Noise reduction with low dose CT data based on a modified ROF model

Yining Zhu; Mengliu Zhao; Yunsong Zhao; Hongwei Li; Peng Zhang

doi:10.1364/OE.20.017987

Noise reduction with low dose CT data based on a modified ROF model

Yining Zhu, Mengliu Zhao, Yunsong Zhao, Hongwei Li, and Peng Zhang

Optics Express
Vol. 20,
Issue 16,
pp. 17987-18004
(2012)
•https://doi.org/10.1364/OE.20.017987

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Yining Zhu, Mengliu Zhao, Yunsong Zhao, Hongwei Li, and Peng Zhang, "Noise reduction with low dose CT data based on a modified ROF model," Opt. Express 20, 17987-18004 (2012)

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Abstract

In order to reduce the radiation exposure caused by Computed Tomography (CT) scanning, low dose CT has gained much interest in research as well as in industry. One fundamental difficulty for low dose CT lies in its heavy noise pollution in the raw data which leads to quality deterioration for reconstructed images. In this paper, we propose a modified ROF model to denoise low dose CT measurement data in light of Poisson noise model. Experimental results indicate that the reconstructed CT images based on measurement data processed by our model are in better quality, compared to the original ROF model or bilateral filtering.

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A. Berrington de Gonzalez, M. Mahesh, K. Kim, M. Bhargavan, R. Lewis, F. Mettler, and C. Land, “Projected cancer risks from computed tomographic scans performed in the united states in 2007,” Arch. Intern Med. 169, 2071 (2009).
[Crossref] [PubMed]
A. Schilham, B. van Ginneken, H. Gietema, and M. Prokop, “Local noise weighted filtering for emphysema scoring of low-dose ct images,” IEEE Trans. Med. Imaging 25, 451–463 (2006).
[Crossref] [PubMed]
M. Tabuchi, N. Yamane, and Y. Morikawa, “Adaptive wiener filter based on gaussian mixture model for denoising chest x-ray ct image,” in IEEE Proceedings of SICE 2007 Annual Conference (IEEE, 2007), pp. 682–689.
[Crossref]
E. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam ct,” J. X-Ray Sci. Technol. 14, 119–139 (2006).
P. La Rivière and D. Billmire, “Reduction of noise-induced streak artifacts in x-ray computed tomography through spline-based penalized-likelihood sinogram smoothing,” IEEE Trans. Med. Imaging 24, 105–111 (2005).
[Crossref] [PubMed]
P. La Rivière, “Penalized-likelihood sinogram smoothing for low-dose ct,” Med. Phys. 32, 1676 (2005).
[Crossref] [PubMed]
T. Li, X. Li, J. Wang, J. Wen, H. Lu, J. Hsieh, and Z. Liang, “Nonlinear sinogram smoothing for low-dose x-ray ct,” IEEE Trans. Nucl. Sci. 51, 2505–2513 (2004).
[Crossref]
J. Wang, T. Li, H. Lu, and Z. Liang, “Penalized weighted least-squares approach to sinogram noise reduction and image reconstruction for low-dose x-ray computed tomography,” IEEE Trans. Med. Imaging 25, 1272–1283 (2006).
[Crossref] [PubMed]
L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[Crossref]
J. Hsieh, Computed tomography: principles, design, artifacts, and recent advances (Society of Photo Optical, 2003), Vol. 114.
K. Lange, R. Carson, and et al., “Em reconstruction algorithms for emission and transmission tomography.” J. Comput. Assist. Tomogr. 8, 306 (1984).
[PubMed]
T. Le, R. Chartrand, and T. Asaki, “A variational approach to reconstructing images corrupted by poisson noise,” J. Math. Imaging Vision 27, 257–263 (2007).
[Crossref]
L. Zhang, L. Zhang, D. Zhang, and H. Zhu, “Computer analysis of images and patterns,” Pattern Recogn. 44, 1990–1998 (2011).
[Crossref]
F. Luisier, T. Blu, and M. Unser, “Image denoising in mixed poisson-gaussian noise,” IEEE Trans. Image Process. 20(3), 696–708 (2011).
[Crossref]
B. Zhang, J. Fadili, and J. Starck, “Wavelets, ridgelets, and curvelets for poisson noise removal,” IEEE Trans. Image Process. 17, 1093–1108 (2008).
[Crossref] [PubMed]
A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Service Center, 1988), p. 327.
H. Lu, T. Hsiao, X. Li, and Z. Liang, “Noise properties of low-dose ct projections and noise treatment by scale transformations,” in in the IEEE Nuclear Science Symposium Conference 2001 Record (IEEE, 2001), Vol. 3, pp. 1662–1666.
A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20, 89–97 (2004).
[Crossref]
T. Chan and P. Mulet, “On the convergence of the lagged diffusivity fixed point method in total variation image restoration,” SIAM J. Numer. Anal. 36, 354–367 (1999).
[Crossref]
M. Unger, T. Pock, and H. Bischof, “Continuous globally optimal image segmentation with local constraints,” in Computer Vision Winter Workshop at Slovenian Pattern Recognition Society, Ljubljana, Slovenia (2008).
T. Goldstein and S. Osher, “The split bregman method for l1 regularized problems,” SIAM J. Imag. Sci. 2, 323–343 (2009).
[Crossref]
L. Rudin and S. Osher, “Total variation based image restoration with free local constraints,” in Proceedings of the IEEE International Conference on Image Processing1994 (IEEE, 1994), vol. 1, pp. 31–35.
G. Gilboa, N. Sochen, and Y. Zeevi, “Estimation of optimal pde-based denoising in the snr sense,” IEEE Trans. Image Process. 15, 2269–2280 (2006).
[Crossref] [PubMed]
B. Wohlberg and Y. Lin, “Upre method for total variation parameter selection,” Tech. Report, Los Alamos National Laboratory (LANL) (2008).
S. Babacan, R. Molina, and A. Katsaggelos, “Parameter estimation in tv image restoration using variational distribution approximation,” IEEE Trans. Image Process. 17, 326–339 (2008).
[Crossref] [PubMed]
H. Gach, C. Tanase, and F. Boada, “2d & 3d shepp-logan phantom standards for mri,” in IEEE 19th International Conference on Systems Engineering 2008 (IEEE, 2008), pp. 521–526.
[Crossref]

