Abstract

In order to reduce the radiation dose of the X-ray computed tomography (CT), low-dose CT has drawn much attention in both clinical and industrial fields. A fractional order model based on statistical iterative reconstruction framework was proposed in this study. To further enhance the performance of the proposed model, an adaptive order selection strategy, determining the fractional order pixel-by-pixel, was given. Experiments, including numerical and clinical cases, illustrated better results than several existing methods, especially, in structure and texture preservation.

© 2016 Optical Society of America

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References

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2014 (1)

2013 (2)

R. H. Chan, A. Lanza, S. Morigi, and F. Sgallari, “An adaptive strategy for the restoration of textured images using fractional order regularization,” Numer. Math. Theor. Meth. Appl. 6(1), 276–296 (2013).

Y. Zhang, W.-H. Zhang, H. Chen, M.-L. Yang, T.-Y. Li, and J.-L. Zhou, “Few-view image reconstruction combining total variation and a high-order norm,” Int. J. Imaging Syst. Technol. 23(3), 249–255 (2013).
[Crossref]

2012 (7)

J. Zhang, Z. Wei, and L. Xiao, “Adaptive fractional-order multiscale method for image denoising,” J. Math. Imaging Vis. 43(1), 39–49 (2012).
[Crossref]

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(23), 7923–7956 (2012).
[Crossref] [PubMed]

Y. Zhang, Y.-F. Pu, J.-R. Hu, and J.-L. Zhou, “A class of fractional order variational image inpainting models,” Appl. Math. Inf. Sci. 6(2), 229–306 (2012).

S. Tang and X. Tang, “Statistical CT noise reduction with multiscale decomposition and penalized weighted least squares in the projection domain,” Med. Phys. 39(9), 5498–5512 (2012).
[Crossref] [PubMed]

Q. Xu, H. Yu, J. Bennett, P. He, R. Zainon, R. Doesburg, A. Opie, M. Walsh, H. Shen, A. Butler, P. Butler, X. Mou, and G. Wang, “Image reconstruction for hybrid true-color micro-CT,” IEEE Trans. Biomed. Eng. 59(6), 1711–1719 (2012).
[Crossref] [PubMed]

E. A. Rashed and H. Kudo, “Statistical image reconstruction from limited projection data with intensity priors,” Phys. Med. Biol. 57(7), 2039–2061 (2012).
[Crossref] [PubMed]

X. Jia, Z. Tian, Y. Lou, J.-J. Sonke, and S. B. Jiang, “Four-dimensional cone beam CT reconstruction and enhancement using a temporal nonlocal means method,” Med. Phys. 39(9), 5592–5602 (2012).
[Crossref] [PubMed]

2011 (7)

Z. Tian, X. Jia, B. Dong, Y. Lou, and S. B. Jiang, “Low-dose 4DCT reconstruction via temporal nonlocal means,” Med. Phys. 38(3), 1359–1365 (2011).
[Crossref] [PubMed]

A. Chambolle and T. Pock, “A first-order primal-dual algorithm for convex problems with applications to imaging,” J. Math. Imaging Vis. 40(1), 120–145 (2011).
[Crossref]

Q. Xu, X. Mou, G. Wang, J. Sieren, E. A. Hoffman, and H. Yu, “Statistical interior tomography,” IEEE Trans. Med. Imaging 30(5), 1116–1128 (2011).
[Crossref] [PubMed]

L. Ritschl, F. Bergner, C. Fleischmann, and M. Kachelriess, “Improved total variation-based CT image reconstruction applied to clinical data,” Phys. Med. Biol. 56(6), 1545–1561 (2011).
[Crossref] [PubMed]

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(18), 5949–5967 (2011).
[Crossref] [PubMed]

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. XRay Sci. Technol. 19(3), 373–384 (2011).
[PubMed]

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

2010 (3)

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(5), 3516–3528 (2010).

