Minimum detection window and inter-helix PI-line with triple-source helical cone-beam scanning

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1 Journal of X-Ray Science and Technology ) IOS Press Minimum detection window and inter-helix PI-line with triple-source helical cone-beam scanning Jun Zhao a,, Ming Jiang b, Tiange Zhuang a and Ge Wang c a Department of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai, , China b LMAM, School of Mathematical Sciences, Peking University, Beijing , China c Department of Radiology, University of Iowa, Iowa City, IA 52242, USA Received 24 October 2005 Revised 21 February 2006 Accepted 2 March 2006 Abstract. In this paper, we propose a helical cone-beam scanning configuration of triple symmetrically located X-ray sources, and study minimum detection windows to extend the traditional Tam-Danielsson window for exact image reconstruction. For three longitudinally displaced scanning helices of the same radius and a source location on any helix, the corresponding minimum detection window is bounded by the most adjacent turns respectively selected from the other two helices. The height of our proposed minimum detector window is only 1/3 of that in the single helix case. Associated with proposed minimum detection windows, we define the inter-helix PI-line and establish its existence and uniqueness property: through any point inside the triple helices, there exists one and only one inter-helix PI-line for any pair of the helices. Furthermore, we prove that cone-beam projection data from such a triple-source helical scan are sufficient for exact image reconstruction. Although there are certain redundancies among those projection data, the redundant part cannot be removed by shrinking the detector window without violating the data sufficiency condition. Those results are important components for development of exact or quasi-exact image reconstruction algorithms in the case of triple-source helical cone-beam scanning in the future. Keywords: Computed tomography CT), cone-beam, triple-sources, helical scanning, minimum detection window, inter-helix PI-line 1. Introduction It is important to develop fast data acquisition methods for computed tomography CT). There are urgent demands for high-speed CT for cardiac imaging, small animal micro-ct, and 4-D CT [1]. One approach to accelerate data acquisition is based on the multiple source-detector strategy. Helical cone-beam CB) CT reconstruction has received intensive attention [2] since the introduction of multi-row CT scanners in Wang et al. proposed an approximate Feldkamp-type spiral conebeam algorithm to reconstruct a long object [3,4]. Kudo et al. developed quasi-exact algorithms to handle truncated CB data acquired with the smallest possible detector [5,6]. In this single helix case, Corresponding author: junzhao@sjtu.edu.cn /06/$ IOS Press and the authors. All rights reserved

2 96 J. Zhao et al. / Minimum detection window and inter-helix PI-line Fig. 1. Illustration of a triple source helical cone-beam scanning configuration. the smallest possible detector is delimited by the Tam-Danielsson window [7,8]. For this scanning configuration, Katsevich recently established exact algorithms of filtered backprojection FBP) type [9, 10]. His algorithms depend on the Tam-Danielsson windows and PI-lines [11]. Hence, it is logical to study the corresponding definitions of those important concepts for multiple-source helical cone-beam scanning. In this paper, we propose a system configuration of triple symmetrically located X-ray sources, and study the minimum detection windows for exact image reconstruction. The structure of the paper is as follows. In Section 2, we define the minimum detection windows and inter-helix PI-lines for the proposed setting and derive the boundary equations of the windows. In Section 3, we establish the existence and uniqueness property of inter-helix PI-lines. In Section 4 we demonstrate that for the triple-helix case, cone-beam projection data from such a triple-source helical scan are sufficient for exact image reconstruction. Although there are certain redundancies among those projection data, those parts cannot be removed by shrinking the detector window without violating the data sufficiency condition. In Section 5 we discuss relevant issues and conclude the paper. 2. Minimum detection windows for triple-helix The minimum detection window for a single helix is called the Tam-Danielsson window [7,8]. It is possible to decrease the sizes of detection windows by increasing the number of sources. In this section, we define minimum detection windows in the case of triple-source configuration. Let fx, y, z) be the density function to be reconstructed. Assume that this function is smooth and vanishes outside the cylinder U r = { x, y, z) x 2 + y 2 r 2,z min z z max }, 0 <r<r 2.1) where r is the radius of the object cylinder, and R is the radius of the spiral cylinder. Consider three equally spaced helices as shown in Fig. 1 H 1 s) =R coss),rsins),hs) 2.2)

