Cone-beam reconstruction for the 2-circles plus line trajectory
Yanbin Lua,*, Jiansheng Yanga, John W. Emersonb, Heng Maoa, Yuanzheng Sia and Ming Jianga a LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China b Department of Statistics, Yale University, New Haven, CT, 06511, USA
Abstract. The 2-circles plus lines trajectory is designed to solve the problem of limited physical space in a multimodality imaging system for the introduction of X-ray cone-beam CT. We apply the backprojection filtration algorithm proposed by Pack et al for X-ray cone-beam data to the standard 2-circles plus lines trajectory. We find that for every point inside the region between the two circles there exists at least one redundantly measured line (a line segment that connects two sources points, R-line), and so the region can be fully covered by R-lines. Furthermore, for about 2/3 of the points inside the region there exist two or more such R-lines. The length of the line curve segment required by the algorithm can be as short as the distance between the two circles when the projections have no truncation. Numerical simulations validate the feasibility of the algorithm and the spiral resolution of cone-beam CT in the imaging system.
Key words: Cone-beam CT, backprojection filtration algorithm, 2-circles plus line trajectory.
1. Introduction [Molecular imaging and multimodal imaging] Imaging modalities can be roughly classified into two categories: those which primarily provide structural information (such as Computed Tomography (CT) and Magnetic Resonance Imaging (MRI)) and those which primarily provide functional information (such as Positron Emission Tomography (PET) and Near Infrared Optical Tomography (OT)). Optical tomography is an emerging functional imaging technique. Its objective is to reconstruct the spatial distribution of optical parameters inside the medium based on measurements of transmitted light intensities on the surface [1]. Depending on the physical problem, OT can fall into many sub-categories, such as diffuse optical tomograph (DOT), florescence molecular tomography (FMT), and bioluminescence tomography (BLT). It has been recognized as a promising molecular imaging modality because of its sensitivity, specificity, low-cost and non-radiation characteristics. However, it often suffers from poor spatial resolution because of the high scattering nature of photons inside the tissue [2]. Thus hybrid systems that utilize multimodality data to combine the advantages of different imaging techniques have been explored. The use of structural information from CT or MRI to penalize the objective function of OT has been studied by many research groups [1, 3, 4]. [The earlier cone-beam CT] The main aim of X-ray CT reconstruction is to recover the object’s absorption coefficient using projection data acquired along a scanning source trajectory. Since Hounsfield reported the first x-ray computed tomography (CT) scanner in 1973 [5], reconstruction methods have made remarkable progress. Tuy proposed an exact reconstruction formulae with Tuy’s data sufficiency condition for cone-beam CT [6]. Related to the Radon transform, Smith’s formula [7] and Grangeat’s formula [8] provided two other classic formulae for cone-beam CT. To solve the long object problem for helical
* Corresponding author: E-mail: [email protected].
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cone-beam CT, Wang et al proposed a Feldkamp-type algorithm [9, 10], and Kudo, Noo and Defrise proposed exact and approximate algorithms in [11-13]. [Recent development of cone-beam CT, helical CT] A series of new cone-beam formulae were developed in the past decade. They can be roughly classified into two categories: filtered-backprojection (FBP) approaches and backprojection-filtration (BPF) approaches. Katsevich reported the first exact shift-invariant FBP reconstruction formula for helical CT using more data than the Tam-Dannielson window [14], and then reduced the requirements in [15]. In his formula, one-dimensional filtrating of a derivative of the projection data was first done along particular lines on the detector, and the filtration results were backprojected to form the image. Its computation is more efficient than previous exact formulae. Zou and Pan then proposed a BPF format formula using data only in the Tam-Dannielson window [16]. They claim that the Hilbert transform of the absorption coefficient along a PI-line (which is also 1D filtrating) can be obtained by backprojection of a derivative of the projection data on this PI-line. Helical CT motivated theoretical research of cone-beam CT. Katsevich developed a general cone-beam FBP formula based on the Grangeat’s formula using a general weight function [17]. Chen et al proposed an alternative derivation of the Katsevich algorithm based on the Tuy’s formula with relaxed Tuy’s data sufficiency condition [18]. Zhao et al gave their proofs for