Grangeat-type helical half-scan computerized tomography algorithm for reconstruction of a short object

Transcription

1 Grangeat-type helical half-scan computerized tomography algorithm for reconstruction of a short object Seung Wook Lee a) CT/Micro-CT Laboratory, Department of Radiology, University of Iowa, Iowa City, Iowa and Department of Nuclear and Quantum Engineering, Korea Advanced Institute of Science and Technology, Daejeon , South Korea Ge Wang b) CT/Micro-CT Laboratory, Department of Radiology, Department of Biomedical Engineering, University of Iowa, Iowa City, Iowa Received 29 January 2003; revised 7 September 2003; accepted for publication 22 September 2003; published 8 December 2003 Currently, cone-beam computerized tomography CT and micro-ct scanners are under rapid development for major biomedical applications. Half-scan cone-beam image reconstruction algorithms assume only part of a scanning turn, and are advantageous in terms of temporal resolution and image artifacts. While the existing half-scan cone-beam algorithms are in the Feldkamp framework, we have published a half-scan algorithm in the Grangeat framework for a circular trajectory Med. Phys. 30, In this paper, we extend our previous work to a helical case without data truncation. We modify the Grangeat s formula for utilization and estimation of Radon data. Specifically, we categorize each characteristic point in the Radon space into singly, doubly, triply sampled, and shadow regions, respectively. A smooth weighting strategy is designed to compensate for data redundancy and inconsistency. In the helical half-scan case, the concepts of projected trajectories and transition points on meridian planes are introduced to guide the design of weighting functions. Then, the shadow region is recovered via linear interpolation after smooth weighting. The Shepp Logan phantom is used to verify the correctness of the formulation, and demonstrate the merits of the Grangeat-type half-scan algorithm. Our Grangeat-type helical halfscan algorithm is not only valuable for quantitative and/or dynamic biomedical applications of CT and micro-ct, but also serves as an intermediate step towards solving the long object problem American Association of Physicists in Medicine. DOI: 0.8/.6255 Key words: half-scan, Grangeat-type reconstruction, cone-beam CT, helical CT I. INTRODUCTION Sixteen-slice helical computerized tomography CT scanners are already commercially available. C-arm systems and several micro-ct systems are also based on cone-beam acquisition and reconstruction. Prototypes of the cone-beam systems were recently reported. 6 With rapid increment in the number of detector rows, the concept of cone-beam CT or volumetric CT will become more and more popular. New applications are made possible by these new fast volumetric imaging technologies, such as for cardiac and lung examinations, CT angiography, and interventional procedures. 5,6 In those applications, high temporal resolution is one of the most important requirements. Half-scan techniques were developed to improve the temporal resolution for axial and spiral CT images. 7,8 While half-scan cone-beam algorithms in the Feldkamp framework have relatively long history, 9 2 the half-scan cone-beam algorithms have been recently developed in the Grangeat framework for the circular scanning geometry by our Laboratory as well by Noo and Heuscher independently.,3 The difference between these results is substantial. That is, our work is in the rebinning framework, 4 while the formulation by Noo and Heuscher is in the filtered backprojection format. 5 This research is a natural extension of our half-scan algorithm from circular to helical scanning geometry to solve the short object problem, which assumes that the object is completely covered by the x-ray cone beam from any source position. Even though the geometry for data truncation along the axial direction is the most practical, research with the short object geometry is not only valuable on its own, such as for micro-ct imaging of spherical samples, but also essential as an intermediate step toward solving the long object problem. 6,7 This paper is organized as follows. In Sec. II A the Grangeat algorithm is briefly reviewed for completeness. In Sec. II B, the rebinning equations for a helical trajectory are introduced. In Sec. II C, the helical half-scan Grangeat formula is described. In Sec. II D the weighting functions are derived. In Sec. II E the interpolation method used in this study is explained. In Sec. II F the implementation procedure is summarized. In Sec. III the results are presented and discussed. Finally, in Sec. IV the paper is concluded. 4 Med. Phys. 3, January Õ2004Õ3 Õ4Õ3Õ$ Am. Assoc. Phys. Med. 4