2011 (2)

L. Zhang, L. Zhang, D. Zhang, and H. Zhu, “Computer analysis of images and patterns,” Pattern Recogn. 44, 1990–1998 (2011).
[Crossref]

F. Luisier, T. Blu, and M. Unser, “Image denoising in mixed poisson-gaussian noise,” IEEE Trans. Image Process. 20(3), 696–708 (2011).
[Crossref]

2009 (2)

A. Berrington de Gonzalez, M. Mahesh, K. Kim, M. Bhargavan, R. Lewis, F. Mettler, and C. Land, “Projected cancer risks from computed tomographic scans performed in the united states in 2007,” Arch. Intern Med. 169, 2071 (2009).
[Crossref] [PubMed]

T. Goldstein and S. Osher, “The split bregman method for l1 regularized problems,” SIAM J. Imag. Sci. 2, 323–343 (2009).
[Crossref]

2008 (2)

S. Babacan, R. Molina, and A. Katsaggelos, “Parameter estimation in tv image restoration using variational distribution approximation,” IEEE Trans. Image Process. 17, 326–339 (2008).
[Crossref] [PubMed]

B. Zhang, J. Fadili, and J. Starck, “Wavelets, ridgelets, and curvelets for poisson noise removal,” IEEE Trans. Image Process. 17, 1093–1108 (2008).
[Crossref] [PubMed]

2007 (1)

