- Email: [email protected]
Metal artifact reduction algorithm based on model images and spatial information
Nuclear Instruments and Methods in Physics Research A 652 (2011) 602–605
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Metal artifact reduction algorithm based on model images and spatial information Jay Wu a, Cheng-Ting Shih b, Shu-Jun Chang c, Tzung-Chi Huang d, Jing-Yi Sun a, Tung-Hsin Wu e,n a
Institute of Radiological Science, Central Taiwan University of Science and Technology, Taichung, Taiwan, ROC Department of Biomedical Engineering and Environmental Sciences, National Tsing-Hua University, Hsinchu, Taiwan, ROC c Health Physics Division, Institute of Nuclear Energy Research, Taoyuan, Taiwan, ROC d Department of Biomedical Imaging and Radiological Science, China Medical University, Taichung, Taiwan, ROC e Department of Biomedical Imaging and Radiological Sciences, National Yang-Ming University, No.155, Sec. 2, Linong Street, Taipei 112, Taiwan, ROC b
a r t i c l e i n f o
a b s t r a c t
Available online 25 January 2011
Computed tomography (CT) has become one of the most favorable choices for diagnosis of trauma. However, high-density metal implants can induce metal artifacts in CT images, compromising image quality. In this study, we proposed a model-based metal artifact reduction (MAR) algorithm. First, we built a model image using the k-means clustering technique with spatial information and calculated the difference between the original image and the model image. Then, the projection data of these two images were combined using an exponential weighting function. At last, the corrected image was reconstructed using the filter back-projection algorithm. Two metal-artifact contaminated images were studied. For the cylindrical water phantom image, the metal artifact was effectively removed. The mean CT number of water was improved from 28.95 7 97.97 to 4.76 7 4.28. For the clinical pelvic CT image, the dark band and the metal line were removed, and the continuity and uniformity of the soft tissue were recovered as well. These results indicate that the proposed MAR algorithm is useful for reducing metal artifact and could improve the diagnostic value of metal-artifact contaminated CT images. & 2011 Elsevier B.V. All rights reserved.
Keywords: Computed tomography Metal artifact k-means clustering
1. Introduction Computed tomography (CT) provides high resolution crosssectional images and has become a daily clinical routine in diagnostic radiology. However, metal objects with high attenuation coefficients in the human body greatly absorb low energy X-ray, causing severe beam hardening and loss of projection data in sinogram. Streak artifacts eventually appear in the reconstructed images, reducing image quality and uniformity [1]. Several metal artifact reduction (MAR) algorithms have been proposed in the last decade. They can be concluded into three categories: interpolation-based sinogram corrections [2], iterative image reconstructions [3,4], and adaptive filtering methods [5]. For the first category, the projection data of the metal objects are replaced by linear [6] or polynomial [1] interpolation of surrounding data in the sinogram. The major drawback is the incomplete artifact correction or even producing secondary artifacts [7]. For the second category, the iterative reconstruction methods usually provide good correction results when the metal objects do not
n
Corresponding author. Tel.: +886 2 28267061. E-mail address: [email protected] (T.-H. Wu).
0168-9002/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2011.01.041
stop all X-rays [8]. However, the relatively high computational complexity of these approaches makes it impractical for clinical practice. As for the third category, adaptive filtering can smooth streak artifacts and reduce image noise, but the adjustment of filter parameters must consider the strength of the artifact, which varies in each case. In this study, we proposed a MAR algorithm based on the model image built by the k-means clustering with spatial information. Forward projections of the model image and the original image were summed together by a weighting function. This MAR algorithm has the advantages of fast computation and simple implementation, and no original sinogram is required.
