Guidance for cone-beam CT design: tradeoff between view sampling rate and completeness of scanning trajectories

Guidance for cone-beam CT design: tradeoff between view sampling rate and completeness of scanning trajectories Guang-Hong Chen a,b, Jeffrey H. Siewerdsen c,shuaileng a, Douglas Moseley c,briane. Nett a, Jiang Hsieh d, David Jaffray c, and C. A. Mistretta a,b a Department of Medical Physics, University of Wisconsin-Madison, WI 53704; b Department of Radiology, University of Wisconsin-Madison, WI 53792; c Ontario Cancer Institute, Princess Margaret Hospital, Canada; d Applied Science Laboratory, GE Healthcare Technologies, Waukesha, Wisconsin ABSTRACT When an imaging task is specified, the design of a cone-beam CT scanner includes specifications of scanning trajectory and the corresponding image reconstruction algorithms, the requirements on detector size, and the requirements on x-ray tubes. Given the limited flat-panel detector readout speed and the need of short scanning time in a clinical setting, the available number of total view angles is normally limited to several hundred. It is known that when all the focal spots are distributed along a circular trajectory, the cone-beam artifacts are present in the reconstructed out-of-plane images when the cone-angle is relatively large. In order to mitigate or eliminate the cone-beam artifacts, the source trajectory should be complete in the sense of satisfying the so-called Tuy data sufficiency condition. However, assuming a constant number of view angles, a complete source trajectory will potentially lead to lower view sampling rate and cause view aliasing artifacts. Therefore, for a given imaging task and given total number of view angles, it is important to study the tradeoff between the view sampling rate and the completeness of the scanning source trajectories. In this paper, we numerically and experimentally studied the above tradeoff. Specifically, numerical simulations were conducted to study this tradeoff using three different source trajectories: (1) a circular trajectory, (2) a helical trajectory, and (3) a two-concentric-orthogonal-circle trajectory. A single x-ray tube and a flat panel imager mounted on an optical bench was utilized to experimentally study the tradeoff between the circular source trajectory and the helical source trajectory. For the complete source trajectories, some novel cone-beam image reconstruction algorithms have been utilized to reconstruct images and compare image quality in numerical simulations and benchtop experiments. 1. INTRODUCTION Recently, computed tomography (CT) has been shifting from a 2-D to a 3-D approach, both in data acquisition and in image reconstruction. However, the most popular modern CT modalities, helical CT and electron beam CT (EBCT), provide limited temporal resolution for an entire organ. The main reason for this drawback is the fact that the Z-coverage of most multi-slice detectors is limited to a few centimeters. At the same time, large spatial coverage requires sequential acquisition of slices as the imaged region of interest (ROI) passes through a relatively narrow region of exposure. As a consequence, the goal of time-resolved cardiac CT and contrast enhanced CT angiography (CTA) is not achieved yet. In order to significantly enlarge the spatial coverage in CT scanning, flat panel detectors were introduced in the CT research field. Recently, several volume CT systems have been developed with a regular angiographic x-ray tube and an opposed flat panel X-ray detector. 1 4, 6 Although these designs increase the useful detection cone angles, they may not provide the optimal source trajectory for dynamic imaging over a large fields of view.in addition, the current flat panel limits the scan time due to the limited data acquisition frame rate. In principle, the feasibility study of a new cone-beam CT will consist of three components: (1) selection of the scanning trajectory and the corresponding image reconstruction algorithms, (2) the required x-ray tube All scientific correspondence should be addressed to G.-H. Chen via email: gchen7@wisc.edu