T. Le, R. Chartrand, and T. Asaki, “A variational approach to reconstructing images corrupted by poisson noise,” J. Math. Imaging Vision 27, 257–263 (2007).
[Crossref]

2006 (4)

G. Gilboa, N. Sochen, and Y. Zeevi, “Estimation of optimal pde-based denoising in the snr sense,” IEEE Trans. Image Process. 15, 2269–2280 (2006).
[Crossref] [PubMed]

J. Wang, T. Li, H. Lu, and Z. Liang, “Penalized weighted least-squares approach to sinogram noise reduction and image reconstruction for low-dose x-ray computed tomography,” IEEE Trans. Med. Imaging 25, 1272–1283 (2006).
[Crossref] [PubMed]

A. Schilham, B. van Ginneken, H. Gietema, and M. Prokop, “Local noise weighted filtering for emphysema scoring of low-dose ct images,” IEEE Trans. Med. Imaging 25, 451–463 (2006).
[Crossref] [PubMed]

E. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam ct,” J. X-Ray Sci. Technol. 14, 119–139 (2006).

2005 (2)

P. La Rivière and D. Billmire, “Reduction of noise-induced streak artifacts in x-ray computed tomography through spline-based penalized-likelihood sinogram smoothing,” IEEE Trans. Med. Imaging 24, 105–111 (2005).
[Crossref] [PubMed]

P. La Rivière, “Penalized-likelihood sinogram smoothing for low-dose ct,” Med. Phys. 32, 1676 (2005).
[Crossref] [PubMed]

2004 (2)

T. Li, X. Li, J. Wang, J. Wen, H. Lu, J. Hsieh, and Z. Liang, “Nonlinear sinogram smoothing for low-dose x-ray ct,” IEEE Trans. Nucl. Sci. 51, 2505–2513 (2004).
[Crossref]

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20, 89–97 (2004).
[Crossref]

1999 (1)

T. Chan and P. Mulet, “On the convergence of the lagged diffusivity fixed point method in total variation image restoration,” SIAM J. Numer. Anal. 36, 354–367 (1999).
[Crossref]

1992 (1)

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[Crossref]

1984 (1)

K. Lange, R. Carson, and et al., “Em reconstruction algorithms for emission and transmission tomography.” J. Comput. Assist. Tomogr. 8, 306 (1984).
[PubMed]

Asaki, T.

T. Le, R. Chartrand, and T. Asaki, “A variational approach to reconstructing images corrupted by poisson noise,” J. Math. Imaging Vision 27, 257–263 (2007).
[Crossref]

Babacan, S.

Berrington de Gonzalez, A.

Bhargavan, M.

Billmire, D.

Bischof, H.

M. Unger, T. Pock, and H. Bischof, “Continuous globally optimal image segmentation with local constraints,” in Computer Vision Winter Workshop at Slovenian Pattern Recognition Society, Ljubljana, Slovenia (2008).

Blu, T.

F. Luisier, T. Blu, and M. Unser, “Image denoising in mixed poisson-gaussian noise,” IEEE Trans. Image Process. 20(3), 696–708 (2011).
[Crossref]

Boada, F.

H. Gach, C. Tanase, and F. Boada, “2d & 3d shepp-logan phantom standards for mri,” in IEEE 19th International Conference on Systems Engineering 2008 (IEEE, 2008), pp. 521–526.
[Crossref]

Carson, R.

K. Lange, R. Carson, and et al., “Em reconstruction algorithms for emission and transmission tomography.” J. Comput. Assist. Tomogr. 8, 306 (1984).
[PubMed]

Chambolle, A.

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20, 89–97 (2004).
[Crossref]

Chan, T.

T. Chan and P. Mulet, “On the convergence of the lagged diffusivity fixed point method in total variation image restoration,” SIAM J. Numer. Anal. 36, 354–367 (1999).
[Crossref]

Chartrand, R.