3 J. Zhao et al. / Minimum detection window and inter-helix PI-line 97 Fig. 2. Illustration of the minimum detection window B 1. H 2 s) = H 3 s) = R cos s),rsin s),h s 2π 3 R cos s),rsin s),h s 4π 3 )) )) 2.3) 2.4) where s is the angle describing the rotation of the sources, 2πh is pitch, and 2π 3 h is the distance along z-axis between any two neighboring helices. These helices represent the scanning trajectories of the three sources relative to the object, respectively. It is assumed that the detector plane is parallel to the axis of the spiral and is tangent to the spiral cylinder at the point opposite to the source Minimum windows and inter-helix PI-lines for the triple source-detector helical configuration Now, consider the triple-helix case as shown in Fig. 1. Note that between the upper turn and lower turn of the Tam-Danielsson window of S 2, there exist two helix turns, one from H 3 s), and the other from H 1 s). Naturally, we chose the upper one from H 3 s) and the lower one from H 1 s) as the upper and lower boundaries of the minimum detection window of S 2, respectively. Similarly we find corresponding boundaries for S 1 and S 3. Denote the detector planes as D 1 s), D 2 s) and D 3 s) which correspond to the sources at H 1 s), H 2 s) and H 3 s) respectively. Note that the distance from H i s) to D i s) is 2R, i = 1, 2, 3. In the following, we assume that i = 1, 2, 3 unless stated otherwise. Definition 1. Define B i as the region of the detector plane D i s 0 ) that is bounded by the CB projections onto D i s 0 ) of the helix turns {H imod3+1 s) s s 0 [ 2π i +1)/3 ), 2π 1 i +1)/3 )]} and { H i+1)mod3+1 s) s s 0 [ 2π i +2)/3 ), 2π 1 i +2)/3 )] } with H i s 0 ) as the vertex. The symbol s 0 is the angular parameter indicating the source position H i s 0 ). An example region B 1 is shown in Fig. 2, where the detector plane is drawn slightly away from the spiral cylinder for a clear presentation. The minimum detection windows for the triple source configuration consists of the regions B 1, B 2 and B 3. We denote the upper and lower boundaries of region B i as Γ top i and Γ bot i respectively. In the detector planed i s), let the d 2,i -axis be parallel to the axis of the spiral and d 1,i -axis be perpendicular to it.

