We propose a multi-pass approach to reduce cone-beam artifacts in a circular orbit cone-beam computed tomography (CT) system. Employing a large 2D detector array reduces the scan time but produces cone-beam artifacts in the Feldkamp, Davis, and Kress (FDK) reconstruction because of insufficient sampling for exact reconstruction. While the two-pass algorithm proposed by Hsieh is effective at reducing cone-beam artifacts, the correction performance is degraded when the bone density is moderate and the cone angle is large. In this work, we treated the cone-beam artifacts generated from bone and soft tissue as if they were from less dense bone objects and corrected them iteratively. The proposed method was validated using a numerical Defrise phantom, XCAT phantom data, and experimental data from a pediatric phantom followed by image quality assessment for FDK, the two-pass algorithm, the proposed method, and the total variation minimization-based iterative reconstruction (TV-IR). The results show that the proposed method was superior to the two-pass algorithm in cone-beam artifact reduction and effectively reduced the overcorrection by the two-pass algorithm near bone regions. It can also be observed that the proposed method produced better correction performance with fewer iterations than the TV-IR algorithm. A qualitative evaluation with mean-squared error, structural similarity, and structural dissimilarity demonstrated the effectiveness of the proposed method.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Technical advances in flat-panel detectors have enabled large volume coverage in a single gantry rotation of a cone-beam computed tomography (CBCT) system [1–3]. The large 2D detector array employed in CBCT systems provides high-resolution images with a field of view over 30 × 30 × 30 cm3 . Thus, CBCT systems have been widely used in various diagnostic imaging processes such as image-guided radiation therapy [5,6], cardiothoracic imaging , endodontic imaging [8,9] and cephalometric analysis . While CBCT systems provide good morphological information on patients with 3D volumetric data, the data acquisition scheme of circular orbit cone-beam geometry does not provide sufficient sampling for exact reconstruction . As a result, reconstructed images using the Feldkamp, Davis, and Kress (FDK) algorithm  contain cone-beam artifacts which become more severe as the cone angle increases. Since the presence of cone-beam artifacts degrades diagnostic performance, appropriate correction methods are necessary.
Several algorithmic approaches have been proposed to reduce the cone-beam artifacts. Grass et al. suggested modified FDK algorithms which rearrange the cone-beam rays into parallel (P-FDK) and tent (T-FDK) geometry to increase object sampling density using a data rebinning process . Chen et al. proposed a weighted filtered backprojection algorithm where the inverse of a cosine function was multiplied for each ray before the backprojection to compensate for the intensity drop of the FDK algorithm . MaaB et al. proposed an iterative approach to reduce cone-beam artifacts where they first estimated a cone-beam artifact image and then subtracted it from the FDK image . The initial corrected FDK image was used to estimate the cone-beam artifact image which was subtracted again from the initially corrected FDK image, and the cone-beam artifacts were reduced by applying this procedure iteratively. While this approach is effective for cone-beam artifact reduction, it requires more than 400 iterations, which is not practical in real clinical applications.
A two-pass algorithm proposed by Hsieh reduces cone-beam artifacts iteratively but requires only one iteration, thus making the computation time tolerable [16,17]. It treats the high-density material as a dominant factor to generate the cone-beam artifacts in the FDK image. Thus, it first segments the high-density material and then regenerates the cone-beam artifacts from the segmented high-density material, after which they are subtracted from the original FDK image. While the two-pass algorithm is very effective with a moderate cone angle (i.e., less than 5 °), the correction performance is degraded when the cone angle is large or the bone density is not high.
In this paper, we propose a multi-pass approach to reduce the cone-beam artifacts which is very effective even with a large cone angle (i.e., more than 10 °). Since the cone-beam artifacts are generated by both high-density material and background material such as soft tissue , we propose a method to handle both effects in cone-beam artifact reduction. The proposed method was validated using a Defrise phantom and XCAT phantom model of the human thorax region. The performance of the proposed method was compared with that of the two-pass algorithm and the total-variation minimization-based iterative reconstruction (TV-IR) algorithm  using mean-squared error (MSE) and the structural similarity index (SSIM) . Experimental validation using a pediatric phantom is also presented with quantitative evaluation using a structural dissimilarity index (DSSIM) .