2. Materials and methods 2.1. Model-based image k-means clustering [9] was used to produce an initial model image by classifying all pixels into several clusters. After inputting the contaminated image and choosing the cluster number, the initial center of the cluster was randomly assigned. The pixel classification was based on the Euclidean distance between the
J. Wu et al. / Nuclear Instruments and Methods in Physics Research A 652 (2011) 602–605
pixel value and each cluster center. The sum of square error (SSE) between the cluster center and its membership was calculated by: SSE ¼
k X
n X
Jxi vi J2 ,
i ¼ 1,. . .,n;
j ¼ 1,. . .,k
603
2.3. Weighting of sinogram After creating the model image, the difference image (Id) was obtained by subtracting the model image (Im) from the original
ð1Þ
j ¼ 0 x A Cj
Pn mðCj 9xi Þxi , vi ¼ Pi n¼ 1 i ¼ 1 mðCj 9xi Þ
j ¼ 1,. . .,k
ð2Þ
where xi is the ith pixel value that belongs to the cluster j and vj is the jth cluster center. m(x) denotes the membership function, Cj is the jth cluster, k is the cluster number, and n is the total pixel number. The iteration was terminated when the change in SSE fell below 0.02. Six clusters were defined, including air, fat, soft tissue, mineral bone, cortical bone, and metal objects. After image clustering, the pixel value was transformed from the original CT number to the value of the cluster center, to which the pixel belonged. 2.2. Spatial information The region severely influenced by streak lines may cause wrong classification during k-means clustering. Therefore, the spatial information of the original image was incorporated [10]. The spatial information was calculated by Ci ¼ argfmaxð9P \ Cj 9Þg,
j ¼ 1,. . .,k
ð3Þ
where P denotes a set of pixels covered by a 9 9 mask centered at the pixel i. 9.9 means the number of elements in the set. This procedure reassigns the pixel i to the cluster Cj with the most members within the mask. By the method, a noise-free and artifact-free model image can be constructed. Original image (Io)
k-means clustering
Spatial information
Model image (Im)
Difference image (Id)
Radon transform
Pd
Po
Pd*
Pm
Weighting function
Corrected sinogram
FBP
Corrected image (Ic) Fig. 1. The flowchart of the proposed MAR algorithm. The k-means clustering with spatial information is applied to the original image (Io) to produce the artifact-free model image (Im). The difference image (Id), model image, and original image are then forward projected to the sinogram space. The corrected sinogram (Pc) is summed by the original sinogram (Po) and the model sinogram (Pm) with an exponential weighting function. Finally, the corrected image (Ic) is reconstructed from the Pc by the filtered back-projection algorithm.
Fig. 2. (a) Original image of the cylindrical water phantom with a 1 cm-radius lead rod inserted at the center, (b) the model image built by the k-means clustering with spatial information, and (c) the final corrected image.
604
J. Wu et al. / Nuclear Instruments and Methods in Physics Research A 652 (2011) 602–605
image (Io). The sinograms of these three images were forward projected by the Radon transform. Then, the corrected sinogram (Pc) was obtained by the weighted combination of the original sinogram (Po) and the model sinogram (Pm): Pc ði,jÞ ¼ wði,jÞ Po ði,jÞ þ ½1wði,jÞ Pm ði,jÞ
at the location shown in Fig. 2(b). The large variance of the CT values near the lead rod was suppressed after correction. Fig. 4(a) shows the original CT image of the pelvis with two gold seeds implanted for brachytherapy. Several bright and dark
ð4Þ
where w(i,j) denotes the pixel-based weighting factor that depends on the local value of the difference sinogram (Pd) forward projected from the difference image. The weighting factor is defined as: wði,jÞ ¼ emPd ði,jÞ
ð5Þ
P d
is the normalized version of Pd, and m is the weighting where parameter that controls the shape of the exponential function and the preservation of details in structure. Subsequently, the corrected image (Ic) was reconstructed from the Pc by the filtered back-projection algorithm. Fig. 1 shows the flowchart of the proposed MAR method.
3. Results and discussion A water phantom image and a pelvic CT image were used to evaluate the performance of the MAR algorithm. Fig. 2(a) shows the original image of the cylindrical water phantom with a 1 cmradius lead rod inserted at the center. The water compartment was covered with several bright and dark streaks, causing nonuniformity in the pixel values. A 100-pixel region of interest (ROI) was drawn at the dark band above the lead rod. The mean CT number was 28.95 797.97. Fig. 2(b) shows the model image built by the k-means clustering with spatial information. Each region of the phantom was successfully classified. Fig. 2(c) shows the corrected image using a weighting parameter m ¼40. The bright and dark streaks were removed and the image uniformity was improved. The ROI drawn at the same location as in Fig. 2(a) had a mean CT number of 4.7674.28, which was close to the CT number of water. Fig. 3 draws the profiles of the original, model, and corrected images of the cylindrical phantom 500 Corrected image Original image Model image
400
300
CT number
200 100 0 -100
-200 -300
0
100
200 300 Position (Pixels)
400
500
Fig. 3. Profiles of the original, model, and corrected images of the cylindrical phantom. The CT values near the center of the phantom had large variance due to metal artifacts caused by the lead rod.
