Journal of X-Ray Science an Technology 12 (2004) 73 82 73 IOS Press A half-scan error reuction base algorithm for cone-beam CT Kai Zeng a, Zhiqiang Chen a, Li Zhang a an Ge Wang a,b a Department of Engineering Physics, Tsinghua University, Beijing, 100084, P.R. China E-mail: zengkai@tsinghua.org.cn, zhangli@nuctech.com, czq@mail.tsinghua.eu.cn b Departments of Raiology, University of Iowa, Iowa City, IA 52242, USA E-mail: ge-wang@uiowa.eu Abstract. The most popular Felkamp algorithm for cone-beam image reconstruction assumes a scanning circle, an performs well only with a small cone angle. In this report, we propose a Felkamp-type algorithm to increase the cone angle by several fols at a raiation ose comparable to that require by the classic Felkamp algorithm. In our scheme, two half-scans of ifferent raii are use. Then, approximate reconstructions from two half-scans are combine to prouce a superior image volume by utilizing a relationship between the half-scan reconstruction error an the scanning locus raius. The merit of this half-scan error reuction base (HERB) algorithm is emonstrate in numerical simulation with the 3D Shepp-Logan phantom. Inex Terms: Compute tomography (CT), cone-beam, half-scan, Felkamp-type algorithm. 1. Introuction In the biomeical fiel, the most popular metho for cone-beam image reconstruction is the socalle Felkamp algorithm, which is base on a scanning circle an reconstructs images efficiently an satisfactorily in the small cone angle case [1]. However, with moerate or large cone angles, the Felkamp algorithm woul prouce significant image artifacts, such as intensity ropping away from the mi-plane. To improve the Felkamp algorithm for practical applications, a number of its variants were evelope [2 6], which may work with a larger cone angle. In this report, we propose a Felkamp-type algorithm to increase the cone angle by several fols at a raiation ose comparable to that require by the classic Felkamp algorithm. In our scheme, two halfscans of ifferent raii are use. Then, approximate reconstructions from two half-scans are combine to prouce a superior image volume by utilizing a relationship between the half-scan reconstruction error an the scanning locus raius. For brevity, this new algorithm is name as a Half-scan Error Reuction Base metho, or HERB metho in short, whose orbit consists of two concentric half-scan arcs. In the following, we will first recall the Felkamp algorithm an its half-scan version. Then, we will escribe a relationship between the reconstruction error an the locus raius, formulate our new algorithm, an evaluate it numerically in comparison with some Felkamp-type algorithms. Finally, we will iscuss a few relevant issues. 0895-3996/04/$17.00 2004 IOS Press an the authors. All rights reserve
74 K. Zeng et al. / A half-scan error reuction base algorithm for cone-beam CT 2. Methos 2.1. Full-scan an half-scan algorithms A cone-beam CT geometry is illustrate in Fig. 1. When a scanning locus is containe in a plane, such as in the circular scanning moe, cone-beam projection ata are insufficient for accurate an reliable reconstruction [9]. Therefore, the Felkamp algorithm can only reconstruct images approximately. It is known that ata acquire along opposite X-rays are reunant on the mi-plane (see Fig. 1), an quite highly correlate off the mi-plane (see [3]). By inserting the Parker weighting formula [7] into the Felkamp algorithm [1], we immeiately obtain a half-scan version of the Felkamp algorithm: f half ( f)= 1 4π 2 where π+2γmax 0 2 φ ( + r ˆx ) P 2 φ (Y ( r),z( r)), (1) Y ( r) = r ŷ /( + r ˆx ), (2) Z( r) = r ẑ/( + r ˆx ), (3) r is the vector from origin to the point to reconstruct, φ enotes the projection angle, P φ (Y, Z) conebeam projection ata acquire by a planar etector, (Y, Z) the etector coorinates (see [1]), the istance from source to the rotation axis, γ max the half fan-angle. P φ (Y,Z)= Y h(y Y )w(φ, Y,Z )P φ (Y,Z ) ( 2 + Y 2 + Z 2, (4) ) 1/2 an w(φ, Y, Z) a cone-beam ata weighting function [8], h is the 1D ramp filter. 