An approximate cone beam reconstruction algorithm for gantry-tilted CT
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1 An approximate cone beam reconstruction algorithm for gantry-tilted CT Ming Yan a, Cishen Zhang ab, Hongzhu Liang a a School of Electrical & Electronic Engineering, Nanyang Technological University, Singapore; b School of Chemical & Biomedical Engineering, Nanyang Technological University, Singapore ABSTRACT FDK algorithm has been known to be a popular 3D approximate computed tomography (CT) reconstruction algorithm. However, it may not provide satisfactory image quality for large cone angle. Recently, it has been improved by performing ramp filtering along the direction tangent to the helix, so to provide improved image quality for large cone angle. In this paper, we present a FDK type approximate reconstruction algorithm for gantry-tilted CT imaging. The proposed method improves FDK algorithm by filtering the projection data along a proper direction. Its filtering direction is determined by CT parameters and gantry-tilted angle. As a result, the proposed gantrytilted reconstruction algorithm can provide more scanning flexibilities in clinical CT scanning and is efficient in computation. The performance of the proposed algorithm is evaluated with Turbell Clock phantom and Thorax phantom compared with gantry tilted FDK algorithm and a popular 2D approximate algorithm. The results show that our new algorithm can achieve better image quality than FDK algorithm and the 2D approximate algorithm for gantry-tilted CT image reconstruction. Keywords: Gantry tilted CT, Helical cone-beam CT, Image reconstruction 1. INTRODUCTION In some applications of clinical CT (computed tomography) scanning process, it is required that the gantry be tilted. For example, in order to avoid exposure of eyes to X-rays, gantry is tilted during the head scanning procedure. To meet special requirements of the gantry tilted CT, a number of algorithms have been developed for gantry tilted MSCT (multislice computed tomography) image reconstruction. 1 4 Among these algorithms, Kachelrieiss and Fuchs 2 developed a gantry tilted reconstruction algorithm based on the earlier developed 2D (2- dimensional) algorithm ASSR. 5 Hein et al. 3 developed a gantry tilted reconstruction algorithm based on the 3D (3-dimensional) FDK algorithm. 7, 8 In the gantry tilted Feldkamp algorithm in Hein et al., 3 the reconstruction plane is perpendicular to the rotating axis of the CT scan. Thus the cone angle increases with the pitch value and slice number which can lead to unavoidable artifacts. Recently, Noo et al. 4 developed a general framework which can be applied to the gantry tilted CT for exact 6 and approximate image reconstruction. In this paper, we present a 3D approximation algorithm for gantry tilted CT image reconstruction based on the FDK algorithm. 7, 8 FDK algorithms for both gantry tilted and non-gantry-tilted MSCT filter projection data horizontally and suffers serious artifacts for the large cone angle. Recently, some improvements have been made for FDK algorithms 9 13 and one improvement is made by filtering projection data along the tangential direction which was first proposed by Yan and Leahy 9 and improved by Sourbelle and Kalender 10 for short scan FDK algorithm. These improved FDK algorithm can obtain better image quality than conventional FDK algorithm. In this paper, we further improve the tangential filtering FDK algorithm to gantry tilted geometry. We provide a general filtering equation such that the helix tangent filtering for non-gantry tilted CT is a special case for this equation with the gantry tilted angle being zero. In the mean time, our algorithm is based on the FDK reconstruction procedure, so it retains the computational efficiency and flexibility of the FDK type Further author information: (Send correspondence to Chishen Zhang) Ming Yan: yanming@pmail.ntu.edu.sg Cishen Zhang: ecszhang@ntu.edu.sg, Telephone: Medical Imaging 2006: Physics of Medical Imaging, edited by Michael J. Flynn, Jiang Hsieh, Proceedings of SPIE Vol. 6142, 61424U, (2006) /06/$15 doi: / Proc. of SPIE Vol U-1