Fig. 4. (a) Original CT image of the pelvis with two gold seeds, (b) the model image built by the k-means clustering with spatial information, and (c) the metalartifact corrected image with improved uniformity and preserved fine structure.
J. Wu et al. / Nuclear Instruments and Methods in Physics Research A 652 (2011) 602–605
700
Table 1 Mean CT values and standard deviations of the four ROIs drawn at different locations.
Corrected image Original image Model image
600 500
Pelvic image
Mean 7 std ROI 1 (dark band)
400 Original image Corrected image
300 CT number
605
200
ROI 2 (proximal region)
61.99 7215.50 34.64 7 29.62 51.48 720.63
38.94 7 9.28
ROI 3 (bright streak)
ROI 4 (distal region)
207.22 7 124.12 22.69 7 36.75 61.18 7 11.85
23.99 7 4.68
100 correction, the mean CT numbers of all ROIs approached the CT number of soft-tissue, and the standard deviations fell below 20%.
0 -100 -200
4. Conclusion
-300 -400 -500
0
100
200 300 Position (Pixels)
400
500
Fig. 5. Profiles of the original, model, and corrected images. The CT numbers of the soft tissue near the gold seeds had large variance ranging from 350 to 600.
streaks covered the region of bladder and the surrounding tissue, causing non-uniformity of the soft tissue. Under the dark band induced by the two metals, the CT number reached 500 that would be classified as air when using the original k-means method. By incorporating the spatial information shown in Fig. 4(b), the streaks and dark bands were classified correctly. Fig. 4(c) shows the corrected image of the pelvis with m ¼7, in that the metal artifacts were removed, the uniformity of the bladder and soft-tissue were improved, and the details of the structure were preserved. Fig. 5 displays the profiles of the original, model, and corrected images at the location shown in Fig. 4(b). In the profile of the original image, the maximum and minimum CT numbers of the soft tissue were about 600 and 350 that had recovered to the CT number of soft-tissue after correction. Table 1 lists the calculated mean CT values and standard deviations at various locations shown in Fig. 4(b). The mean CT values of ROI 1 (dark band) and ROI 3 (bright streak) were 61.99 and 207.22, which were fairly discrepant from the CT number of the surrounding soft tissue. The ROI 2 and ROI 4, representing the proximal and distal regions of the soft tissue, were also slightly influenced by the metal artifacts, showing large standard deviations. After
This study proposed a MAR algorithm based on the model image built by k-means clustering with spatial information. Results show that this algorithm is useful for reducing metal artifacts in CT images while preserving the detail structures. The proposed MAR algorithm has the advantages of fast computation and simple implementation, and no original sinogram is required. Therefore, it could be applied to clinical practice to improve the diagnostic value of the metal-artifact contaminated CT images.
Acknowledgement This study was financially supported by the National Science Council of Taiwan (NSC99-2314-B-010 -043-MY3) and (NSC 962320-B-166-001). References [1] G.H. Glover, N.J. Pelc, Med. Phys. 8 (1981) 799. [2] M. Yazdia, L. Gingras, L. Beaulieu, Int. J. Radiat. Oncol. Biol. Phys. 62 (2005) 1224. [3] B. De Man, J. Nuyts, P. Dupont, G. Marchal, P. Suetens, IEEE Trans. Med. Imaging 20 (2001) 999. [4] W. Ge, D.L. Snyder, J.A. O’Sullivan, M.W. Vannier, IEEE Trans. Med. Imaging 15 (1996) 657. [5] J. Hsieh, Med. Phys. 25 (1998) 2139. [6] W.A. Kalender, R. Hebel, J. Ebersberger, Radiology 164 (1987) 576. [7] D. Prell, Y. Kyriakou, M. Beister, W.A. Kalender, Phys. Med. Biol. 54 (2009) 6575. [8] F. Natterer, The Mathematics of Computerized Tomography, Wiley, New York, 1986. [9] J.A. Hartigan, M.A. Wong, J. R. Stat. Soc. Ser. C–Appl. Stat. 28 (1979) 100. [10] K.S. Chuang, H.L. Tzeng, S. Chen, J. Wu, T.J. Chen, Comput. Med. Imaging Graph. 30 (2006) 9.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Metal artifact reduction algorithm based on model images and spatial information Jay Wu a, Cheng-Ting Shih b, Shu-Jun Chang c, Tzung-Chi Huang d, Jing-Yi Sun a, Tung-Hsin Wu e,n a