2 5 S. W. Lee and G. Wang: Helical half-scan Grangeat formula 5 FIG.. Grangeat s cone-beam geometry. The cone-beam projection and the first derivative of Radon transform are linked. where is the angle between SO and SC. By substituting d d d d r cos, we have Rfn /2 /2 0 f n,r,dr cos d. Here, r is cancelled out and the cone-beam projection can be directly utilized as Rfn /2 Xfn, /2 cos d. 7 The two angular variables, and can be changed to the variables on the detector plane, s and t, with the following geometrical relationships: sso tan, II. MATERIALS AND METHODS A. Grangeat framework. Grangeat s derivation For cone-beam CT, it is instrumental to connect conebeam data to three-dimensional 3D Radon data. Smith, Tuy, and Grangeat independently established such connections. 4,8,9 Grangeat s formulation is geometrically attractive and becomes popular. Here, we review Grangeat s derivation process. It largely follows the notations defined in Ref. 20. In cone-beam geometry as in Fig., the 3D Radon transform at the characteristic point C is defined by tsc D tan. Differentiating the two yields ds SO cos 2 d, dt SA cos d. Finally, we have Rfn SO cos 2 s SA Xfsn,tdt /2 Rfn /2 f n,r,r drd, 0 which means the plane integration of the gray triangle with one of the vertices being the source point S. The plane is normal to the vector n. Any point on the plane is represented with the polar coordinate (r,) on itself. A cone-beam projection is the line integration from S to any point A on the detector plane and is mathematically represented by Xfn, 0 f n,r,dr. As you can notice, there is r in Eq., which prevents us from using the measured cone-beam projection. To eliminate this, the first derivative with respect to is applied to Eq. and we have Rfn f n,r,r drd. 3 0 /2 /2 To first order, we have the following relation at any point on the plane, (n,r,): dr cos d, Cone-beam data to derivative data in the Radon domain Figure is transformed to Fig. 2 to show the geometrical relationship between the detector plane and the meridian plane in the Radon domain. Accordingly, Eq. 2 can be rewritten as Rfn Rf n cos 2 s SO D SA Xfsn,t,n dt Xw f sn,t,n dt, cos 2 s 3 where Xfs(n ),t,(n ) is the detector value, which is defined by the distance s from the detector center O D along the line t perpendicular to O D C D on the detector plane D extending an angle from the x axis, SO D the distance between the source and the detector center, SA the distance between the source and an arbitrary point A along t, and

3 6 S. W. Lee and G. Wang: Helical half-scan Grangeat formula 6 FIG. 3. Helical half-scan geometry. FIG. 2. Meridian and detector planes. a Relationship between meridian and detector planes, b meridian plane, and c detector plane. the angle between SO D and SC. Given a characteristic point C on a meridian plane M in the Radon domain, the plane orthogonal to the vector n is determined. Then, the intersection points of the plane with the source trajectory a () can be found, and the detector planes D specified, on which the line integration should be performed. Let C D denote the intersection of the detector plane D with the ray that comes from S and goes through C. The position C D can be described by a vector sn D. To compute the derivative of Radon value at C, the line integration is performed along t, which is orthogonal to the vector sn D. Once the first derivative of Radon is calculated, the original function f can be reconstructed with the following 3D Radon inversion formula: 2,22 f x /2 8 2 / Rfx n n sin d d. B. Rebinning equations for a helical trajectory 4 The geometry for a helical half-scan is shown in Fig. 3, where (x,y,z) is the reference coordinate system, (y,z) denotes a meridian plane at from the x axis, and the source vertex,, ranges from 0 to 2 m. If the helical pitch is denoted by h, the source trajectory can be parametrized as a SO cos,so sin,hz 0. 5 To compute the derivative of the Radon value at n,we must first calculate the line integration point C D on a detector plane, through which a line integration is performed along the line normal to OC D. There is a geometrical relationship between the characteristic point,, and the line integration point (s,,). In other words, we can find (s,,) from,,. The 3D Radon value at a characteristic point,, is the integration of an object on a plane satisfying the following equation: n x. 6 The intersection of the plane with the source trajectory is found by replacing x with a () to solve n a, 7 where n (sin cos,sin sin,cos ) denotes a unit vector towards the corresponding characteristic point, and a () (R cos,r sin,hz 0 ) the source position. The equation can be written as R sin cos cos sin sin hz 0 cos. 8 While we can calculate the line integration point analytically in a circular trajectory case, we can only do it numerically in a helical trajectory case. Once we acquire, we have line integration point (,s) on the detector plane D, where /2. The equations are expressed as 6 cot tan, 9 sin s hz 0cos sin. 20 cos Therefore, the Radon value at the characteristic point defined by,, can be calculated by the integration along the line represented by (s,,) according to Eqs C. Grangeat-type helical half-scan formula With a circular half-scan, there are three types of regions: shadow, singly, and doubly sampled regions, respectively. With a helical half-scan, in addition to those three types of regions, there may be triply sampled regions as well. They are schematically illustrated in Fig. 4. Hence, the Grangeat-

