Abstract

This work presents a novel computed tomography (CT) reconstruction method for the few-view problem based on fractional calculus. To overcome the disadvantages of the total variation minimization method, we propose a fractional-order total variation-based image reconstruction method in this paper. The presented model adopts fractional-order total variation instead of traditional total variation. Different from traditional total variation, fractional-order total variation is derived by considering more neighboring image voxels such that the corresponding weights can be adaptively determined by the model, thus suppressing the over-smoothing effect. The discretization scheme of the fractional-order model is also given. Numerical and clinical experiments demonstrate that our method achieves better performance than existing reconstruction methods, including filtered back projection (FBP), the total variation-based projections onto convex sets method (TV-POCS), and soft-threshold filtering (STH).

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. A. Andersen and A. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imaging 6, 81–94 (1984).
    [CrossRef]
  4. A. P. Dempster, N. M. Laired, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Royal Stat. Soc. B 39, 1–38 (1977).
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    [CrossRef]
  6. E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
    [CrossRef]
  7. E. Y. Sidky, C. M. 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).
  8. E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
    [CrossRef]
  9. Y.-L. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Trans. Image Process. 9, 1723–1730 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. Z. Tian, X. Jia, K. Yuan, T. Pan, and S. B. Jiang, “Low-dose CT reconstruction via edge-preserving total variation regularization,” Phys. Med. Biol. 56, 5949–5967 (2011).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  17. J. Zhang, Z. Wei, and L. Xiao, “Adaptive fractional-order multi-scale method for image denoising,” J. Math. Imaging Vis. 43, 39–49 (2012).
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  21. Y. Zhang, Y.-F. Pu, J.-R. Hu, Y. Liu, Q.-L. Chen, and J.-L. Zhou, “Efficient CT metal artifacts reduction based on fractional-order curvature diffusion,” Comput. Math. Method Med. 2011, 173748 (2011).
    [CrossRef]
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    [CrossRef]
  23. K. B. Oldham and J. Spanier, The Fractional Calculus (Academic, 1974).
  24. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
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  26. R. L. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. 12, 252–255 (1985).
    [CrossRef]

2013

Z. Chen, X. Jin, L. Li, and G. Wang, “A limited-angle CT reconstruction method based on anisotropic TV minimization,” Phys. Med. Biol. 58, 2119–2141 (2013).
[CrossRef]

2012

Y. Liu, J. Ma, Y. Fan, and Z. Liang, “Adaptive-weighted total variation minimization for sparse data toward low-dose x-ray computed tomography image reconstruction,” Phys. Med. Biol. 57, 7923–7956 (2012).
[CrossRef]

J. Zhang, Z. Wei, and L. Xiao, “Adaptive fractional-order multi-scale method for image denoising,” J. Math. Imaging Vis. 43, 39–49 (2012).
[CrossRef]

Y. Zhang, Y.-F. Pu, J.-R. Hu, and J.-L. Zhou, “A class of fractional-order variational image inpainting models,” Appl. Math. Inform. Sci. 6, 229–306 (2012).

2011

Y. Zhang, Y.-F. Pu, J.-R. Hu, Y. Liu, and J.-L. Zhou, “A new CT metal artifacts reduction algorithm based on fractional-order sinogram inpainting,” J. X-Ray Sci. Technol. 19, 373–384 (2011).

Y. Zhang, Y.-F. Pu, J.-R. Hu, Y. Liu, Q.-L. Chen, and J.-L. Zhou, “Efficient CT metal artifacts reduction based on fractional-order curvature diffusion,” Comput. Math. Method Med. 2011, 173748 (2011).
[CrossRef]

J. Zhang and Z. Wei, “A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising,” Appl. Math. Model. 35, 2516–2528 (2011).
[CrossRef]

Z. Tian, X. Jia, K. Yuan, T. Pan, and S. B. Jiang, “Low-dose CT reconstruction via edge-preserving total variation regularization,” Phys. Med. Biol. 56, 5949–5967 (2011).
[CrossRef]

2010

H. Yu and G. Wang, “A soft-threshold filtering approach for reconstruction from a limited number of projections,” Phys. Med. Biol. 55, 3905–3916 (2010).
[CrossRef]