Institute of Radiological Science, Central Taiwan University of Science and Technology, Taichung, Taiwan, ROC Department of Biomedical Engineering and Environmental Sciences, National Tsing-Hua University, Hsinchu, Taiwan, ROC c Health Physics Division, Institute of Nuclear Energy Research, Taoyuan, Taiwan, ROC d Department of Biomedical Imaging and Radiological Science, China Medical University, Taichung, Taiwan, ROC e Department of Biomedical Imaging and Radiological Sciences, National Yang-Ming University, No.155, Sec. 2, Linong Street, Taipei 112, Taiwan, ROC b
a r t i c l e i n f o
a b s t r a c t
Available online 25 January 2011
Computed tomography (CT) has become one of the most favorable choices for diagnosis of trauma. However, high-density metal implants can induce metal artifacts in CT images, compromising image quality. In this study, we proposed a model-based metal artifact reduction (MAR) algorithm. First, we built a model image using the k-means clustering technique with spatial information and calculated the difference between the original image and the model image. Then, the projection data of these two images were combined using an exponential weighting function. At last, the corrected image was reconstructed using the filter back-projection algorithm. Two metal-artifact contaminated images were studied. For the cylindrical water phantom image, the metal artifact was effectively removed. The mean CT number of water was improved from 28.95 7 97.97 to 4.76 7 4.28. For the clinical pelvic CT image, the dark band and the metal line were removed, and the continuity and uniformity of the soft tissue were recovered as well. These results indicate that the proposed MAR algorithm is useful for reducing metal artifact and could improve the diagnostic value of metal-artifact contaminated CT images. & 2011 Elsevier B.V. All rights reserved.
Keywords: Computed tomography Metal artifact k-means clustering
1. Introduction Computed tomography (CT) provides high resolution crosssectional images and has become a daily clinical routine in diagnostic radiology. However, metal objects with high attenuation coefficients in the human body greatly absorb low energy X-ray, causing severe beam hardening and loss of projection data in sinogram. Streak artifacts eventually appear in the reconstructed images, reducing image quality and uniformity [1]. Several metal artifact reduction (MAR) algorithms have been proposed in the last decade. They can be concluded into three categories: interpolation-based sinogram corrections [2], iterative image reconstructions [3,4], and adaptive filtering methods [5]. For the first category, the projection data of the metal objects are replaced by linear [6] or polynomial [1] interpolation of surrounding data in the sinogram. The major drawback is the incomplete artifact correction or even producing secondary artifacts [7]. For the second category, the iterative reconstruction methods usually provide good correction results when the metal objects do not
n
Corresponding author. Tel.: +886 2 28267061. E-mail address: [email protected] (T.-H. Wu).
0168-9002/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2011.01.041
stop all X-rays [8]. However, the relatively high computational complexity of these approaches makes it impractical for clinical practice. As for the third category, adaptive filtering can smooth streak artifacts and reduce image noise, but the adjustment of filter parameters must consider the strength of the artifact, which varies in each case. In this study, we proposed a MAR algorithm based on the model image built by the k-means clustering with spatial information. Forward projections of the model image and the original image were summed together by a weighting function. This MAR algorithm has the advantages of fast computation and simple implementation, and no original sinogram is required.