configurations, and (3) the required detector. In particular, the scanning trajectory will significantly affect the efficiency of usage of the available x-ray radiation, and thus may change the practical specifications on tube requirements. Image reconstruction algorithms, which are inherently scanning trajectory dependent, impose restrictions on the required detector size. In a flat panel based cone-beam CT scanner, with approximately ten seconds data acquisition time, only 300 views of cone-beam projections may be acquired. It is important to study how to distribute these 300 focal spots along a scanning trajectory. Simple distribution along a circle allows for high sampling rate, but it suffers cone-beam artifacts since the trajectory does not fulfill the so-called Tuy data sufficiency condition. When a complete scanning trajectory is selected, the view sampling density is lower and thus the reconstructed images may suffer the view sampling artifacts. Therefore, it is fundamentally important to numerically and experimentally study the interplay between the view sampling density and the completeness of scanning trajectories. This paper concentrates on studying the tradeoffs between view sampling density and completeness of the scanning trajectory. Specifically, we numerically distributed the assumed 300 focal spots of cone-beam projections along three source trajectories: (1) a circular trajectory, (2) a helical trajectory, and (3) a two-concentric-orthogonal-circle trajectory to study the cone-beam image artifacts due to completeness of a scanning trajectory and the streaking artifacts caused by the view sampling density. We also conducted experimental studies to assess the tradeoff between cone-beam artifacts and streaking artifacts using 320 views of cone-beam projection data acquired from a helical trajectory and along a circular trajectory using a bench-top CT system. The manuscript layout is as follows. Numerical simulations were conducted (Section 2) to study the tradeoff between view sampling rate and completeness of the scanning trajectories. Initial experiments were conducted (Section 3) to study the above tradeoff by implementing a single circle scanning trajectory and a helical trajectory using a bench top cone-beam CT system. Experimental results are presented in Section 4. Finally, some discussion and conclusions are presented in Section 5. 2. NUMERICAL SIMULATIONS When the available views of cone-beam projections are about 300, the view angles are significantly undersampled even for a fan-beam CT scanner. In most of clinical fan-beam CT systems and helical/spiral CT systems, the number of view angles is about 1000 per rotation. In order to reconstruct CT images without significant view sampling artifacts, we need a view sampling criterion to guide us. Mathematically, it is much harder to formulate an explicit sampling criterion in the cone-beam CT than in the fan-beam CT case. To the best of the authors knowledge, we do not have an explicit criterion on the view sampling rate for cone-beam CT yet. In fan-beam medical CT, the scanning trajectory is a circle, it is known that such a circular trajectory will provide complete projection information to reconstruct an image object that is located inside the circle. Therefore, completeness of the source trajectory is never an issue. However, in cone-beam CT, in order to perfectly reconstruct a volume of interest (VOI), we need to acquire projection information in a fully threedimensional space. When the scanning is conducted along a curve, this curve must satisfy the so-called Tuy data sufficiency condition in order to provide exact reconstruction. If a scanning source trajectory does not satisfy the Tuy data sufficiency condition, cone-beam artifacts are present in the reconstructed images due to the incompleteness of projection information. The severity of cone-beam artifacts depends on the degree of incompleteness of the scanning trajectory. One common example of an incomplete scanning trajectory is the circular trajectory, it is well known that cone-beam artifacts are present for the reconstructed image points that are not co-planar with the scanning source trajectory. Therefore, in cone-beam CT, in order to select an optimal scanning trajectory, it is important to investigate the view sampling artifacts and cone-beam artifacts simultaneously. In particular, when the available number of cone-beam projections is small, it is crutial to study how to optimally distribute the focal spots along a scanning trajectory. In the following, we numerically study three well-known scanning source trajectories. One is the circular source trajectory, one is the helical trajectory, the other one is a trajectory consisting of two-concentric-circles.

