scans. We show below that, when applied to data acquired in full- and short-circular cone-beam scans, the algorithm in Eq. (6) and the FFBP algorithm [1] yield ...
New Filtered-backprojection-based Algorithms for Image reconstruction in Fan-beam Scans Dan Xia, Yu Zou, Lifeng Yu and Xiaochuan Pan Department of Radiology, The University of Chicago 5841 S Maryland Avenue, Chicago, IL 60637 A BSTRACT Fan-beam scan is a configuration widely used in clinical computed tomography (CT) because it is easy to implement and control. Filtered-backprojection(FBP)-based algorithms have been developed previously for exact reconstruction of the entire image or an ROI within the image from the data acuqired in fanbeam scans. Recently, we have developed a 3D FBP-based algorithm for image reconstruction on PI-line segments in a helical cone-beam scan. In this work, we demonstrated that the 3D FBP-based algorithm indeed provided a rather general FBPbased formulation for image reconstruction. Based upon this formulation we derived new FBP-based algorithms for image reconstruction in the fan-beam scans, which can be interpreted as the special case of the helical scan. Existing algorithms corresponding to the new derived algorithms were identified. We also performed a preliminary numerical study to verify our theoretical results. The results in the work can readily be generalized to other non-circular trajectories. Index Terms: computed tomography (CT), image reconstruction, fan-beam I
I NTRODUCTION
Most of clinical used computed tomography systems adopt fan-beam scanning configuration for data aquisition because it is easy to implement and control. A fan-beam scan that covers an angular range of 2π is referred to as a full scan. When it covers an angular range of π plus the fan-angle, it is referred to as a short scan. Furthermore, we refer to a circular scan covering an angular range smaller than that in a short scan as a reduced scan. Algorithms have been proposed for image reconstruction from the data aquired in the fan-beam scan[1, 2, 3]. Among them, the filtered-backprojection(FBP)-based algorithms are of particular interest theoretically and are most widely used in practice. The fan-beam FBP (FFBP) algorithm [1] has been developed for image reconstruction in full- and short-circular fan-beam scans, and in the last few years, FBP-based algorithms were proposed for reconstructing exactly 2D ROIs in a reduced-circular fanbeam scan [4] . Recently, we have developed a 3D FBP-based algorithm for image reconstruction on PI-line segments in a helical cone-beam
scan [5]. Because the fan-beam scan can be interpretted as the special case of a helical cone-beam scan by letting pitch and cone-angle be zero, we can derive new FBP-based algorithms for image reconstruction from the data acquired in full-, shortand reduced-circular fan-beam scans based upon our 3D FBPbased algorithm. Furthermore, we demonstrate that these newly derived algorithms yield reconstructions mathematically identical to those produced by the corresponding existing algorithms. We also perform a preliminary numerical study to verify our theoretical results in each of the cases. We organize the paper as follows. In Sec. II, we derive new algorithms for image reconstruction in the fan-beam scan, and show that these algorithms produce images that are mathematically identical to those reconstructed with most of the existing algorithms. The results of our preliminary numerical study are presented in Sec. III. Finally, in Sec. IV, we make remarks on the implications and possible generalization of our theoretical and numerical results. II
N EW FBP- BASED ALGORITHMS FOR FAN - BEAM SCANS
As will be described below, based upon our 3D FBP algorithm for image reconstruction in helical cone-beam CT [6, 7], we derive fan-beam FBP algorithms for exact image reconstruction. Furthermore, most of existing algorithms corresponding to the derived algorithms were identified. A PI-line segments in a fan-beam scan The concept of PI-line segments plays an important role in the algorithm development for image reconstruction in our 3D reconstruction algorithms[6, 7], and it can readily be generalized to a fan-beam scan. In the fan-beam case, a PI-line segment is a straight line segment joining two points labeled by the rotation angles λ1 and λ2 on the trajectory, as shown in Fig. 1a. We use xπ to denote the coordinate of a point on the PI-line segment and refer (xπ , λ1 , λ2 ) as the PI-line coordinates. Let λmin and λmax denote rotation angles at the starting and ending points of the trajectory. The relationship between the fixed coordinates (x, y) and the PI-line coordinates (xπ , λ1 , λ2 ) is determined by x = R [(1 − t) cosλ1 + t cosλ2 ] y
= R [(1 − t) sinλ1 + t sinλ2 ],
0-7803-8701-5/04/$20.00 (C) 2004 IEEE
(1)
u λ1
R
λ min
λ max
r
y
λ2
xπ
w
x
a
a
b
Fig. 1. (a) A PI-line segment is a straight line segment joining two points, labeled by λ1 and λ2 on the trajectory with a starting angle λmin and an ending angle λmax . ( b) Illustration of the fixed-coordinate system {x, y} with its origin on the center of rotation of the source, and the rotation-coordinate system {u, v} with its origin on the source point. The radius of the trajectory is R.