Katsevich’s formula and extended Zou and Pan’s formula to any interior point on an R-line on a continuous scanning curve in the Tuy framework [19]. In this case, a redundantly measured line (R-line) is a line segment connecting two separate source positions. Ye et al proposed a generalized BPF formula for reconstruction on R-lines for a single smoothing scanning curve [20]. Ye and Wang proposed a FBP formula and gave suggestions of filtering directions for a smooth curve [21]. Pack et al proposed BPF reconstruction formulae for both a single smoothing curve and multiple smooth curves [22]. Reconstructions by their formulae could be more flexible by choosing R-lines and M-lines in the image space. A measured line (M-line) is a radial line starting from a source position such that for every point in the intersection of this M-line and the object there is at least one R-line passing through it. Pack and Noo also developed FBP reconstruction formula for multiple smooth curves [23]. All these formulae demanded that for every interior point there existed an R-line passing through it. [Our work] In this paper we study a special scanning trajectory for X-ray cone-beam CT in a multimodality imaging system called the Kodak In-Vivo Imaging System FX (Carestream Health, Woodbridge, CT). The system can provide 2D X-ray images, florescence images, bioluminescence images and radioisotopic images, but the precise 3D reconstruction images can’t be achieved from these images directly. In order to introduce the X-ray cone-beam CT, an innovation of the system is proposed here by implementing object movement mechanisms (such as rotation and translation mechanisms) into the system. Because the physical space in the imaging cabinet is very limited, we apply the general BPF algorithm for multiple smooth curves proposed by Pack et al [22] to a special scanning trajectory: 2-circles plus line trajectory. This 2-circles plus line trajectory consists of two parallel circles with radius R and a line orthogonal to the plane of the circles. It can be described as
R cos, Rsin, z1 if [0,2 ) 2 (1) a R,0, z1 z2 z1 if [2 ,4 ) 2 R cos, Rsin, z2 if [4 ,6 ) We find that the two-circles plus line trajectory has nice geometry properties for reconstruction. It is proved that for every point inside the region between the two circles, there exists at least one R-line. This means that the region between two circles can be fully covered by R-lines. Assuming
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data sufficiency for reconstruction, that region can be fully reconstructed by using the general algorithm on many-curve source trajectory. Finally, the number of R-lines for each point can be determined. [Organization of paper] The paper is organized as follows. In Section 2, we review the BPF algorithm in [22] and derive it for the 2-circles plus line trajectory. The geometry property of the 2-circles plus line trajectory is described, including the existence and calculation of R-lines. Section 3 shows numerical simulations and Section 4 discusses characteristics of this trajectory and the data requirement of the algorithm. Section 5 concludes the paper.
2. Method
2.1 Notations
Let be a bounded convex neighbourhood of the object in R3 . The 3D source trajectory a() consists of a union of smooth curves. One example is the 2-circles plus line trajectory defined by (1). Two assumptions of the trajectory are needed here: (1) a() locates outside ; (2) For interior source points of every single smooth curve, the tangent vector a'() da()/ d is bounded, continuous and nonzero. Let r be a 3D point and f (r) : R3 R be the 3D function of the objects’ absorption coefficient of X-ray. f (r) equals to zero for r . The cone-beam projection
at a() is defined as g(,) f (a() t)dt (2) 0 where S 2 is the unit vector in R3 . The local coordinate can be introduced into the detector plane as show in Figure 1. Let the orthogonal projection of a() onto the detector plane be the origin of the detector plane and ew perpendicular to the detector plane. eu and ev are orthogonal unit vectors on the detector plane.
Figure 1. Local orientation on detector plane. The origin of the detector plane locates at the orthogonal projection of a() onto the detector plane. ew perpendicular to the detector plane. eu and ev are orthogonal unit vectors on the detector plane.
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2.2 General formula of Pack et.al
The idea behind BPF algorithms is that the backprojection of derivatives of the projection data can be linearly transferred to the Hilbert transform of f (r) on a line segment. Here we introduce the formula for multiple source curves based on [22]. First, two important concepts must be introduced. As shown in Figure 2, redundantly measured lines (R-lines) are line segments that connect two different source positions, and measured lines (M-lines) are radial lines from the source trajectory.