2 µ β (a) (b) Figure 1. Geometry for gantry tilted multislice CT scanner. algorithms. Simulation results can show that the proposed algorithm can provide considerably improved image quality under large cone angle. The rest of this paper is organized as follows. Section II presents the geometric scheme and the proposed method is presented. In section III, results of simulation and evaluation of the proposed algorithm are presented. 2. METHOD The method for improving FDK algorithm with filtering projection data along the helix tangent direction was developed by Yan and Leahy 3. We further develop this method by taking the gantry tilted angle into account. With the filtered projection data, Feldkamp algorithm is applied to reconstruct images on a sequence of planes Geometry for gantry tilted MSCT The MSCT scanning set up consists of an X-ray source and a detector array forming a source-detector framework. The geometry of the source-detector framework is shown in Figure 1 (a), where the detector array is a rectangular surface, C s denotes the X-ray source, a coordinate on the detector plane is l u. The plane formed by C s and l is called the centre plane. The geometry of the centre plane is shown in Fig. 1 (b), β is the projection angle and l is one coordinate on the detector plane. The distance from the X-ray source to the rotating axis is R f and the distance from the rotating axis to the detector is R d. For gantry tilted CT scanner, the gantry is tilted around x-axis and the tilted angle is µ. There exists a temporal coordinate system x y z for the tilted gantry. The centre plane lies on x y and z is perpendicular to the centre plane. The origin of the temporal coordinate system o is the intersection between the centre plane and z-axis and the z coordinate of o is z o. The axis x is parallel to the x axis and the angle between z and z is µ as shown in Fig. 1 (a) The relationship between the temporal coordinate system x y z and the global coordinate system is x y = cosµ sin µ x y = T µ x y (1) z 0 sinµ cos µ z z o z z o where z o = z 0 + psm 2π (β β 0) Proc. of SPIE Vol U-2
3 µ Figure 2. Illustration of conventional filter direction and tangential filter direction. The projection angle β 0 is the initial projection angle and z 0 is the initial z position of the X-ray source. In the the temporal coordinate system, because the X-ray source is on the x y plane and the projection angle β is defined in the x y plane as shown in Fig. 1 (b), the X-ray source position is x s y s = R f cos β R f sin β (2) z s 0 With the transformation between the temporal coordinate system x y z and the global coordinate system s y z defined in (1), the x-ray source trajectory in global coordinate system can be expressed as C s (β) = x s y s z s = z 0 + psm 2π R f cos β R f cos µ sin β (3) (β β 0)+R f sin µ sin β where S is the slice thickness, p is the pitch value of the spiral cone beam scanning, M is the number of detector slice Projection data set reformation In conventional FDK algorithms, the ramp filtering is performed on detector rows and the filtering direction is parallel to the l axis on the detector surface. Motivated by the technique of tangent filtering for non-gantry tilted CT, we propose in this paper that the ramp filtering is performed in the tangential direction of the X-ray source trajectory as shown in Fig. 2. For this purpose, the rebinning method is used to reform a projection data set where rows of the reformed projection data set are parallel to the tangential direction of the X-ray source trajectory. Proc. of SPIE Vol U-3
4 + ηζ σ η + ξ + (a) ς η η γ η (b) Figure 3. Illustration of transformation between s l u and ξ η ζ. Recalling the definition of the s l u and ξ η ζ coordinate systems, each detector cell is at (R d +R f,l,u) T in the s l u coordinate system and the corresponding projection ray on this cell is denoted by D(β,l,u). Referring to Fig. 3 (a), the vector η is the projection of the axis η on the x y plane. The angle between η and axis l on the detector array surface is denoted by σ and given by σ = arcsin h sin µ sin β Rf 2 + h2 sin 2 µ (4) The angle between η and the x y plane is γ and because η is the projection of η on x y plane, the angle γ is the angle between η and η as shown in Fig. 3 (b) and given by γ =arctan h cos µ (5) Rf 2 +2hR f sin µ cos β + h 2 For filtering the projection data along the tangential direction of the X-ray source trajectory, we introduce a virtual detector array surface in the rotating coordinate system ξ η ζ. The virtual detector array surface is of the same size and shape of the real detector array surface and is place on the plane ξ =(R f + R d ), as shown in Fig. 3 (a), with the rows being parallel to the η axis. The transformation between the two coordinate systems s l u to ξ η ζ can enable formation of the projection data on the virtual detector surface from the projection data set on the real detector array. This transformation can be carried out as follows. First, a temporal coordinate ξ η u is obtained by rotating s l with σ degree around u. Then the ξ η ζ coordinate is obtained by rotating ξ η u with γ degree around ξ. As a result, the transformation matrix between ξ η ζ Proc. of SPIE Vol U-4