2.1. A review of the two-pass algorithm
The basic assumption of the two-pass algorithm is that the cone-beam artifacts are predominantly generated by high-density material (e.g., high-density bone) [16, 17]. Thus, the two-pass algorithm segments the high-density material from the FDK image by simple thresholding and regenerates the cone-beam artifacts using the segmented high-density material. Next, a corrected image is acquired by subtracting the generated cone-beam artifacts from the original FDK image. However, when the bone density is low or the X-ray energy used for data acquisition is high, the density difference between bone and soft tissue is reduced. In this scenario, using a simple threshold would not be effective to segment bone material in the presence of the cone-beam artifacts, which degrades the performance of the two-pass algorithm.
Figure 1 shows the noiseless simulation results of the two-pass algorithm with a Defrise phantom containing moderate-density (i.e., 0.4 cm−1) and high-density (i.e., 0.8 cm−1) bone material. Note that the density of the background tissue was 0.18 cm−1, and the maximum cone angle was ± 11.5 °. In both cases, a threshold value of 65% was selected for each bone density (i.e., 0.26 cm−1 for moderate-density and 0.52 cm−1 for high-density) to segment the bone material. It can be observed that the two-pass algorithm reduced the cone-beam artifacts effectively for a volume with a cone angle of less than 5 °, as indicated by the green boxes in Fig. 1(c), but the performance was degraded as the cone angle increased, as shown by the yellow boxes in Fig. 1(c). It is also evident that more residual errors remained after the two-pass algorithm for the Defrise phantom with moderate-density bone material because the bone-segmentation errors became higher than those of the Defrise phantom with high-density bone material. These segmentation errors propagated through each correction step of the two-pass algorithm and thus higher segmentation errors introduced more residual errors after applying the two-pass algorithm.
2.2. The proposed method
The basic idea of the proposed method is to regenerate the cone-beam artifacts from less-dense bone objects. Consider a bone Defrise phantom embedded within a uniform soft tissue cylinder phantom. Since the reconstruction process is linear, the cone-beam artifacts can be decomposed as a summation of the bone- and soft-tissue-induced cone-beam artifacts (Fig. 2(a)) or generated purely from the less-dense bone objects since the uniform soft tissue cylinder phantom does not produce cone-beam artifacts (Fig. 2(b)). By assuming that the density of the soft tissue around the bone objects does not vary rapidly, this approach helps to prevent the overcorrection by the conventional two-pass algorithm, especially when the bone density is moderate. A schematic diagram of the proposed method is depicted in Fig. 3 and the details of each step are as follows:
- (Step 1) Set the FDK reconstructed image ffdk as a start image. Subsequently, classify bone fbone,τ(1) and soft tissue ftissue,τ(1) images using a simple threshold value τ(1), which is set to 65% of the mean value of the bone density calculated by averaging the upper 10% of the high-density voxels within the bone regions.
- (Step 2) Calculate the mean value of soft tissue component μτ(1) in ftissue,τ(1).
- (Step 3) Apply a 3D median filter  to fbone,τ(1) to reduce bone-segmentation errors. After that, subtract μτ(1) from fbone,τ(1) to obtain less-dense bone objects fbone_less,τ(1). By subtracting the calculated scalar value μτ(1) from the segmented bone results fbone,τ(1), the proposed algorithm minimizes soft-tissue-induced cone-beam artifacts. Next, conduct a forward projection on fbone_less,τ(1) and perform FDK reconstruction to generate fbone_less_fdk,τ(1).
- (Step 4) Calculate errors ferrors,τ(1) by subtracting fbone_less_fdk,τ(1) from fbone_less,τ(1), and acquire artifacts-corrected image fτ(1) by subtracting ferrors,τ(1) from the original ffdk image.