Y.-F. Pu, J.-L. Zhou, and X. Yuan, “Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement,” IEEE Trans. Image Process. 19, 491–511 (2010).
[CrossRef]

2008

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
[CrossRef]

2007

D. J. Brenner and E. J. Hall, “Computed tomography—an increasing source of radiation exposure,” New Engl. J. Med. 357, 2277–2284 (2007).
[CrossRef]

J. Bai and X. Feng, “Fractional-order anisotropic diffusion for image denoising,” IEEE Trans. Image Process. 16, 2492–2502 (2007).
[CrossRef]

2006

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[CrossRef]

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

E. Y. Sidky, C. M. 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).

2004

M. Lysaker, S. Osher, and X.-C. Tai, “Noise removal using smoothed normals and surface fitting,” IEEE Trans. Image Process. 13, 1345–1357 (2004).
[CrossRef]

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[CrossRef]

2000

Y.-L. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Trans. Image Process. 9, 1723–1730 (2000).
[CrossRef]

T. F. Chan, A. Marquina, and P. Mulet, “High-order total variation based image restoration,” SIAM J. Sci. Comput. 22, 503–516 (2000).
[CrossRef]

1985

R. L. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. 12, 252–255 (1985).
[CrossRef]

1984

A. Andersen and A. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imaging 6, 81–94 (1984).
[CrossRef]

1977

A. P. Dempster, N. M. Laired, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Royal Stat. Soc. B 39, 1–38 (1977).

1970

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef]

Andersen, A.

A. Andersen and A. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imaging 6, 81–94 (1984).
[CrossRef]

Bai, J.

J. Bai and X. Feng, “Fractional-order anisotropic diffusion for image denoising,” IEEE Trans. Image Process. 16, 2492–2502 (2007).
[CrossRef]

Bender, R.

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef]

Bovik, A. C.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[CrossRef]

Brenner, D. J.

D. J. Brenner and E. J. Hall, “Computed tomography—an increasing source of radiation exposure,” New Engl. J. Med. 357, 2277–2284 (2007).
[CrossRef]

Candes, E. J.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

Chan, T. F.

T. F. Chan, A. Marquina, and P. Mulet, “High-order total variation based image restoration,” SIAM J. Sci. Comput. 22, 503–516 (2000).
[CrossRef]

Chen, Q.-L.

Y. Zhang, Y.-F. Pu, J.-R. Hu, Y. Liu, Q.-L. Chen, and J.-L. Zhou, “Efficient CT metal artifacts reduction based on fractional-order curvature diffusion,” Comput. Math. Method Med. 2011, 173748 (2011).
[CrossRef]

Chen, Z.

Z. Chen, X. Jin, L. Li, and G. Wang, “A limited-angle CT reconstruction method based on anisotropic TV minimization,” Phys. Med. Biol. 58, 2119–2141 (2013).
[CrossRef]

Dempster, A. P.

A. P. Dempster, N. M. Laired, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Royal Stat. Soc. B 39, 1–38 (1977).

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[CrossRef]

Fan, Y.

Y. Liu, J. Ma, Y. Fan, and Z. Liang, “Adaptive-weighted total variation minimization for sparse data toward low-dose x-ray computed tomography image reconstruction,” Phys. Med. Biol. 57, 7923–7956 (2012).
[CrossRef]

Feng, X.

J. Bai and X. Feng, “Fractional-order anisotropic diffusion for image denoising,” IEEE Trans. Image Process. 16, 2492–2502 (2007).
[CrossRef]

Gordon, R.

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef]

Hall, E. J.

D. J. Brenner and E. J. Hall, “Computed tomography—an increasing source of radiation exposure,” New Engl. J. Med. 357, 2277–2284 (2007).
[CrossRef]

Herman, G. T.

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef]

Hu, J.-R.

Y. Zhang, Y.-F. Pu, J.-R. Hu, and J.-L. Zhou, “A class of fractional-order variational image inpainting models,” Appl. Math. Inform. Sci. 6, 229–306 (2012).

Y. Zhang, Y.-F. Pu, J.-R. Hu, Y. Liu, and J.-L. Zhou, “A new CT metal artifacts reduction algorithm based on fractional-order sinogram inpainting,” J. X-Ray Sci. Technol. 19, 373–384 (2011).

