February  2020, 14(1): 117-132. doi: 10.3934/ipi.2019066

Nonlocal TV-Gaussian prior for Bayesian inverse problems with applications to limited CT reconstruction

1. 

School of Mathematical Sciences and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

2. 

School of Mathematical Sciences, Tongji University, Shanghai 200082, China

3. 

Department of Mathematical Sciences, University of Liverpool, Liverpool L69 6ZL, United Kingdom

4. 

School of Mathematical Sciences, MOE-LSC, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: Xiaoqun Zhang

Received  April 2019 Revised  August 2019 Published  November 2019

Fund Project: The work is supported by NSFC grants (No. 11771288, 11301337, 91630311) and National key research and development program No. 2017YFB0202902.

Bayesian inference methods have been widely applied in inverse problems due to the ability of uncertainty characterization of the estimation. The prior distribution of the unknown plays an essential role in the Bayesian inference, and a good prior distribution can significantly improve the inference results. In this paper, we propose a hybrid prior distribution on combining the nonlocal total variation regularization (NLTV) and the Gaussian distribution, namely NLTG prior. The advantage of this hybrid prior is two-fold. The proposed prior models both texture and geometric structures present in images through the NLTV. The Gaussian reference measure also provides a flexibility of incorporating structure information from a reference image. Some theoretical properties are established for the hybrid prior. We apply the proposed prior to limited tomography reconstruction problem that is difficult due to severe data missing. Both maximum a posteriori and conditional mean estimates are computed through two efficient methods and the numerical experiments validate the advantages and feasibility of the proposed NLTG prior.

Citation: Didi Lv, Qingping Zhou, Jae Kyu Choi, Jinglai Li, Xiaoqun Zhang. Nonlocal TV-Gaussian prior for Bayesian inverse problems with applications to limited CT reconstruction. Inverse Problems & Imaging, 2020, 14 (1) : 117-132. doi: 10.3934/ipi.2019066
References:
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R. N. Bracewell and A. C. Riddle, Inversion of fan-beam scans in radio astronomy, Astrophysical Journal, 150 (1967).  doi: 10.1086/149346.  Google Scholar

[2]

K. BrediesK. Kunisch and T. Pock, Total generalized variation, SIAM J. Imaging Sci., 3 (2010), 492-526.  doi: 10.1137/090769521.  Google Scholar

[3]

A. BuadesB. Coll and J. M. Morel, A review of image denoising algorithms, with a new one, Multiscale Model. Simul., 4 (2005), 490-530.  doi: 10.1137/040616024.  Google Scholar

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K. ChoiJ. WangL. ZhuT. S. SuhS. Boyd and L. Xing, Compressed sensing based cone-beam computed tomography reconstruction with a first-order method, Med. Phys., 37 (2010), 5113-5125.  doi: 10.1118/1.3481510.  Google Scholar

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F. R. K. Chung, Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, 92, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/cbms/092.  Google Scholar

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S. L. CotterG. O. RobertsA. M. Stuart and D. White, MCMC methods for functions: Modifying old algorithms to make them faster, Statist. Sci., 28 (2013), 424-446.  doi: 10.1214/13-STS421.  Google Scholar

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M. DashtiK. J. H. LawA. M. Stuart and J. Voss, MAP estimators and their consistency in Bayesian nonparametric inverse problems, Inverse Problems, 29 (2013), 27pp.  doi: 10.1088/0266-5611/29/9/095017.  Google Scholar

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A. A. Efros and T. K. Leung, Texture synthesis by non-parametric sampling, IEEE International Conference on Computer Vision, Greece, 1999, 1033-1038. Google Scholar

[10]

A. ElmoatazO. Lezoray and S. Bougleux, Nonlocal discrete regularization on weighted graphs: A framework for image and manifold processing, IEEE Trans. Image Process., 17 (2008), 1047-1060.  doi: 10.1109/TIP.2008.924284.  Google Scholar