where t ∈ [0, 1] is related to xπ through 1 (2) xπ = (t − ) |r0 (λ1 ) − r0 (λ2 )|, 2 and r0 (λ), denoting the location of the source point, can be expressed as r0 (λ) = (R cosλ, R sinλ)T
(3)
b
c
d
Fig. 2. Illustration of the intersecting and non-intersecting PI-line segment sets within a circular trajectory: (a) A point at r within a circular trajectory can belong to an infinite number of PI-line segments, (b) a set of parallel PI-line segments, (c) a set of converging PI-line segments, and (d) a set of combinations of parallel and coverging PI-line segments.
The algorithm in Eq. (6) reconstructs an image only on an individual PI-line segment. In the case of a fan-beam scan, as shown in Fig. 2a, any point within a circular trajectory can belong to an infinite number of PI-line segments. This is because, as Eq. (1) indicates, for two given variables x and y, one needs to solve for three variables λ1 , λ2 and t. Fortunately, it can be observed in Fig. 2b-2d that one can always identify sets of PI-line segments that do not intersect with each other and thus that can fill completely the area enclosed by a circular trajectory. Therefore, when the algorithm in Eq. (6) is used to reconstruct an image on this type of non-intersecting PI-line segments, it in effect reconstructs the whole image.
in the fixed-coordinate system. It is also beneficial to introduce a rotation-coordinate system {u, w} for characterizing data on the detector. We assume that the origin of the rotation-coordinate system is on the source point r0 (λ). In the fixed-coordinate system, as shown in Fig. 1b, for a rotation angle λ, the unit C Image reconstruction in full- and short-circular fan-beam vectors along the u- and w-axis can be written as ˆeu (λ) = scans (−sinλ, cosλ)T and ˆew (λ) = (cosλ, sinλ)T . Furthermore, the We show below that, when applied to data acquired in fullfixed and rotation coordinates, (x, y) and (u, w) are related through and short-circular cone-beam scans, the algorithm in Eq. (6) x = −u sinλ + (w + R) cosλ and the FFBP algorithm [1] yield mathematically identical imy = u cosλ + (w + R) sinλ. (4) ages. Consider the image reconstruction on a PI-line segment specified by λ1 and λ2 . In full-scan case, as displayed in Fig. Without loss of generality, consider a line detector that is always 3a, the image on this PI-line segment can be reconstructed from parallel to ˆeu (λ) and that has a distace S from the source. We data acquired over the portion of the circular trajectory either on use ud as the coordinate of a point on the detector, and one can the right-hand-side (i.e., from λ1 to λ2 ) or on the left-hand-side readily show that (i.e., from λ2 to λ1 ) of the PI-line segment. Averaging this two S (5) reconstructions, one can subsequently obtain a new reconstrucud = − u. w tion formula on the PI-line segment, which can be expressed as B The fan-beam FBP algorithm for image reconstruction on a PI-line segment Based upon our 3D FBP algorithm for image reconstruction in helical cone-beam CT, we obtain an algorithm, in terms of the notations defined above, for image reconstruction on a PIline segment specified by λ1 and λ2 in a circular fan-beam scan: f (r) = − 2π1 2 − 2π1 2
λ2 (r)
2
(ud ) dλ |rA − r0 (λ)|2 dud A(ud ) λ1 ( r)
| r − r0 (λ)|
R ud −ud
dud P (ud ,λ) R ud −ud A2 (ud ) λ2 (r) P (ud ,λ) , A(ud ) λ1 ( r)
(6) where P (ud , λ) denotes the projection data, and P (ud , λ) can be calculated by the following equation: P (ud , λ)
0 (λ) ˆ (u , λ) = −[ drdλ · β]P d ∂P (u ,λ) d r0 (λ) +[ dλ · ˆeu (λ)]A(ud ) ∂ud . d
(7)
f (r) =
1 2
2π 0
2
S √ dλ [R−r·ˆ ew (λ)]2 [
R S 2 +u2d
P (ud , λ)] ∗ n(ud ),
(8) where the notation “*” denotes a 1D convolution and n(ud ) denotes the spatial function of the ramp filter. It can be seen that the expression in Eq. (8) is exactly the same as that of the FFBP algorithm. Therefore, our algorithm in Eq. (6) yields a combined reconstruction that is mathematically identical to that obtained with the FFBP algorithm when applied to data acquired in a full-circular fan-beam scan. In a short fan-beam scan, the FFBP algorithm has been extended to reconstruct an image from the data acquired only over π plus the fan-beam angle. In this case, a short-scan data should be normalized appropriately by a weighting function ω(ud , λ), which satisfies the condition [8]
0-7803-8701-5/04/$20.00 (C) 2004 IEEE
λmax
λ2
λ2
λmin λ1
λ1
a
b
Fig. 3. Illustration of data redundancy with respect to a specific PI-line. According to Eq. (6), our algorithm requires data only over [λ1 , λ2 ] for image reconstruction on the PI-line segment specified by λ1 and λ2 . (a) In full scan case, the image on the PI-line segment can be reconstructed from data over the portion of the trajectory on the right-hand-side or on the left-hand-side of the PI-line segment. (b) In a reduced scan over [λmin , λmax ], data redundancy occurs because the algorithm in Eq. (6) reconstructs the image on the PI-line segment without using data over [λmin , λ1 ) and over (λ2 , λmax ].
