RECENTLY, biomedical imaging applications of

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1 1190 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 24, NO. 9, SEPTEMBER 2005 A General Exact Reconstruction for Cone-Beam CT via Backprojection-Filtration Yangbo Ye*, Shiying Zhao, Hengyong Yu, Ge Wang, Fellow, IEEE Abstract In this paper, we prove a generalized backprojectionfiltration formula for exact cone-beam image reconstruction with an arbitrary scanning locus. Our proof is independent of the shape of the scanning locus, as long as the object is contained in a region where there is a chord through any interior point. As special cases, this generalized formula can be applied with cone-beam scanning along nonstard spiral saddle curves, as well as in an -PI window setting. The algorithmic implementation numerical results are described to support the correctness of our general claim. Index Terms Backprojection-filtration, computed tomography (CT), cone-beam, exact reconstruction, filtered-backprojection. I. INTRODUCTION RECENTLY, biomedical imaging applications of cone-beam computed tomography (CT) call for more flexible scanning loci. Specifically, we are working on significant development along this direction for electron-beam CT/micro-CT [17], bolus-chasing CT angiography [18], other applications. Approximate reconstruction algorithms of Feldkamp-type for cone-beam scanning along nonstard spirals were already reported [15], [16]. An exact reconstruction formula in the filtered-backprojection format was proved for cone-beam helical scanning with constant or variable pitch [4] [6], [8]. A backprojection-filtration counterpart of the Katsevich formula was proposed as well [26], [27], in which derivatives of cone-beam data are first backprojected on a PI-line, a Hilbert transform is then performed along the PI-line to produce the reconstruction results. This formulism was also extended to the case of cone-beam helical scanning with variable pitch [28]. It is highly desirable to extend these important results further to allow more general scanning patterns. In this paper, we will give a new much more general proof of the backprojection-filtration formula proposed by Zou Manuscript received November 14, 2004; revised June 9, This work was supported in part by a Carver Scientific Research Initiative grant, in part by a University of Iowa Mathematical Physical Sciences Funding Program award, in part by the National Institutes of Health (NIH) under NIBIB Grant EB NIBIB Grant EB The Associate Editor responsible for coordinating the review of this paper recommending its publication was C. Crawford. Asterisk indicates corresponding author. *Y. Ye is with the Departments of Mathematics Radiology, University of Iowa, Iowa City, IA USA ( yangbo-ye@uiowa.edu). S. Zhao H. Yu are with the Department of Radiology, University of Iowa, Iowa City, IA USA ( shiying-zhao@uiowa.edu; hengyong-yu@ uiowa.edu). G. Wang is with the Departments of Radiology, Biomedical Engineering, Mathematics, University of Iowa, Iowa City, IA USA ( ge-wang@ ieee.org). Digital Object Identifier /TMI Pan [26], [27], describe the algorithmic steps associated with the generalized backprojection-filtration formula, report numerical simulation results with the 3D Shepp-Logan head phantom the Defrise disk phantom. The main difference between our proof the original proof by Zou Pan [25] is that they used a geometric argument based on the shape of the involved helix, while our proof is analytic, applies to quite general scanning loci. Our generalized backprojection-filtration formula will be exact with scanning loci of the following types. (i) Our basic setting is a smooth curve (ii) (1.1) with a chord through a given point with two end points on the curve [Fig. 1(a)]. Let be an object function of compact support not in touch with the curve. Our goal is to prove the exactness of the backprojection-filtration formula for reconstruction of at any given point (Theorem 3.1). As a special case, let (1.2) be a nonstard spiral with variable radius variable pitch function. Assume that for. We will consider a region inside the spiral such that for any point, there is at least one PI-line through. Here a PI-line goes through two points on the spiral, with [Fig. 1(b)]. This region is also referred to as a region of PI-lines. We note that the union of all PI-lines defines such a region. Uniqueness of PI-lines is not essential for the proposed reconstruction formula, although this uniqueness would minimize the redundancy in cone-beam data acquisition. (iii) By taking for some constants, the curve described by (1.2) becomes an elliptic spiral with a variable pitch, or with a constant pitch when for some constant. (iv) By setting to a constant, (1.2) becomes a helix with a variable pitch, studied by [8], [28]. Equation (1.2) becomes a stard helix if we further assume for some constant. (v) If two points on the nonstard or stard spiral are apart for more then one turn, say, the chord /$ IEEE