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(2), 491–511 (2010).
[Crossref] [PubMed]

F. Li, M. K. Ng, and C. Shen, “Multiplicative noise removal with spatially varying regularization parameters,” SIAM J. Imaging Sci. 3(1), 1–20 (2010).
[Crossref]

2009 (1)

A. Manduca, L. Yu, J. D. Trzasko, N. Khaylova, J. M. Kofler, C. M. McCollough, and J. G. Fletcher, “Projection space denoising with bilateral filtering and CT noise modeling for dose reduction in CT,” Med. Phys. 36(11), 4911–4919 (2009).
[Crossref] [PubMed]

2008 (2)

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53(17), 4777–4807 (2008).
[Crossref] [PubMed]

Y. Pu, W. Wang, J. Zhou, H. Jia, and Y. Wang, “Fractional derivative detection of digital image texture details and implementation of fractional derivative filter,” Sci. in China Series F: Information Sci. 51(9), 1319–1339 (2008).
[Crossref]

2007 (3)

J. Bai and X. C. Feng, “Fractional-order anisotropic diffusion for image denoising,” IEEE Trans. Image Process. 16(10), 2492–2502 (2007).
[Crossref] [PubMed]

D. J. Brenner and E. J. Hall, “Computed tomography--an increasing source of radiation exposure,” N. Engl. J. Med. 357(22), 2277–2284 (2007).
[Crossref] [PubMed]

J.-B. Thibault, K. D. Sauer, C. A. Bouman, and J. Hsieh, “A three-dimensional statistical approach to improved image quality for multislice helical CT,” Med. Phys. 34(11), 4526–4544 (2007).
[Crossref] [PubMed]

2006 (6)

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (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. Tech. (Paris) 14(2), 119–139 (2006).

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(10), 1272–1283 (2006).
[Crossref] [PubMed]

M. Ortigueira, “A coherent approach to non integer order derivatives,” Signal Process. 86(10), 2505–2515 (2006).
[Crossref]

G. Gilboa, N. Sochen, and Y. Y. Zeevi, “Variational denoising of partly textured images by spatially varying constraints,” IEEE Trans. Image Process. 15(8), 2281–2289 (2006).
[Crossref] [PubMed]

2005 (1)

P. J. La Rivière, “Penalized-likelihood sinogram smoothing for low-dose CT,” Med. Phys. 32(6), 1676–1683 (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(5), 2505–2513 (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(4), 600–612 (2004).
[Crossref] [PubMed]

2003 (1)

M. Lysaker, A. Lundervold, and X.-C. Tai, “Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time,” IEEE Trans. Image Process. 12(12), 1579–1590 (2003).
[Crossref] [PubMed]

2002 (1)

I. A. Elbakri and J. A. Fessler, “Statistical image reconstruction for polyenergetic X-ray computed tomography,” IEEE Trans. Med. Imaging 21(2), 89–99 (2002).
[Crossref] [PubMed]

2000 (2)

Y.-L. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Trans. Image Process. 9(10), 1723–1730 (2000).
[Crossref] [PubMed]

T. F. Chan, A. Marquina, and P. Mulet, “High-order total variation based image restoration,” SIAM J. Sci. Comput. 22(2), 503–516 (2000).
[Crossref]

1993 (1)

C. Bouman and K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2(3), 296–310 (1993).
[Crossref] [PubMed]

1992 (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1), 259–268 (1992).
[Crossref]

1985 (1)

R. L. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. 12(2), 252–255 (1985).
[Crossref] [PubMed]

1977 (1)

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

1970 (1)

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(3), 471–481 (1970).
[Crossref] [PubMed]

Bai, J.

J. Bai and X. C. Feng, “Fractional-order anisotropic diffusion for image denoising,” IEEE Trans. Image Process. 16(10), 2492–2502 (2007).
[Crossref] [PubMed]

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(3), 471–481 (1970).
[Crossref] [PubMed]

Bennett, J.

Q. Xu, H. Yu, J. Bennett, P. He, R. Zainon, R. Doesburg, A. Opie, M. Walsh, H. Shen, A. Butler, P. Butler, X. Mou, and G. Wang, “Image reconstruction for hybrid true-color micro-CT,” IEEE Trans. Biomed. Eng. 59(6), 1711–1719 (2012).
[Crossref] [PubMed]

Bergner, F.

L. Ritschl, F. Bergner, C. Fleischmann, and M. Kachelriess, “Improved total variation-based CT image reconstruction applied to clinical data,” Phys. Med. Biol. 56(6), 1545–1561 (2011).
[Crossref] [PubMed]

Bouman, C.

C. Bouman and K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2(3), 296–310 (1993).
[Crossref] [PubMed]

Bouman, C. A.

J.-B. Thibault, K. D. Sauer, C. A. Bouman, and J. Hsieh, “A three-dimensional statistical approach to improved image quality for multislice helical CT,” Med. Phys. 34(11), 4526–4544 (2007).
[Crossref] [PubMed]

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(4), 600–612 (2004).
[Crossref] [PubMed]

Brenner, D. J.