T. Le, R. Chartrand, and T. Asaki, “A variational approach to reconstructing images corrupted by poisson noise,” J. Math. Imaging Vision 27, 257–263 (2007).
[Crossref]

Fadili, J.

B. Zhang, J. Fadili, and J. Starck, “Wavelets, ridgelets, and curvelets for poisson noise removal,” IEEE Trans. Image Process. 17, 1093–1108 (2008).
[Crossref] [PubMed]

Fatemi, E.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[Crossref]

Gach, H.

H. Gach, C. Tanase, and F. Boada, “2d & 3d shepp-logan phantom standards for mri,” in IEEE 19th International Conference on Systems Engineering 2008 (IEEE, 2008), pp. 521–526.
[Crossref]

Gietema, H.

Gilboa, G.

G. Gilboa, N. Sochen, and Y. Zeevi, “Estimation of optimal pde-based denoising in the snr sense,” IEEE Trans. Image Process. 15, 2269–2280 (2006).
[Crossref] [PubMed]

Goldstein, T.

T. Goldstein and S. Osher, “The split bregman method for l1 regularized problems,” SIAM J. Imag. Sci. 2, 323–343 (2009).
[Crossref]

Hsiao, T.

H. Lu, T. Hsiao, X. Li, and Z. Liang, “Noise properties of low-dose ct projections and noise treatment by scale transformations,” in in the IEEE Nuclear Science Symposium Conference 2001 Record (IEEE, 2001), Vol. 3, pp. 1662–1666.

Hsieh, J.

T. Li, X. Li, J. Wang, J. Wen, H. Lu, J. Hsieh, and Z. Liang, “Nonlinear sinogram smoothing for low-dose x-ray ct,” IEEE Trans. Nucl. Sci. 51, 2505–2513 (2004).
[Crossref]

J. Hsieh, Computed tomography: principles, design, artifacts, and recent advances (Society of Photo Optical, 2003), Vol. 114.

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Service Center, 1988), p. 327.

Kao, C.

E. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam ct,” J. X-Ray Sci. Technol. 14, 119–139 (2006).

Katsaggelos, A.

Kim, K.

La Rivière, P.

P. La Rivière, “Penalized-likelihood sinogram smoothing for low-dose ct,” Med. Phys. 32, 1676 (2005).
[Crossref] [PubMed]

Land, C.

Lange, K.

K. Lange, R. Carson, and et al., “Em reconstruction algorithms for emission and transmission tomography.” J. Comput. Assist. Tomogr. 8, 306 (1984).
[PubMed]

Le, T.

T. Le, R. Chartrand, and T. Asaki, “A variational approach to reconstructing images corrupted by poisson noise,” J. Math. Imaging Vision 27, 257–263 (2007).
[Crossref]

Lewis, R.

Li, T.

T. Li, X. Li, J. Wang, J. Wen, H. Lu, J. Hsieh, and Z. Liang, “Nonlinear sinogram smoothing for low-dose x-ray ct,” IEEE Trans. Nucl. Sci. 51, 2505–2513 (2004).
[Crossref]

Li, X.

T. Li, X. Li, J. Wang, J. Wen, H. Lu, J. Hsieh, and Z. Liang, “Nonlinear sinogram smoothing for low-dose x-ray ct,” IEEE Trans. Nucl. Sci. 51, 2505–2513 (2004).
[Crossref]

Liang, Z.

T. Li, X. Li, J. Wang, J. Wen, H. Lu, J. Hsieh, and Z. Liang, “Nonlinear sinogram smoothing for low-dose x-ray ct,” IEEE Trans. Nucl. Sci. 51, 2505–2513 (2004).
[Crossref]

Lin, Y.

B. Wohlberg and Y. Lin, “Upre method for total variation parameter selection,” Tech. Report, Los Alamos National Laboratory (LANL) (2008).

Lu, H.