4 98 J. Zhao et al. / Minimum detection window and inter-helix PI-line Fig. 3. Vertical plane intersecting with the helices and the detection windows. Shrinking the detection windows would violate the data sufficiency condition [15]. For example, the projection data on a vertical plane are incomplete for calculating the 3D Radon transform of the plane, as shown in Fig. 3. Points on the upper and lower boundaries of a region B i are strongly related to the so-called inter-helix PI-lines. In the triple-helix case, we should generalize the concept of PI-lines. For that purpose, we introduce the inter-helix PI-lines as follows. Definition 2. An inter-helix PI-line for H i and H imod3+1 is defined as a line such that: 1) it intersects at one point and H imod3+1 at another point; 2) the absolute difference of the s-parameter values at both intersection points are less than 2π. By the definitions of the region B i and the inter-helix PI-line, the line connecting H i s) to a point on the boundaries of the region B i corresponds to an inter-helix PI-line. The set of all inter-helix PI-lines belonging to the same projection is called an inter-helix PI-surface. The upper and lower inter-helix PI-surfaces are denoted as Si U H i s)) and Si L H is)) respectively. C i H i s)) is a truncated cone bounded by Si U H i s)) and Si L H is)). In the following, we will demonstrate an important property of points on an inter-helix PI-segment. We assume the sources are arranged as shown in Fig. 1. More general cases can be studied similarly. Assume that the object is translating upwards and rotating counter-clockwise when observed from the top, with the source and the detector being fixed. In Figs 1 and 4, an inter-helix PI-line is shown for H 2 and H 3, containing three object points P 1 P P 2 in two positions where the line enters the truncated cone C 3 H 3 s)) and exits the truncated cone C 2 H 2 s)), respectively. The rotation angular difference around z-axis of the two positions is π +2γ, where γ is specified in Fig. 4. The end points of the inter-helix PI-segment are moving along the helix H 3 and the helix H 2, respectively. One end point will coincide with the source S 3. The other will coincide with the source S 2. We find that the position of S 3 at entrance of C 3 H 3 s)) and the position of S 2 at exit of C 2 H 2 s)) are exactly 180 degrees apart as seen from any point on this inter-helix PI-segment. Since the inter-helix PI-line is chosen arbitrarily, the property holds for all points of the object which belong to inter-helix PI-lines. We will prove that there is an inter-helix PI-line for any pair of helices through any point in the object. The same statement is true for the other two kind of inter-helix PI-lines. Thus, we have

5 J. Zhao et al. / Minimum detection window and inter-helix PI-line 99 Fig. 4. Top view for demonstrating the inter-helix PI-line property. Property 1. The position of S i at entrance of C i H i s)) and the position of S i+1)mod3+1 at exit of C i+1)mod3+1 Hi+1)mod3+1 s) ) are exactly 180 degrees apart as seen from any point in the object Boundary equations of the minimum detection windows In this subsection, we will derive the boundary equations of the minimum detection windows. Similar to the case of a single helix [10,11], we can derive the equations of the boundaries of B i. First we derive the equations of the boundaries of B 1 with the source H 1 s 0 ). Let E = R cos s),rsin s),h s 2π )) 3 be any point on the upper turn of the minimum detection window. The upper turn is selected from H 2. Let Ê be the CB projection of E. Obviously, Ê is on Γtop 1. In Fig. 5a), the triangle ΔEOS is isosceles. Then, we get cos α =sin s s 0 2 and sin α =cos s s 0 2. Hence d 1,1 s) =2R tan α =2R cot s s 0 2 From Fig. 5b), we have = 2R sins s 0) 1 coss s 0 ). 2.5) h s 2π ) 3 hs0 d 2,1 s) = R cos s 0 cos s) 2R cos s 0 + d 1,1 s)sins ) Therefore, d 2,1 s) = h s s 0 2π ) 3 2R cos s0 d 1,1 s)sins 0 ). 2.7) R cos s 0 cos s) Subsituting Eqs 2.5) into 2.7) yields d 2,1 s) = 2h s s 0 2π ) 3 1 cos s s 0 ). 2.8)

6 100 J. Zhao et al. / Minimum detection window and inter-helix PI-line Fig. 5. Different Geometric views for the computation of Γ top 1. a) Top view, b) Side view, c) View for the computation of Δ. Therefore, we have the equations for the upper boundary Γ top 1 of B 1 : d 1,1 s) = 2R sins s 0) 1 coss s 0 ),d 2,1s) = 2 s s0 2π ) 3 h 1 coss s 0 ), Δ s s 0 2π Δ, 2.9) where Δ=2cos 1 r/r) is determined by the ratio of the radii of the object cylinder and the spiral cylinder see Fig. 5c)). The portion of Γ top 1 outside the range Δ s s 0 2π Δ records zero CB data. Similarly, we find Γ top i : d 1,i s)= 2R sins s 0) 1 coss s 0 ), d 2,i s)= 2 s s 0 2π 3 +2π i +1)/3 )) h, Δ 2π i +1)/3 ) s s 0 2π 1 coss s 0 ) 1 i +1)/3 ) Δ, and Γ bot i : d 1,i s)= 2R sins s 0) 1 coss s 0 ), d 2,i s)= 2 s s 0 4π 3 +2π i +2)/3 )) h, 1 coss s 0 )