Figure 2. Illustration of R-lines and M-lines. R-lines are line segments that connect two different source positions, and M-lines are radial lines from the source trajectory.
[General formula] The differentiated backprojection (DBP) at r from source position a(1) to a(2 ) on one single curve can be achieved by the backprojection of the derivates of projections
2 D a'() D * * B(r,1 ,2 ) 2 g,u ,v d 1 u 2 2 2 a() r ew D u v (3) g ,( ,r) g ,( ,r) 2 2 1 1 r a(2 ) r a(1) where D is the distance from a() to the origin of the detector, u*(,r) and v*(,r) are the local coordinates of the intersection point of the detector and the line connecting r and a() . r a()e _ r a()e u*(,r) D u , v*(, x) D v . (4) r a()e r a()e w w The relation between DBP and the Hilbert transform of f (r) is given below. For a fixed point r , suppose there exists N source intervals [a(i ),a(i )] for i 1,...,N on the trajectory such that (1) each interval belongs to only one smoothing curve and (2) r , a( ) and a( ) are i i1 r a() on the same line for i 1,...,N 1. Let (,r) be the unit vector from a() to r . r a() i Define k (,r) ( ,r) and k ( ,r) (,r) for i 2,...,N , and q k for 1 1 N i i1 i i j1 j i 1,...,N . Let K(t,s,) be the Hilbert transform of f (r) along a line passing through s R3 with direction
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1 K(t,s,) f (s t')dt' . (5) (t t') The relation between DBP and the Hilbert transform of f (r) is given by 1 N 2K(0,r,( ,r)) q 1 q B(r,, ) N N . (6) i1 i i i 0 qN 1 From (5) and (6), the Hilbert transform of f (r) on the M-line from a(N ) with the direction (N ,r) can be obtained. Once the Hilbert transform of the intersection of this M-line and have been calculated, Tricomi formula can be used for finite inverse Hilbert transform [24]. The computing details can be found in [22]. Using the inverse Hilbert transform, we will get the value of f (r) along this line. If can be covered by such M-lines, then f (r) can be obtained all over .
2.3 M-line reconstruction formula for the 2-circles plus line trajectory
In this part we give the reconstruction formula for 2-circles plus line trajectory in R3 . This trajectory consists of two parallel circle curves and a line curve segment orthogonal to the plane of the circles as shown in Figure 3.
Figure 3. Illustration of the 2-circles plus line trajectory. a( ) and a( ) are the circle curves on z z and C1 C2 1 z z2 separately, and a(L ) is the line curve segment.
The two parallel circle curves have the same radius. The line curve segment is parallel to the line connecting the centres of the two circles. The line segment and the two circles intersect at the endpoints of the line curve segment. The line connecting the centres of the two circles is denoted as z axis, and the two circle curves locate at z z1 and z z2 . The line curve segment is on the xz plane. The trajectory can be described as (1). M-line reconstruction requires that for each point in the intersection of the M-line and , there exists at least one R-line passing through this point. In this case we choose M-lines that start from the line curve segment with direction parallel to the circle curves, such as the line from a(1 ) with direction in Figure 4. There are two kinds of R-lines. The first kind of R-line connects source positions on one of the two circles and the line, such as a(2 ) a(3 ) in Figure 4 (a). The second kind
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of R-line connects source positions on the two circles, such as a(2 ) a(3 ) in Figure 4(b). The M-line reconstruction formula for the 2-circles plus line trajectory has two cases because of the different choices of R-lines. Case 1: Suppose r is on the first kind of R-line, as shown in Figure 4(a). It is easy to verify k 1, k 1 and k 1, thus q 1, q 1 and q 1. Since q 1, the Hilbert transform 1 2 3 1 2 3 3 of f (r) follows from (6). 1 3 K(0,r,) K(0,r,( ,r)) K(0,r,(,r)) q B(r,, ) . (7) 3 1 2 i1 i i i Case 2: Suppose r is on the second kind of R-line, as shown in Figure 4(b). Then q1 1,
q2 1, q3 1 and q4 1, so 1 4 K(0,r,) K(0,r,( ,r)) K(0,r,(,r)) q B(r,, ) . (8) 4 1 i1 i i i 2 Then we can get f (r) along the M-lines using the inverse Hilbert transform. If is covered by such M-lines, then f (r) can be obtained all over .