T. Li, X. Li, J. Wang, J. Wen, H. Lu, J. Hsieh, and Z. Liang, “Nonlinear sinogram smoothing for low-dose x-ray ct,” IEEE Trans. Nucl. Sci. 51, 2505–2513 (2004).
[Crossref]

Luisier, F.

F. Luisier, T. Blu, and M. Unser, “Image denoising in mixed poisson-gaussian noise,” IEEE Trans. Image Process. 20(3), 696–708 (2011).
[Crossref]

Mahesh, M.

Mettler, F.

Molina, R.

Morikawa, Y.

M. Tabuchi, N. Yamane, and Y. Morikawa, “Adaptive wiener filter based on gaussian mixture model for denoising chest x-ray ct image,” in IEEE Proceedings of SICE 2007 Annual Conference (IEEE, 2007), pp. 682–689.
[Crossref]

Mulet, P.

T. Chan and P. Mulet, “On the convergence of the lagged diffusivity fixed point method in total variation image restoration,” SIAM J. Numer. Anal. 36, 354–367 (1999).
[Crossref]

Osher, S.

T. Goldstein and S. Osher, “The split bregman method for l1 regularized problems,” SIAM J. Imag. Sci. 2, 323–343 (2009).
[Crossref]

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[Crossref]

L. Rudin and S. Osher, “Total variation based image restoration with free local constraints,” in Proceedings of the IEEE International Conference on Image Processing1994 (IEEE, 1994), vol. 1, pp. 31–35.

Pan, X.

E. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam ct,” J. X-Ray Sci. Technol. 14, 119–139 (2006).

Pock, T.

Prokop, M.

Rudin, L.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[Crossref]

Schilham, A.

Sidky, E.

E. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam ct,” J. X-Ray Sci. Technol. 14, 119–139 (2006).

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Service Center, 1988), p. 327.

Sochen, N.

G. Gilboa, N. Sochen, and Y. Zeevi, “Estimation of optimal pde-based denoising in the snr sense,” IEEE Trans. Image Process. 15, 2269–2280 (2006).
[Crossref] [PubMed]

Starck, J.

B. Zhang, J. Fadili, and J. Starck, “Wavelets, ridgelets, and curvelets for poisson noise removal,” IEEE Trans. Image Process. 17, 1093–1108 (2008).
[Crossref] [PubMed]

Tabuchi, M.

Tanase, C.

H. Gach, C. Tanase, and F. Boada, “2d & 3d shepp-logan phantom standards for mri,” in IEEE 19th International Conference on Systems Engineering 2008 (IEEE, 2008), pp. 521–526.
[Crossref]

Unger, M.

Unser, M.

F. Luisier, T. Blu, and M. Unser, “Image denoising in mixed poisson-gaussian noise,” IEEE Trans. Image Process. 20(3), 696–708 (2011).
[Crossref]

van Ginneken, B.

Wang, J.

T. Li, X. Li, J. Wang, J. Wen, H. Lu, J. Hsieh, and Z. Liang, “Nonlinear sinogram smoothing for low-dose x-ray ct,” IEEE Trans. Nucl. Sci. 51, 2505–2513 (2004).
[Crossref]

Wen, J.

T. Li, X. Li, J. Wang, J. Wen, H. Lu, J. Hsieh, and Z. Liang, “Nonlinear sinogram smoothing for low-dose x-ray ct,” IEEE Trans. Nucl. Sci. 51, 2505–2513 (2004).
[Crossref]

Wohlberg, B.

B. Wohlberg and Y. Lin, “Upre method for total variation parameter selection,” Tech. Report, Los Alamos National Laboratory (LANL) (2008).

Yamane, N.

Zeevi, Y.

G. Gilboa, N. Sochen, and Y. Zeevi, “Estimation of optimal pde-based denoising in the snr sense,” IEEE Trans. Image Process. 15, 2269–2280 (2006).
[Crossref] [PubMed]

Zhang, B.