- (Step 5) Repeat (Step 1) ∼ (Step 4) until the MSE of the ferrors,τ(i) converges. ferrors,τ(i) represents the estimated cone-beam artifacts at each iteration, and thus there are no reference images for it. The MSE of ferrors,τ(i) increases at each iteration if the correction performance is improved. Note that the MSE of ferrors,τ(i) reaches the maximum value if the cone-beam artifacts are not present in the corrected image, fτ(i). The threshold value for bone-segmentation increases linearly with each iteration to improve the segmentation performance, which is critical for reducing the cone-beam artifacts effectively. Note that the threshold value is fixed after the 3rd iteration to prevent oversegmentation of the bone material.
2.3. Selection of threshold value, τ(i)
We selected the threshold values τ(i), experimentally, which minimized the bone-segmentation errors. To examine the effect of the threshold values on the errors, we segmented the bone regions using different thresholds from the reconstructed Defrise image and calculated the NMSE using reference bone regions in the Defrise phantom. Figure 4 shows the NMSE for Defrise phantoms containing various bone densities. It can be observed that the moderate-density bone (i.e., 0.4 cm−1) shows the smallest NMSE in the 55% to 75% threshold regions. Thus, we selected the initial threshold value τ(1) at 65% of the bone density. At each iteration, we increased the threshold value linearly (i.e., by 2.5% at each iteration) but not over the threshold value of 70% for the bone density to avoid oversegmentation of the bone regions (i.e., up to the 3rd iteration).
Note that the small NMSE regions increased as the bone density increased, which provided more tolerance when selecting threshold values after each iteration.
To validate the proposed algorithm, we used a Defrise phantom and an XCAT phantom  developed by Segars. The Defrise phantom consisted of five ellipsoids separated by 5.1 cm along the z-direction within a uniform cylinder background, as shown in Fig. 5. The attenuation coefficients of the ellipsoids and the cylinder (treated as moderate-density bone and normal soft tissue material) were 0.4 and 0.18 cm−1, respectively. We also generated a thorax XCAT phantom with monochromatic 60 keV energy, as shown in Fig. 6, where the attenuation coefficients of the bone and soft tissue ranged from 0.3572 to 0.4508 cm−1 and 0.2020 to 0.2578 cm−1, respectively. The detailed specifications of each phantom are summarized in Table 1.
Both phantoms were constructed as 480 × 480 × 540 matrices with a voxel size of 0.517 × 0.517 × 0.517 mm3, where 2 × 2 × 2 subvoxels per voxel were allocated to avoid discretization errors. Projection data were acquired using the 3D forward projector developed by Gao . We added uniform noise of 10,000 photons per detector pixel generated by Poisson statistics. Note that the noise level was matched to that of the experimental data. After that, a Hanning weighted ramp filter was applied to the projection data. The FDK algorithm based on the voxel-driven backprojection using linear interpolation was used for the reconstruction and 2 × 2 binning mode with a detector pixel size of 0.776 × 0.776 mm2 was used to reduce the computational complexity. Note that the corresponding cone angle (± 11.5 °) is similar to the breast cone-beam CT system , where the 300 × 400 mm2 flat panel detector is shifted along the z-direction for data acquisition. To avoid the object truncation error along the z-direction, we reconstructed the image with 480 × 480 × 640 voxels and extracted the 480 × 480 × 540 voxels with a volume size of 248 × 248 × 279 mm3. We also reconstructed the images using the TV-IR algorithm, implemented using the gradient projection Bazilai-Borwein (GPBB) algorithm , which provided faster convergence speed than the adaptive steepest descent projection onto convex sets (ASD-POCS) algorithm . The constrained total-variation (TV) minimization for iterative reconstruction is as follows:Eq. (1) was selected as 0.001 since it produced the smallest MSE value in the range 0.001 to 0.005. The simulation geometry is shown in Fig. 7 and the simulation parameters are summarized in Table 2.
