (ω, θ). (5) Using the inverse Radon transform in Fourier space, one can easil obtain the following image reconstruction formula [9,, ]: (, )= π + π dθ dω [ iπsgn(ω) ] Θ(ω, θ)e iπω(cos θ+ sin θ). () Therefore, after the data are acquired from view angles in the angular range [, π], a one-dimensional Fourier transform is performed with respect to the Radon distance ρ to obtain Θ(ω, θ). A filtering kernel iπsgn(ω) is used to filter the data. Finall, an integral over the view angles is performed to reconstruct images... Effective Fan-Beam Data Acquisition for Differential Phase Retrieval Method Note that the image reconstruction formula (), used b Pfeiffer et al. [9], makes sense onl when the acquired data are written in terms of the Radon distance ρ and the view angle θ. Namel, it is onl convenient for the parallel-beam data acquisition geometr.

4 X-Ra Optics and Instrumentation For the effective fan-beam data acquisition geometr as shown in Figure, the X-ra tube and detector are assumed to rotate around the image object along a circle with radius R and source-to-detector distance D. The acquired data are naturall labeled b the angular position, t, of the X-ra focal spot and the detector position which is labeled b a projection distance u. We refer to this measured data as Θ(t, u) in this paper. The half width of the detector is u m, which implies that the fan angle γ m = tan (u m /D). For fan-beam data, one wa to reconstruct an image is to rebin the acquired data into parallel-beam projection data and then utilize (7) to reconstruct the image. In this paper, we develop new image reconstruction formulae which directl reconstruct images from the acquired data Θ(t, u) without requiring the rebinning step... Fan-Beam Reconstruction Formula for Full Scan Case In order to directl reconstruct the image without the rebinning step, we rewrite () in the real domain b taking the inverse Fourier transform: (, ) = π + π dθ dρ cosθ + sin θ ρ Θ(ρ, θ). (7) This formula can simpl be derived b noting that the product in Fourier space is a convolution in real domain. This formula dictates how each individual projection datum Θ(ρ, θ) contributes to the final image. Note that we have etended the integral limit from [, π] to[, π] and introduced an etra factor of / to account for the redundanc. We net derive image reconstruction formulae for the fan-beam data acquisition geometr. Each individual projection datum Θ(t, u) can also be relabeled in terms of the Radon distance ρ and angle θ b the following nonlinear coordinate transforms: ρ = Ru u + D, θ = π + t + arctan u D. () This transform introduces an etra Jacobian factor: RD dρ dθ = ( u + D ) / du dt, cosθ + sin θ ρ = (9) L(, ; t) [ ] u U(, ; t), u + D where the functions L(, ; t)andu(, ; t)aregivenb L(, ; t) = R cos t sin t, sin t cos t U(, ; t) = D R cos t sin t. () Therefore, (7) can now be written as (, ) = π R π dt L(, ; t) F[ t, U(, ) ], () where the filtered function F[t, U(, )] is given b + F(t, U) = du Θ(t, u). () u U In this equation, the preweighted projection data Θ(t, u) is defined as Θ(t, u) = D u Θ(t, u). () + D Equations () () are the main results of the paper. The give a direct image reconstruction from the measured projection data Θ(t, u). The algorithm for fan-beam DPC- CT is summarized in the following steps: () preweight the acquired projection data using (); () filter the preweighted data using the Hilbert filtering kernel in (); () back-project the filtered data to reconstruct an image using (). Note that these formulae require that the data are acquired along a full circle. This is referred to as the full scan data acquisition mode. In the net subsection, we etend the above result to the short scan case, where the data are acquired from less than a full circle... Fan-Beam Reconstruction Formula for Short Scan Case In absorption X-ra CT, it has been proven that the data are sufficient to reconstruct the entire image object when the angular range is greater than + fan angle, where the fan angle is defined as the entire angle subtended b the detector from the source position. The same condition is also true for differential phase-contrast CT, we refer to this case as the short scan mode. When the short scan data acquisition mode is utilized, some incident ras will have a conjugate ra, while others will be measured onl once. Therefore, a weighting function is needed for the redundantl measured data. It is not difficult to convince oneself that the following two data points should be considered as the same: (t, u) ( t = t + π arctan ( u D ), u = u ). () Thus, similar to absorption CT [], the following weighting function can be utilized to weight the data before the filtering step: ( ) sin π t, t [, γ m γ ), γ m γ W(t, u) =, t [ γ m γ, π γ ), ), t [ ] π γ, π + γ m, ( sin π π t + γ m γ m +γ (5) where the variable γ is defined as γ = arctan(u/d). Thus, b incorporating the normalized weighting function W(t, u) into (), the preweighted projection data become D Θ(t, u) = W(t, u) u Θ(t, u). () + D