2.2. Relationship between reconstruction error an scanning raius Although the Felkamp-type [9] (or Raon-type) algorithms are rather ifferent from the Grangeat-type algorithms in terms of formulation, the Felkamp-type reconstruction is equivalent to the Grangeat-type reconstruction in the circular scanning case if the missing ata in the Raon space is assume to be zero [10]. Therefore, the Felkamp algorithm can be analyze in the Raon space. The Raon transform is well known (Fig. 2(a)): Rf(ρ, n) = f( r) r, (5) r P (ρ, n) where f( r) is a function on R 3, P (ρ, n) is a plane efine by r n = ρ. The inverse Raon transform is given as follows (Fig. 2(b)): f( r) = 1 2 Rf 8π 2 ( r n, n) n, (6) s 2 ρ2 where n is a vector on the unit sphere S 2, r n = ρ, As shown in Fig. 3, ue to the limitation with the imaging geometry, part of Raon information is not embee in cone-beam ata collecte along a scanning circle, which forms a shaow zone [11]. Let us
K. Zeng et al. / A half-scan error reuction base algorithm for cone-beam CT 75 Detector Plane z ( Y, Z ) P ( Y, Z ) φ y y r! O φ Opposite Rays x x Source Trajectory (a) Source 1 Two Half-Scans 2 Source (b) Fig. 1. (a) Arrangement of a cone-beam tomography system. The istance from source to rotation axis is. The opposite rays are partially reunant. The coorinate system is the same as [1]. (b) HERB s scan geometry. assume a spherical object support of raius R centere at the origin of the reconstruction system. Then, the shaow zone is a pair of cone-shape regions. Then, in the shaow zone all the ata in Raon space can be assume to be zero. The inverse 3D Raon transform can be reformulate as ( f( r) = 1 2 ) ( Rf 8π 2 ρ n V S ρ 2 ( r n, n) n + 1 2 ) Rf 8π 2 ρ n V S ρ 2 ( r n, n) n (7) Hence, the Felkamp reconstruction is equivalent to the secon term of Formula Eq. (7). In other wors, the reconstruct error must be f error ( r) = 1 2 Rf 8π 2 ( r n, n) n. (8) ρ n V S ρ2 When the cone angle is small (i.e. Vs is very small compare with the whole object), we can approximately have [13]: f error ( r) =const( r, f( r)) SPH 1, (9)
76 K. Zeng et al. / A half-scan error reuction base algorithm for cone-beam CT z n!! P( ρ, n) ρ O y x (a) SPH 1 Raon Shell efine by r! Point to be reconstructe r! SPH 2 Shaow Zone Vs (Fig. 3)! ( n, ρ) Support of an object with raius R (b) Fig. 2. 3D Raon transform an a shaow zone in the Raon space. (a) 3D Raon transform efine as planar integrals, an (b) a shaow zone associate with a circular cone beam scanning locus. where SPH 1 is the area of the Raon shell in the shaow zone (V s ), an the iameter of the shell is enote by r. Similarly, SPH 2 is the counterpart of SPH 1 outsie the shaow zone (V S ). Therefore, when we reconstruct f( r) twice with raii 1 an 2 for any fixe point, we have f error,1 ( r)/f error,2 ( r) SPH 1,1 /SP H 1,2. (10) Finally, we can erive the following formula [13]: f error,1 ( r)/f error,2 ( r) 2 1 / 2 2 (11) Alternatively, we can achieve the above relationship more intuitively. As the orbit raius increases, the volume of the shaow zone ecreases accoringly. Eviently, the less the missing ata is, the less the reconstruction error will be. Furthermore, because the shaow zone consists of two opposite cones with a common vertex at the origin of the Raon space, it shoul generally be of the same importance in terms of image reconstruction as compare to any ata zone of the same configuration of a ifferent
K. Zeng et al. / A half-scan error reuction base algorithm for cone-beam CT 77 V s z c S 1 (a) S 1 z c V s (b) Fig. 3. Characteristics of the haow zone in the Raon space. Parameters of the shaow zone: volume V s, bottom area S 1, an height z c, an a zoome shaow zone consisting of two opposite cones. orientation in the case of a somehow spherical object, at least for low frequency components in the spherical harmonic omain. Therefore, the reconstruction error shoul be approximately proportional to the volume of the shaow zone. Mathematically, we approximately have the following relationship: f error ( r) V S, where r enotes the vector from the origin to the point to be reconstructe, an Vs the volume of the shaow zone. The volume of the shaow zone V s can be estimate as follows. As shown in Fig. 3, let O enote a shaow cone vertex, z c the height an S 1 the bottom area. It is trivial to prove that V S =2/3 S 1 z c. If 2R <<, it can be shown that S 1 2 Hence, V s 2 (12) (13)