We acquired the projection data using a benchtop CBCT system, which consisted of a generator (Indico 100, CPI Communication & Medical Products Division, Georgetown Ontario, Canada), a tungsten target X-ray source (Varian G-1592, Varian X-ray Product, Salt Lake City, UT) with a 0.6 × 0.6 mm2 focal spot and a 300 × 400 mm2 flat-panel detector (PaxScan 4030CB, Varian Medical Systems, Salt Lake City, UT) with an anti-scatter grid (Philips Medical Systems, Best, the Netherlands). The projection data for the pediatric phantom (Model 715, CIRS, Almeda Ave, VA) shown in Fig. 8 were acquired using a 90 kVp X-ray spectrum with a current of 8 mA. The FDK reconstruction was performed on 480 × 480 × 343 matrices with a voxel size of 0.517 × 0.517 × 0.517 mm3. TV-IR was also conducted to reconstruct images using experimental data with the 0.001 value of λ after 50 iterations. Due to the usage of an anti-scatter grid, the maximum cone angle in the experimental system was set to ± 7.5 °, which was smaller than the simulation. The details of the experimental parameters are reported in Table 3.
2.6. Image quality evaluation
To compare image quality using FDK, the two-pass algorithm, the proposed method, and the TV-IR algorithm, we set five regions of interest (ROIs) for the simulation phantoms, as shown in Figs. 5 and 6. The MSE and SSIM between reference and the reconstructed images using FDK and the two-pass, the proposed, and the TV-IR algorithms were calculated for the quantitative analysis of the simulation results.
MSE is defined as
SSIM is defined as
For the experimental data, we also used 5 ROIs of the reconstructed images. Since the reference image was not available for the experimental data, DSSIM was used and defined as
Figure 9 shows the coronal slice of the reference, the FDK volume, and the corrected volume using the two-pass algorithm, the proposed method after the 1st and 5th iterations, and the TV-IR algorithm for the Defrise phantom. The initial threshold value τ(1) to segment bone material was 0.26 cm−1 for the two-pass and proposed methods and was increased to 0.27 cm−1 and 0.28 cm−1 for the 2nd and 3rd iterations, respectively, for the proposed method. After that, 0.28 cm−1 was used as the threshold value for further iterations. Figure 10 shows the MSE of ferrors,τ(i) as a function of iteration number in which the proposed method converged after the 5th iteration. Note that the MSE of ferrors,τ(i) became higher when the corrected image using the proposed method was close to the reference image. It can be observed that the corrected image using the two-pass algorithm reduced the cone-beam artifacts effectively for the volume with small cone angles, but its correction performance became degraded as the cone angle increased because the segmentation errors for the bone material increased in the volume with large cone angles. As a result, residual artifacts still remained near the upper (lower) slab of the Defrise phantom. In contrast, the proposed method provided improved correction performance even after the 1st iteration, as shown in Fig. 9(d). Since the residual artifacts of the proposed method were much less than the two-pass algorithm, segmentation errors of bone material became lower after each iteration, and thus correction performance increased, as demonstrated in Fig. 10. The TV-IR algorithm also provided better correction performance than the two-pass algorithm, as shown in Figs. 9(c) and 9(f). However, it required many iterations. In contrast, the proposed method achieved higher image quality with fewer iterations than the TV-IR algorithm.
Figure 11 shows a comparison of the vertical profiles marked by a yellow line in Fig. 9(a) for the reference Defrise phantom and the corrected images using the two-pass algorithm, the proposed method, and the TV-IR algorithm. An intensity drop and shape distortion can be observed in the upper and lower slabs of the original FDK image. While the two-pass algorithm recovered the intensity drop to a certain degree, undershooting near the edge of the upper and lower slabs was present and blurring of the image along the z-direction still remained. In contrast, the proposed method recovered the intensity drop and shape distortion while reducing the undershooting effectively. Tables 4 and 5 report the respective MSE and SSIM values for the five ROIs in Fig. 9(a). Both metrics indicate that the proposed algorithm provides better correction performance than the two-pass and TV-IR algorithms, especially for the volumes with large cone angles (ROIs 1 and 5). It is also evident that the correction performance increased as the iteration continued.