Y. Zhang, Y.-F. Pu, J.-R. Hu, Y. Liu, Q.-L. Chen, and J.-L. Zhou, “Efficient CT metal artifacts reduction based on fractional-order curvature diffusion,” Comput. Math. Method Med. 2011, 173748 (2011).
[CrossRef]

Jia, X.

Z. Tian, X. Jia, K. Yuan, T. Pan, and S. B. Jiang, “Low-dose CT reconstruction via edge-preserving total variation regularization,” Phys. Med. Biol. 56, 5949–5967 (2011).
[CrossRef]

Jiang, S. B.

Z. Tian, X. Jia, K. Yuan, T. Pan, and S. B. Jiang, “Low-dose CT reconstruction via edge-preserving total variation regularization,” Phys. Med. Biol. 56, 5949–5967 (2011).
[CrossRef]

Jin, X.

Z. Chen, X. Jin, L. Li, and G. Wang, “A limited-angle CT reconstruction method based on anisotropic TV minimization,” Phys. Med. Biol. 58, 2119–2141 (2013).
[CrossRef]

Kak, A.

A. Andersen and A. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imaging 6, 81–94 (1984).
[CrossRef]

Kao, C. M.

E. Y. Sidky, C. M. 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).

Kaveh, M.

Y.-L. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Trans. Image Process. 9, 1723–1730 (2000).
[CrossRef]

Laired, N. M.

A. P. Dempster, N. M. Laired, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Royal Stat. Soc. B 39, 1–38 (1977).

Li, L.

Z. Chen, X. Jin, L. Li, and G. Wang, “A limited-angle CT reconstruction method based on anisotropic TV minimization,” Phys. Med. Biol. 58, 2119–2141 (2013).
[CrossRef]

Liang, Z.

Y. Liu, J. Ma, Y. Fan, and Z. Liang, “Adaptive-weighted total variation minimization for sparse data toward low-dose x-ray computed tomography image reconstruction,” Phys. Med. Biol. 57, 7923–7956 (2012).
[CrossRef]

Liu, Y.

Y. Liu, J. Ma, Y. Fan, and Z. Liang, “Adaptive-weighted total variation minimization for sparse data toward low-dose x-ray computed tomography image reconstruction,” Phys. Med. Biol. 57, 7923–7956 (2012).
[CrossRef]

Y. Zhang, Y.-F. Pu, J.-R. Hu, Y. Liu, Q.-L. Chen, and J.-L. Zhou, “Efficient CT metal artifacts reduction based on fractional-order curvature diffusion,” Comput. Math. Method Med. 2011, 173748 (2011).
[CrossRef]

Y. Zhang, Y.-F. Pu, J.-R. Hu, Y. Liu, and J.-L. Zhou, “A new CT metal artifacts reduction algorithm based on fractional-order sinogram inpainting,” J. X-Ray Sci. Technol. 19, 373–384 (2011).

Lysaker, M.

M. Lysaker, S. Osher, and X.-C. Tai, “Noise removal using smoothed normals and surface fitting,” IEEE Trans. Image Process. 13, 1345–1357 (2004).
[CrossRef]

Ma, J.

Y. Liu, J. Ma, Y. Fan, and Z. Liang, “Adaptive-weighted total variation minimization for sparse data toward low-dose x-ray computed tomography image reconstruction,” Phys. Med. Biol. 57, 7923–7956 (2012).
[CrossRef]

Marquina, A.

T. F. Chan, A. Marquina, and P. Mulet, “High-order total variation based image restoration,” SIAM J. Sci. Comput. 22, 503–516 (2000).
[CrossRef]

Mulet, P.

T. F. Chan, A. Marquina, and P. Mulet, “High-order total variation based image restoration,” SIAM J. Sci. Comput. 22, 503–516 (2000).
[CrossRef]

Oldham, K. B.

K. B. Oldham and J. Spanier, The Fractional Calculus (Academic, 1974).

Osher, S.

M. Lysaker, S. Osher, and X.-C. Tai, “Noise removal using smoothed normals and surface fitting,” IEEE Trans. Image Process. 13, 1345–1357 (2004).
[CrossRef]

Pan, T.