4 7 S. W. Lee and G. Wang: Helical half-scan Grangeat formula 7 If we assume that there is no motion or data inconsistency during the scan, we can simply average Radon data from different vertices. In this case, the weighting functions become discontinuous. However, in real situations, motion effects and data inconsistency should be taken into account. Then, the discontinuous weighting functions could cause artifacts. Therefore, the smooth weighting functions are needed for satisfactory image quality. The next section will be devoted to this purpose. D. Weighting functions The concepts of projected trajectory and transition points are needed before our weighting functions are designed. Hence, we first introduce them in Secs. II D and II D 2. Then, we design the weighting functions in Sec. II D 3. FIG. 4. Classification of the Radon space. a Shadow region, b singly sampled region, c doubly sampled region, and d triply sampled region. type helical half-scan formula must be in the following format: 3 Rfn g g R g f n, 2 where g denotes the weighting functions, g is a group identifier to be explained in detail later, and R g f n SO cos 2 s SA Xfs,t,dt. 22 The form is basically the same as that with a circular halfscan trajectory but three weighting functions and corresponding Radon values are needed for each characteristic point, while we only need two weighting functions in a circular half-scan case. In Eq. 2, the value of the group identifier g is determined according to the following criteria: g, 0 t, 2, t t2, 3, t2 2 m, 23 where t and t2 are tangential vertices and will be explained in detail in Sec. II D 2. This means that the calculated Radon data must belong to one of the three groups depending on the. Regarding the weighting functions, they must satisfy 3 g g. 24. Projected trajectory on a meridian plane at The projected trajectory on a meridian plane at is useful for design of the weighting functions. Several projected trajectories on different meridian planes are shown in Fig. 5. The source trajectory of Eq. 5 is first transformed from (x,y,z) to(x,y,z) coordinate systems, as shown in Fig. 3. Therefore, for a given meridian plane at, the projected trajectory on this plane is described as yso sin 2, zhz Dashed lines in Fig. 6a represent the planes normal to the meridian plane. The integration results on the planes corresponding to each line must be equal to the Radon value at,,. In the Grangeat framework, the derivative of Radon data is calculated from the detector planes as denoted by the colored lines associated with the intersected vertex points. Geometrically, there can be maximally three intersection points. For a given,, the number of intersecting points determines the degree of redundancy and depends on. 2. Transition points on the projected trajectory Every projected trajectory has two end points expressed by e y,z SO sin 0 2,0z 0, e 2 y,z SO sin 2 m 2, h2 m z For a given on a meridian plane, there may be planes normal to and tangent to the projected trajectory. The tangential points can be analytically specified by t y,z SO sin t 2,h t z 0, 28 t 2 y,z SO sin t2 2,h t2 z 0, where t, t2 are calculated in the following. 29

5 8 S. W. Lee and G. Wang: Helical half-scan Grangeat formula 8 FIG. 5. Projected trajectories on meridian planes of a 90, b 0, c 30, d 50, e 70, f 90, g 20, h 230, and i 250. A projected trajectory on a meridian plane is determined as yso sin 2, zhz 0. To express as a function of z, we obtain 30 3 zz 0, 32 h substitute Eq. 32 into Eq. 30, and have yso sin zz 0 h Then, we compute the derivative and set it to the slope of the colored line, dy dz h SO cos zz 0 h 2 cot. 34 In other words, zh cos h SO cot 2 z 0. Then, we have 35