2. Materials and methods 2.1. Model-based image k-means clustering [9] was used to produce an initial model image by classifying all pixels into several clusters. After inputting the contaminated image and choosing the cluster number, the initial center of the cluster was randomly assigned. The pixel classification was based on the Euclidean distance between the
J. Wu et al. / Nuclear Instruments and Methods in Physics Research A 652 (2011) 602–605
pixel value and each cluster center. The sum of square error (SSE) between the cluster center and its membership was calculated by: SSE ¼
k X
n X
Jxi vi J2 ,
i ¼ 1,. . .,n;
j ¼ 1,. . .,k
603
2.3. Weighting of sinogram After creating the model image, the difference image (Id) was obtained by subtracting the model image (Im) from the original
ð1Þ
j ¼ 0 x A Cj
Pn mðCj 9xi Þxi , vi ¼ Pi n¼ 1 i ¼ 1 mðCj 9xi Þ
j ¼ 1,. . .,k
ð2Þ
where xi is the ith pixel value that belongs to the cluster j and vj is the jth cluster center. m(x) denotes the membership function, Cj is the jth cluster, k is the cluster number, and n is the total pixel number. The iteration was terminated when the change in SSE fell below 0.02. Six clusters were defined, including air, fat, soft tissue, mineral bone, cortical bone, and metal objects. After image clustering, the pixel value was transformed from the original CT number to the value of the cluster center, to which the pixel belonged. 2.2. Spatial information The region severely influenced by streak lines may cause wrong classification during k-means clustering. Therefore, the spatial information of the original image was incorporated [10]. The spatial information was calculated by Ci ¼ argfmaxð9P \ Cj 9Þg,
j ¼ 1,. . .,k
ð3Þ
where P denotes a set of pixels covered by a 9 9 mask centered at the pixel i. 9.9 means the number of elements in the set. This procedure reassigns the pixel i to the cluster Cj with the most members within the mask. By the method, a noise-free and artifact-free model image can be constructed. Original image (Io)
k-means clustering
Spatial information
Model image (Im)
Difference image (Id)
Radon transform
Pd
Po
Pd*
Pm
Weighting function
Corrected sinogram
FBP
Corrected image (Ic) Fig. 1. The flowchart of the proposed MAR algorithm. The k-means clustering with spatial information is applied to the original image (Io) to produce the artifact-free model image (Im). The difference image (Id), model image, and original image are then forward projected to the sinogram space. The corrected sinogram (Pc) is summed by the original sinogram (Po) and the model sinogram (Pm) with an exponential weighting function. Finally, the corrected image (Ic) is reconstructed from the Pc by the filtered back-projection algorithm.
Fig. 2. (a) Original image of the cylindrical water phantom with a 1 cm-radius lead rod inserted at the center, (b) the model image built by the k-means clustering with spatial information, and (c) the final corrected image.
604
J. Wu et al. / Nuclear Instruments and Methods in Physics Research A 652 (2011) 602–605
image (Io). The sinograms of these three images were forward projected by the Radon transform. Then, the corrected sinogram (Pc) was obtained by the weighted combination of the original sinogram (Po) and the model sinogram (Pm): Pc ði,jÞ ¼ wði,jÞ Po ði,jÞ þ ½1wði,jÞ Pm ði,jÞ
at the location shown in Fig. 2(b). The large variance of the CT values near the lead rod was suppressed after correction. Fig. 4(a) shows the original CT image of the pelvis with two gold seeds implanted for brachytherapy. Several bright and dark
ð4Þ
where w(i,j) denotes the pixel-based weighting factor that depends on the local value of the difference sinogram (Pd) forward projected from the difference image. The weighting factor is defined as: wði,jÞ ¼ emPd ði,jÞ
ð5Þ
P d
is the normalized version of Pd, and m is the weighting where parameter that controls the shape of the exponential function and the preservation of details in structure. Subsequently, the corrected image (Ic) was reconstructed from the Pc by the filtered back-projection algorithm. Fig. 1 shows the flowchart of the proposed MAR method.
3. Results and discussion A water phantom image and a pelvic CT image were used to evaluate the performance of the MAR algorithm. Fig. 2(a) shows the original image of the cylindrical water phantom with a 1 cmradius lead rod inserted at the center. The water compartment was covered with several bright and dark streaks, causing nonuniformity in the pixel values. A 100-pixel region of interest (ROI) was drawn at the dark band above the lead rod. The mean CT number was 28.95 797.97. Fig. 2(b) shows the model image built by the k-means clustering with spatial information. Each region of the phantom was successfully classified. Fig. 2(c) shows the corrected image using a weighting parameter m ¼40. The bright and dark streaks were removed and the image uniformity was improved. The ROI drawn at the same location as in Fig. 2(a) had a mean CT number of 4.7674.28, which was close to the CT number of water. Fig. 3 draws the profiles of the original, model, and corrected images of the cylindrical phantom 500 Corrected image Original image Model image
400
300
CT number
200 100 0 -100
-200 -300
0
100
200 300 Position (Pixels)
400
500
Fig. 3. Profiles of the original, model, and corrected images of the cylindrical phantom. The CT values near the center of the phantom had large variance due to metal artifacts caused by the lead rod.