Z. Tian, X. Jia, K. Yuan, T. Pan, and S. B. Jiang, “Low-dose CT reconstruction via edge-preserving total variation regularization,” Phys. Med. Biol. 56, 5949–5967 (2011).
[CrossRef]

Pan, X.

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
[CrossRef]

E. Y. Sidky, C. M. 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).

Pu, Y.-F.

Y. Zhang, Y.-F. Pu, J.-R. Hu, and J.-L. Zhou, “A class of fractional-order variational image inpainting models,” Appl. Math. Inform. Sci. 6, 229–306 (2012).

Y. Zhang, Y.-F. Pu, J.-R. Hu, Y. Liu, and J.-L. Zhou, “A new CT metal artifacts reduction algorithm based on fractional-order sinogram inpainting,” J. X-Ray Sci. Technol. 19, 373–384 (2011).

Y. Zhang, Y.-F. Pu, J.-R. Hu, Y. Liu, Q.-L. Chen, and J.-L. Zhou, “Efficient CT metal artifacts reduction based on fractional-order curvature diffusion,” Comput. Math. Method Med. 2011, 173748 (2011).
[CrossRef]

Y.-F. Pu, J.-L. Zhou, and X. Yuan, “Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement,” IEEE Trans. Image Process. 19, 491–511 (2010).
[CrossRef]

Romberg, J.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

Rubin, D. B.

A. P. Dempster, N. M. Laired, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Royal Stat. Soc. B 39, 1–38 (1977).

Sheikh, H. R.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[CrossRef]

Siddon, R. L.

R. L. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. 12, 252–255 (1985).
[CrossRef]

Sidky, E. Y.

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
[CrossRef]

E. Y. Sidky, C. M. 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).

Simoncelli, E. P.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[CrossRef]

Spanier, J.

K. B. Oldham and J. Spanier, The Fractional Calculus (Academic, 1974).

Tai, X.-C.

M. Lysaker, S. Osher, and X.-C. Tai, “Noise removal using smoothed normals and surface fitting,” IEEE Trans. Image Process. 13, 1345–1357 (2004).
[CrossRef]

Tao, T.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

Tian, Z.

Z. Tian, X. Jia, K. Yuan, T. Pan, and S. B. Jiang, “Low-dose CT reconstruction via edge-preserving total variation regularization,” Phys. Med. Biol. 56, 5949–5967 (2011).
[CrossRef]

Wang, G.

Z. Chen, X. Jin, L. Li, and G. Wang, “A limited-angle CT reconstruction method based on anisotropic TV minimization,” Phys. Med. Biol. 58, 2119–2141 (2013).
[CrossRef]

H. Yu and G. Wang, “A soft-threshold filtering approach for reconstruction from a limited number of projections,” Phys. Med. Biol. 55, 3905–3916 (2010).
[CrossRef]

Wang, Z.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[CrossRef]

Wei, Z.

J. Zhang, Z. Wei, and L. Xiao, “Adaptive fractional-order multi-scale method for image denoising,” J. Math. Imaging Vis. 43, 39–49 (2012).
[CrossRef]

J. Zhang and Z. Wei, “A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising,” Appl. Math. Model. 35, 2516–2528 (2011).
[CrossRef]

Xiao, L.

J. Zhang, Z. Wei, and L. Xiao, “Adaptive fractional-order multi-scale method for image denoising,” J. Math. Imaging Vis. 43, 39–49 (2012).
[CrossRef]

You, Y.-L.

Y.-L. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Trans. Image Process. 9, 1723–1730 (2000).
[CrossRef]

Yu, H.

H. Yu and G. Wang, “A soft-threshold filtering approach for reconstruction from a limited number of projections,” Phys. Med. Biol. 55, 3905–3916 (2010).
[CrossRef]

Yuan, K.

Z. Tian, X. Jia, K. Yuan, T. Pan, and S. B. Jiang, “Low-dose CT reconstruction via edge-preserving total variation regularization,” Phys. Med. Biol. 56, 5949–5967 (2011).
[CrossRef]

Yuan, X.

Y.-F. Pu, J.-L. Zhou, and X. Yuan, “Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement,” IEEE Trans. Image Process. 19, 491–511 (2010).
[CrossRef]

Zhang, J.