6 9 S. W. Lee and G. Wang: Helical half-scan Grangeat formula 9 FIG. 6. Transition points for design of the weighting functions on meridian planes. a Parameters of the projected trajectory on a meridian plane, b case i, c case ii, andd case viii. cos h SO cot The solution of this equation is meaningful only when 0 2 m, resulting in up to two solutions: t and t2. It is assumed that t2 is greater than t if it exists. Recall that these t and t2 are used for grouping vertices in Sec. II C. In terms of the above end points and tangential points, we can find the transition points as follows: e e, e2 e 2, t t, t2 t 2, 37 where (y,z)(sin,cos ). As shown in Fig. 6a, we

7 0 S. W. Lee and G. Wang: Helical half-scan Grangeat formula 0 can identify the type of a region according to the following criterion: t2 e2, indicating a doubly sampled region; e2 e, a singly sampled region; e t, a doubly sampled region. 3. Smooth weighting functions Weighting functions are designed in reference to e, e2, t, t2, which are functions of and. Mathematically speaking, there are up to two possibilities when we have neither t nor t2, which are e e2 and e e2. Similarly, there are up to six cases when we have only either t or t2. Furthermore, there are up to 24 cases when we have both t and t2. However, all the statements are not meaningful after our case-by-case inspection. It is found that there exist only cases. For example, in absence of t and t2, we always have the case of e e2, and never have the case of e e2. Similarly, it is impossible to have the cases of e t e2 and e2 t e in absence of t2. Those cases are i e e2 in absence of t and t2 ; ii t e e2 in absence t2 ; iii e2 e t in absence of t2 ; iv t e2 e in absence of t2 ; v e e2 t in absence of t2 ; vi t2 e2 e t ; vii t e e2 t2 ; viii e t2 t e2 ; ix t2 e e2 t ; x t2 e t e2 ; xi e t2 e2 t. In addition to our geometric analysis on projected trajectories on the meridian plane, we did exclusive numerical simulation to confirm that there are only the above meaningful cases. For all the,, we calculated the transition points, sorted the values, and decided which one of the mathematically possible 32 cases it belongs to. Once a specific case was found, we set the flag from that case on. After this kind of numerical verification, we eliminated the cases that never happened. Also, we repeated the simulation with respect to representative combinations of imaging parameters, including the source-to-origin distance, helical pitch and cone angle (2 m ). Finally, it was confirmed that there are indeed only the above cases that are feasible. For each of those meaningful cases, smooth weighing functions are designed in terms of e, e2, t, t2. The two general requirements are that the sum of the weights for each characteristic point be one, and 2 the weight profile along the direction be continuous. To further understand the design processing, some representative illustrations are considered helpful. Figure 6b represents the case where there is neither t nor t2 for given,, and only group exists. Therefore,, and 2 and 3 are set to zero. Figure 6c corresponds to the case where there is one tangential point, and two groups are available. The trajectory from e to t belongs to group, while that from t to e 2 belongs to group 2. The weighting function for group,, is designed to change gradually from to 0 along the projected trajectory from t to e, while the weighting function for group 2, 2, increases from 0 to in the same interval, and stays constant between e and e2. 3 is set to zero since there is no group 3. Of course, the sum of the weighting functions should be made one. Figure 6d illustrates the case where there are two tangential points, and we have group between e and t, group 2 between t and t 2, and group 3 between t e and e 2. The weighting function for the group,, should be one between e to t2, and smoothly change from to 0 along the projected trajectory from t to e. The weighting function for the group 3, 3, should smoothly change from 0 to along the trajectory from t2 to t, and be 0 between t and e2. The weighting function for the group 2, 2, is designed to smoothly increase from 0 at t2 until it reaches the middle point between t2 and t, and decrease to 0 at t. Some representative distributions of the weighting functions are included in Fig. 7. The weighting functions are formulated as follows, keyed to each of the feasible cases. i e e2 in absence of t and t2,, 2 0, 3 0. ii t e 2 e2 in absence of t2, cos t 2 e t e, t, 0, e e2, 2 sin t e t, t e,, e e2, iii e2 e t in absence of t2, 0, e2 e, sin 2 2 e t e, e t, 2, e e2, 3 0. cos 2 2 e t e, e2 t, iv t e2 e in absence of t2, sin 2 2 t e2 t, t e2,, e2 e, 2 cos t e2 t, t e2, 0, e2 e,

8 S. W. Lee and G. Wang: Helical half-scan Grangeat formula FIG. 7. a d Region map and weighting functions for meridian plane at 90, e h 70, and i l 250. a, e, i Region map, the brightest is for triply sampled zones, and the darkest for shadow zones. b, f, j The first weighting distribution; c, g, k the second weighting distribution; and d, h, l the third weighting distribution. v e e2 t in absence of t2, e e2, cos , e e2, e2 t e2, e2 t, sin sin 2 2 e t e, e t, t2 e2 t2, t2 e2,, e2 e, 3 0. sin 2 2 vi t2 e2 e t, e2 t e2, e2 t, 0, t2 e2, 0, e2 e, cos cos 2 2 e t e, e t, t2 e2 t2, t2 e2, 0, e2 e, 0, e t.