The two-concentric-circle trajectory is a composite of two circular trajectories. Both helical and two-concentriccircle trajectories enable exact volumetric reconstruction, but the circular trajectory is incomplete. In our numerical simulations, a flat-panel detector has been assumed. The source and detector rigidly rotate around the iso-center (an extension of the so-called third generation scanning geometry in fan-beam CT). A fixed number of views of cone-beam projections were acquired (300) from each trajectory. In the helical case, in order to avoid data truncation, two helical turns were assumed in the simulations. The parameters utilized in simulations are summarized in Table 1. Parameter Value Radius 3 Magnification 2 Half fan angle 18 Half cone angle 18 Detector matrix 400 400 Detector sub-sampling 3 3 Number of monoenergetic photons 2 10 5 Table 1. A summary of the parameters used in the numerical simulations of the under-sampled cone-beam acquisition. A modified Shepp-Logan phantom with both low contrast and high contrast insertions has been utilized in the numerical simulations. The parameters of the phantom was given the Appendix (Table 2). Poisson noise was added to the projection data to simulate the quantum statistics of the CT system. The noise level is dictated by the assumption that the entrance x-ray photon number in each projection is 2 10 5. The standard Feldkamp image reconstruction algorithm 7 was utilized to reconstruct images from conebeam projection data acquired from a circular trajectory. A mathematically exact and shift-invariant filtered backprojection (FBP) image reconstruction algorithm 8 developed at the University of Wisconsin-Madison has been utilized to reconstruct images using cone-beam projection data acquired from the two-concentric-circle source trajectory. For the helical trajectory, a state-of-the-art mathematically exact FBPD (filtering the backprojection image of differentiated projection data) 11 13 was utilized in the image reconstruction. 3. INITIAL PHANTOM EXPERIMENTS USING A BENCH TOP SYSTEM Phantom experiments were conducted to study the tradeoff between cone-beam artifacts and view sampling artifacts. The bench top cone-beam CT system (Figure 1) consists of a CsI flat-panel detector with an active area of 39.7 29.8cm 2 and sampling matrix of 1024 768(PaxScan4030A, Varian Medical System), an x-ray tube with nominal focal spot sizes of 0.4/0.8mm (Rad94, Varian Medical System), and a rotary stage to hold the phantom. The x-ray beam filtration includes a 0.5mm Al inherent filtration and 1mm Cu added filtration. The source to detector distance (SDD) is 155cm and source to iso-center distance (SID) is 100cm. A physical phantom with both high contrast objects and low contrast insertions has been constructed to demonstrate the cone-beam artifacts (Defrise Phantom-type high contrast objects) and the low contrast detectability. The phantom has a water dish in the central plane with six plastic insertions with different linear attenuation coefficients including: teflon (1000 HU), polypropylene (130 HU), Acrylic (110 HU), High-density polyethylene (90 HU), Nylon (75 HU), and polystyrene(? HU). Four acrylic disks were placed parallel to the scanning plane of the circular source trajectory (Defrise type phantom). Additional structure is provided by several plastic rods and two wires orientated vertically between the acrylic disks.

Figure 1. Photograph of the flat panel based cone-beam CT bench top system housed at Princess Margaret Hospital in Toronto. In order to implement a circular trajectory, the source and detector positions are fixed. The phantom is rotated 360 degrees over which 320 views of projection data are collected with a detector readout at 1 frame per second. In order to acquire cone-beam projection data along a helical trajectory, the motions of the x-ray source, flat-panel detector, and the rotary stage have been synchronized. When the source moved upward vertically with a velocity v t, the flat-panel detector was accurately prescribed to move downward with a velocity v d = v t /1.55. The factor 1.55 was utilized to take the magnification into account. The height of the phantom has been adjusted using additional spacers so that the phantom is positioned within the reconstructable field of view with the helical trajectory. The rotation of the phantom was synchronized with the vertically uniform motion of the source and the detector. Thus, an effective helical trajectory was implemented which enabled the reconstruction of a phantom with height of 12cm without truncation in the projection data. The physical pitch of the implemented helix was 12cm. The phantom was scanned at 100kV p, 2mA and 7.5ms for both the circular and helical trajectories. 4. NUMERICAL RESULTS AND EXPERIMENTAL RESULTS 4.1. Numerical results Numerical simulations were conducted in order to obtain a qualitative measure of the trade-off between streak artifacts due to sparse sampling along the trajectory and cone-beam artifacts due to the utilization of a trajectory which does not satisfy Tuy s data sufficiency condition for exact reconstruction. Thus, thethree sample trajectories used here were the single circle, two concentric-orthogonal circles and the helical trajectory. In each of these cases a truly cone-beam acquistition was used with a very large cone-angle in order to highlight the effects of data sufficiency on the reconstruction quality. The number of total focal spots used in each of these reconstructions was kept constant at 300 in accordiance with the initial design paramaters of the Z- Scanner. In the circular case 300 focal spots were distributed over 2π. In the two-orthogonal circle case, 150 focal spots were distributed over 2π in each trajectory. Finally, in the helical case 300 focal spots were distributed over a 4π angular range, where the helical pitch was appoximately equal to the height of the phantom. Thus, the single circle achieves the highest in-plane source sampling rate, but it does not satisfy the Tuy condition for exact reconstruction. Two other trajectories have been examined here which each has different sampling properties.