J. Zhang, Z. Wei, and L. Xiao, “Adaptive fractional-order multi-scale method for image denoising,” J. Math. Imaging Vis. 43, 39–49 (2012).
[CrossRef]

J. Zhang and Z. Wei, “A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising,” Appl. Math. Model. 35, 2516–2528 (2011).
[CrossRef]

Zhang, Y.

Y. Zhang, Y.-F. Pu, J.-R. Hu, and J.-L. Zhou, “A class of fractional-order variational image inpainting models,” Appl. Math. Inform. Sci. 6, 229–306 (2012).

Y. Zhang, Y.-F. Pu, J.-R. Hu, Y. Liu, and J.-L. Zhou, “A new CT metal artifacts reduction algorithm based on fractional-order sinogram inpainting,” J. X-Ray Sci. Technol. 19, 373–384 (2011).

Y. Zhang, Y.-F. Pu, J.-R. Hu, Y. Liu, Q.-L. Chen, and J.-L. Zhou, “Efficient CT metal artifacts reduction based on fractional-order curvature diffusion,” Comput. Math. Method Med. 2011, 173748 (2011).
[CrossRef]

Zhou, J.-L.

Y. Zhang, Y.-F. Pu, J.-R. Hu, and J.-L. Zhou, “A class of fractional-order variational image inpainting models,” Appl. Math. Inform. Sci. 6, 229–306 (2012).

Y. Zhang, Y.-F. Pu, J.-R. Hu, Y. Liu, and J.-L. Zhou, “A new CT metal artifacts reduction algorithm based on fractional-order sinogram inpainting,” J. X-Ray Sci. Technol. 19, 373–384 (2011).

Y. Zhang, Y.-F. Pu, J.-R. Hu, Y. Liu, Q.-L. Chen, and J.-L. Zhou, “Efficient CT metal artifacts reduction based on fractional-order curvature diffusion,” Comput. Math. Method Med. 2011, 173748 (2011).
[CrossRef]

Y.-F. Pu, J.-L. Zhou, and X. Yuan, “Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement,” IEEE Trans. Image Process. 19, 491–511 (2010).
[CrossRef]

Appl. Math. Inform. Sci.

Y. Zhang, Y.-F. Pu, J.-R. Hu, and J.-L. Zhou, “A class of fractional-order variational image inpainting models,” Appl. Math. Inform. Sci. 6, 229–306 (2012).

Appl. Math. Model.

J. Zhang and Z. Wei, “A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising,” Appl. Math. Model. 35, 2516–2528 (2011).
[CrossRef]

Comput. Math. Method Med.

Y. Zhang, Y.-F. Pu, J.-R. Hu, Y. Liu, Q.-L. Chen, and J.-L. Zhou, “Efficient CT metal artifacts reduction based on fractional-order curvature diffusion,” Comput. Math. Method Med. 2011, 173748 (2011).
[CrossRef]

IEEE Trans. Image Process.

Y.-F. Pu, J.-L. Zhou, and X. Yuan, “Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement,” IEEE Trans. Image Process. 19, 491–511 (2010).
[CrossRef]

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[CrossRef]

J. Bai and X. Feng, “Fractional-order anisotropic diffusion for image denoising,” IEEE Trans. Image Process. 16, 2492–2502 (2007).
[CrossRef]

Y.-L. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Trans. Image Process. 9, 1723–1730 (2000).
[CrossRef]

M. Lysaker, S. Osher, and X.-C. Tai, “Noise removal using smoothed normals and surface fitting,” IEEE Trans. Image Process. 13, 1345–1357 (2004).
[CrossRef]

IEEE Trans. Inf. Theory

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[CrossRef]

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Figures (23)

Fig. 1.
Fig. 1.

Position illustration for ux,y.

Fig. 2.
Fig. 2.

Fan-beam CT geometry configuration.

Fig. 3.
Fig. 3.

Fractional-order differentials masks of four directions. (a) Mx+α, (b) Mxα, (c) My+α, and (d) Myα.

Fig. 4.
Fig. 4.

Original images of modified Shepp–Logan and abdomen phantoms.

Fig. 5.
Fig. 5.