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A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari and D. B. Rubin, Bayesian Data Analysis, Texts in Statistical Science Series, CRC Press, Boca Raton, FL, 2014.  Google Scholar

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G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul., 6 (2007), 595-630.  doi: 10.1137/060669358.  Google Scholar

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G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028.  doi: 10.1137/070698592.  Google Scholar

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[21]

M. Lassas and S. Siltanen, Can one use total variation prior for edge-preserving Bayesian inversion?, Inverse Problems, 20 (2004), 1537-1563.  doi: 10.1088/0266-5611/20/5/013.  Google Scholar

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J. Li, A note on the Karhunen-Loève expansions for infinite-dimensional Bayesian inverse problems, Statist. Probab. Lett., 106 (2015), 1-4.  doi: 10.1016/j.spl.2015.06.025.  Google Scholar

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[24]

J. LiuH. DingS. MolloiX. Zhang and H. Gao, TICMR: Total image constrained material reconstruction via nonlocal total variation regularization for spectral CT, IEEE Trans. Medical Imaging, 35 (2016), 2578-2586.  doi: 10.1109/TMI.2016.2587661.  Google Scholar

[25]

Y. LouX. ZhangS. Osher and A. Bertozzi, Image recovery via nonlocal operators, J. Sci. Comput., 42 (2010), 185-197.  doi: 10.1007/s10915-009-9320-2.  Google Scholar

[26]

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F. Natterer, The Mathematics of Computerized Tomography, John Wiley & Sons, Ltd., Chichester, 1986. doi: 10.1137/1.9780898719284.  Google Scholar

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X. PanE. Y. Sidky and M. Vannier, Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?, Inverse Problems, 25 (2009), 36pp.  doi: 10.1088/0266-5611/25/12/123009.  Google Scholar

[29]

G. Peyré, Image processing with nonlocal spectral bases, Multiscale Model. Simul., 7 (2008), 703-730.  doi: 10.1137/07068881X.  Google Scholar

[30]

G. Peyré, S. Bougleux and L. Cohen, Non-local regularization of inverse problems, in ECCV 2008: Computer Vision, Lecture Notes in Computer Science, 5304, Springer, Berlin, Heidelberg, 2008, 57-68. doi: 10.1007/978-3-540-88690-7_5.  Google Scholar

[31]

E. T. Quinto, Singularities of the X-ray transform and limited data tomography in R2 and R3, SIAM J. Math. Anal., 24 (1993), 1215-1225.  doi: 10.1137/0524069.  Google Scholar

[32]

J. Radon, Uber die bestimmug von funktionen durch ihre integralwerte laengs geweisser mannigfaltigkeiten, Berichte Saechsishe Acad. Wissenschaft. Math. Phys., Klass, 69 (1917).   Google Scholar

[33]

L. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[34]

T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods in Banach Spaces, Radon Series on Computational and Applied Mathematics, 10, Walter de Gruyter GmbH & Co. KG, Berlin, 2012. doi: 10.1515/9783110255720.  Google Scholar

[35]

W. P. SegarsG. SturgeonS. MendoncaJ. Grimes and B. M. W. Tsui, 4D XCAT phantom for multimodality imaging research, Med. Phys., 37 (2010), 4902-4915.  doi: 10.1118/1.3480985.  Google Scholar

[36]

E. Y. Sidky and X. Pan, Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization, Phys. Med. Biol., 53 (2008).  doi: 10.1088/0031-9155/53/17/021.  Google Scholar

[37]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559.  doi: 10.1017/S0962492910000061.  Google Scholar

[38]

S. J. Vollmer, Dimension-independent MCMC sampling for inverse problems with non-gaussian priors, SIAM/ASA J. Uncertain. Quantif., 3 (2015), 535-561.  doi: 10.1137/130929904.  Google Scholar

[39]