Figure 2. Comparison of the images reconstructed in the XY plane using 300 total focal spots. From left to right the images are: the phantom, the circular trajectory, the two concentric circle trajectory and the helical trajectory. Figure 3. Comparison of the images reconstructed in the YZ plane using 300 total focal spots. From left to right the images are: the phantom, the circular trajectory, the two concentric circle trajectory and the helical trajectory. The FDK algorithm 7 processes the cone-beam projection data using a ramp filter and the two-concentric circle algorithm 8 and the FBPD helical reconstruction 11 13 used in this section each perform a discrete differentiation operation followed by a Hilbert filter. The filtering kernels must be carefully matched to generate equivalent spatial resolution before a complete quantitative comparison can be made on the noise properties of the algorithms. This step has not been performed yet; thus, the attempt to compare the noise properties is not supported in the images presented here. An additional complication in the comparison is the fact that the exact helical algorithm used here for this high-pitch case has low dose efficiency since only data in the 1 PI segments has been used in reconstructing each single point. Note that if a helical trajectory were chosen for one may investigate approximate or quasi-exact algorithms which increase the dose efficiency of the system. Despite the lack of exact correspondence in the resolution/noise properties of the algorithms one may still glean useful information about the inherent differences in the streak artifacts present using each trajectory. Since the artifact levels are not isotropic images are presented from three othogonal planes through the object. Each image is displayed in a window of [0.95 1.05] which corresponds to a window width of 100 HU. The first example is the XY plane which is the central plane of the FDK algorithm and the central plane of one of the circles in the two-orthogonal circle trajectory (Figure 2 ). The streak artifacts are of the highest magnitude in the helical reconstruction for this plane due to the low z-sampling in this very high pitch acquisition. Since this is the central plane of the FDK reconstruction and the circular trajectory has the highest angular sampling the circular FDK provides the best image quality in this plane. The second example is the YZ plane, which is orthogonal to the single circle and contains one of the two-orthogonal circles (Figure 3). In this plane the streak artifacts are most apparent in the two-orthogonal circle reconstruction since the backprojection from the YZ plane yields significant streaking due to the presence of only 150 focal points in that circle and the presence of a large high contrast object lying directly in this plane. However, note that the two-orthogonal-circle reconstruction enables visualization of low contrast objects which are not seen in the FDK reconstruction. Finally, an example from the XZ plane which is orthogonal to both circles in the

Figure 4. Comparison of the images reconstructed in the XZ plane using 300 total focal spots. From left to right the images are: the phantom, the circular trajectory, the two concentric circle trajectory and the helical trajectory. orthogonal-circle trajectory is given below (Figure 4). In this case there are few streaking artifacts visible in the two-orthogonal reconstruction since the backprojection is directed from the two orthogonal planes and thus most of the streaks obliquely intersect this plane. Note, the visibility of the two low contrast objects away from the central plane in the two-orthogonal reconstruction. These objects are obscured by the cone-beam artifacts in the case of the reconstruction from the central plane. Most of the reconstructed planes will be intermediate to these three orthogonal planes and thus these planes demonstrate the extreme cases in terms of streaking and cone-beam artifacts. Obviously further quantitative studies are required before any definite conclusions can be drawn about the ideal source trajectory. It is clear from these simulations that the ideal trajectory will be dependent upon the imaging task and the size of the object of interest. Also these initial simulations have assumed no data truncation and in the case that the detector size is limited by fiscal considerations additional simulations will be necessary in order to determine the optimal source trajectory. 