Figure 12(a) shows the coronal and sagittal slices of the reference XCAT phantom. For a visual inspection, we zoomed in on regions containing bony structures indicated by yellow boxes, within which the cone-beam artifacts can be clearly observed in the FDK image (Fig. 12(b)), showing a distorted anatomical structure and an intensity drop in both slices. Figure 12(c) shows the corrected image using the two-pass algorithm, where 0.27 cm−1 was selected as a threshold to segment the bone material. It is evident that the two-pass algorithm improved the sharpness of the bones but generated additional streak artifacts near the bone regions. Note that the intensity drop for the bone object was the most significant within ROI 5, and thus simple thresholding was not effective for segmenting the bone objects. This is due to the small intensity difference between the bone and the surrounding tissue which degraded the performance of the two-pass algorithm due to the increased bone-segmentation errors. Figures 12(d) and 12(e) show the corrected image of the proposed method after the 1st and 4th iterations, respectively. For each iteration, we increased the threshold value linearly from 0.27 cm−1 to 0.29 cm−1 until the 3rd iteration, after which we used 0.29 cm−1 as the threshold value. It is clear that the correction performance was improved as the iteration continued and became saturated after the 4th iteration, as shown in Fig. 13. Moreover, the proposed method provided better performance in cone-beam artifact reduction than the TV-IR algorithm with fewer iterations, as shown in Figs. 12(e) and 12(f).
Tables 6 and 7 summarize the respective MSE and SSIM of the ROIs between the reference and corrected images using the two-pass method, the proposed method and the TV-IR algorithm, thereby quantitatively demonstrating the effectiveness of the proposed method. Unlike the Defrise phantom results, the two-pass algorithm was not effective at reducing the cone-beam artifacts for the XCAT phantom because the presence of complex structures and intensity variation around the bone material introduced large errors in the bone-segmentation. In contrast, the proposed method reduced the cone-beam artifacts effectively, and its performance was improved by each iteration because of the augmentation in the bone-segmentation performance. Figure 14 shows the vertical profiles of the coronal and sagittal slices in Fig. 12. It is clear that the overcorrection caused by the two-pass algorithm was reduced significantly when using the proposed method.
The experimental results with the pediatric phantom data are shown in Fig. 15, where we zoomed in on the regions containing bony structures. Note that the cone angle of this region ranged from 4 ° to 7.5 ° and the cone-beam artifacts in FDK image can be observed clearly in Fig. 15(a). While the two-pass algorithm increased the sharpness of bone structures (Fig. 15(b)), residual artifacts still remained. In contrast, the proposed method effectively reduced cone-beam artifacts in Fig. 15(c) and, especially by the 5th iteration as shown in Fig. 15(d). Note that the initial threshold value for the segmented bone material was 0.26 cm−1 for both the two-pass and proposed methods; it was increased to 0.27 cm−1 and 0.28 cm−1 for the 2nd and 3rd iterations, respectively, and then fixed at 0.28 cm−1 for the proposed method. Note that five iterations were sufficient for the convergence to occur with the proposed method, as shown in Fig. 16, and it performed better than the TV-IR algorithm with fewer iterations (Figs. 15(d) and 15(e)). Table 8 summarizes the DSSIM values of each ROI region in Fig. 15, which were calculated using the FDK and corrected images.
It can be observed that the proposed method provided higher DSSIM values compared to the two-pass and TV-IR algorithm, indicating its better performance in cone-beam artifact reduction. The vertical profiles in Fig. 17 also show that the two-pass algorithm caused overcorrection (e.g., undershooting near the bone material), while the proposed method reduced the cone-beam artifacts effectively without any residual errors.