D. J. Brenner and E. J. Hall, “Computed tomography--an increasing source of radiation exposure,” N. Engl. J. Med. 357(22), 2277–2284 (2007).
[Crossref] [PubMed]

Butler, A.

Q. Xu, H. Yu, J. Bennett, P. He, R. Zainon, R. Doesburg, A. Opie, M. Walsh, H. Shen, A. Butler, P. Butler, X. Mou, and G. Wang, “Image reconstruction for hybrid true-color micro-CT,” IEEE Trans. Biomed. Eng. 59(6), 1711–1719 (2012).
[Crossref] [PubMed]

Butler, P.

Q. Xu, H. Yu, J. Bennett, P. He, R. Zainon, R. Doesburg, A. Opie, M. Walsh, H. Shen, A. Butler, P. Butler, X. Mou, and G. Wang, “Image reconstruction for hybrid true-color micro-CT,” IEEE Trans. Biomed. Eng. 59(6), 1711–1719 (2012).
[Crossref] [PubMed]

Candès, E. J.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Chambolle, A.

A. Chambolle and T. Pock, “A first-order primal-dual algorithm for convex problems with applications to imaging,” J. Math. Imaging Vis. 40(1), 120–145 (2011).
[Crossref]

Chan, R. H.

R. H. Chan, A. Lanza, S. Morigi, and F. Sgallari, “An adaptive strategy for the restoration of textured images using fractional order regularization,” Numer. Math. Theor. Meth. Appl. 6(1), 276–296 (2013).

Chan, T. F.

T. F. Chan, A. Marquina, and P. Mulet, “High-order total variation based image restoration,” SIAM J. Sci. Comput. 22(2), 503–516 (2000).
[Crossref]

Chen, H.

Y. Zhang, W.-H. Zhang, H. Chen, M.-L. Yang, T.-Y. Li, and J.-L. Zhou, “Few-view image reconstruction combining total variation and a high-order norm,” Int. J. Imaging Syst. Technol. 23(3), 249–255 (2013).
[Crossref]

Chen, Q.-L.

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

Dempster, A. P.

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

Doesburg, R.

Q. Xu, H. Yu, J. Bennett, P. He, R. Zainon, R. Doesburg, A. Opie, M. Walsh, H. Shen, A. Butler, P. Butler, X. Mou, and G. Wang, “Image reconstruction for hybrid true-color micro-CT,” IEEE Trans. Biomed. Eng. 59(6), 1711–1719 (2012).
[Crossref] [PubMed]

Dong, B.

Z. Tian, X. Jia, B. Dong, Y. Lou, and S. B. Jiang, “Low-dose 4DCT reconstruction via temporal nonlocal means,” Med. Phys. 38(3), 1359–1365 (2011).
[Crossref] [PubMed]

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

Elbakri, I. A.

I. A. Elbakri and J. A. Fessler, “Statistical image reconstruction for polyenergetic X-ray computed tomography,” IEEE Trans. Med. Imaging 21(2), 89–99 (2002).
[Crossref] [PubMed]

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(23), 7923–7956 (2012).
[Crossref] [PubMed]

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1), 259–268 (1992).
[Crossref]

Feng, X. C.

J. Bai and X. C. Feng, “Fractional-order anisotropic diffusion for image denoising,” IEEE Trans. Image Process. 16(10), 2492–2502 (2007).
[Crossref] [PubMed]

Fessler, J. A.

I. A. Elbakri and J. A. Fessler, “Statistical image reconstruction for polyenergetic X-ray computed tomography,” IEEE Trans. Med. Imaging 21(2), 89–99 (2002).
[Crossref] [PubMed]

Fleischmann, C.

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Y. Zhang, Y.-F. Pu, J.-R. Hu, Y. Liu, Q.-L. Chen, and J.-L. Zhou, “Efficient CT metal artifact reduction based on fractional-order curvature diffusion,” Comput. Math. Methods Med. 2011, 173748 (2011).
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Figures (9)