B. Zhang, J. Fadili, and J. Starck, “Wavelets, ridgelets, and curvelets for poisson noise removal,” IEEE Trans. Image Process. 17, 1093–1108 (2008).
[Crossref] [PubMed]

Zhang, D.

L. Zhang, L. Zhang, D. Zhang, and H. Zhu, “Computer analysis of images and patterns,” Pattern Recogn. 44, 1990–1998 (2011).
[Crossref]

Zhang, L.

L. Zhang, L. Zhang, D. Zhang, and H. Zhu, “Computer analysis of images and patterns,” Pattern Recogn. 44, 1990–1998 (2011).
[Crossref]

Zhu, H.

L. Zhang, L. Zhang, D. Zhang, and H. Zhu, “Computer analysis of images and patterns,” Pattern Recogn. 44, 1990–1998 (2011).
[Crossref]

Arch. Intern Med. (1)

IEEE Trans. Image Process. (4)

F. Luisier, T. Blu, and M. Unser, “Image denoising in mixed poisson-gaussian noise,” IEEE Trans. Image Process. 20(3), 696–708 (2011).
[Crossref]

B. Zhang, J. Fadili, and J. Starck, “Wavelets, ridgelets, and curvelets for poisson noise removal,” IEEE Trans. Image Process. 17, 1093–1108 (2008).
[Crossref] [PubMed]

G. Gilboa, N. Sochen, and Y. Zeevi, “Estimation of optimal pde-based denoising in the snr sense,” IEEE Trans. Image Process. 15, 2269–2280 (2006).
[Crossref] [PubMed]

IEEE Trans. Med. Imaging (3)

IEEE Trans. Nucl. Sci. (1)

T. Li, X. Li, J. Wang, J. Wen, H. Lu, J. Hsieh, and Z. Liang, “Nonlinear sinogram smoothing for low-dose x-ray ct,” IEEE Trans. Nucl. Sci. 51, 2505–2513 (2004).
[Crossref]

J. Comput. Assist. Tomogr. (1)

K. Lange, R. Carson, and et al., “Em reconstruction algorithms for emission and transmission tomography.” J. Comput. Assist. Tomogr. 8, 306 (1984).
[PubMed]

J. Math. Imaging Vision (2)

T. Le, R. Chartrand, and T. Asaki, “A variational approach to reconstructing images corrupted by poisson noise,” J. Math. Imaging Vision 27, 257–263 (2007).
[Crossref]

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20, 89–97 (2004).
[Crossref]

J. X-Ray Sci. Technol. (1)

E. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam ct,” J. X-Ray Sci. Technol. 14, 119–139 (2006).

Med. Phys. (1)

P. La Rivière, “Penalized-likelihood sinogram smoothing for low-dose ct,” Med. Phys. 32, 1676 (2005).
[Crossref] [PubMed]

Pattern Recogn. (1)

L. Zhang, L. Zhang, D. Zhang, and H. Zhu, “Computer analysis of images and patterns,” Pattern Recogn. 44, 1990–1998 (2011).
[Crossref]

Physica D (1)

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[Crossref]

SIAM J. Imag. Sci. (1)

T. Goldstein and S. Osher, “The split bregman method for l1 regularized problems,” SIAM J. Imag. Sci. 2, 323–343 (2009).
[Crossref]

SIAM J. Numer. Anal. (1)

T. Chan and P. Mulet, “On the convergence of the lagged diffusivity fixed point method in total variation image restoration,” SIAM J. Numer. Anal. 36, 354–367 (1999).
[Crossref]

Other (8)

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Service Center, 1988), p. 327.

J. Hsieh, Computed tomography: principles, design, artifacts, and recent advances (Society of Photo Optical, 2003), Vol. 114.

B. Wohlberg and Y. Lin, “Upre method for total variation parameter selection,” Tech. Report, Los Alamos National Laboratory (LANL) (2008).