ω(ud , λ) + ω(−ud , λ + π − 2 arctan( uSd ))
=
1.
(9)
Following the same procedure for showing the equivalence between the combined reconstruction of our algorithm in Eq. (6) and the reconstruction of FFBP algorithm, one can readily show that, our algorithm in Eq. (6) yields a combined reconstruction that is mathematically identical to that obtained with the FFBP algorithm when applied to data acquired in a short fan-beam scan. D Image reconstruction in reduced-circular fan-beam scans In the case of a reduced-circular fan-beam scan, the starting and ending rotation angles satisfy λmax − λmin < π + 2γm , and 2γm is the fan-angle. Let us consider a PI-line segment specified by λ1 and λ2 , where λmin < λ1 ≤ λ2 < λmax . Obviously, the algorithm in Eq. (6) reconstructs the image on the PI-line segment from data only over [λ1 , λ2 ]. Therefore, the reducedscan data contain redundant information with respect to this PIline segment because data over [λmin , λ1 ) and (λ2 , λmax ] are not used, as shown in Fig. 3b. We have shown that our algorithm in Eq. (6) can be relate to the algorithm that was obtained previously by Noo et al (2002) for exploiting such redundant information, which can be written as f (r) =
1 2π
λmax λmin
dλ R− r ·ˆ ew [ω(ud , λ) g(ud , λ)],
(10)
where the filtered fan-beam data g(ud , λ) can be written as g(ud , λ)
=
dud R ud −ud
√
S S 2 +ud 2
dP (ud ,λ) dλ
βˆ
.
(11)
Following the procedure similar to that in Appendix A in Ref. [5], Eq. (6) can be re-expressed as f (r) = − 2π1 2
λ2 λ1
dλ R− r ·ˆ ew (λ)
g(ud , λ),
(12)
Noticing that the filtered fan-beam data g(ud , λ) satisfies the symmetry condition
g(ud , λ)
r ·ˆ eu (λ) = g(−ud , λ + π − 2tan−1 R− r ·ˆ ew (λ) ),
one can readily show that Eq. (12) is mathematically equivalent to the Eq. (10). So our formula (6) yields the same reconstruction as that yields by Noo’s algorithm mathematically when applied to the reduced scan. In realistic fan-beam imaging situations, the samples on a detector are generally much denser than the angular samples. In order to fully utilize these information, we have previously reexpressed such an angular derivative of the data function P (ud , λ) in Eq. (11) in terms of the detector coordinates ud [5]. Using such an expression for the angular derivative of the data function in Eq. (11), we obtain another algorithm for image reconstruction in a reduced fan-beam scan,
(13)
f (r) = − 2π1 2 − 2π1 2 − 2π1 2
λmax
dud Pω (ud ,λ) R ud −ud A2 (ud ) dud P (ud ,λ) dω d) dλ |rA(u − r0 (λ)| dλ R ud −ud A(ud ) dud P (ud ,λ) λmax A(ud ) ω , | r − r0 (λ)| R ud −ud A(ud ) λ min λ λmin max λmin
2
(ud ) dλ |rA − r (λ)|2 0
(14) where ω indicates the weighting function satisfying Eq. (9), and 0 (λ) Pω (ud , λ) = − drdλ · βˆ ω P (ud , λ) (ud ,λ) 0 (λ) + drdλ · ˆeu (λ) A(ud ) ω ∂P∂u . d (15) III
N UMERICAL S TUDIES
We have performed a computer-simulation study to demonstrate quantitatively the theoretical results discussed in Secs. C and D. In the numerical study, we considered a fan-beam configuration in which the trajectory has a radius of R = 29.2 cm and the source-detector distance is S = 54.9 cm. The low constrast 2D Shepp-Logan phantom was used. The detector array consists of 256 detection elements with a size of 0.78 mm. We generated data at 300 projection views over [0, 2π] on the circular trajectory. The entire set of data was used to simulate a full scan and a portion of the full scan data over the angular range [0, π + 2γm ] and [0, π] were used to simulate short and reduced scans, respectively. We also generated noisy data by adding Gaussian noise to the simulated noiseless data. Because the lowest contrast level in the Shepp-Logan phantom is about 1%, the Gaussian noise with a standard deviation of 0.063% was used. From the noiseless data acquired in full and short scans, we used our algorithm in Eq. (6) and the conventional FFBP algorithm to reconstruct images, which are displayed in Fig. 4. We also have conducted a numerical study for a reduced-circular fan-beam scan. From the noiseless data, we used our algorithm in Eq. (6) and Noo’s algorithm in Eq. (10) to reconstruct the ROIs, which are shown in the Fig. (5). The results in Figs. 4 and 5 verify our theoretical prediction that, under ideal condi-