78 K. Zeng et al. / A half-scan error reuction base algorithm for cone-beam CT That is, f error ( r) 2 (14) 2.3. Half-scan error reuction base (HERB) algorithm The main iea of the HERB approach is to perform two or more full-scan an/or half-scan reconstructions corresponing to ifferent scanning locus raii, an combine reconstructe images for a reconstruction more accurate than any of the irect results. Without loss of generality, we focus on use of two half-scans acquire with ifferent scanning raii. In this context, we reconstruct each voxel twice from each of the two half-scan atasets, estimate the ifference between the real value an the reconstructe values, an arrive at a more accurate voxel value as follows: f( r) =f full,1 ( r)+err estimate (f full,1 ( r),f full,2 ( r), 1, 2 ). (15) Base on the relationship Eq. (11), it can be reaily shown that: f error,1 ( r) f error,2 ( r) = 2 1 Thus we have: 2 1 2 1 2 2 2 2 = f( r) f full,1( r) f( r) f full,2 ( r), = f( r) f full,1( r) f full,2 ( r) f full,1 ( r), Err estimate ( r) =(f( r) f full,1 ( r)) = (f full,2 ( r) f full,1 ( r)) 2 1 2 1, (16) 2 2 f( r) =(f full,2 ( r) f full,1 ( r)) 2 1 2 1 + f 2 full,2 ( r) (17) 2 where f full,1 is the reconstruction result obtaine by applying the Felkamp algorithm with a full-scan of raius 1,f full,2 the counterpart with a full-scan of raius 2, an f HERB the output of our HERB algorithm. It is well known that the Felkamp algorithm is essentially fan-beam reconstruction after cone-beam ata are approximately correcte to fan-beam ata, an in the fan-beam case the half-scan reconstruction is equal to the full-scan reconstruction. Therefore, the Felkamp-type reconstruction shoul not iffer much in the full-scan an half-scan cases, at least uner the assumption of a stationary object an a moerate cone angle. That is, we typically have f half ( r) f full ( r). By the above argument, we immeiately have a half-scan version of Formula Eq. (17), which is the main result of this report: f HERB ( r) =(f half,2 ( r) f half,1 ( r)) 2 1 2 1 + f 2 half,2 ( r), (19) 2 where the variables are similarly efine as for Formula Eq. (17). (18)
K. Zeng et al. / A half-scan error reuction base algorithm for cone-beam CT 79 Table 1 Simulation parameters for evaluation of the HERB algorithm. Distance measures are in millimeter Scanning raius(mm) 1 300 2 400 Cone angle (egree) 1 tg 1 (128/300) = 23 2 tg 1 (128/400) = 18 Fan angle (egree) 1 tg 1 (128/300) = 23 2 tg 1 (128/400) = 18 1 1.3 2 1 2 2 Number of projections per full-scan 360 Detector size (Height With)(mm 2 ) 256 256 Detector number 256 256 Reconstruction gri (mm 3 ) 256 256 256 (a) (b) (c) () Fig. 4. Ieal an reconstructe slices of the 3D Shepp-Logan phantom at Y = 0.25 using the half-scan Felkamp algorithm an our HERB algorithm, respectively. (a) Phantom, (b) Full-Scan ( = 400 mm), (c) T-FDK ( = 400 mm), an () HERB. The grayscale was mappe to [0,255] from the original interval [1.00.1.05]. 3. Results In the numerical simulation, the 3D Shepp-Logan phantom was use to evaluate the HERB algorithm against the Felkamp-type algorithms in the full- an half-scan formats, respectively. Noise-free conebeam projections were analytically synthesize on a panel etector. The simulation parameters are summarize in Table 1. Representative reconstruction results are shown in Figs 4 an 5. It can be observe that the traitional Felkamp-type reconstructions suffer from a significant intensity ropping away from the mi-plane, which is a well-known rawback of the Felkamp algorithm. On the other han, our HERB reconstruction improve the image quality remarkably even compare with T-FDK [2], which gives better image quality than FDK an P-FDK [3]. Specifically, the HERB algorithm suppresse this sort of artifacts, even for a fairly large cone angle, which is several times larger than that with the traitional Felkamp-type reconstruction for a similar image quality. However, there still are substantial image artifacts when the cone angle is large.