(a) (b) Figure 4. Illustration of R-lines and M-lines of the 2-circles plus line trajectory. M-lines start from the line curve segment source position with direction parallel to the circle curves. There are two kinds of R-lines: (a) a(1 ) connecting source positions on one of the two circle curves and the line curve segment, and (b) connecting source positions on the two circle curves.
2.4 Geometry properties
The reconstruction formula is based on both R-lines and M-lines as mentioned previously. So it is important to investigate the existence of the R-lines on this trajectory.
2.4.1 The existence of R-lines Suppose the 2-circles plus line trajectory is defined by (1). We use r (x, y, z) below for convenience. The region inside the trajectory can be defined as 2 2 RCPL (x, y, z) : x y R, z1 z z2.
Theorem 1: RCPL can be fully covered by R-lines of the 2-circles plus line trajectory. That is, for every point r inside RCPL , there exists at least one R-line passing through r .
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In order to prove theorem 1, the region covered by the two different kinds of R-lines will be proposed separately. The region covered by the first kind of R-lines will be given by Lemma 1, 2 and Lemma 2’. The region covered by the second kind of R-lines is given by Lemma 3 and 4. We consider the intersection of the R-lines with a circular region R (x, y, z) : x2 y2 R, z z with z (1 t)z tz , 0 t 1. z0 0 0 1 2 The R-lines from a fixed source position on the line curve segment to the circle curve segment on
z z1 are described in Lemma 1. Lemma 1: For a fixed point a( ) (R,0, z ) on the line trajectory with z z z , let L L 0 L 2 L1 be the surface which consists of the R-lines from a(L ) to the circle curve segment on z z1 . The z z z z intersection of and R is a circle centred at 2 1 tR,0, z with radius 1 t 2 1 R . L1 z0 0 zL z1 zL z1
Proof: For a point a(B ) Rcos(B ),Rsin(B ),z1 on the circle curve segment on z z1 , denote z z (1 t)z tz z z z t' 0 1 1 2 1 2 1 t . (9) zL z1 zL z1 zL z1
The R-line a(B ) a(L ) intersects at x 1 t'Rcos( ),Rsin( ),z t' R,0, z 0 B B 1 L . (10) t'R,0, z0 1 t'Rcos(B ),sin(B ),0 z z 2 1 Since B [0,2) , the intersection curve is a circle centred at tR,0, z0 with radius zL z1
z2 z1 1 t'R 1 t R . zL z1
The existence of R-lines from the line curve segment to the circle curve segment on z z1 is described by Lemma 2. Lemma 2: For every point inside R (x, y, z) : (x tR)2 y2 (1t)2 R2, z z , there exists at L1 0 least one R-line of the first kind. Proof: For any point (x, y, z) R , if we can find such that there exists a circle on R L1 L z0 z z z z 2 1 2 1 containing (x, y, z) with centre tR,0, z0 and radius 1 t R , then by Lemma 1 the zL z1 zL z1 point belongs to one R-line from a(L ) . It is equivalent to solve the equation z z z z (x 2 1 tR)2 y2 (1 2 1 t)2 R2 (11) zL z1 zL z1
z2 z1 z0 z1 with constrain condition z0 zL z2 . Let t' t , (11) is equivalent to zL z1 zL z1 (x t'R) 2 y2 (1 t')2 R2 (12) with constrain condition 1 t' t . So it follows that R2 (x2 y2 ) t' . (13) 2(R2 xR) Since (x, y, z) belonging to R satisfies (x tR)2 y2 (1 t)2 R2 , which is equivalent to L1
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R2 (x2 y2) 2(R2 xR)t . It is equivalent to t' t . It is easy to verify that t'1 by (13). Then 2(R2 xR) we have z z (z z ) / t' z (z z ) . So the point (x, y, z) belongs to one L 1 0 1 1 0 1 R2 (x2 y2 ) R-line from a( ) (R,0, z ) by Lemma 1. Since (x, y, z) is chosen arbitrarily from R , Lemma L L L1 2 is proved.