2 YE et al.: A GENERAL EXACT RECONSTRUCTION FOR CONE-BEAM CT VIA BACKPROJECTION-FILTRATION 1191 Fig. 1. Various cone-beam scanning patterns. (a) A general scanning locus with a point r to be reconstructed on a chord L, (b) a nonstard spiral with a Pl-line, (c) a nonstard spiral with a 2-Pl- line, (d) a saddle curve with a t-line. connecting them is no longer a PI-line in the original sense. The corresponding reconstruction of at becomes a solution to the -PI-windows problem, studied extensively by [13], [1], [17] in the stard helical case. The generalized backprojection-filtration formula in Theorem 3.1 will also cover this case. We call such an an -PI-line [Fig. 1(c)] define a region of -PI-lines. (vi) We can also deal with stard nonstard saddle curves. [12] designed reconstruction algorithms for saddle curves, where a chord is named as a -line [Fig. 1(d)]. Our Theorem 3.1 is valid for such a cone-beam scanning locus as well. We will call the union of all -lines the region of -lines. The results in this paper were reported at the SPIE s 49th Annual Meeting (Denver, CO, August 6, 2004), with the full paper published in the corresponding proceedings [19]. II. NOTATION Consider the scanning locus (1.1) an object function with compact support such that for any in a neighborhood of the curve. We assume that is continuous smooth. For any unit vector,define the cone-beam projection of from the source point on the locus in the direction by (2.1) Note that this integral is actually taken over a finite interval, because the function is compactly supported. In the computation below, we will take the unit vector as the one pointing to from the source point on the curve (2.2) Recall in our setting (1.1), is a point on the chord connecting the two points on the locus, where are the corresponding parameters of these two points along the curve. We also need to use a unit vector along (2.3)

3 1192 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 24, NO. 9, SEPTEMBER 2005 III. EXACT RECONSTRUCTION FORMULA First, let us define the integral kernel (3.1) of exist everywhere are absolutely integrable in. Then (3.4) which can also be written as the product of two delta functions [see (5.1)]. As to be explained in Section V, is the kernel function of the Hilbert transform along the chord passing through the point. Therefore the meaning of is that it defines an integral transform from a suitable function on to (3.2) In other words, the integral transform on the left-h side of (3.2) is by definition a twisted Fourier inverse transform of the Fourier transform of. Note that, in general the order of integration on the right-h side of (3.2) cannot be interchanged. Given an object function with compact support which vanishes in a neighborhood of the curve (1.1), let us consider a special function which is given by following expression: (3.3) We remark that one of the two terms under the differentiation vanishes at fixed, because either or points away from the object. Adding the term in (3.3) is indeed a mathematical trick to extend the finite integral over in (4.8) to an infinite integral. Note that, because of the upper lower limits from the chord of, our function also depends on. As pointed out by [26], [27],, is the weighted cone-beam backprojection of the derivative of the cone-beam data for the point, back-projected over the PI segment of point. It is the subject matter of [26], [27] that one can recover by applying the integral transform (3.2) to in (3.3), for cone-beam projections along a scanning helix. In other words, they formulated (3.4) below for the helix. What we will contribute in this paper is to prove (3.4) for an arbitrary smooth curve (i) as defined in (1.1), valid for other smooth scanning loci (ii) through (vi). Theorem 3.1: Consider a smooth curve (1.1) a point on a chord from to. Let be an object function of compact support such that it vanishes in a neighborhood of the curve. Assume that the fifth partial derivatives where the integral transform is defined in (3.2), given by (3.3), by (2.1), by (2.2). We remark that for stard or nonstard spirals (the cases (ii), (iii), (iv) described above), Theorem 3.1 enables an exact reconstruction of if its support is contained in a region of PI-lines. If the region does not completely cover the support of, the theorem is still applicable for the exact reconstruction inside the region. When Theorem 3.1 is applied to the cases (v) ( -PI-windows) (vi) (saddle curves), the same discussion holds for a region of -PI-lines the region of -lines. Note that in Theorem 3.1 we may simply assume that is a smooth function of compact support, all its partial derivatives exist are continuous. The theorem is formulated in the present stronger form because we will need this in a subsequent work. We plan to study exact reconstruction for certain discontinuous object functions. By Theorem 3.1, this is reduced to reconstruction of functions with jumps in the function /or its partial derivatives of up to the fourth order. The required detector area can be deduced based on the backprojection-filtration structure of (3.4). Specifically, consider a chord that intersects a volume of interest spans a piece of a scanning locus. In order to invert the Hilbert transform along the chord, we need backprojected data