H. Gach, C. Tanase, and F. Boada, “2d & 3d shepp-logan phantom standards for mri,” in IEEE 19th International Conference on Systems Engineering 2008 (IEEE, 2008), pp. 521–526.
[Crossref]

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Fig. 1:

The probability density functions of distributions with different means

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Fig. 2:

The denoised raw data as well as the reconstructed images. In the left column, (a) and (b) are the ideal data and noise data respectively, from (c) to (e) are data filtered by bilateral filtering, ROF model and Poisson-ROF model respectively. In the right column, from (a’) to (e’)are the corresponding reconstructed images.

Download Full Size | PPT Slide | PDF

Fig. 3:

The profiles of the reconstructed CT image: (a) is the 256th row of the images; (b) is the 256th column of the images.

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Fig. 4:

The high dose CT image of volcanic rock

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Fig. 5:

(a) is CT image reconstructed without noise reduction in 25% dose of high dose data; from (b) to (d) are reconstructed results from raw data filtered by bilateral filtering, ROF model and Poisson-ROF model.

Download Full Size | PPT Slide | PDF

Fig. 6:

(a) is CT image reconstructed without noise reduction in 12.5% dose of high dose data; from (b) to (d) are reconstructed results from raw data filtered by bilateral filtering, ROF model and Poisson-ROF model.

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Fig. 7:

The first row of (a) an (b) are region A of corresponding images in Fig. 5 and Fig. 6; the second row of (a) an (b) are region B of corresponding images in Fig. 5 and Fig. 6.

Download Full Size | PPT Slide | PDF

Fig. 8:

Analysis of edge blurring in the reconstructed images

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Fig. 9:

The high dose CT image of line-pairs phantom

Download Full Size | PPT Slide | PDF

Fig. 10:

(a) is CT image reconstructed from the raw data using 12.5% dose; (b) is CT image reconstructed from the same raw data processed by the Poisson-ROF model.

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Tables (2)

Table 1: SNR, NMAD and NRMSE for low dose raw data

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Table 2: SNR, NMAD and NRMSE for CT images reconstructed from the corresponding (denoised) raw data

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Equations (32)

Equations on this page are rendered with MathJax. Learn more.

I^{d} ~ Poisson {I_{0} exp (- \int_{x \in L} f (x) d x)}

p_{i} = - log (I_{i}^{d} / I_{0}) = \int_{L_{i}} f (x) d x

p_{i} \approx p_{i}^{Ideal} + n

σ_{i}^{2} = f_{i} e^{\frac{μ_{i}}{T}}

{\begin{array}{l} min \int_{Ω} | \nabla u | d Ω \\ \frac{1}{| Ω |} \int_{Ω} {(u - f)}^{2} d Ω = σ^{2} \end{array}

min_{u} {J (u) + \frac{1}{2 λ} \int_{Ω} {(u - f)}^{2} d Ω}

J (u) = \int_{Ω} \sqrt{{(\frac{\partial u}{\partial x})}^{2} + {(\frac{\partial u}{\partial y})}^{2}} d Ω

p^{n + 1} = \frac{p^{n} + τ (\nabla (div (p^{n})) - f / λ)}{1 + τ | \nabla (div (p^{n})) - f / λ |}

u^{n + 1} = (f - λ div (p^{n + 1}))

f (x) = u (x) + n

\frac{1}{| Ω |} \int_{Ω} {(u - f)}^{2} d Ω = σ^{2}

\frac{1}{| Ω |} \int_{Ω} \frac{1}{σ^{2}} {(u - f)}^{2} d Ω = 1 .