9 2 S. W. Lee and G. Wang: Helical half-scan Grangeat formula 2 vii t e e2 t2, cos 2 2 t e t, t e, 0, e e2, 0, e2 t2, 2 sin 2 2 t e t, t e,, e e2, cos , t e, e2 t2 e2, e2 t2, 0, e e2, 2 2, t2 e, 2, e e2, 2, e2 t, 2 2, t2 e, cos2 e 2 2 e2 e e2, e, 0, e2 t. x t2 e t e2, 0, t2 e, sin 2 e t e, e e t e /2, sin 2 2 viii e t2 t e2, e2 t2 e2, e2 t2., e t2, sin 2 e t e, e t e /2 t, 0, t e2, 2 2, t2 e, cos 2 t2 t t2, t2 t2 t t2 /2, 2 cos2 e t e, e e t e /2, 0, t2 t t2 /2 t, 0, t e2, 2 0, e t2, 0, e t e /2 t, 0, t e2, 3 2, t2 e, sin 2 t2 t t2, t2 t2 t t2 /2, 2 cos2 e t e, e e t e /2, sin 2 t2 t t2, t2 t t2 /2 t, 0, t e2, 3 0, e t2, 0, t2 t2 t t2 /2, cos 2 t2 t t2, t2 t t2 /2 t,, t e2. ix t2 e e2 t, 0, t2 e, sin2 e 2 2 e2 e e2, e, 2, e2 t, cos 2 e t e, e t e /2 t,, t e2. xi e t2 e2 t,, e t2, cos 2 t2 e2 t2, t2 t2 e2 t2 /2, 2 cos2 t2 e2 t2, t2 e2 t2 /2 e2, 2, e2 t, 2 0, e t2, 0, t2 t2 e2 t2 /2,

10 3 S. W. Lee and G. Wang: Helical half-scan Grangeat formula 3 7 Apply the weighting functions to the derivative of Radon data using Eq Repeat steps 7 until all the measurable characteristic points are done. 9 Estimate the Radon data in the shadow zone using the linear interpolation method in this study. 0 Use the two stage parallel-beam backprojection algorithm as defined by Eq. 4 to reconstruct an image volume. Note that results from steps 2 4 can be precalculated and stored for computational efficiency. Similarly, the rebinning coefficients from step 5 can be found beforehand. FIG. 8. First derivative Radon data of the 3D Shepp Logan phantom after data filling. a Zero padding, b linear interpolation. 2 cos2 t2 e2 t2, t2 e2 t2 /2 e2, 2, e2 t, 3 0, e t2, E. Interpolation sin 2 t2 e2 t2, t2 t2 e2 t2 /2, sin 2 t2 e2 t2, t2 e2 t2 /2 e2, 0, e2 t. As we did in the circular scanning case,,23 in this study we continue using the linear interpolation strategy to estimate missing data. The boundaries of the shadow zone were numerically determined. Then, the shadow zones were linearly interpolated along the direction using the measured boundary values. To demonstrate this interpolation idea graphically, Fig. 8 shows the derivatives of Radon data with no interpolation and with linear interpolation, respectively. F. Implementation procedure To summarize, the Grangeat-type half-scan algorithm can be implemented in the following steps. Specify a characteristic point,,, where the derivative of Radon data can be calculated. 2 Calculate t and t2. 3 Calculate t, t2, e, e2. 4 Calculate smooth weighting functions (,,), 2 (,,), and 3 (,,). 5 Determine line integration points for the given characteristic point according to the rebinning Eqs Calculate the derivatives of Radon data using Eq. 3, and store them according to their group membership as determined by Eq. 23. III. RESULTS We developed a software simulator in the IDL language Research Systems Inc., Boulder, Colorado for Grangeattype image reconstruction. In the implementation of the Grangeat-type formula, the numerical differentiation was performed with a built-in function based on 3-point Lagrangian interpolation. The source-to-origin distance was set to 5. The number of detectors per cone-beam projection was 256 by 256. The size of the 2D detector plane was 3.3 by 3.3. The helical pitch was 2. The full-cone angle was about 30 degree. The scan range was from 0 to 2 m. The number of projections was 20. The number of meridian planes was 80. The numbers of radial and angular samples were 256 and 360, respectively. Each reconstructed image volume had dimensions of 4. by 4. by 4., and contained 256 by 256 by 256 voxels. One might use the same number of projections above and below each and every slice so that all slices have the same image quality. Our preferred approach uses an asymmetric number of slices above and below the slice except for the z 0 slice. If we use the symmetric half-scan helical Feldkamp, 0 which has symmetric projections for any z location, all slices would have similar image quality. However, it means that each slice is reconstructed in a different time window. The primary purpose of our work is to develop half-scan algorithms in the Grangeat framework with superior temporal characteristics temporal resolution and temporal consistency; the latter requires that a whole volume of interest is reconstructed with projections in the same time window and less image artifacts which is achieved by appropriate data handling in the shadow zone. The reason that we use the asymmetric projections is to maintain this temporal consistency. We can also use half-scan helical Feldkamp methods in the same way. The 3D Shepp Logan phantom was used in the numerical simulation as shown in Table I. Figure 9 shows the derivatives of Radon data of the Shepp Logan phantom, the weighting functions for each group, and the combined data. Figure 0 presents typical reconstructed slices of the Shepp Logan phantom. With the helical half-scan Feldkamp method Fig. 0a and the helical half-scan Grangeat and zeropadding method Fig. 0b, the low intensity drop was away from the center plane. However, this type of artifact