along the piece of where the object is nonzero (not only over the section of that intersects the region of interest ). The derivatives of cone-beam data collected at any source position along that piece of the curve only need to be recorded if they can be backprojected on. When this principle is applied in the stard helical scanning case, we have the well-known Tam-Danielsson window [3]. In the case of nonstard spirals, we obtain a generalized Tam-Danielsson window. Similarly, we can formulate the detector requirement in other cases of scanning loci. IV. PROOF OF THEOREM 3.1 With (3.2), the right-h-h side of (3.4) is as shown in (4.1) at the bottom of the page. We will first compute the expression on the right-h side of (4.1) contributed by. The computation of the expression contributed by is similar. Denote by (4.2) (4.1)

4 YE et al.: A GENERAL EXACT RECONSTRUCTION FOR CONE-BEAM CT VIA BACKPROJECTION-FILTRATION 1193 the Fourier transform of. Then respect to in (4.5). Since the innermost integral in (4.5) is dominated by, we can further interchange the integrals with respect to in (4.5). Now the innermost integral becomes Changing variables from to, we get (4.6) (4.3) by Fourier s inversion formula. We take the derivative under the inner integral to get (4.6) becomes an integral of to, which equals integrated from (4.4) As we have assumed that the fifth partial derivatives of are absolutely integrable in, its Fourier transform is bounded by, by the Riemann-Lebesgue lemma. Consequently, the inner integral on the right-h side of (4.3) is dominated by, while the inner integral on the right-h side of (4.4) is dominated by a convergent integral. This proved that it is legitimate to interchange the order of differentiation the inner integral in (4.3). Hence, (4.4) is valid. Substituting (4.4) to the right-h side of (4.1), the expression on the right-h side of (4.1) contributed by becomes (4.7) Although this change of variables is not one-to-one, one can still prove its validity by decomposing into subintervals on each of which is monotone. An alternate argument is that the integral (4.6) can be put in the form of the work of a conservative force on a given path in space, which yields (4.7) as the difference between the values taken by the potential of the force at the end points of the integration path. Using (4.7) (4.5), we now get the expression on the right-h side of (4.1) contributed by as (4.8) (4.5) where we have changed variables to using (2.2). As is of compact support, we can differentiate the right-h side of (4.2) under the integration. Hence, is a smooth function, its partial derivatives are also bounded by. Consequently, the function is bounded by, the innermost integral in (4.5) converges absolutely. We next observe that by setting. From the contribution of of (4.1) equals, we can see that to the right-h side (4.9) Therefore, the right-h side of (4.1) is now equal to the sum of (4.8) (4.9): (4.10) The integral on the left-h side is indeed the Fourier inversion of, hence equals. Therefore, the innermost integral in (4.5) as a function of is of compact support for, due to the compactness of the support of. Note that in (4.5) the integral with respect to is taken over a compact set. Therefore, we can interchange the integrals with To change the order of the integrals in (4.10), we notice that, by the Fourier inversion formula, the innermost integral in (4.10) equals We now consider two cases. When, the above function is of compact support with respect

5 1194 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 24, NO. 9, SEPTEMBER 2005 to the variable. Therefore, the singularity of the integrant in (4.10) occurs only at. This singularity is nevertheless removable, because in a neighborhood of, the relevant integrant can be estimate by its derivative, shown in the first equation at the bottom of the page, which is integrable. When or, the integral with respect to appears to be improper because the support for is. To treat these cases, take a small consider the region for.for, the two innermost integrals in (4.10) are equal to the second equation shown at the bottom of the page. As the function is of compact support, this integral has a bound independent of. Also note that the expression inside the square brackets in the above equation tends to 0 when. By a similar argument as above, we know that the singularity at of the integr on the right-h side of the above equality is again removable when. Consequently Similar arguments apply to the same integral taken over. When, the integral with respect to in (4.10) converges uniformly with respect to. This allows us to interchange the order of integration get the third equation shown at the bottom of the page. Taking, we obtain a legitimate change of integration order in (4.10) with respect to. Similarly, we can further interchange the order of integrations in (4.10) with respect to. First, we can change variables from to with, for these three-dimensional (3-D) integrals. Then, we can change variables from to with. Consequently, (4.10) becomes (4.11), shown at the bottom of the page, by the Fourier inversion formula. The innermost integral on the right-h side of (4.11) can be written as shown in the fifth (4.11)