7 J. Zhao et al. / Minimum detection window and inter-helix PI-line 101 Fig. 6. Boundary curves of the minimum windows in the cases of single and triple sources dashed and solid curves), respectively. Δ 2π i +2)/3 ) s s 0 2π 1 i +2)/3 ) Δ. Figure 6 shows a typical representation of Γ top i and Γ bot i. For comparison, Fig. 6 also shows the dashed curves, which correspond to the boundaries of the Tam-Danielsson window in the single helix case. Because of the symmetry of the helices, the shapes of the minimum detection windows B 1, B 2 and B 3 are the same. 3. Proof of existence and uniqueness of inter-helix PI-line The existence and uniqueness of the PI-line in the case of a single helix with constant pitch were proved in Danielsson et al. [7] and Defrise et al. [13]. A proof for a more general case of single helix with variable pitch is given in Ye et al. [12]. In the following, we prove the existence and uniqueness of the inter-helix PI-lines analytically. Theorem 1. Any point Ax 0,y 0,z 0 )=r 0 cos μ 0,r 0 sin μ 0,z 0 ) U r belongs to one and only one inter-helix PI-line for H i and H imod3+1. First, we prove Theorem 1 when i =1. An inter-helix PI-segment from H 1 s 1 ) to H 2 s 2 ) with 0 <s 2 s 1 < 2π can be written as x = Rt 12 cos s 1 + R1 t 12 )coss 2 3.1) y = Rt 12 sin s 1 + R1 t 12 )sins 2 3.2) z = t 12 hs 1 +1 t 12 )h s 2 2π ) 3.3) 3 with t 12 [0, 1]. To find a unique inter-helix PI-segment that passes through a given point A is equivalent to seeking a unique triplet s 1,s 2,t 12 ) with t 12 0, 1) and 0 <s 2 s 1 < 2π such that x 0 = RT t cos s 1 + R1 t 12 )coss 2, 3.4) y 0 = RT t sin s 1 + R1 t 12 )sins 2, 3.5)

8 102 J. Zhao et al. / Minimum detection window and inter-helix PI-line and z 0 = t 12 hs 1 +1 t 12 )h s 2 2π 3 ) 3.6) By Eqs 3.1), 3.2), 3.4) and 3.5), we obtain r 0 cos μ 0 = Rt 12 cos s 1 + R1 t 12 )coss 2 3.7) r 0 sin μ 0 = Rt 12 sin s 1 + R1 t 12 )sins 2 3.8) with 0 <s 2 s 1 < 2π. For each s 1 R, we solve the above two equations for s 2 and t 12. The solution of Eqs 3.7) and 3.8) is unique, we get t 12 = t 12 x 0,y 0,s 1 )= s 2 = s 2 x 0,y 0,s 1 ) is determined by R 2 r 2 0 2R[R r 0 cosμ 0 s 1 )]. 3.9) cos s 2 s 1 2 = r 0 sinμ 0 s 1 ) R 2 + r 2 0 2Rr 0 cosμ 0 s 1 ) 3.10) By Eq. 3.6), we have zx 0,y 0,s 1 )=t 12 x 0,y 0,s 1 )hs 1 +1 t 12 x 0,y 0,s 1 ))h s 2 x 0,y 0,s 1 ) 2π 3 ). 3.11) The inter-helix PI-segment Eqs 3.1) 3.3) contains A if and only if zx 0,y 0,s 1 )=z ) Theorem 1 when i =1is equivalent to the claim that Eq. 3.12) has one and only one solution for s 1. Differentiating Eq. 3.12) with respect to s 1, we get dz = h s 1 s 2 + 2π ) dt12 + t t 12 ) ds ) ) ds 1 3 ds 1 ds 1 From Eqs 3.9) and 3.10), we have dt 12, ds 2 ds 1 = t 12 1 t ) = t 12 cot s 2 s 1 ds 1 2 Substituting Eq. 3.14) into Eq. 3.13) yields: dz = 2π ds 1 3 ht 12 cot s 2 s 1 +2ht 12 1 s 2 s 1 cot s 2 s We can get dz > 0 ds 1 ). 3.15) 3.16)