11 4 S. W. Lee and G. Wang: Helical half-scan Grangeat formula 4 TABLE I. Parameters of the phantoms used in our numerical simulation. (a,b,c) denote the semiaxes of ellipsoids (x,y,z) the center coordinates of each ellipsoid, and are the same as in Fig.. The actual density at a position is determined by summing the densities of the ellipsoids covering that point. Phantom a b c x y z Density Shepp-Logan FIG. 9. First derivative Radon data of the 3D Shepp Logan phantom at 90 for a group, b group 2, and c group 3. d, e, f Weighting functions for each group and g combined data.

12 5 S. W. Lee and G. Wang: Helical half-scan Grangeat formula 5 FIG. 0. Reconstructed images of the 3D Shepp Logan phantom. a Helical half-scan Feldkamp, b Grangeattype helical half-scan reconstruction with zero padding, b Grangeat-type helical half-scan reconstruction with linear interpolation. The same projection range has been used to reconstruct the image volume in the same time window. First row, vertical slice at y 0.242; second row, vertical slice at x0.0369; and third row, transverse slice at z0.37. The contrast range is.005,.05. was essentially eliminated with the helical half-scan Grangeat and linear interpolation method Fig. 0c. IV. DISCUSSIONS AND CONCLUSION In the writing period of our first draft, it came to our attention that Noo and Heuscher recently published a halfscan cone-beam reconstruction paper in a SPIE conference. 3 The similarity between our work and their paper is that both the groups considered the half-scan cone-beam reconstruction using the Grangeat method. However, their work is in the filtered backprojection framework, 5 while ours is in the rebinning framework. 4 We believe that both half-scan algorithms are complementary. It seems that the data filling mechanism is more flexible in our rebinning framework. Noo and Heuscher suggested that the parallel-beam approximation of cone-beam projection data be used to estimate missing data, which is done in the spatial domain. This kind of spatial domain processing is also allowed in our framework. In addition to the spatial domain approximation, the Radon domain estimation, such as linear interpolation, spline interpolation and knowledge-based interpolation, can be done in our framework as well. However, in the filtered backprojection framework, each frame of cone-beam projection data can be processed as soon as it is acquired, a desirable property for practical implementation. Clearly, a systematic comparison of the two algorithms is worth further investigation. The helical scanning geometry studied in this paper is to solve the short object problem, which assumes that the object is completely covered by the x-ray cone beam from any source position. Even though research with the short object geometry is valuable on its own, such as for micro-ct imaging of spherical samples, the geometry for data truncation along the axial direction is the most practical. Therefore, the extension of our Grangeat-type helical half-scan CT work to the long object case is an important future topic. In conclusion, we have formulated a Grangeat-type halfscan algorithm in the helical scanning case to solve the short object problem. The smooth half-scan weighting functions have been designed to compensate for data redundancy and inconsistency. Numerical simulation results have verified the correctness of our formulation, and demonstrated the merits of our algorithm. We believe that the Grangeat-type half-scan algorithm is promising for quantitative and dynamic biomedical applications of CT and micro-ct.