6 YE et al.: A GENERAL EXACT RECONSTRUCTION FOR CONE-BEAM CT VIA BACKPROJECTION-FILTRATION 1195 equation at the bottom of the previous page. Therefore, (4.11), the right-h side of (4.1), becomes (4.12) Note that (4.12) can also be deduced from (4.11) directly using a distribution formula Now we need to compute the inner integral in (4.12). Since is on the chord between, are of opposite signs, while the sign of the former is indeed. It is a stard formula of the Fourier transforms that (4.13) We finally remark that this integral formula is valid as distributions as well as in the regular sense. The integral on the left-h side of (4.13) converges, not absolutely but conditionally. Using (4.13), (4.12) is immediately simplified to which completes the proof of Theorem 3.1. V. IMPLEMENTATION AND SIMULATION Following the steps of [26], as shown in Fig. 2, for a chord specified by, a local Cartesian coordinate system can be defined based on the chord. The -axis is made along the chord its origin is located at the middle of the chord segment. Then, the - -axes are assigned perpendicular to the chord. Note that the selection of the - -axes is arbitrary, only the component is used in the reconstruction. After appropriate translation rotation transforms, any in the global coordinate system can be turned into the chord-based local coordinate system.in the local coordinate system, for any on the chord, we have. Hence, (3.1) can be simplified as follows [26]: (5.1) Obviously, will be zero if is not on the chord. Therefore, (3.1) means a Hilbert transform along the chord determined by. Denote the object function the weighted backprojection at on the chord as, respectively. Substituting (5.1) into (3.4), we have (5.2) Fig. 2. Coordinate systems variables for backprojection-filtration cone-beam reconstruction based on a chord, such as a P1-, n-pi-, or t-iine. where represents the Hilbert transform operator. Because has a finite support on the PI-segment, we have [26] (5.3) where denote the minimal maximal values of the finite support interval, respectively. Using the method proposed by [10], one can recover from within. Based on the above analysis, we can implement the general backprojection-filtration formula for exact image reconstruction in the following four steps: Step 1) Data Differentiation: Compute the derivatives of cone-beam data with respect to variable while keeping in (2.1) unchanged, using the chain rule or the derivative definition of [9], [24]. Step 2) Weighted Backprojection: Calculate the weighted 3-D backprojection of the cone-beam data derivatives onto the chord-segment specified by, using (3.3) [26]. Note that if there are multiple chords now through any point inside the object, we may perform multiple reconstructions at that point. Step 3) Inverse Hilbert Filtering: Reconstruct the object function by solving (5.3), using the method of [10]. Step 4) Image Rebinning: Remap the reconstructed image from the chord-based coordinate system into the global coordinate system, using an appropriate interpolation method. To validate the correctness of the generalized backprojection-filtration formula the associated algorithm, we implemented it in MATLAB, evaluated it using the 3D Shepp-Logan head phantom the Defrise disk phantom.

7 1196 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 24, NO. 9, SEPTEMBER 2005 Fig. 3. Generalized reconstruction of the 3D Shepp-Logan phantom via backprojection-filtration from cone-beam data collected along an elliptical spiral locus. Representative slices were reconstructed at (a) x = 0cm, (b) y = 02:5cm (c) z = 02:5cm(a display window [1:0; 1:05]), (d) a profile comparison along the white line in (c), where the dotted curve is for the original while the solid curve is for the reconstruction. In the preliminary simulation, an elliptic spiral was defined as, where,. The phantom was located in the center of the global coordinate system within a spherical support of radius 10 cm. As shown in Fig. 2, a rectangular planar detector was used, consisting of elements each being The detector was made perpendicular to the line that intersects the -axis orthogonally links the X-ray source the detector center, the short side of the detector parallel to the -axis. The distance from the detector to the -axis was 75 cm. Then, 1200 cone-beam projections were uniformly acquired for. The final reconstruction was done on a grid. Figures 3 4 present typical slices of the reconstructed phantoms, which are in an excellent agreement with the corresponding slices of the ideal counterparts. VI. DISCUSSIONS AND CONCLUSION Computationally speaking, the backprojection-filtration method is not as efficient as the filtered-backprojection method. Since the Hilbert filtering is performed along various chords such as PI-, -PI-, or -lines, the weighted backprojection must be performed upon these chords. Then, a rebinning step is needed to have the final reconstruction in the Cartesian coordinates. This rebinning step requires a larger amount of intermediate data rather extensive interpolative processing.