5 X-Ra Optics and Instrumentation (a) (b) (c) (d) Figure : Numerical results for short scan data acquisition mode. (a) Reconstructed image. (b), (c), and (d) Image intensit plots corresponding to lines, and, respectivel. The solid line represents the reconstructed image, and the dashed line represents the ground truth. The abscissas of (b), (c), and (d) are normalized to.5 m. The image reconstruction formulae () () together with () can be used to reconstruct the image.. Numerical Simulations and Results In order to validate both the full scan and short scan DPC- CT reconstruction formulae derived in this paper, numerical simulations were conducted using the fan-beam data acquisition geometr as shown in Figure. The mathematical phantom consisted of a uniform elliptical disk and two uniform circular disks. The projection data were numericall generated using Snell s law []. The radius of the circular disks was.7 m. The semimajor ais and semiminor ais of the elliptical disk were.5 m and.75 m, respectivel. The decrement of the real part of the refractive inde was assumed to be. and.5 for the circular and elliptical disks, respectivel. The radius of the source trajector R was assumed to be. m. The distance from the source to the detector D was selected to be. m. The sampling rate of the view angle was Δt =.5.Thedetector was assumed to be collinear, with total width of. m. The detector sampling rate Δu was. m/ mm. The imagematriceswere Full Scan Data Acquisition Mode A fan angle of was used in order to cover the entire image object. The view angle sampling range was [, ]. The

6 X-Ra Optics and Instrumentation (a) (b) (c).5.5 (d) Figure 5: The reconstruction result for angular range less than a short scan. Figure (a) is the reconstructed image. Figures (b), (c), and (d) are profile plots corresponding to lines,, and, respectivel, in Figure (a). The solid line represents the reconstructed image, and the dashed line represents the ground truth. The abscissas of (b), (c), and (d) are normalized to.5 m. results are presented in Figure. From Figure (a) and the corresponding image intensit plots, it is evident that the image was accuratel reconstructed using the new algorithm... Short Scan Data Acquisition Mode In the short scan mode, all parameters sta the same as the full scan mode with the following eceptions: () the decrement of the real part of the refractive inde of the circular objects is changed to, and () the angular range of the data acquisition is reduced from to,which is sufficient for reconstruction according to X-ra absorption CT image reconstruction theor []. The numerical results are presented in Figure. Again, the image is faithfull reconstructed with ver high accurac as indicated in image intensit plots... Super Short Scan Data Acquisition Mode In this subsection, we numericall demonstrate that the angular range for data acquisition should be at least π +fan angle for the presented new algorithm. When the scanning angular range is shorter than that, significant image artifacts ma occur. In numerical simulations, all of the parameters are the same as the short scan case ecept that the source angle is reduced to from to, which is less than a short scan. As shown in Figure 5, image artifacts appear in the reconstructed image, and signal drop is clearl observed