0-7803-8701-5/04/$20.00 (C) 2004 IEEE
cases. As expected, these algorithms in their discrete forms respond differently to data inconsistencies such as noise. We have conducted a preliminary numerical study, in which the quantitative results corroborate with our theoretical analysis. It should also be pointed out that, although our FBP-based algorithms for image reconstruction on PI-line segments is formulated based only upon circular trajectories, they can readily be generalized to other trajectories such as the elliptical trajectory. a
b
c
d
Fig. 4. Noiseless (upper row) and noisy (lower row) images reconstructed by use of our alogrithm in Eq. (6) (panels (a) and (b)) and by use of FFBP algorithm (panels (c) and (d)) from the data acquired in a full scan (panels (a) and (c)) and the data acquired in a short scan ( panels (b) and (d)). The display grey scale is [1.0,1.04].
V
ACKNOWLEDGMENTS
This work was supported in part by National Institutes of Health grant EB00225. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of the National Institutes of Health. The authors gratefully acknowledge the use of the Chiba City Linux cluster in the Mathematics and Computer Science Division of Argonne National Laboratory. R EFERENCES
a
b
c
Fig. 5. Noiseless (upper row) and noisy (lower row) images reconstructed by use of our alogrithms in Eqs. (6) (panel a) and (14) (panel b) and by use of Noo’s algorithm (panel c) in the reduced scan. The display grey scale is [1.0,1.04].
tions, our new algorithms yield images identical to those obtained with the existing algorithms. When the discrete data contain significant inconsistency such as noise, we expect that the these algorithms respond differently to such inconsistency. This observation is verified by the images in Figs. 4 and 5, which were reconstructed from noisy full scan data by use of our algorithm and the FFBP algorithm , respectively. IV
C ONCLUSION
In this work, based upon our recent results on image reconstruction in a helical cone-beam scan, we have derived new FBP-based algorithms for image reconstruction from data acquired in fan-beam scans. In particular, we have made an observation that a circular fan-beam scanning configuration can be obtained from a helical trajectory by letting the pitch and cone-angle be zero. Therefore, starting out from our FBP-based algorithm for image reconstruction on PI-segments in a helical cone-beam scan, we developed algorithms for image reconstruction on PI-line segments from data acquired with full, short and reduced fan-beam scan. Furthermore, we have identified the existing counterpart algorithms for our derived algorithms, and demonstrated that the newly derived algorithms and their existing counterparts yield mathematically identical images in most
[1] A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, IEEE Press, New York, 1988. [2] X. Pan, “Optimal noise control in and fast reconstruction of fan-beam computed tomography image,” Med. Phys., vol. 26, pp. 689–697, 1999. [3] X. Pan and L. Yu, “Image reconstruction with shift-variant filtration and its implication for noise and resolution properties in fan-beam computed tomography,” Med. Phys., vol. 30, pp. 590–600, 2003. [4] F. Noo, M. Defrise, R. Clackdoyle, and H. Kudo, “Image reconstruction from fan-beam projections on less than a short scan,” Phys. Med. Biol., vol. 47, pp. 2525–2546, 2002. [5] Y. Zou and X. Pan, “Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT,” Phys. Med. Biol., vol. 49, pp. 2717–2731, 2004. [6] Y. Zou and X. Pan, “Exact image reconstructin on PI-line from minimum data in helical cone-beam CT,” Phys. Med. Biol., vol. 49, pp. 941–959, 2004. [7] Y. Zou and X. Pan, “An extended data function and its backprojection onto PI-lines in helical cone-beam CT,” Phys. Med. Biol., (submitted), 2004. [8] D. L. Parker, “Optimal short scan convolution reconstruction for fan-beam CT,” Med. Phys., vol. 9, pp. 245–257, 1982.
0-7803-8701-5/04/$20.00 (C) 2004 IEEE