8 YE et al.: A GENERAL EXACT RECONSTRUCTION FOR CONE-BEAM CT VIA BACKPROJECTION-FILTRATION 1197 Fig. 4. Generalized reconstruction of the Defrise disk phantom via backprojection-filtration from cone-beam data collected along an elliptical spiral locus. A central coronal slice was reconstructed at (a) x =0cm(a display window [0; 1]), (b) a profile comparison along the white line in (a), where the dotted curve is for the original while the solid curve is for the reconstruction. Hence, we need reconstruct an intermediate backprojected image in the chord-segment-based coordinate system before rebinning. Since the additional rebinning step is subject to some loss in spatial resolution, the intermediate image should be typically 2 4 times larger than the final reconstructed image for image quality comparable to that with the FBP algorithm. The greatest computational expense is actually due to extensive backprojections onto PI lines, because the amount of backprojections in BPF algorithms is far greater than in FBP algorithms. The resultant computational overhead is the main factor that makes the backprojection-filtration method significantly slower than the filtered-backprojection method. By the way, if we use the method suggested by [26] to avoid the computation of data derivatives respective to, the image quality will be improved at an additional computational cost. While for a stard spiral there is a unique PI-line through any interior point [3], in a general case a chord through a point may not have such a uniqueness property (cf. [20] [21]). When there are chords through a given, the generalized formula (3.4) can be applied to any one of the chords the corresponding arc. In the case of noise-free data, this must produce the exact reconstruction. However, in the case of noisy data the reconstruction result at depends on which of the chords is in use. A simplest scheme to utilize the data redundancy is to average all the reconstruction results for noise reduction. Theoretically speaking, an optimal solution to this signal estimation problem should be statistically determined, which is considered beyond the scope of the present paper. Our analytic proof of the generalized backprojection-filtration formula has not only confirmed the exactness of the backprojection-filtration methodology for helical cone-beam CT but also revealed its validity for exact cone-beam reconstruction with rather general scanning loci. As proposed by [26] [27], the exact reconstruction from data within the Tam-Danielsson window is doable using their backprojection-filtration formula. According to our above analysis, the exact image reconstruction from data within the generalized Tam-Danielsson window in the nonstard spiral case is also doable. Here by the generalized Tam-Danielsson window we mean the minimum detector window bounded by the upper lower turns of a nonstard spiral. Furthermore, in the case of other types of scanning loci such as saddle curves [12], the generalized formula in Theorem 3.1 should hold as well, as long as the detection window is adapted to contain the needed data. Finally, this generalized formula actually allows exact reconstruction on a chord from transversely truncated data if sufficient data are collected for the chord based inverse Hilbert transform [10]. The full conference paper [19] was published by us early August last year in which a complete proof of the backprojection-filtration (BPF) reconstruction formula with a general scanning trajectory was given in essentially the same way as in this paper. Later on, [27] was published on the BPF ( FBP as well) reconstruction formula in the setting of a stard helical cone-beam scanning, with the understing that the results can also be extended to general, smooth trajectories. Then, two other papers were independently published on the same subject [11], [25]. In conclusion, we have extended the backprojection-filtration formula proposed by [26] from stard helical cone-beam scans to quite general cone-beam scans, allowing nonstard spirals saddle curves, virtually any other smooth scanning loci. Our proof is analytic, being independent of the specific shape of the scanning locus only subject to the condition that the object is contained in a region of chords such as PI-, -PI-, or -lines. The numerical simulation results obtained using the generalized formula are in an excellent agreement with the theoretical prediction.