4.2. Experimental Results As discussed above we are going to compare two of the possible trajectories for cone-beam reconstruction and assess the severity of cone-beam artifacts and the streaking artifacts. Here we compare the results from the single circular acquisition using the FDK reconstruction to the results from the high pitch helical acquisition using the exact helical FBPD image reconstruction algorithm. Each acquisition has the same number of total focal spot positions. As discussed above in order to reconstruct one given point the FBPD algorithm uses 11, 12 only the data from a single PI-line segment of the helical trajectory. Thus, although the algorithm is exact for this under-sampled high pitch helix case the dose efficiency of this algorithm is low. Therefore, in this comparison we expect that the contrast to noise level for the helical reconstruction will be significantly lower than the FDK reconstruction. However, note that other approximate reconstructions may be explored in order to increase the dose efficiency of the helical acquisition while still maintaining a low level of cone-beam artifacts. The experimental comparison will be demonstrated using three orthogonal planes which have been reconstructed with both the circular trajectory (FDK) and the helical trajectory (FBPD). The axial slice (central transverse slice) contains the water bath and several plastic inserts as discussed above (Figure 5-6). Each of these algorithms provides exact reconstruction of this slice, and examination of the images confirms that there are no geometrical distortions of the image objects. Note that the FDK case utilizes projection information from all 320 view angles, while the helical FBPD algorithm uses only approximately one quarter of the total view angles for each image voxel. Thus, is expected that the contrast to noise ratio (CNR) of the FDK should be roughly twice as high as the helical case. Keep in mind that this dose efficiency is highly algorithm dependent and may be further improved with alternative reconstruction algorithms. Also note that, despite the fact that only roughly 80 view angles were utilized to reconstruct each image point in the helical case, that there are no structurized streaking artifacts due to this vast undersampling. In order to assess the severity of the cone-beam artifacts present for each of these scanning configurations the sagital/coronal images were also reconstructed (Figure 7-8). A comparison of the Defrise type phantom

(a) (b) Figure 5. A comparison of the axial slices of through phantom reconstructed using the circular FDK (a) and the helical FBPD (b) where the display window is [-1500 1000] HU. demonstrates that the helical FBPD more faithfully reconstructs the acrylic disks than the FDK algorithm. These disks are reconstructed with uniform thickness in the helical case, whereas in the FDK case the conebeam artifacts artificially widen the thickness of the center of the outer most acrylic disks. As discussed above the CNR of the helical FBPD reconstruction is lower, but this is not intrinsic to this trajectory rather this is a due to the low dose efficiency of the current FBPD algorithm. 5. CONCLUSIONS In conclusion, numerical simulations have been conducted to study the tradeoff between the view sampling rate and the completeness of the scanning trajectory. The numerical results demonstrate that the cone-beam artifacts due to the incompleteness of the scanning trajectory may be significantly mitigated by utilizing a complete source trajectory. However, the view sampling streaking artifacts may manifest in some images slices when an image slice is co-planar with a significant portion of the scanning trajectory due to the lower sampling rate. This is explicitly shown in the results from the two-orthogonal-circle configuration. The view sampling streaking artifacts may be significantly alleviated when a source trajectory is designed such that there are no image slices which are co-plane with scanning trajectory, such as a helical trajectory. However, a disadvantage of the helical source trajectory is that the radiation dose efficiency is low in the current PI-line based image reconstruction algorithms. Thus, the CNR is significantly lower in the reconstructed images. This dictates that higher radiation doses are required. Initial experiments were conducted to further verify the numerical studies. In our initial experimental studies, two source trajectory were utilized. One the the circular trajectory, the other the a helical source trajectory. The experimental results are consistent with the numerical simulations. For any given imaging task in order to quantitatively design and measure a figure of merit of each scanning configurations, more numerical simulations and physical phantom experiments are required and additional source trajectories should be included.

(a) (b) Figure 6. A comparison of the axial slices using the circular FDK (a) and the helical FBPD (b) in a smaller display window for visualization of the low contrast objects [-580 345] HU. (a) (b) Figure 7. The reconstruction results using the circular FDK (a) and the helical FBPD (b) for the images [-1210 360].

(a) (b) Figure 8. The reconstruction results using the circular FDK (a) and the helical FBPD (b) for the images [-1210, 360].

Acknowledgement This work was partially supported by National Institute of Health grants 1R21 EB001683-01, 1R21 CA109992-01, T32CA009206-24 a startup grant from University of Wisconsin-Madison, and a grant from GE Healthcare Technologies. We acknowledge the use of open source code including Agner Fog s random number generator and the Blitz++ templated array library.