Results of the modified S–L phantom without noise added from different methods and projection numbers. The 1st row shows the results from FBP, the 2nd row is from TV-POCS, the 3rd row is from STH, and the last row is from FTV-POCS. The columns from left to right, respectively, indicate 20, 40, and 60 projection-views. The gray scale window is [0, 0.5].

Fig. 6.
Fig. 6.

Results of the modified S–L phantom without noise added from different methods and projection numbers. The 1st row shows the results from TV-POCS, the 2nd row is from STH, and the last row is from FTV-POCS. The columns from left to right indicate 20, 40, and 60 projection views, respectively. The gray scale window is [0.1, 0.3].

Fig. 7.
Fig. 7.

Zoomed parts indicated by the white arrows. The 1st row is from TV-POCS, the 2nd row is from STH, and the 3rd row is from FTV-POCS. The 1st and 2nd columns are from 20 views, the 3rd and 4th columns are from 40 views, and the 5th and 6th columns are from 60 views.

Fig. 8.
Fig. 8.

Horizontal profiles (128th row) of the results reconstructed by different methods from 20 projection views without noise. (a) The complete profiles. (b)–(d) are the enlarged parts in (a), indicated by black boxes.

Fig. 9.
Fig. 9.

Vertical profiles (128th column) of the results reconstructed by different methods from 20 projection views without noise. (a) The complete profiles. (b)–(d) are the enlarged parts in (a), indicated by black boxes.

Fig. 10.
Fig. 10.

Results of the modified S–L phantom with Poisson noise added from different methods and projection numbers. The 1st row shows the results from FBP, the 2nd row is from TV-POCS, the 3rd row is from STH, and the last row is from FTV-POCS. The columns from left to right indicate 20, 40, and 60 projection-views, respectively. The gray scale window is [0, 0.5].

Fig. 11.
Fig. 11.

Results of the modified S–L phantom with Poisson noise added from different methods and projection numbers. The 1st row shows the results from TV-POCS, the 2nd row is from STH, and the last row is from FTV-POCS. The columns from left to right indicate 20, 40, and 60 projection-views, respectively. The gray scale window is [0.1, 0.3].

Fig. 12.
Fig. 12.

Zoomed parts indicated by the white boxes. The 1st row is from TV-POCS, the 2nd row is from STH, and the 3rd row is from FTV-POCS. The 1st column is from 20 views, the 2nd column is from 40 views, and the 3rd column is from 60 views.

Fig. 13.
Fig. 13.

Horizontal profiles (128th row) of the results reconstructed by different methods from 20 projection views with Poisson noise added. (a) The complete profiles. (b)–(d) are the enlarged parts in (a), indicated by black boxes.

Fig. 14.
Fig. 14.

Vertical profiles (128th column) of the results reconstructed by different methods from 20 projection views with Poisson noise added. (a) The complete profiles. (b)–(d) are the enlarged parts in (a), indicated by black boxes.

Fig. 15.
Fig. 15.

Results of the abdomen phantom without noise added from different methods. (a) FBP, (b) TV-POCS, (c) STH, and (d) FTV-POCS. The number of projection-views is 21. The gray scale window is [0,1].

Fig. 16.
Fig. 16.

Results of the abdomen phantom without noise added from different methods. (a) FBP, (b) TV-POCS, (c) STH, and (d) FTV-POCS. The number of projection-views is 39. The gray scale window is [0,1].

Fig. 17.
Fig. 17.

Results of the abdomen phantom with Poisson noise added from different methods. (a) FBP, (b) TV-POCS, (c) STH, and (d) FTV-POCS. The projection-view number is 21. The gray scale window is [0,1].

Fig. 18.
Fig. 18.

Results of the abdomen phantom with Poisson noise added from different methods. (a) FBP, (b) TV-POCS, (c) STH, and (d) FTV-POCS. The projection-view number is 39. The gray scale window is [0,1].

Fig. 19.
Fig. 19.

Two CT images from a chest study. (a) A normal dose dataset. (b) A low-dose dataset. The window level and window width are 50 and 240 HU, respectively.

Fig. 20.
Fig. 20.

Results of few-view reconstruction from 44 noiseless projections over 360°. (a) FBP, (b) TV-POCS, (c) STH, and (d) FTV-POCS. The window level and window width are 50 and 240 HU, respectively.