9 1198 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 24, NO. 9, SEPTEMBER 2005 ACKNOWLEDGMENT The authors are deeply grateful for Dr. Noo the anonymous reviewers for their rigorous checking, constructive comments, valuable advice, which have made this paper much stronger than the initial submission. REFERENCES [1] C. Bontus, T. Kohler, R. Proksa, A quasiexact reconstruction algorithm for helical CT using a 3-PI acquisition, Med. Phys., vol. 30, no. 9, pp , [2] R. Bracewell, The Fourier Transform Its Applications, 3rd ed. New York: McGraw-Hill, [3] P. E. Danielsson, P. Edholm, M. Seger, Toward exact 3d-reconstruction for helical cone-beam scanning of long objects. A new detector arrangement a new completeness condition, in Proc Int. Meeting Fully Three-Dimensional, Pittsburgh, PA, 1997, pp [4] A. Katsevich, Theoretically exact filtered backprojection-type inversion algorithm for spiral CT, SIAM J. Appl. Math., vol. 62, pp , [5], A general scheme for constructing inversion algorithms for cone beam CT, Int. J. Math. Math. Sci., vol. 21, pp , [6], Improved exact FBP algorithm for spiral CT, Adv. Appl. Math., vol. 32, pp , 2004a. [7], On two versions of a 3 algorithm for spiral CT, Phys. Med. Biol., vol. 49, pp , [8] A. Katsevich, S. Basu, J. Hsieh, Exact filtered backprojection reconstruction for dynamic pitch helical cone beam computed tomography, Phys. Med. Biol., vol. 49, pp , [9] F. Noo, J. Pack, D. Heuscher, Exact helical reconstruction using native cone beam geometries, Phys. Med. Biol., vol. 48, pp , [10] F. Noo, R. Clackdoyle, J. D. Pack, A two-step Hilbert transform method for 2D image reconstruction, Phys. Med. Biol., vol. 49, pp , [11] J. Pack, F. Noo, R. Clackdoyle, Cone-beam reconstruction using the backprojection of locally filtered projections, IEEE Trans. Med. Imag., vol. 24, no. 1, pp , Jan [12] J. D. Pack, F. Noo, H. Kudo, Investigation of a saddle trajectory for cardiac CT imaging in cone beam geometry, in Proc. VIIth Int. Conf. Fully 3D Reconstruction In Radiology Nuclear Medicine, Saint Malo, France, Jun. Jul. 29 4, [13] R. Proksa, T. Kohler, M. Grass, J. Timmer, The n-pi-method for helical cone-beam CT, IEEE Trans. Med. Imag., vol. 19, no. 9, pp , Sep [14] K. C. Tam, S. Samarasekera, F. Sauer, Exact cone-beam CT with a spiral scan, Phys. Med. Biol., vol. 43, pp , [15] G. Wang, T. H. Lin, P. C. Cheng, D. M. Shinozaki, H. G. Kim, Scanning cone-beam reconstruction algorithms for x-ray microtomography, Proc. SPIE, vol. 1556, pp , Jul [16] G. Wang, T. H. Lin, P. C. Cheng, D. M. Shinozaki, A general conebeam reconstruction algorithm, IEEE Trans. Med. Imag., vol. 12, no. 3, pp , Sep [17] G. Wang Y. Ye, Nonstard Spiral Cone-Beam Scanning Methods, Apparatus, Applications, Provisional Patent Application 60/588682, Filing date: July 16, [18] G. Wang M. W. Vannier, Bolus-Chasing Angiography With Adaptive Real-Time Computed Tomography, U.S. Patent , Mar. 18, [19] Y. Ye, S. Zhao, Y. Yu, G. Wang, Exact reconstruction for cone-beam scanning along nonstard spirals other curves, in Proc. SPIE, vol. 5535, Development in X-Ray Tomography IV, U. Bonse, Ed., Aug. 6, 2004, pp [20] Y. Ye, J. Zhu, G. Wang, Minimum detection windows, PI-line existence uniqueness for helical cone-beam scanning of variable pitch, Med. Phys., vol. 31, no. 3, pp , 2004a. [21], Geometric studies on variable radius spiral cone-beam scanning, Med. Phys., vol. 31, no. 6, pp , 2004b. [22] H. Yu, Y. Ye, G. Wang, Katsevich-type algorithms for variable radius spiral cone-beam CT, in Proc. SPIE, vol. 5535, Development in X-Ray Tomography IV, U. Bonse, Ed., Aug. 6, 2004, pp [23] H. Yu G. Wang, Studies on artifacts of the Katsevich algorithm for spiral cone-beam CT, in Proc. SPIE, vol. 5535, Development in X-Ray Tomography IV, U. Bonse, Ed., Aug. 6, 2004, pp [24], Studies on implementation of the Katsevich algorithm for spiral cone-beam CT, J. X-ray Science Technol., vol. 12, pp , 2004c. [25] T. L. Zhuang, S. Leng, B. E. Nett, G. H. Chen, Fan-beam cone-beam image reconstruction via filtering the backprojection image of differentiated data, Phys. Med. Biol., vol. 49, pp , [26] Y. Zou X. Pan, Exact image reconstruction on PI lines from minimum data in helical cone-beam CT, Phys. Med. Biol., vol. 49, pp , 2004a. [27], An extended data function its generalized backprojection for image reconstruction in helical cone-beam CT, Phys. Med. Biol., vol. 49, pp. N383 N387, 2004b. [28] Y. Zou, X. Pan, D. Xia, G. Wang, PI-line- based image reconstruction in helical cone-beam computed tomography with a variable pitch, Med. Phys., vol. 32, no. 8, pp , 2005.