5 and s l u denoted by T is given by T = cos σ sin σ 0 0 cosγ sin γ sin σ cos σ 0 0 sin γ cos γ = cos σ sin σ 0 sin σ cos γ cos σ cos γ sin γ sin σ sin γ cos σ sin γ cos γ (6) It is shown in (4), that when the gantry tilted angle µ = 0, the angle σ = 0. Then the transformation function T is identical to the transformation matrix for non-gantry tilted CT proposed by Yan and Leahy. 9 Thus the standard tangential filtering for non-gantry tilted CT can be considered as a special case of the proposed method when the gantry tilted angle is zero. Each projection ray radiates on a real detector cell at the actual detector on a point ((R f + R d ),l,u) T in the s l u coordinate also radiates on a point and this projection ray also radiates on a point ((R f + R d ),η,ζ) T on the virtual detector array. The point ((R f + R d ),l,u) T in the s l u coordinate is ( ξ, η, ζ) T in the ξ η ζ coordinate given by ξ = T (R f + R d ) l (7) η ζ u In the ξ η ζ coordinate, the projection ray passing through ( ξ, η, ζ) T radiates on the virtual detector array at ((R f + R d ),η,ζ) T with η = R f + R d η (8) ξ ζ = R f + R d ζ (9) ξ This reforms a projection data set on the virtual detector array from the projection data on the real detector array. Let the reformed data set on the virtual detector array be denoted by D(β,η,ζ), which will be used for the tangential direction filtering Reconstruction and interpolation In order to reconstruct images on horizontal planes, we first reconstruct images on a sequences of intermediate planes perpendicular to the rotating axis. Let ψ = {P i : i =1, 2, 3,,n} be the set of these intermediate planes. Let the intersection of the tilted plane P i and the z-axis be at point o i =(0, 0,z i ) T. z i = z 0 + i z, for i =1, 2, 3,, n with z >0 being a constant. The projection angle for X-ray source position with its z coordinate equal to z i is β i and β i = β 0 + 2πi z psm. The intermediate plane P i is: z = z i + y tan µ (10) The ramp filtering is along the η direction which is also helix tangent direction, g is the ramp filter and D is projection data on the visual detector: D(β,η D(β,ξ,ζ),ζ) = g(η η ξ )dη (11) 2 + Rf 2 + ζ2 For a point (x, y, z) T on the plane P i, it can be reconstructed with this equation: f i (x, y, z) = 1 2 βi+β m/2 β i β m/2 R f R 2 f +2hR f sin µ cos β + h 2 D(β,ξ,ζ) (x cos β + y sin β R f ) 2 dβ (12) where β m is the length of projection angle interval. Given the reconstructed image of the tilted planes P i for i =1, 2, 3,,nand a point (x, y, z) on the plane parallel to the x y plane where the image value is to be obtained by interpolation, the z-positions on the tilted planes P i, i =1, 2, 3,, n,at(x, y) T and denoted by z pi (x, y) can be determined by Eq. (10). Thus there Proc. of SPIE Vol U-5
6 exist two points (x, y, z Pj ) T and (x, y, z Pk ) T on two tilted planes P j and P k, respectively, at the upper and lower sides of (x, y, z) T, respectively, which are on two tilted plans and are closest to (x, y, z) T. Using the obtained attenuation functions f j (x, y, z Pj )andf k (x, y, z Pk ) for the tilted plans P j and P k, respectively, the interpolated attenuation function can be obtained using the following interpolation formula f(x, y, z) = f j(x, y, z Pj (x, y)(z Pk z) + z Pk z Pj f k (x, y, z Pk (x, y))(z z Pj ). (13) z Pk z Pj The image reconstruction of the proposed TPFR algorithm is thus completed. 4n (a) (b) (c) (d) Figure 4. Images for the plane z=0mm of Clock phantom generated with the proposed algorithm, gantry tilted Feldkmp and gantry tilted ASSR with S=1mm, p=1.0, M=96, µ =10 ( ). 3. SIMULATION AND RESULTS We use Turbell Clock phantom and thorax phantom to evaluate the performance of the GOTPFR algorithm. Parameters of our simulation are: R f =570mm, θ fan =49.8, S =1mm, p =1.0, projection number per rotation N p = 1024 and number of projection rays in the fan N f = 800. There are three approximate reconstruction algorithms are compared: the proposed algorithm, Gantry tilted FDK and Gantry tilted ASSR 2 which was presented by Kachelreiss in Proc. of SPIE Vol U-6
7 (a) (b) (c) Figure 5. Plot of image centre vertical line (y = 0.0mm) of the the reconstructed (solid line) and original images(dashed line). Results obtained from the simulation of Turbell Clock phantom are shown in Fig. 4, where Fig. 4 (a) is the original Turbell Clock phantom, Fig. 4 (b) is the image reconstructed by our proposed algorithm, Fig. 4 (c) is reconstructed by the gantry tilted FDK algorithm and Fig. 4 (d) shows the image reconstructed by the gantry tilted ASSR. The results demonstrate that the image reconstructed by our proposed algorithm contains fewer artifacts than that of the other two existing algorithms for gantry tilted CT. In the simulation of Turbell Clock phantom, the centre vertical lines