80 K. Zeng et al. / A half-scan error reuction base algorithm for cone-beam CT (a) (b) Fig. 5. Intensity profiles along the central certical line in the selecte slice of the 3D Shepp-Logan phantom. (a) Entire profiles, an (b) zoome protions of the profiles. 4. Discussions an conclusion Our HERB algorithm took roughly the same computational time as the Felkamp algorithm, an was easily implemente base on a Felkamp program. Its primary isavantage is that it requires two half-scans, an may increase the raiation ose. However, such a ose increment is not always necessary. For example, for a given ose requirement we can istribute it in the two half-scans, then use a smart filtering scheme to maintain an aequate signal to noise ratio in a specific application. Furthermore, in many inustrial CT applications the ose level is not a critical concern. In these contexts the ose with our new algorithm shoul be tolerate since the cone angle can be now mae several times larger than
K. Zeng et al. / A half-scan error reuction base algorithm for cone-beam CT 81 that of the Felkamp algorithm for a given image quality. Base on our arguments for the relationship between the reconstruction error an the scanning raius, it is warne that the HERB approach will not perform well when the attenuation coefficient varies greatly. In other wors, the HERB algorithm is more suitable to reconstruct low contrast structures, which are common in applications of non-estructive evaluation an meical imaging. This is because Raon ata of low contrast structures vary smoothly, while that of asymmetric high-contrast objects may change rapily. As a result, the relationship Eq. (10) may fail to estimate the reconstruction error in that high-contrast case [13]. As shown in Fig. 5(a,b), a slight overcompensation happene near the region with sharp changes in attenuation coefficient. To overcome this rawback we may apply an aaptive factor instea of a irect combination of the relevant terms to control this magnitue of compensation, which is left for future research. There are some artifacts associate with half-scan in HERB when the cone angle is large; for example, larger than 15, as shown in Fig. 5(). Such artifacts are cause by the non-exactness of the algorithm. Specifically, the larger cone angle, the less reunant the opposite X-rays are. These artifacts are not significant when the cone angle is small; for example, about 10. Although our work has targete the imaging geometry of two half-scans of ifferent raii, the results can be extene in several ways. Base on other error analysis results, such as [12], we can vary the helical scanning pitch to estimate an compensate for the reconstruction error. The iea can be also applie in the isplace etector configuration in both circular an helical moes [14,15]. In aition to the stanar Felkamp algorithm, we can also implement the error reuction mechanism in other moifie Felkamp algorithms to achieve similar enhancement, such as P-FDK [3] an T-FDK [2]. In 2001, Katsevich propose exact cone-beam algorithms [16], which is consiere a major avancement relative to the earlier spiral cone-beam algorithms. However, the Katsevich metho is for exact reconstruction, which cannot be use in the case of circular scanning where cone-beam ata are incomplete. Therefore, from a practical viewpoint our work shoul be of some value. On the other han, some insights an ieas suggeste by Katsevich s results may be aapte in our further stuies for better approximate reconstruction. In conclusion, we have propose a Felkamp-type algorithm to increase the cone angle by several fols at a raiation ose comparable to that require by the Felkamp algorithm. The merit of this half-scan error reuction base (HERB) algorithm has been numerically emonstrate. Further refinements an extensions are unerway. References [1] L.A. Felkamp, L.C. Davis an J.W. Kress, Practical cone-beam algorithm, J. Opt. Soc. Am. A 1 (1984), 612 619. [2] M. Grass, etc, 3D cone-beam CT reconstruction for circular trajectories, Phys. Me. Biol 45 (2000), 329 347. [3] H. Turbell, Cone-beam reconstruction using filtere backprojection, Linkoping Stuies in Science an Technology, Thesis no 672, 2001. [4] G. Wang, T.H. Lin, P.C. Cheng an D.M. Shinozaki, A general cone-beam reconstruction algorithm, IEEE Trans on Me Imaging 17 (1993), 361 370. [5] F. Noo an M. Defrise, Single-slice rebinning metho for helical cone-beam CT, Phys. Me. Biol. 44 (1999), 561 570. [6] F. Noo, H. Kuo an M. Defrise, Approximate Short-scan Filtere-backprojection for Helical CB Reconstruction, IEEE Nuclear Science Symposium 3 (1998), 2073 2077. [7] D.L. Parker, Optimal short scan convolution reconstruction for fan beam CT, Me. Phys. 9 (1982), 254 257. [8] Y. Liu, H. Liu an G. Wang, Half-scan cone-beam CT fluoroscopy with multiple x-ray sources, Me. Phys 28 (2001), 1466 1471. [9] P. Grangeat, Mathematical framework of cone beam 3-D reconstructn via the first erivative of the raon transform, in: Mathematical Methos in tomography, G.T. Herman, A.K. Louis an F. Natterer, es, Springer Verlag, 1990.
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