9 J. Zhao et al. / Minimum detection window and inter-helix PI-line 103 Fig. 7. Horizontal plane intersecting the three helices. Note that as a function of s 1, zx 0,y 0,s 1 ) is continuous and lim z =, lim z = 3.17) s 1 s 1 Therefore, Eq. 3.12) has one and only one solution for s 1. This completes our proof of Theorem 1 when i =1. The proofs for i =2or i =3can be similarly done and skipped for brevity. 4. Orlov s theorem and sufficiency condition It is noted that the CB projection data collected in the above system configuration are complete for a given plane that passes through the object for the recovery of the Radon transform derivative associated with that plane. The geometric sense of the completeness is that the 3D Radon transform of the plane can be calculated from a combination of several projections which covers the plane in a seamless fashion, or in other words, constitutes a triangulation of the plane [5,6]. Figure 3 shows a vertical plane which intersects the object and the helices, and its triangulation- pattern. The projection data are complete without redundancy. However, in certain planes the CB projection data are redundant. An example is shown in Fig. 7. The points inside the triangle ΔH 1 H 2 H 3 are shot three times by the sources. Because the CB projection data are complete, exact or quasi-exact reconstruction algorithms of Grangeat-type and Katsevich-type can be developed [5,6,9,10]. The results will be generalized filtered-backprojection or backprojected filtration algorithms in the single helix case [5,6,9,10,16 18]. However, the redundancy of the CB projection data must be carefully dealt with in developing the algorithms. Another perspective to study the sufficiency condition for the present setting is based on Orlov s theorem [14] as Danielsson [7] did in the single helix case. If an object is irradiated with parallel beams, and the set of projection directions can be mapped onto the surface of the unit sphere to form a certain curve or curves. Then, Orlov s theorem can be rewritten as follows [14]. Theorem 2. Orlov s theorem) If any plane through the center of the unit sphere intersects the direction curve or curves on the surface of the sphere, then the projections are complete.

10 104 J. Zhao et al. / Minimum detection window and inter-helix PI-line Fig. 8. 3D view for proving Lemma 1. One corollary for the present configuration is as follows: Corollary 1. If a direction curve joins diametrically opposite points, then the projections are complete. Proof. Clearly, any plane through the center of the unit sphere intersects the direction curve joining diametrically opposite points. By Theorem 2, the conclusion follows immediately. With the geometry of the Tam-Danielsson window in the single helix case, any point in the object satisfies the Orlov s sufficiency condition. To discuss the case of the triple helices, we need some necessary notations. As shown in Fig. 8, let U be an arbitrary curve on the surface of a unit sphere. The two end points of U are denoted as P b and P e, respectively. Q b and Q e are the symmetric points of P b and P e with respect to the center of the unit sphere, respectively. Through Q b and Q e, there exists one and only one great circle on the sphere. There are two arcs connecting Q b and Q e. We select the shorter one and denote it as U. Lemma 1. Any plane that intersects U and passes through the center of the unit sphere must intersect U. Proof. If a plane passes through the center of the unit sphere and intersects U, there are two cases: only one end point is on the plane, or the two end points are in the opposite sides of the plane. Case 1: Only one end point on the plane. Without loss of generality, we assume that Q b is on the plane. Clearly, line Q b P b lies in the plane. Therefore, the plane intersects U at P b. Case 2: Two end points in the opposite sides of the plane. It is clear that the line Q b P b passes through the plane. P b is in the opposite side of the plane with respect to Q b. Similarly, P e is in the opposite side of the plane with respect to Q e. Note that Q b and Q e are in the opposite sides of the plane, thus P b and P e are in the opposite sides of the plane. Since U is continuous, then the plane intersects U. Next, we discuss the direction curves on the surface of the unit sphere for a point in the object assuming the geometry of the minimum detection windows in the triple-helix case. As shown in Fig. 3, let M be a point in the object and also on the z-axis. Let the object be translated upwards and rotated counterclockwise when seen from the top around the z-axis, with the source and the detector being fixed with R = 3πh 6. The direction curves on the surface of the unit sphere for M are shown in Fig. 9. There