The existence of R-lines from the line curve segment to the circle curve segment on z z2 is described by Lemma 2’. Lemma 2’: For every point inside R (x, y, z):(x (1t)R)2 y2 t2R2, z z , there exists at L2 0 least one R-line of the first kind.
Proof: considering the intersection of R-lines from the line and the circle on z z2 , the conclusion can be proved similarly. The region covered by the second kind of R-lines is given by Lemma 3 and 4. We also consider the intersection of the R-lines with R . The R-lines from a fixed source position on the circle curve z0 segment on z z2 to the circle curve segment on z z1 are described in Lemma 3.
Lemma 3: For a fixed point a(A) (Rcos(A),Rsin(A),z2) , let C be the surface which consists of the R-lines from a(A) to the circle curve segment on z z1 . The intersection of C and R is a circle centred at (tRcos( ),tRsin( ),z ) with radius (1 t)R . z0 A A 0 Proof: For a point a( ) (Rcos( ),Rsin( ),z ) , the R-line a( ) a( ) must intersects R B B B 1 A B z0 at
x0 1 tRcos(B ),Rsin(B ),z1 t R cos(A ),Rsin(A ),z2
tRcos(A ),tRsin(A ),(1 t)z1 tz2 1 tRcos(B ),sin(B ),0. (14)
tRcos(A ),tRsin(A ),z0 1 tRcos(B ),sin(B ),0
Since A is fixed and B belongs to [0,2 ) , the intersection points consist a circle centred at
tRcos(A),tRsin(A),z0 with radius (1 t)R .
The existence of R-lines from the circle curve segment on z z2 to the circle curve segment on
z z1 is described by Lemma 4. 2 2 2 2 2 Lemma 4: Define RC (x, y, z):(1 2t) R x y R , z z0. For the inner points of RC , there exist two R-lines of the second kind. For the boundary points of RC , there exist only one R-line of the second kind.
Proof: For any (x, y, z) RC , if we can find A such that there exists a circle containing (x, y, z) with centre tRcos(A),tRsin(A) and radius (1 t)R , then by Lemma 3 the point belongs to one
R-line from a(A) . Let Rx tRcos(A) and Ry tRsin(A) , then it is equivalent to solve the equations 2 2 2 2 (x Rx ) (y Ry ) (1 t) R . (15) 2 2 2 2 Rx Ry t R From (12) we have 2 2 2 Rx x Ry y x y (1 2t)R / 2. (16) Letting r2 x2 y2 , the above equations have real solutions when
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2 4r2t 2r2R2 r2 (1 2t)R2 / 4 0 . (17) The inequality is equivalent to r2 R2 r2r2 (1 2t)2 R2 0. Note that 2 2 2 2 2 2 2 2 0 r R and r (1 2t) R (1 2t) R x y R , , (18) 2 2 2 2 2 2 2 2 2 2 0 r R or r (1 2t) R x y R or x y (1 2t) R 2 2 2 2 2 2 For any (x, y, z) RC , we have (1 2t) R x y r R which exactly lead to 0 . Then
(Rx ,Ry ) can be solved from (15). Two different solutions of (15) lead to two different A , which means two different R-lines. Only one solution of (15) leads to only one A , which means one R-line. So Lemma 4 is proved. Now we proof Theorem 1. Note that R R R R . By Lemma 2, Lemma 2’ and Lemma z0 C L1 L2 4, for every point in R there exists at least one R-line passing through that point. Since z can be z0 0 chosen arbitrarily, Theorem 1 is proved. Figure 5 shows the region covered by R , R , R when C L1 L2 t 1/ 2 .
Figure 5. Illustration of R R R R . The region inside the two dotted circles illustrate R (the larger one) z0 C L1 L2 L1 and R (the smaller one), and the region between two solid circles is R when 0 t 1/ 2 . L2 C
2.4.2 The Number of R-lines for Points in RCPL
In this chapter the number of R-lines for points in RCPL is given. Considering the circular region R (x, y, z) : x2 y2 R, z z z0 0
with z0 (1 t)z1 tz2 for any 0 t 1/ 2 . From Theorem 1, we already knew that R R R R . By Lemma 2, Lemma 2’ and Lemma 4, we can obtain the number of R-lines z0 C L1 L2 for points in R as shown in z0 Table 1. It is easy to obtain similar results when 1/ 2 t 1 due to symmetry. When t 1/ 2 , we have R R and R R . For every point outside R R , there exist two R-lines of the z0 C L1 L2 L1 L2 second kind. For every point inside R (R ) , there exist two R-lines of both the first kind and the L1 L2
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second kind. The centre of symmetry OC 0,0,(z1 z2 )/ 2 has infinite R-lines. For any
[0,2) , the line connects a() and OC must intersect the upper circle curve segment at one a(') satisfied ' 4 .