4. Discussion and conclusion
In this work, we proposed a multi-pass approach to reduce cone-beam artifacts with a large cone angle. In the presence of moderate-density bone objects in a volume with a large cone angle, FDK algorithm incurred the severe intensity drops for the bone objects. As a result, the bone-segmentation performance, which is essential for the two-pass algorithm, was degraded, especially in the presence of background material. Since the cone-beam artifacts are generated by both bone and background material, both effects should be considered together for the effective cone-beam artifact reduction. In the proposed method, we assumed that the attenuation coefficients of the background material around the bone material vary slowly, and thus treated the cone-beam artifacts as if they were generated purely from the less-dense bone material. This procedure significantly reduced the bone-segmentation errors, thereby demonstrating its superior performance in cone-beam artifact reduction over the two-pass algorithm even after one iteration. Since the presence of undershooting near the bone structures after the two-pass algorithm introduced bone-segmentation errors, the two-pass algorithm with additional iterations produced overcorrected results, as shown in Fig. 18, where the exampled XCAT images of the reference, FDK, and the two-pass algorithm after the 1st and 4th iterations are compared.
Although not presented in this paper, we examined the robustness of the proposed method for different initial threshold values, anatomical structures, and imaging conditions. With 60%, 65%, and 70% initial threshold values for the thorax XCAT phantom, it was observed that using a higher threshold required more iterations to converge. However, six iterations were sufficient for all cases to produce comparable image quality for different initial threshold values. It was also observed that the proposed method effectively reduced the cone-beam artifacts for head and thorax XCAT phantoms under three different X-ray energy conditions (40, 60, and 80 keV).
The limitation of the proposed method is the computation time due to the iterative procedures, which could be solved using GPU-based parallel computing system. In our GPU implementation (NVIDIA Tesla S2050), the computation time was around 20 seconds for each iteration, and we showed that 4 ∼ 5 iterations were sufficient to effectively reduce cone-beam artifacts. A more optimized GPU-based implementation would reduce the computation time even further, making it acceptable in real clinical applications. Note that even one iteration of the proposed method produced a better performance in cone-beam artifact reduction compared to the two-pass algorithm.
In conclusion, we proposed a new method to reduce cone-beam artifacts based on an iterative procedures. The simulation and experimental results demonstrated that the proposed method achieved superior performance over the two-pass algorithm, especially for volumes containing moderate-density bone objects in large cone angle circumstances. A comparison with the TV-IR algorithm also showed that the proposed method achieved better performance in cone-beam artifact reduction with fewer iterations.
Ministry of Science and ICT (MSIT), Korea, under the “ICT Consilience Creative Program” (IITP-2018-2017-0-01015) supervised by the Institute for Information & communications Technology Promotion (IITP); Bio & Medical Technology Development Program of the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (NRF-2018R1A1A1A05077894, 2018M3A9H6081483, 2017M2A2A4A01070302, 2017M2A2A6A01019663, and 2017M2A2A6A02087175).
2. J. Siewerdsen, D. Moseley, S. Burch, S. Bisland, A. Bogaards, B. Wilson, and D. 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. 32, 241–254 (2005). [CrossRef] [PubMed]
3. R. Gupta, A. C. Cheung, S. H. Bartling, J. Lisauskas, M. Grasruck, C. Leidecker, B. Schmidt, T. Flohr, and T. J. Brady, “Flat-panel volume ct: fundamental principles, technology, and applications,” Radiographics 28, 2009–2022 (2008). [CrossRef] [PubMed]