7 X-Ra Optics and Instrumentation 7 in image intensit plots. This numerical eperiment also indicates that the short scan angular range is necessar for the presented fan-beam DPC-CT reconstruction.. Discussion In this section, several aspects of the newl developed image reconstruction algorithm will be discussed. First, fan-beam DPC-CT image reconstruction and fan-beam absorption CT image reconstruction algorithms are compared. Net, the differences between the fan-beam and known parallel-beam reconstruction algorithms are discussed. Finall, the angular range requirements for fan-beam DPC-CT data acquisition are eamined... Comparison with Fan-Beam Absorption CT Image Reconstruction Algorithm (a) (b) In this paper, the fan-beam image reconstruction algorithm was derived using a collinear detector geometr. It is eas to change to the curved detector case using the relation γ = arctan(u/d). For absorption CT, fanbeam image reconstruction algorithms have been derived for both full and short scan cases []. The DPC-CT fanbeam image reconstruction algorithm is different from its absorption counterpart in three was. First, in DPC-CT, the preweighting factor is D /(D + u ). For absorption CT, the preweighting factor is a square root, that is, D /(D + u ) = D/ D + u. Second, the distance weighting function in the backprojection step in DPC-CT is R/L(, ; t). While for absorption CT, the weighting function is R /L (, ; t), the square of the DPC-CT epression. Finall, the filtering kernel for DPC-CT image reconstruction is a Hilbert filter, whereas a ramp filter is used in absorption CT image reconstruction... Parallel-Beam Approimation When the source to detector distance is much larger than the size of the image object, one can epect that the difference between the parallel-beam and fan-beam image reconstruction methods is negligible. The numerical results demonstrate that when the fan angle is smaller than 5, the parallel-beam approimation is valid, and parallel-beam image reconstruction can be utilized to reconstruct DPC-CT images. However, for larger fan angles, a fan-beam image reconstruction algorithm, such as the one presented in this paper, is needed for an accurate image reconstruction. In the first numerical eperiment, we applied the parallel-beam reconstruction algorithm [9,, ] to the case of a fan angle, the same angle used in the reconstruction presented in Figure. The parallel-beam reconstruction algorithm generates significant image artifacts as shown in Figure (a). In the second numerical eperiment, the radius of the source trajector is increased to m and the fan angle of the divergent beam is reduced to in order to cover the entire image object. As shown in Figure (b), the image qualit is improved comparing with Figure (a), but blurring still eists at the (c) Figure : The parallel-beam reconstruction result for fan-beam projection data. (a), (b), (c), and (d) correspond to fan angles,,5, and.5,respectivel. boundaries. When the source to detector distance increases such that the fan angle is reduced to 5, as shown in Figure (c), the image artifacts become increasingl negligible, indicating that the parallel beam approimation is becoming more acceptable. In the last numerical eperiment, the fan angle of the divergent beam is reduced to.5 for the same image object. In this case, as shown in Figure (d), the parallel-beam reconstruction method is sufficient to reconstruct DPC-CT images. Using the eperimental data presented in [9], the fan angle can be estimated to be less than.5. Thus, parallel-beam reconstruction would be sufficient to reconstruct near artifact-free DPC-CT images [9]. 5. Conclusions In this paper, an image reconstruction formula for fanbeam DPC-CT was derived and validated using numerical simulations. It has been demonstrated that when the image object is relativel large, the fan angle must increase to cover the entire image object, and a parallel-beam approimation cannot be directl applied to reconstruct images. In this case, the fan-beam image reconstruction formula presented in this paper can be directl applied to accuratel reconstruct DPC- CT images. The presented image reconstruction formula is also useful for other DPC-CT imaging method []provided that the divergent beam data acquisition geometr is utilized. As a final remark, as shown in Figure 5, one limitation of the presented algorithm in this paper is that the data acquisition angular range must be longer than π + fan (d)

8 X-Ra Optics and Instrumentation angle. Otherwise, significant image artifacts are observable in the reconstructed image. In order to enable accurate image reconstruction when the angular range is shorter than π + fan angle, some other new image reconstruction must be developed [5, ]. Note that it is straightforward to etend the fan-beam reconstruction algorithms such as the one presented in this paper and others [5, ]tocone-beamdata acquisition sstems using the so-called FDK approimation [7], where the onl change is to replace the fan-beam data b the projected cone-beam data, that is, the multiplication of cone-beam data and a cosine weighting factor. Acknowledgments