13 6 S. W. Lee and G. Wang: Helical half-scan Grangeat formula 6 ACKNOWLEDGMENTS We would like to thank Dr. Dominic J. Heuscher and Dr. Frederic Noo for sending us their paper. 3 This work was supported in part by the NIH Grant Nos. R0 DC03590 and EB a Electronic mail: swlee@ct.radiology.uiowa.edu b Electronic mail: ge@ct.radiology.uiowa.edu S. W. Lee and G. Wang, A Grangeat-type half-scan algorithm for conebeam CT, Med. Phys. 30, D. W. Holdsworth, Micro-CT in small animal and specimen imaging, Trends Biotechnol. 20, S34 S G. Wang and M. W. Vannier, Micro-CT scanners for biomedical applications: an overview, Adv. Imaging 6, R. Ning, X. Tang, D. Conover, and R. Yu, Flat panel detector-based cone beam computed tomography with a circle-plus-two-arcs data acquisition orbit: Preliminary phantom study, Med. Phys. 30, W. A. Kalender, Computed Tomography: Fundamentals, System Technology, Image Quality, Applications Publicis MCD Verlag, Munich, G. Wang, C. R. Crawford, and W. A. Kalender, Multirow detector and cone-beam spiral/helical CT, IEEE Trans. Med. Imaging 9, D. L. Parker, Optimal short scan convolution reconstruction for fanbeam CT, Med. Phys. 9, C. R. Crawford and K. F. King, Computed tomography scanning with simultaneous patient translation, Med. Phys. 7, G. T. Gullberg and G. L. Zeng, A cone-beam filtered backprojection reconstruction algorithm for cardiac single photon emission computed tomography, IEEE Trans. Med. Imaging, G. Wang, Y. Liu, T. H. Lin, and P. C. Cheng, Half-scan cone-beam x-ray microtomography formula, Scanning 6, S. Zhao and G. Wang, Feldkamp-type cone-beam tomography in the wavelet framework, IEEE Trans. Med. Imaging 9, Y. Liu, H. Liu, Y. Wang, and G. Wang, Half-scan cone-beam CT fluoroscopy with multiple x-ray sources, Med. Phys. 28, F. Noo and D. J. Heuscher, Image reconstruction from cone-beam data on a circular short-scan, Proc. SPIE 4684, P. Grangeat, Mathematical framework of cone beam 3D reconstruction via the first derivative of the Radon transform, in Mathematical Methods in Tomography, edited by G. T. Herman, A. K. Louis, and F. Natterer, Lecture Notes in Mathematics 497 Springer-Verlag, Berlin, 99, pp M. Defrise and R. Clack, A Cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection, IEEE Trans. Med. Imaging 3, Y. Weng, G. L. Zeng, and G. T. Gullberg, A reconstruction algorithm for helical cone-beam SPECT, IEEE Trans. Nucl. Sci. 40, H. Kudo, F. Noo, and M. Defrise, Cone-beam filtered-backprojection algorithm for truncated helical data, Phys. Med. Biol. 43, B. D. Smith, Image reconstruction from cone-beam projections: Necessary and sufficient conditions and reconstruction methods, IEEE Trans. Med. Imaging MI-4, H. K. Tuy, An inversion formula for cone-beam reconstruction, SIAM Soc. Ind. Appl. Math. J. Appl. Math. 43, C. Jacobson, Ph.D. dissertation thesis, Linkoeping University, S. R. Deans, The Radon Transform and Some of its Applications Wiley- Interscience, New York, F. Natterer, The Mathematics of Computerized Tomography Society for Industrial and Applied Mathematics, Philadelphia, S. W. Lee, G. Cho, and G. Wang, Artifacts associated with implementation of the Grangeat formula, Med. Phys. 29,