4. W. A. Kalender, “X-ray computed tomography,” Phys. Med. Biol. 51, R29 (2006). [CrossRef]
5. D. A. Jaffray, J. H. Siewerdsen, J. W. Wong, and A. A. Martinez, “Flat-panel cone-beam computed tomography for image-guided radiation therapy,” Int. J. Radiat. Oncol. Biol. Phys. 53, 1337–1349 (2002). [CrossRef] [PubMed]
6. M. Oldham, D. Létourneau, L. Watt, G. Hugo, D. Yan, D. Lockman, L. H. Kim, P. Y. Chen, A. Martinez, and J. W. Wong, “Cone-beam-ct guided radiation therapy: A model for on-line application,” Radiother. Oncol. 75, 271E1 (2005). [CrossRef] [PubMed]
7. T. Flohr, M. Prokop, C. Becker, U. Schoepf, A. Kopp, R. White, S. Schaller, and B. Ohnesorge, “A retrospectively ecg-gated multislice spiral ct scan and reconstruction technique with suppression of heart pulsation artifacts for cardio-thoracic imaging with extended volume coverage,” Eur. Radiol. 12, 1497–1503 (2002). [CrossRef] [PubMed]
8. S. Patel, A. Dawood, T. P. Ford, and E. Whaites, “The potential applications of cone beam computed tomography in the management of endodontic problems,” Int. Endod. J. 40, 818–830 (2007). [CrossRef] [PubMed]
9. D. A. Tyndall and S. Rathore, “Cone-beam ct diagnostic applications: caries, periodontal bone assessment, and endodontic applications,” Dental Clin. North Am. 52, 825–841 (2008). [CrossRef]
10. B. Hassan, P. van der Stelt, and G. Sanderink, “Accuracy of three-dimensional measurements obtained from cone beam computed tomography surface-rendered images for cephalometric analysis: influence of patient scanning position,” Eur. J. Orthod. 31, 129–134 (2008). [CrossRef] [PubMed]
11. A. C. Kak, M. Slaney, and G. Wang, “Principles of computerized tomographic imaging,” Med. Phys. 29, 107 (2002). [CrossRef]
12. L. A. Feldkamp, L. Davis, and J. W. Kress, “Practical cone-beam algorithm,” J. Opt. Soc. Am. A 1, 612–619 (1984). [CrossRef]
13. M. Grass, T. Köhler, and R. Proksa, “3d cone-beam ct reconstruction for circular trajectories,” Phys. Med. Biol. 45, 329 (2000). [CrossRef]
15. C. Maaß, F. Dennerlein, F. Noo, and M. Kachelrieß, “Comparing short scan ct reconstruction algorithms regarding cone-beam artifact performance,” in IEEE Nuclear Science Symposuim & Medical Imaging Conference, (IEEE, 2010), pp. 2188–2193. [CrossRef]
16. J. Hsieh, “Two-pass algorithm for cone-beam reconstruction,” in Medical Imaging 2000: Image Processing, vol. 3979 (International Society for Optics and Photonics, 2000), pp. 533–541. [CrossRef]
17. J. Hsieh, “A practical cone beam artifact correction algorithm,” in 2000 IEEE Nuclear Science Symposium. Conference Record (Cat. No. 00CH37149), vol. 2 (IEEE, 2000), pp. 15–71.
18. P. Forthmann, M. Grass, and R. Proksa, “Adaptive two-pass cone-beam artifact correction using a fov-preserving two-source geometry: A simulation study,” Med. Phys. 36, 4440–4450 (2009). [CrossRef] [PubMed]
19. J. C. Park, B. Song, J. S. Kim, S. H. Park, H. K. Kim, Z. Liu, T. S. Suh, and W. Y. Song, “Fast compressed sensing-based cbct reconstruction using barzilai-borwein formulation for application to on-line igrt,” Med. Phys. 39, 1207–1217 (2012). [CrossRef] [PubMed]
20. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Proc. 13, 600–612 (2004). [CrossRef]
21. A. Loza, L. Mihaylova, N. Canagarajah, and D. Bull, “Structural similarity-based object tracking in video sequences,” in 2006 9th International Conference on Information Fusion, (IEEE, 2006), pp. 1–6.
22. K. K. V. Toh, H. Ibrahim, and M. N. Mahyuddin, “Salt-and-pepper noise detection and reduction using fuzzy switching median filter,” IEEE Trans. Cons. Electron. 54, 1956–1961 (2008). [CrossRef]
25. K. Yang, A. L. Kwan, S.-Y. Huang, N. J. Packard, and J. M. Boone, “Noise power properties of a cone-beam ct system for breast cancer detection,” Med. Phys. 35, 5317–5327 (2008). [CrossRef]
26. E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777 (2008). [CrossRef]