Table 1. The number of R-lines for points in R when 0 t 1/ 2 . z0 Number of R-lines Number of R-lines Region Region Description of the first kind of the second kind R / R 2 2 2 2 L1 C x y (1 2t) R 1 0 2 2 2 2 x y (1 2t) R , x 1 2t R 1 1 RC x 1 2t R, y 0 2 1 (x tR)2 y2 (1 t)2 R2 R / R z0 L1 0 2 x2 y2 R2 (x tR)2 y2 (1 t)2 R2 (R \ R ) R 2 2 2 2 L1 L2 C (x (1 t)R) y t R 1 2 x2 y2 (1 2t)2 R2
R R R 2 2 2 2 L1 L2 C (x (1t)R) y t R 2 2
z z Theorem 2: For points which satisfy x2 y2 (1 2t)2 R2 ( t 1 ) there exists only one z2 z1
R-line (except for the original). The volume of these points is 1/3 of the volume of RCPL . For Other 2/3 points, there exist more than two R-lines. The conclusion can be obtained immediately by Table 1 for 0 t 1/ 2 . Due to symmetry, it is easy to obtain similar results when 1/ 2 t 1.
2.4.3 Calculation of R-lines
Here we give a way to calculate the R-line for points inside RCPL . We propose a method based on 2.4.2.
z z1 Step 1: for points on Rz , compute t ; z2 z1 Step 2: calculate whether the point belongs to R , R or R . If it belongs to R , solve C L1 L2 C
(Rx ,Ry ) from (15). Then endpoints of the R-line are
a(A) (Rx /t, Ry /t, z2 ) 1 . (19) a(B ) (x, y, z) (Rx tx, Ry ty,tz2 tz) 1 t z z If it belongs to R , t' 2 1 t can be got from (13), then z is obtained. The endpoint of L1 L zL z1
R-line on the line curve segment is a(L ) (R,0, zL ) . By the proportion t' , it is easy to get the other t' endpoint a( ) (x, y, z) (R x,y, z z) . If it belongs to R , the computation is similar. If B 1 t' L L2 it belongs to R R (R ) , both of the R-lines can be chosen. C L1 L2
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2.5 The procedure of the algorithm
The algorithm can be implemented as follows. Step1: Choose suitable M-lines, for example, M-lines which can cover . Step2: Reconstruction on M-lines. For any M-line, Step2-1: Discretization. Discrete the intersection of the M-line and by determined sampling interval. Step2-2: Backprojection. For every discrete point on the M-line, Step2-2-1: Compute the endpoint of the R-line using the method in Section 2.4.3. Step2-2-2: Determine the backprojection interval [i ,i ],i 1,...,N following Figure 4. Step2-2-3: Compute the weighted backprojection by (3) for every interval. Step2-2-4: Compute the Hilbert transform along the M-line by (7) or (8). Step 2-3: Filtering. Reconstruct f (x) on the M-line by performing the finite inverse Hilbert transform. Step3: Interpolation. Obtain the reconstruction result in the nature coordinate system using linear interpolation.