Fig. 21.
Fig. 21.

Difference images relative to the reference image. (a) FBP, (b) TV-POCS, (c) STH, and (d) FTV-POCS.

Fig. 22.
Fig. 22.

Results of few-view reconstruction from 44 noisy projections over 360°. (a) FBP, (b) TV-POCS, (c) STH, and (d) FTV-POCS. The window level and window width are 50 and 240 HU, respectively.

Fig. 23.
Fig. 23.

Difference images relative to the reference image. (a) FBP, (b) TV-POCS, (c) STH, and (d) FTV-POCS.

Tables (3)

Tables Icon

Table 1. Main Steps of FTV-POCS

Tables Icon

Table 2. Qualitative Evaluation Results of the Noiseless Reconstruction

Tables Icon

Table 3. Qualitative Evaluation Results of the Reconstruction with Noise Added

Equations (22)

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u=(Δxu,Δyu),
Δxu=ux,yux1,y,Δyu=ux,yux,y1.
Au=f,
minuTVsubject tou0,Au=f,
minuFTVsubject tou0,Au=f,
Dαs(x)=limh0+k=0K1(1)kCkαs(xkh)hα,α>0,
Δαs(x)=k=0K1(1)kCkαs(xk).
αu=(Δxαu,Δyαu),
Δxαu=(Δxαu)x,yM,N=k=0K1(1)kCkαuxk,y,Δyαu=(Δyαu)x,yM,N=k=0K1(1)kCkαux,yk,x=1,2,,M,y=1,2,,N.
Δxαuux,yαux1,y+(α(α1)2)ux2,y++(1)n(Γ(α+1)Γ(n+1)Γ(αn+1))uxn,y++(1)K1(Γ(α+1)Γ(K)Γ(αK+2))uxK+1,y,
Δyαuux,yαux,y1+(α(α1)2)ux,y2++(1)n(Γ(α+1)Γ(n+1)Γ(αn+1))ux,yn++(1)K1(Γ(α+1)Γ(K)Γ(αK+2))ux,yK+1.
Δx+αuux,y+αux+1,y(α(α1)2)ux+2,y(1)n(Γ(α+1)Γ(n+1)Γ(αn+1))ux+n,y(1)K1(Γ(α+1)Γ(K)Γ(αK+2))ux+K1,y,
Δy+αuux,y+αux,y+1(α(α1)2)ux,y+2(1)n(Γ(α+1)Γ(n+1)Γ(αn+1))ux,y+n(1)K1(Γ(α+1)Γ(K)Γ(αK+2))ux,y+K1.
{C0α=1C1α=αC2α=α(α1)2Cnα=(1)n(Γ(α+1)Γ(n+1)Γ(αn+1))CK1α=(1)K1(Γ(α+1)Γ(K)Γ(αK+2)).
Δx+αux,y=Mx+α*ux,y,Δxαux,y=Mxα*ux,y,Δy+αux,y=My+α*ux,y,Δy+αux,y=Mx+α*ux,y,
uFTV=((Mx+α*ux,y)2/2+(Mxα*ux,y)2/2+(My+α*ux,y)2/2+(Myα*ux,y)2/2)1/2.
μx,y=((Mx+α*ux,y)2/2+(Mxα*ux,y)2/2+(My+α*ux,y)2/2+(Myα*ux,y)2/2+ε)1/2,
ξx,y=uFTVux,y=Mxα*ux,yμx,yMx+α*ux,yμx,y+Myα*ux,yμx,yMy+α*ux,yμx,yMx+α*ux,yμx+1,y+Mxα*ux,yμx1,yMy+α*ux,yμx,y+1+Myα*ux,yμx,y1.
ujp+1=ujp+ωA+,ji=1IAi,jAi,+(fi0fi(up)),Ai,+=j=1JAi,j,fori=1,,I,A+,j=i=1IAi,j,forj=1,,J,f(up)=Aup,
α=1+curvcurv+|u|,
RMSE=((x,y(fx,yfx,y*)2)/I)1/2,
SSIM(f,f*)=2μfμf*(2σff*+c2)(μf2+μf*2+c1)(σf2+σf*2+c2),

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