\frac{1}{| Ω |} \int_{Ω} \frac{1}{u} {(u - f)}^{2} d Ω = 1 .

min_{u} {J (u) + \frac{1}{2 λ} \int_{Ω} \frac{1}{u} {(u - f)}^{2} d Ω},

\frac{1}{σ} (f (x) - u (x))

\frac{1}{\sqrt{u (x)}} (f (x) - u (x))

P {f (x) = k} = \frac{e^{- u (x)} {(u (x))}^{k}}{k!}, k \in ℕ

w (x) = \frac{f (x) - u (x)}{\sqrt{u (x)}}

P {f (x) = s} = {\begin{array}{r} 0, & s ⩽ 0 \\ \frac{e^{- u (x)} {(u (x))}^{k}}{k!}, & s \in (k - 1, k], k \in ℕ . \end{array}

P {w (x) = s} = {\begin{array}{r} 0, s ⩽ - \frac{u (x)}{\sqrt{u (x)}}, \\ \frac{e^{- u (x)} {(u (x))}^{[u (x) + s \sqrt{u (x)}]}}{[u (x) + s \sqrt{u (x)}]!}, s > - \frac{u (x)}{\sqrt{u (x)}} . \end{array}

σ^{2} = f_{N F}

min_{u} {J (u) + \frac{1}{2 λ} \int_{Ω} \frac{1}{f_{N F}} {(u - f)}^{2} d Ω}

J (u) = \int_{Ω} \sqrt{{(\frac{\partial u}{\partial x})}^{2} + {(\frac{\partial u}{\partial y})}^{2}} d Ω

0 = - div (\frac{\nabla u}{| \nabla u |}) + \frac{1}{λ f_{N F}} (u - f)

\bar{u} (x) = \frac{\sum_{ε \in W} s (u (ε), u (x)) c (ε, x) u (ε)}{\sum_{ε \in W} s (u (ε), u (x)) c (ε, x)}

\begin{array}{r} s (u (ε), u (x)) = e^{- \frac{{| u (ε) - u (x) |}^{2}}{2 σ_{r}^{2}}} \\ c (ε, x) = e^{- \frac{{| ε - x |}^{2}}{2 σ_{d}^{2}}} \end{array}

\begin{array}{l} min_{{u, v}} J (u) + \frac{1}{2 θ} \int_{Ω} {(u - v)}^{2} d Ω \\ + \frac{1}{2 λ} \int_{Ω} \frac{1}{f_{N F}} {(v - f)}^{2} d Ω \end{array}

\frac{1}{θ} (v - u) + \frac{1}{λ \bar{u}} (v - f) = 0 \Rightarrow v = \frac{λ u \bar{u} + θ f}{λ \bar{u} + θ}

λ = - \frac{\int_{Ω} < div (p), \frac{1}{u} (u - f) > d Ω}{\int_{Ω} {| div (p) |}^{2} d Ω}

S N R = - 10 {log}_{10} (\int_{Ω} {(f_{result} - f_{N F})}^{2} d Ω / \int f_{N F}^{2} d Ω)

N M A D = \int_{Ω} | (f_{result} - f_{N F}) / f_{N F} | d Ω

N R M S E = \int_{Ω} {(f_{result} - f_{N F})}^{2} / σ_{N F}^{2} d Ω

	Noisy Data	Bilateral Filtering	ROF	Poisson-ROF
SNR	39.242	46.919	43.519	55.808
NMAD	0.008753	0.003268	0.002881	0.002146
NRMSE	0.09641	0.04112	0.0609	0.02352

	Noisy Data	Bilateral Filtering	ROF	Poisson-ROF
SNR	10.338	19.021	13.821	25.982
NMAD	0.3752	0.1167	0.1421	0.0821
NRMSE	0.3541	0.1303	0.2371	0.0939

	Noisy Data	Bilateral Filtering	ROF	Poisson-ROF
SNR	39.242	46.919	43.519	55.808
NMAD	0.008753	0.003268	0.002881	0.002146
NRMSE	0.09641	0.04112	0.0609	0.02352

	Noisy Data	Bilateral Filtering	ROF	Poisson-ROF
SNR	10.338	19.021	13.821	25.982
NMAD	0.3752	0.1167	0.1421	0.0821
NRMSE	0.3541	0.1303	0.2371	0.0939