3. Experiments [Introduction of simulation technique (R)] This section presents two numerical experiments which demonstrate the feasibility of the algorithm proposed for the 2-circles plus line trajectory. The algorithm is implemented in the R statistical programming environment [25]. It uses the bigmemory package to handle multiple-gigabyte matrices and to allow the algorithm to succeed even when the matrices exceed available RAM [26]. [Simulation of Shepp-Logon phantom] In the first experiment we use a 3D low contrast Shepp-Logan phantom with ten ellipsoids. Table 2 shows the parameters of the phantom. The support of the phantom is a cylinder with radius 1 and length 2. A rectangular detector plane consists of 600×600 detection pixels with a size of 0.0042. The radius R of the circle curve segment is 10. The two circles are located at z 1 and z 1 respectively. The length of the line curve segment is 2. The distance from the source to the detector plane is the same as the radius of the circle curve segment. Projections along the circle curve segment are acquired for every 0.5 degree. Projections along the line segment are acquired with sampling intervals of 0.01. We use the method mentioned in [22] for the finite inverse Hilbert transform. The sampling interval of the M-lines is 0.002. The reconstruction pixel is 0.005. M-lines from the line curve segment perpendicular to the z-axis are chosen in the reconstruction. Figure 6 shows the reconstruction images of the slices x 0 and z 0.25 respectively; the top slices are the original images while the bottom slices are reconstructed from noise-free data. We use the grey scale window [1.01, 1.03] to make low contrast features visible.
Table 2. Parameters of the Shepp-Logan phantom. A, B, C are the lengths of half-axes of the ellipsoid. , are the coordinates of the centre of the ellipsoid. φ is the rotation angle around -axis, is the x0 y0 and z0 μ incremental density.
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No. A B C x0 y0 z0 φ μ 1 0.6900 0.9200 0.9000 0.000 0.000 0.000 0 2.00 2 0.6624 0.8740 0.8800 0.000 0.000 0.000 0 -0.98 3 0.4100 0.1600 0.2100 -0.220 0.000 -0.250 180 -0.02 4 0.3100 0.1100 0.2200 0.220 0.000 -0.250 72 -0.02 5 0.2100 0.2500 0.5000 0.000 0.350 -0.250 0 0.02 6 0.0460 0.0460 0.0460 0.000 0.100 -0.250 0 0.02 7 0.0460 0.0230 0.0200 -0.080 -0.650 -0.250 0 0.01 8 0.0460 0.0230 0.0200 0.060 -0.650 -0.250 90 0.01 9 0.0560 0.0400 0.1000 0.060 -0.105 0.625 90 0.02 10 0.0560 0.0560 0.1000 0.000 0.100 0.625 0 -0.02
[Simulation of Line-pair phantom] In the second experiment we use a line-pair phantom with more than 1300 small cuboids to show the spiral resolution that the X-ray cone-beam CT in the multimodality imaging system introduced in Section 1 may achieve. The parameters are chosen to be the same as the real geometry parameters of the system. The phantom is a cuboid of 60mm*60mm*120mm. There are several thin slices inside the phantom whose heights are 5mm. The slices z 0 mm, z 21.9 mm, z 38 mm and y 0 mm are used. Each slice consists of line pairs of various width and directions, and the length of the line pairs is 3mm. We choose widths of 0.7mm, 0.5mm, 0.3mm, 0.2mm and 0.1mm for the simulations, compared with the pixel size 0.1mm of the reconstruction image. Each group of line pairs consists of 6 lines with the same width and interval (for example, 0.3mm). The values of the points on a single line are the same, and the values of the points on different lines are from 0.02 to 2.
Figure 6. The cross-sections of the Shepp-Logan phantom at x 0 (left) and z 0.25 (right). The top slices are the original images while the bottom slices are reconstructed from noise-free data. We use the grey scale window [1.01, 1.03].
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The support of the phantom is a cylinder with radius of 30mm and length of 120mm. A rectangular detector plane consists of 800×1600 detection pixels with a size of 0.1mm2. The radius R of the circle curve segment is 400mm. The two circles are located at z 60mm respectively. The length of the line curve segment is 120mm. Projections along the circle curve segment are acquired for every 0.5 degree. Projections along the line segment are acquired with a sampling interval of 0.5mm. The sampling interval of the M-lines is 0.05mm. The reconstruction pixel is 0.1mm. Figure 7 shows the reconstruction image of z 0 mm; the left image is the full reconstruction image, and the right image is the top left part of the reconstruction image. The values in the images are the width of the corresponding line pairs.
(a) (b) Figure 7. The pseudo-color reconstruction images of z 0 mm. (a) the reconstruction image and (b) the top left part of the reconstruction image. The values in the image are the width of the corresponding line pairs (unit: mm). The image window is [0.05, 2].