Fig. 4. (a) Original CT image of the pelvis with two gold seeds, (b) the model image built by the k-means clustering with spatial information, and (c) the metalartifact corrected image with improved uniformity and preserved fine structure.
J. Wu et al. / Nuclear Instruments and Methods in Physics Research A 652 (2011) 602–605
700
Table 1 Mean CT values and standard deviations of the four ROIs drawn at different locations.
Corrected image Original image Model image
600 500
Pelvic image
Mean 7 std ROI 1 (dark band)
400 Original image Corrected image
300 CT number
605
200
ROI 2 (proximal region)
61.99 7215.50 34.64 7 29.62 51.48 720.63
38.94 7 9.28
ROI 3 (bright streak)
ROI 4 (distal region)
207.22 7 124.12 22.69 7 36.75 61.18 7 11.85
23.99 7 4.68
100 correction, the mean CT numbers of all ROIs approached the CT number of soft-tissue, and the standard deviations fell below 20%.
0 -100 -200
4. Conclusion
-300 -400 -500
0
100
200 300 Position (Pixels)
400
500
Fig. 5. Profiles of the original, model, and corrected images. The CT numbers of the soft tissue near the gold seeds had large variance ranging from 350 to 600.
streaks covered the region of bladder and the surrounding tissue, causing non-uniformity of the soft tissue. Under the dark band induced by the two metals, the CT number reached 500 that would be classified as air when using the original k-means method. By incorporating the spatial information shown in Fig. 4(b), the streaks and dark bands were classified correctly. Fig. 4(c) shows the corrected image of the pelvis with m ¼7, in that the metal artifacts were removed, the uniformity of the bladder and soft-tissue were improved, and the details of the structure were preserved. Fig. 5 displays the profiles of the original, model, and corrected images at the location shown in Fig. 4(b). In the profile of the original image, the maximum and minimum CT numbers of the soft tissue were about 600 and 350 that had recovered to the CT number of soft-tissue after correction. Table 1 lists the calculated mean CT values and standard deviations at various locations shown in Fig. 4(b). The mean CT values of ROI 1 (dark band) and ROI 3 (bright streak) were 61.99 and 207.22, which were fairly discrepant from the CT number of the surrounding soft tissue. The ROI 2 and ROI 4, representing the proximal and distal regions of the soft tissue, were also slightly influenced by the metal artifacts, showing large standard deviations. After
This study proposed a MAR algorithm based on the model image built by k-means clustering with spatial information. Results show that this algorithm is useful for reducing metal artifacts in CT images while preserving the detail structures. The proposed MAR algorithm has the advantages of fast computation and simple implementation, and no original sinogram is required. Therefore, it could be applied to clinical practice to improve the diagnostic value of the metal-artifact contaminated CT images.
Acknowledgement This study was financially supported by the National Science Council of Taiwan (NSC99-2314-B-010 -043-MY3) and (NSC 962320-B-166-001). References [1] G.H. Glover, N.J. Pelc, Med. Phys. 8 (1981) 799. [2] M. Yazdia, L. Gingras, L. Beaulieu, Int. J. Radiat. Oncol. Biol. Phys. 62 (2005) 1224. [3] B. De Man, J. Nuyts, P. Dupont, G. Marchal, P. Suetens, IEEE Trans. Med. Imaging 20 (2001) 999. [4] W. Ge, D.L. Snyder, J.A. O’Sullivan, M.W. Vannier, IEEE Trans. Med. Imaging 15 (1996) 657. [5] J. Hsieh, Med. Phys. 25 (1998) 2139. [6] W.A. Kalender, R. Hebel, J. Ebersberger, Radiology 164 (1987) 576. [7] D. Prell, Y. Kyriakou, M. Beister, W.A. Kalender, Phys. Med. Biol. 54 (2009) 6575. [8] F. Natterer, The Mathematics of Computerized Tomography, Wiley, New York, 1986. [9] J.A. Hartigan, M.A. Wong, J. R. Stat. Soc. Ser. C–Appl. Stat. 28 (1979) 100. [10] K.S. Chuang, H.L. Tzeng, S. Chen, J. Wu, T.J. Chen, Comput. Med. Imaging Graph. 30 (2006) 9.