S. J. Vollmer, Dimension-independent MCMC sampling for inverse problems with non-Gaussian priors, SIAM/ASA J. Uncertain. Quantif., 3 (2015), 535-561.  doi: 10.1137/130929904.  Google Scholar

[40]

G. Wang and H. Yu, The meaning of interior tomography, Phys. Med. Biol., 58 (2013), R161-186.  doi: 10.1088/0031-9155/58/16/R161.  Google Scholar

[41]

J. P. WardM. LeeJ. C. Ye and M. Unser, Interior tomography using 1D generalized total variation. Part Ⅰ: Mathematical foundation, SIAM J. Imaging Sci., 8 (2015), 226-247.  doi: 10.1137/140982428.  Google Scholar

[42]

J. YangH. YuM. Jiang and G. Wang, High-order total variation minimization for interior tomography, Inverse Problems, 26 (2010), 29pp.  doi: 10.1088/0266-5611/26/3/035013.  Google Scholar

[43]

Z. YaoZ. Hu and J. Li, A TV-Gaussian prior for infinite-dimensional Bayesian inverse problems and its numerical implementations, Inverse Problems, 32 (2016), 19pp.  doi: 10.1088/0266-5611/32/7/075006.  Google Scholar

[44]

X. ZhangM. BurgerX. Bresson and S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, SIAM J. Imaging Sci., 3 (2010), 253-276.  doi: 10.1137/090746379.  Google Scholar

[45]

X. Zhang and T. F. Chan, Wavelet inpainting by nonlocal toral variation, Inverse Probl. Imaging, 4 (2010), 191-210.  doi: 10.3934/ipi.2010.4.191.  Google Scholar

[46]

D. Zhou and B. Schölkopf, Regularization on discrete spaces, in Joint Pattern Recognition Symposium, Lecture Notes in Computer Science, 3663, Springer, Berlin, Heidelberg, 2005,361-368. doi: 10.1007/11550518_45.  Google Scholar

[47]

Q. ZhouW. LiuJ. Li and Y. M. Marzouk, An approximate empirical Bayesian method for large scale linear Gaussian inverse problems, Inverse Problems, 34 (2018), 18pp.  doi: 10.1088/1361-6420/aac287.  Google Scholar

show all references

References:
[1]

R. N. Bracewell and A. C. Riddle, Inversion of fan-beam scans in radio astronomy, Astrophysical Journal, 150 (1967).  doi: 10.1086/149346.  Google Scholar

[2]

K. BrediesK. Kunisch and T. Pock, Total generalized variation, SIAM J. Imaging Sci., 3 (2010), 492-526.  doi: 10.1137/090769521.  Google Scholar

[3]

A. BuadesB. Coll and J. M. Morel, A review of image denoising algorithms, with a new one, Multiscale Model. Simul., 4 (2005), 490-530.  doi: 10.1137/040616024.  Google Scholar

[4]

C. Chang and C. Lin, LIBSVM: A library for support vector machines, ACM Transac. Intelligent Systems Technology (TIST), 2 (2011).  doi: 10.1145/1961189.1961199.  Google Scholar

[5]

K. ChoiJ. WangL. ZhuT. S. SuhS. Boyd and L. Xing, Compressed sensing based cone-beam computed tomography reconstruction with a first-order method, Med. Phys., 37 (2010), 5113-5125.  doi: 10.1118/1.3481510.  Google Scholar

[6]

F. R. K. Chung, Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, 92, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/cbms/092.  Google Scholar

[7]

S. L. CotterG. O. RobertsA. M. Stuart and D. White, MCMC methods for functions: Modifying old algorithms to make them faster, Statist. Sci., 28 (2013), 424-446.  doi: 10.1214/13-STS421.  Google Scholar

[8]

M. DashtiK. J. H. LawA. M. Stuart and J. Voss, MAP estimators and their consistency in Bayesian nonparametric inverse problems, Inverse Problems, 29 (2013), 27pp.  doi: 10.1088/0266-5611/29/9/095017.  Google Scholar