4. Discussion The reconstruction algorithm on the 2-circles plus line trajectory has three characteristics. First, the existence of R-lines by theorem 1 ensures that region inside the trajectory can be fully reconstructed. Theorem 2 shows that about 2/3 points inside the trajectory have at least two R-lines. So the choice of R-lines is feasible in the backprojection step. Second, this algorithm requires a very short source movement distance along the line curve segment when the projection data for the 2-circles plus line trajectory is sufficient for reconstruction. For example, in the second experiment the length of the line curve segment (or the distance of the two circles) is just the length of the line-pair phantom along the z-axis. Then the required translation length of the object movement mechanisms will be reduced to a proper level in the imaging cabinet of the Kodak multimodality imaging system, solving the problem of limited space in the introduction of X-ray cone-beam CT into the system. Third, the 2-circles plus line trajectory can be easily extended to N-circles plus line trajectory, defined by
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Rcos(),Rsin(),zn if [(4n 4),(4n 2) ) a() (20) R,0, zn tn ()(zn1 zn ) if [(4n 2),4n ) ( (4n 2) ) with t () for n 1,...,N 1. The N-circles plus line trajectory consists of a series of n 2 parallel circles with radius R and a line orthogonal to the plane of the circles. Since it can be divided into a series of 2-circles plus line curves, the reconstruction algorithm remains the same. The geometric properties are also similar to the 2-circles plus line trajectory. [Detector analysis] Katsevich has proposed a filtered backprojection algorithm for one circle plus line trajectory [27]. Suppose the region of interest (ROI) is defined as{(x, y, z) : x2 y2 r2,H z H}. The length of the line curve segment in his algorithm is at least 4H , more than two times the length of the ROI, 1 r / R while the length of the line curve segment can be just the length of the cylinder 2H for the 2-circles plus line trajectory. The requirement of the detector size of our algorithm is similar to that of the circle plus line trajectory. In Katsevich’s algorithm, the detector size of the circular scan should r H satisfy u and v , and the detector size of the line scan should satisfy 1 r / R2 1 r / R r and H 1 , a little larger than that of the circular scan; it is much u v 2 1 r / R2 1 r / R 1 r / R more suitable for shorter objects. A sufficient detector size for the 2-circles plus line trajectory is H 1 r / R and v . Note that the requirements of the detector size are just 1 r / R1 r / R2 a little bigger than that of the circular scan of the circle plus line trajectory when r / R is small. So the algorithm based on the 2-circles plus line trajectory could offer a much shorter distance of source movement along the line curve segment than that of the circle plus line trajectory with the similar size of projections, but projections along two circle curves are required. In Section 3, we simulated the spiral resolution of the X-ray cone-beam CT in the Kodak multimodality imaging system by the line pair phantom. The line pairs of 0.3mm can be distinguished clearly in Figure 7(b). This result shows that the spiral resolution of the system could achieve 3.3 LP/mm. The object movement mechanisms, including one rotation mechanism and one translation mechanism of high accuracy, have been employed for data acquisition of the 2-circles plus lines trajectory. The mechanisms are also convenient if it is necessary to implement circle or helical trajectories.
5. Conclusion [Conclusion] We have given a backprojection filtration algorithm for the 2-circles plus lines trajectory which could solve the problem of limited space in the Kodak multimodality imaging system for the introduction of X-ray cone-beam CT. The geometric properties of the trajectory are first proposed. The existence of R-lines ensures that the region inside the trajectory can be fully reconstructed, and there are at least two R-lines for about 2/3 points inside the trajectory. The algorithm can offer a shorter distance of source movement along the line curve segment with sufficient projection data. Numerical simulations demonstrate the feasibility of the algorithm and the spiral resolution of the X-ray cone-beam CT in the multimodality imaging system under construction. [Prospect]
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Once the cone-beam CT is introduced into the system, the 3D X-ray reconstruction images can be obtained. At that time, 3D reconstruction of multimodality imaging will be studied.
Acknowledgments This work is partially supported by NBRPC (973 Program, 2011CB809105), NSFC (60325101, 60872078), Key Laboratory of Machine Perception (Ministry of Education) of Peking University, Microsoft Research of Asia. We would like to thank Prof. Tie Zhou and Yu Zhou for valuable discussions.
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