for images reconstructed by the proposed algorithm Fig. 5 (a), gantry tilted FDK algorithm Fig. 5 (b) and gantry tilted ASSR Fig. 5 (c). In this figure, the dashed lines represent the original image and the solid lines represent reconstructed profiles. The results show that the profile of the proposed algorithm as shown in the left hand side figure has smaller variance than the other two results. The proposed algorithm can reconstruct more accurate image than the other two algorithms. (a) (b) (c) (d) Figure 6. Images for the plane z=0mm of Thorax phantom generated with the proposed algorithm, gantry tilted FDK algorithm and gantry tilted ASSR algorithm with M=96, µ =10 ( ). A thorax phantom is further simulated which is designed by referring to human thorax consisting of many important organs and often scanned in CT examination. These organs include lungs, heart, aorta, ribs, spine, Proc. of SPIE Vol U-7
8 sternum, and shoulders. The phantom definitions are obtained from a world phantom database FORBILD. This kind of body section phantoms are more close to the real patient CT scan, thus they are feasible for examining The simulation results are shown in Fig. 6, where Fig. 6 (a) is the original thorax phantom, Fig. 6 (b) shows the image reconstructed by the proposed algorithm, Fig. 6 (c) is the image reconstructed by the gantry tilted FDK algorithm and Fig. 6 (d) is reconstructed by the gantry tilted ASSR algorithm. The lower row is zoomed reconstructed image. It is shown that there are obvious artifacts are around ribs in the images reconstructed by conventional FDK algorithm and ASSR algorithm. In contrast, the proposed algorithm provides better image quality and more accurate reconstruction. 4. CONCLUSIONS This paper presents an approximate algorithm for gantry tilted helical MSCT image reconstruction. It is based on the idea of filtering the 3D projection data and the filtering direction is varying and dependent on the CT parameters and gantry-tilted angle. As a result, the proposed gantry-tilted reconstruction algorithm can provide more scanning flexibility in clinical CT scanning and is efficient in computation. In comparison with the existing 2D and 3D algorithms for gantry tilted CT image reconstruction, our proposed algorithm can provide improved image quality. The performance of the proposed algorithm is evaluated with Turbell Clock phantom and Thorax phantom in comparison with the recent gantry tilted FDK algorithm and gantry tilted ASSR algorithm. The improved performance and image quality of the proposed algorithm have been shown. REFERENCES 1. J. Hsieh, Tomographic reconstruction for tilted helical multislice CT, IEEE Trans. Med. Imag., vol. 19, pp , Sept M. Kachelreiss,T. Fuchs, Advanced single-slice rebinning for tilted spiral cone-beam CT, Med. Phys. Vol. 28 No. 6, , I. Hein, K. Taguchi, M. D. Silver, M. Kazama, I. Mori, Feldkamp-based cone-beam reconstruction for gantry-tilted helical multislice CT, Med. Phys. Vol. 30, No. 12, , F. Noo, M. Defrise,H. Kudo, General Reconstruction Theory for Multislice X-ray Computed Tomography With a Gantry Tilt, IEEE Trans. Med. Imag., vol. 23, No. 9, M. Kachelreiss, S.Schaller, W.A. Kalender, Advanced single slice rebinning in cone-beam spiral CT, Med. Phys. Vol. 27, No. 4, , A. I. Katsevich, Theoretically exact filtered backprojection-type inversion algorithm for spiral CT, J. Appl. Math. Vol.62, No. 6, , L. A. Feldkamp, L. C. Davis, and J. W. Kress, Practical cone-beam algorithm, J. Opt. Soc. Am. A 1, , G. Wang, T. Lin, P. Cheng, D. M. Schiozaki, A General Cone-Beam Reconstruction Algorithm, IEEE Transactions on Medical Imaging, Vol, 12, No.3, , X. Yan and R. M. Leahy, Cone-beam tomography with circular, elliptical and spiral orbits, Phys. Med. Biol. 37, , K. Sourbelle and W. A. Kalender, Generalisation of Feldkamp reconstruction for clinical spiral cone beam CT, Proceedings of the 2003 Meeting on Fully 3D Image Reconstruction in Radiology and Nuclear Medicine, France. 11. M. Kachelreiss, M. Knaup, and W. A. Kalender, Extended parallel backprojection for standard threedimensional and phase-correlated four-dimensional axial and spiral cone-beam CT with arbitrary pitch, arbitrary cone-angle, and 100% dose usage, Med. Phys. Vol. 31, No. 6, , J. Hu, K. Tam, J. Qi, An approximate short scan helical FDK cone beam algorithm based on nutating curved surfaces satisfying the Tuys condition, Med. Phys. 32, , M. Yan, C. Zhang, Tilted plane feldkamp type reconstruction algorithm for spiral cone beam CT, Med. Phys.,Vol. 32, No.11, , D. L. Parker, Optimal short scan convolution reconstruction for fan beam CT, Med. Phys. Vol. 9, No.9, ,1982. Proc. of SPIE Vol U-8
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