(x, y, z) β a b c ρ (0,0,0) 0.69.92.9 2.0 (0,0,0) 0.6624.874.88 -.98 (-.22,0,-.25) 108.41.16.21 -.02 (.22,0,-.25) 72.31.11.22 -.02 (0,0.35,-.25) 0.21.25.50.02 (0,.1,-.25) 0.046.046.046.02 (-.08,-.65,-.25) 0.046.023.02.01 (.06,-.650,-.25) 90.046.023.02.01 (.06,-.105,.625) 90.056.040.10.02 (0,.1,.625) 0.056.056.1 -.02 (-0.290, 0.200, 0.000) 0.0000 0.0667 0.0033 0.0033 1.0000 (-0.290, 0.050, 0.000) 0.0000 0.0667 0.0033 0.0033 1.0000 (-0.290, 0.125, 0.000) 56.3100 0.1133 0.0033 0.0033 1.0000 (-0.160, 0.125, 0.000) 0.0000 0.0467 0.0033 0.0033 1.0000 (-0.050, 0.050, 0.000) 0.0000 0.0567 0.0033 0.0033 1.0000 (-0.050, 0.125, 0.000) 0.0000 0.0567 0.0033 0.0033 1.0000 (-0.050, 0.200, 0.000) 0.0000 0.0567 0.0033 0.0033 1.0000 (-0.092, 0.163, 0.000) 0.0000 0.0033 0.0467 0.0033 1.0000 (-0.007, 0.087, 0.000) 0.0000 0.0033 0.0467 0.0033 1.0000 ( 0.108, 0.050, 0.000) 0.0000 0.0567 0.0033 0.0033 1.0000 ( 0.065, 0.125, 0.000) 0.0000 0.0033 0.1000 0.0033 1.0000 ( 0.108, 0.200, 0.000) 0.0000 0.0567 0.0033 0.0033 1.0000 ( 0.225, 0.125, 0.000) 67.0000 0.1033 0.0033 0.0033 1.0000 ( 0.295, 0.125, 0.000) -67.0000 0.1033 0.0033 0.0033 1.0000 ( 0.260, 0.115, 0.000) 0.0000 0.0400 0.0033 0.0033 1.0000 ( 0.355, 0.125, 0.000) 0.0000 0.0033 0.1000 0.0033 1.0000 ( 0.398, 0.125, 0.000) -65.0000 0.1033 0.0033 0.0033 1.0000 ( 0.440, 0.125, 0.000) 0.0000 0.0033 0.1000 0.0033 1.0000 ( 0.000, 0.000,-0.400) -20.0000 0.0500 0.1000 0.2000 1.0000 ( 0.200, 0.200,-0.700) 45.0000 0.2000 0.3000 0.100 0.0100 (-0.100,-0.300,-0.300) 0.0000 0.1000 0.4000 0.0500 0.1000 ( 0.200, -0.200,-0.700) 45.0000 0.2000 0.3000 0.100-0.0100 ( 0.00, 0.050,-0.800) 0.0000 0.0400 0.0400 0.040 0.0100 (-0.100,-0.00,0.300) 0.0000 0.4000 0.0500 0.0500 0.100 (0.440,0.00,0.300) 0.0000 0.0100 0.0100 0.0100 0.500 (0.480,0.00,0.300) 0.0000 0.0100 0.0100 0.0100 0.500 (0.520,0.00,0.300) 0.0000 0.0100 0.0100 0.0100 0.500 (0.560,0.00,0.300) 0.0000 0.0100 0.0100 0.0100 0.500 (0.60,0.00,0.300) 0.0000 0.0100 0.0100 0.0100 0.500 (0.440,0.00,0.340) 0.0000 0.0100 0.0100 0.0100 0.500 (0.480,0.00,0.340) 0.0000 0.0100 0.0100 0.0100 0.500 (0.520,0.00,0.340) 0.0000 0.0100 0.0100 0.0100 0.500 (0.560,0.00,0.340) 0.0000 0.0100 0.0100 0.0100 0.500 (0.60,0.00,0.340) 0.0000 0.0100 0.0100 0.0100 0.500 (0.440,0.00,0.00) 0.0000 0.0100 0.0100 0.0100 0.500 (0.480,0.00,0.0) 0.0000 0.0100 0.0100 0.0100 0.500 (0.520,0.00,0.0) 0.0000 0.0100 0.0100 0.0100 0.500 (0.560,0.00,0.0) 0.0000 0.0100 0.0100 0.0100 0.500 (0.60,0.040,0.0) 0.0000 0.0100 0.0100 0.0100 0.500 (0.440,0.040,0.0) 0.0000 0.0100 0.0100 0.0100 0.500 (0.480,0.040,0.0) 0.0000 0.0100 0.0100 0.0100 0.500 (0.520,0.040,0.0) 0.0000 0.0100 0.0100 0.0100 0.500 (0.560,0.040,0.0) 0.0000 0.0100 0.0100 0.0100 0.500 (0.60,0.040,0.0) 0.0000 0.0100 0.0100 0.0100 0.500 Table 2. A modified Shepp-Logan phantom, which contains high contrast opacified vessel like structures as well as small high contrast spheres and several additional low contrast objects. The ellipsoids above the horizontal line are given in the standard phantom. The convention of ellipoid defintions follows those given by Kak and Slaney. 10