11 J. Zhao et al. / Minimum detection window and inter-helix PI-line 105 Fig. 9. Direction curves on the surface of the unit sphere in the triple helix case. a) 3D view showing symmetric points on the unit sphere, b) 3D view for proving Theorem 3. are three disconnected direction curves. Let T i be the direction curve corresponding to the angles of exposure by S i. The beginning and end points of T i are Pi b and P i e, respectively. By Property 1, we find that P3 b and P 2 e, P 2 b and P 1 e, P 1 b and P 3 e are symmetric on the surface of the sphere with respect to the center of the sphere, respectively. Theorem 3. Given the geometry of the minimum detection windows in the triple-helix case, any point in the object satisfies the Orlov s sufficiency condition, if the point is treated as a single independent object. Proof. Given an arbitrary point Ax 0,y 0,z 0 ) U r, if the point is treated as a single independent object. The density function can be redefined as { fx0,y gx, y, z) = 0,z 0 ), x,y,z) {x 0,y 0,z 0 )} 0, x,y,z) R 3 4.1) \{x 0,y 0,z 0 )} If Ax 0,y 0,z 0 ) is illuminated by a source, there is only one ray in the corresponding truncated cone-beam through Ax 0,y 0,z 0 ). From Eq. 4.1), we can consider the ray as a virtual parallel beam with the same direction. The role of the virtual parallel beam and the role of truncated cone-beam are essentially the same towards Ax 0,y 0,z 0 ) under the condition of Eq. 4.1) up to a scaling factor). Without loss of generality, consider the direction curves on the surface of the unit sphere in Fig. 9b). Through P 2 b and P3 e, there exists one and only one great circle on the sphere. There are two arcs connecting P 2 b and P3 e. Select the shorter one and denote it as T 1. Then, T 3 T 1 T2 is a continuous curve joining the symmetirc points P3 b and P 2 e. As seen in the proof of Corollary 1, any plane through the center of the unit direction sphere intersects the direction curve joining symmetric points. Thus, any plane through the center of the unit sphere intersects the curve T 3 T 1 T2. By Lemma 1, any plane through the center of the unit direction sphere intersects T 3 T1 T2. By Theorem 2, we conclude that the projections are complete. 5. Discussion and conclusion Using a similar method, we can prove that in the double-helix case with a similar geometric configuration, the projections are incomplete for exact reconstruction. By the similar geometric configuration,