Fig. 1
Fig. 1 Fractional-order derivative masks of (a) x and (b) y directions.
Fig. 2
Fig. 2 Reconstructed FORBILD abdomen phantom image for comparison. (a) Reference, (b) FBP, (c) TV-SIR, (d) EPTV, and (e) AFTV-SIR. All the images are displayed with the same window.
Fig. 3
Fig. 3 Performance comparison of four algorithms on the reconstruction of detailed ROIs marked in Fig. 2(a) with four different indexes. The corresponding algorithms are illustrated in figure legend.
Fig. 4
Fig. 4 Comparison of the target profile 1 marked in Fig. 2(a) for different methods.
Fig. 5
Fig. 5 Comparison of the target profile 2 marked in Fig. 2(a) for different methods.
Fig. 6
Fig. 6 Reconstructed patient image for comparison. (a) Reference, (b) FBP, (c) TV-SIR, (d) EPTV, and (e) AFTV-SIR. All the images are displayed with the same window.
Fig. 7
Fig. 7 Performance comparison of four algorithms on the reconstruction of detailed ROIs marked in Fig. 5(a) with four different indexes. The corresponding algorithms are illustrated in figure legend.
Fig. 8
Fig. 8 Comparison of the target profile 1 marked in Fig. 5(a) for different methods.
Fig. 9
Fig. 9 Comparison of the target profile 2 marked in Fig. 5(a) for different methods.

Tables (1)

Tables Icon

Table 1 Main steps of AFTV-SIR

Equations (21)

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I= I 0 exp( l μdl ),
Y i Poisson( b i exp( j=1 J a ij μ j )+ r i ),i=1,2,...,I,j=1,2,...,J,
L(μ)= i=1 I ( b i exp( j=1 J a ij μ j )+ r i Y i log( b i exp( j=1 J a ij μ j )+ r i ) ) .
P(μ|Y)= P(Y|μ)P(μ) P(Y) .
μ * =arg min μ0 E(μ)=arg min μ0 { L(μ)-logP(μ) }.
E(μ)= i=1 I w i 2 ( j=1 J a ij μ j l ^ i ) 2 +R(μ),
min μ0 i=1 I w i 2 ( j=1 J a ij μ j l ^ i ) 2 +λ μ AFTV ,
μ j t = μ j t1 i=1 I ( a ij w i ( j=1 J a ij μ j t1 l ^ i ) ) +λ μ AFTV μ j | μ= μ t1 i=1 I a ij 2 w i γ ij +λ 2 μ AFTV μ j 2 | μ= μ t1 ,
( α m,n u ) m,n =( ( Δ x α m,n u) m,n , ( Δ y α m,n u) m,n ),
( Δ x α m,n u ) m,n = k=0 K1 C k α m,n u mk,n , ( Δ y α m,n u ) m,n = k=0 K1 C k α m,n u m,nk , m=1,2,,M,n=1,2,,N, α m,n R + ,
Δ x α m,n u m,n = M x α m,n u m,n , Δ y α m,n u m,n = M y α m,n u m,n ,
μ AFTV = ( ( Δ x α j μ j ) 2 + ( Δ y α j μ j ) 2 ) 1/2 1 .
μ AFTV μ j = υ j μ j ,
υ j μ j = υ m,n μ m,n = M x α m,n μ m,n + M y α m,n μ m,n ξ m,n α m,n M x α m+1,n μ m+1,n ξ m+1,n α m+1,n M y α m,n+1 μ m,n+1 ξ m,n+1 α m,n+1 .
2 μ AFTV μ j 2 = 2 υ j μ j 2 ,
2 υ j μ j 2 = 2 υ m,n μ m,n 2 = 2 ( ξ m,n α m,n ) 2 ( M x α m,n μ m,n + M y α m,n μ m,n ) 2 ( ξ m,n α m,n ) 3 + ( ξ m+1,n α m+1,n ) 2 ( M x α m+1,n μ m+1,n ) 2 ( ξ m+1,n α m+1,n ) 3 + ( ξ m,n+1 α m,n+1 ) 2 ( M y α m,n+1 μ m,n+1 ) 2 ( ξ m,n+1 α m,n+1 ) 3 .
α={ 1+ 2 5π arctan( P v TV min( P v TV )),if P v TV < σ ^ 2 1.2+ 4 5π arctan( P v TV σ ^ 2 ),if P v TV σ ^ 2 ,
RMSE= (( j ( μ j μ j * ) 2 )/J) 1/2 ,
PSNR=10 log 10 ( max( μ j ) ) 2 j ( μ j μ j * ) /J .
CC= j ( μ j μ ¯ )( μ j * μ ¯ * ) j ( μ j μ ¯ ) 2 j ( μ j * μ ¯ * ) 2 ,
SSIM(μ, μ * )= 2 μ ¯ μ ¯ * (2 σ μ μ * + c 2 ) ( μ ¯ 2 + μ ¯ * 2 + c 1 )( σ μ 2 + σ μ * 2 + c 2 ) ,

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