The authors would like to thank Nicholas Bevins and Joseph Zambelli for editorial assistance in editing this paper. This work is partiall supported b the National Institutes of Health through the Grant: REB 57. References [] U. Bonse and M. Hart, An X-ra interferometer, Applied Phsics Letters, vol., no., pp. 55 5, 95. [] A. Momose, T. Takeda, Y. Itai, and K. Hirano, Phase-contrast X-ra computed tomograph for observing biological soft tissues, Nature Medicine, vol., no., pp. 7 75, 99. [] T. J. Davis, D. Gao, T. E. Gureev, A. W. Stevenson, and S. W. Wilkins, Phase-contrast imaging of weakl absorbing materials using hard X-ras, Nature, vol. 7, no. 55, pp , 99. [] D. Chapman, W. Thomlinson, R. E. Johnston, et al., Diffraction enhanced X-ra imaging, Phsics in Medicine and Biolog, vol., no., pp. 5 5, 997. [5]S.W.Wilkins,T.E.Gureev,D.Gao,A.Pogan,andA.W. Steveson, Phase-contrast imaging using polchromatic hard X-ras, Nature, vol., no. 7, pp. 5, 99. [] K. A. Nugent, T. E. Gureev, D. F. Cookson, D. Pagnin, and Z. Barnea, Quantitative phase imaging using hard X-ras, Phsical Review Letters, vol. 77, no., pp. 9 9, 99. [7] P. Cloetens, W. Ludwig, J. Baruchel, et al., Holotomograph: quantitative phase tomograph with micrometer resolution using hard snchrotron radiation X-ras, Applied Phsics Letters, vol. 75, no. 9, pp. 9 9, 999. [] C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, Differential X-ra phase contrast imaging using a shearing interferometer, Applied Phsics Letters, vol., no. 7, pp. 7 9,. [9] A. Momose, S. Kawamoto, I. Koama, Y. Hamaishi, K. Takai, and Y. Suzuki, Demonstration of X-ra Talbot interferometr, Japanese Journal of Applied Phsics, vol., part, no. 7B, pp. L L,. [] T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, X-ra wavefront analsis and optics characterization with a grating interferometer, Applied Phsics Letters, vol., no. 5, Article ID 5, pages, 5. [] A. Momose, W. Yashiro, T. Takeda, Y. Suzuki, and T. Hattori, Phase tomograph b X-ra Talbot interferometr for biological imaging, Japanese Journal of Applied Phsics, vol. 5, part, no. A, pp. 55 5,. [] H. F. Talbot, Fact relating to optical science, Philosophical Magazine, vol. 9, pp. 7,. [] J. P. Guiga, On Fresnel diffraction b one-dimensional periodic objects, with application to structure determination of phase objects, Journal of Modern Optics, vol., no. 9, pp. 77, 97. [] V. Arrizón and J. Ojeda-Castañeda, Irradiance at Fresnel planes of a phase grating, Journal of the Optical Societ of America A, vol. 9, no., pp., 99. [5] O. Kafri and I. Glatt, The Phsics of Moire Metrolog, Wile- Interscience, New York, NY, USA, 99. [] T. Weitkamp, A. Diaz, C. David, et al., X-ra phase imaging with a grating interferometer, Optics Epress, vol., no., pp. 9, 5. [7] F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Phase retrieval and differential phase-contrast imaging with lowbrilliance X-ra sources, Nature Phsics, vol., no., pp. 5,. [] F. Pfeiffer, C. Grünzweig, O. Bunk, G. Frei, E. Lehmann, and C. David, Neutron phase imaging and tomograph, Phsical Review Letters, vol. 9, no., Article ID 555,. [9] F. Pfeiffer, C. Kottler, O. Bunk, and C. David, Hard X- ra phase tomograph with low-brilliance sources, Phsical Review Letters, vol. 9, no., Article ID 5, 7. [] A. C. Kak and M. Slane, Principles of Computerized Tomographic Imaging, IEEE Press, New York, NY, USA, 9. [] G. W. Faris and R. L. Ber, Three-dimensional beamdeflection optical tomograph of a supersonic jet, Applied Optics, vol. 7, no., pp. 5 5, 9. [] Z.-F. Huang, K.-J. Kang, Z. Li, et al., Direct computed tomographic reconstruction for directional-derivative projections of computed tomograph of diffraction enhanced imaging, Applied Phsics Letters, vol. 9, no., Article ID, pages,. [] M. Born and E. Wolf, Principles of Optics, Pergamon Press, New York, NY, USA, th edition, 9. [] D. L. Parker, Optimal short scan convolution reconstruction for fan beam CT, Medical Phsics, vol. 9, no., pp. 5 57, 9. [5] G.-H. Chen and Z. Qi, Image reconstruction for fan-beam differential phase contrast computed tomograph, Phsics in Medicine and Biolog, vol. 5, no., pp. 5 5,. [] Z. Qi and G.-H. Chen, A local region of interest image reconstruction via filtered backprojection for fan-beam differential phase-contrast computed tomograph, Phsics in Medicine and Biolog, vol. 5, no., pp. N7 N, 7. [7] L. A. Feldkamp, L. C. Davis, and J. W. Kress, Practical conebeam algorithm, Journal of the Optical Societ of America A, vol., no., pp. 9, 9.