12 106 J. Zhao et al. / Minimum detection window and inter-helix PI-line we mean that the loci of the double-source are symmetrically arranged double helices, a minimum detection window is bounded by different helices, and the height of the minimum detection window is a half of that of the traditional Tam-Danielsson window. Of course, if we make the height of the detection window sufficiently large, a complete projection dataset will be produced to allow the exact reconstruction. However, in this case the completeness and redundancy of the data must be region-wise analyzed, and are considered beyond the scope of this paper. In conclusion, we have proposed a helical cone-beam scanning configuration of triple symmetrically located X-ray sources, and studied it s the associated minimum detection windows as an extension of the traditional Tam-Danielsson window. Specifically, we have defined the inter-helix PI-lines and established their existence and uniqueness property: through any point inside the triple helices, there exists one and only one inter-helix PI-line for any pair of the helices. Furthermore, we have proved that cone-beam projection data from such a triple-source helical scan are sufficient for exact image reconstruction. Those results are important components for development of exact or quasi-exact image reconstruction algorithms in the future. Acknowledgments This work is partially supported by National Science Foundation of China ), and NIH/NIBIB grants EB and EB M. Jiang was supported in part by the National Basic Research Program of China 2003CB716101), National Science Foundation of China , , ) and Engineering Research Institute of Peking University. References [1] M. Endo, S. Mori, T. Tsunoo, S. Kandatsu, S. Tanada, H. Aradate, Y. Saito, H. Miyazaki, K. Satoh, S. Matsusita and M. Kusakabe, Development and performance evaluation of the first model of 4-D CT-scanner, IEEE Trans. on Nuclear Science ), [2] G. Wang, C.R. Crawford and W.A. Kalender, Multi-row-detector and cone-beam spiral/helical CT, IEEE Trans. Med. Imaging ), [3] G. Wang, T.H. Lin, P.C. Cheng, D.M. Shinozaki and H.G. Kim, in: Scanning cone-beam reconstruction algorithms for x-ray microtomography, Vol. 1556), G.S. Kino, ed., Proc. SPIE, Scanning Microscopy Instrumentation), 1991, pp , [4] G. Wang, T.H. Lin, P.C. Cheng and D.M. Shinozaki, A general cone-beam reconstruction algorithm, IEEE Trans. on Medical Imaging ), [5] H. Kudo, F. Noo and M. Defrise, Cone-beam filtered-backprojection algorithm for truncated helical data, Phys. Med. Biol ), [6] H. Kudo, F. Noo and M. Defrise, Quasi-exact filtered backprojection algorithm for long-object problem in helical cone-beam tomography, IEEE Trans. Med. Imaging ), [7] P.E. Danielsson, P. Edholm, J. Eriksson and S.M. Magnusson, Towards exact reconstruction for helical cone-beam scanning of long object, in: A new detector arrangement and a new completeness condition Proc, D.W. Townsend and P.E Kinahan, eds, 1997 Meeting on Fully 3D Image Reconstruction in Radiology and Nuclear Medicine Pittsburgh, PA), 1997, pp [8] K.C. Tam, Three-dimensional computerized tomography scanning method and system for large objects with small area detector, US patent US , [9] A. Katsevich, A general scheme for constructing inversion algorithms for cone beam CT, Int l J. of Math. and Math. Sci ), [10] A. Katsevich, Improved exact FBP Algorithm for spiral CT, Adv. Appl. Math ), [11] P.E. Danielsson, Technique and arrangement for tomographic imaging, US patent US , [12] Y. Ye, J. Zhu and G. Wang, Minimum detection windows, PI-line existence and uniqueness for helical cone-beam scanning of variable pitch, Medical Physics 313) 2004),

13 J. Zhao et al. / Minimum detection window and inter-helix PI-line 107 [13] M. Defrise, F. Noo and H. Kudo, A solution to the long-object problem in helical cone-beam tomography, Physics in Medicine and Biology ), [14] S.S. Orlov, Theory of three-dimensional reconstruction. I. Conditions for a complete set of projections, Sov. Phys. Crystallogr. 1975), [15] P. Grangeat, Mathematical framework of cone-beam 3D reconstruction via the first derivative of the Radon transform, in: Mathematical Methods in Tomography, G.T. Herman, A.K. Louis, and F. Natterer, eds, Berlin: Springer, 1991, pp [16] Y. Zou and X. Pan, Exact image reconstruction on PI-lines in helical cone beam CT, Physics in Medicine and Biology ), [17] Y. Ye and G. Wang, Cone beam reconstruction along a 3D spiral of a variable radius and/or variable pitch, Proceedings of SPIE 5535 August 2004). [18] H. Yu, Y. Ye and G. Wang, Katsevich-Type Algorithms for Variable Radius Spiral Cone-Beam CT, Proceedings of SPIE 5535 August 2004).

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