REFERENCES 1. D. A. Jaffray and J. H. Siewerdsen, Cone-beam computed tomography with a flat-panel imager: initial performance characterization, Med. Phys., Vol. 27, 1311-132(2000). 2. B. Chen and R. Ning, Cone-beam volume CT breast imaging: feasibility study, Med. Phys. 29, 755770 (2002). 3. J. M. Boone, T. R. Nelson, K. K. Lindfors, and J. A. Seibert, Dedicated breast CT: radiation dose and image quality evaluation, Radiology, Vol. 221, 657667 (2001). 4. J.H. Siewerdsen, D. J. Moseley, S. Burch, S. K. Bisland, and A. Bogaards, B. C. Wilson, and D. A. Jaffray, Volume CT with a flat-panel detector on a mobile, isocentric C-arm: Pre-clinical investigation in guidance of minimally invasive surgery, Med. Phys., Vol. 32, 241-254(2005). 5. J.H. Siewerdsen and D. A. Jaffray, Optimization of X-ray imaging system (with specific application to flat-panel cone-beam computed tomography), Med. Phys., Vol.28, 220-231(2001). 6. S. Glick, S. Vedantham, and A. Karellas, Investigation of optimal kvp settings for CT mammography using a flat panel detector, Proc. SPIE 4682, 392402 (2003). 7. L. A. Feldkamp, L. C. Davis, and J. W. Kress, Practical cone-beam algorithm, J. Opt. Soc. Am. A 1, 612619 (1984). 8. T. L. Zhuang, B. E. Nett, S. Leng, and G.-H. Chen, Image reconstruction via filtered backprojection for cone-beam data from two orthogonal circles using an equal weighting scheme, Proc. 8 th Internationsal Meeting on Fully three-dimensional image reconstruction in radioogy and nuclear Medicines, Salt Lake City, 198-202 (2005). 9. A. Katsevich, Analysis of an exact inversion algorithm for spiral cone-beam CT, Phys. Med. Biol. 47, 25832597 (2002). 10. A.C.Kak and Malcolm Slaney, Principles of Computerized Tomographic Imaging,IEEE, Bellingham, WA. (1988). 11. Y. Zou and X. Pan, Exact image reconstruction on PI-lines from minimum data in helical cone beam CT, Phys. Med. Biol. 49, 941959 (2004). 12. Tingliang Zhuang, Shuai Leng, Brian E. Net, and Guang-Hong Chen, Fan-beam and conebeam image reconstruction via filtering the backprojection image of differentiated projection data, Phys. Med. Biol. 49, 54895503 (2004). 13. J. D. Pack, F. Noo, and Rolf Clackdoyle, Cone-beam reconstruction using the backprojection of locally filtered projections, IEEE Trans. Med. Imaging 24(1), 116 (2005).