[9]

A. A. Efros and T. K. Leung, Texture synthesis by non-parametric sampling, IEEE International Conference on Computer Vision, Greece, 1999, 1033-1038. Google Scholar

[10]

A. ElmoatazO. Lezoray and S. Bougleux, Nonlocal discrete regularization on weighted graphs: A framework for image and manifold processing, IEEE Trans. Image Process., 17 (2008), 1047-1060.  doi: 10.1109/TIP.2008.924284.  Google Scholar

[11]

G. B. Folland, Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1999.  Google Scholar

[12]

A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari and D. B. Rubin, Bayesian Data Analysis, Texts in Statistical Science Series, CRC Press, Boca Raton, FL, 2014.  Google Scholar

[13]

G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul., 6 (2007), 595-630.  doi: 10.1137/060669358.  Google Scholar

[14]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028.  doi: 10.1137/070698592.  Google Scholar

[15]

T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM J. Imaging Sci., 2 (2009), 323-343.  doi: 10.1137/080725891.  Google Scholar

[16]

T. Heuẞer, M. Brehm, S. Marcus, S. Sawall and M. Kachelrieẞ, CT data completion based on prior scans, IEEE Nuclear Science Symposium and Medical Imaging Conference Record (NSS/MIC), Anaheim, CA, 2012, 2969-2976. doi: 10.1109/NSSMIC.2012.6551679.  Google Scholar

[17]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Applied Mathematical Sciences, 160, Springer-Verlag, New York, 2005. doi: 10.1007/b138659.  Google Scholar

[18]

H. KimJ. ChenA. WangC. ChuangM. Held and J. Pouliot, Non-local total-variation NLTV minimization combined with reweighted L1-norm for compressed sensing CT reconstruction, Phys. Med. Biol., 61 (2016).  doi: 10.1088/0031-9155/61/18/6878.  Google Scholar

[19]

S. KindermannS. Osher and P. W. Jones, Deblurring and denoising of images by nonlocal functionals, Multiscale Model. Simul., 4 (2005), 1091-1115.  doi: 10.1137/050622249.  Google Scholar

[20]

E. Klann, A Mumford-Shah-like method for limited data tomography with an application to electron tomography, SIAM J. Imaging Sci., 4 (2011), 1029-1048.  doi: 10.1137/100817371.  Google Scholar

[21]

M. Lassas and S. Siltanen, Can one use total variation prior for edge-preserving Bayesian inversion?, Inverse Problems, 20 (2004), 1537-1563.  doi: 10.1088/0266-5611/20/5/013.  Google Scholar

[22]

J. Li, A note on the Karhunen-Loève expansions for infinite-dimensional Bayesian inverse problems, Statist. Probab. Lett., 106 (2015), 1-4.  doi: 10.1016/j.spl.2015.06.025.  Google Scholar

[23]

J. LiuH. DingS. MolloiX. Zhang and H. Gao, Nonlocal total variation based spectral CT image reconstruction, Med. Phys., 42 (2015), 3570-3570.   Google Scholar

[24]

J. LiuH. DingS. MolloiX. Zhang and H. Gao, TICMR: Total image constrained material reconstruction via nonlocal total variation regularization for spectral CT, IEEE Trans. Medical Imaging, 35 (2016), 2578-2586.  doi: 10.1109/TMI.2016.2587661.  Google Scholar

[25]

Y. LouX. ZhangS. Osher and A. Bertozzi, Image recovery via nonlocal operators, J. Sci. Comput., 42 (2010), 185-197.  doi: 10.1007/s10915-009-9320-2.  Google Scholar

[26]

F. LuckaS. PursiainenM. Burger and C. H. Wolters, Hierarchical Bayesian inference for the EEG inverse problem using realistic FE head models: Depth localization and source separation for focal primary currents, Neuroimage, 61 (2012), 1364-1382.  doi: 10.1016/j.neuroimage.2012.04.017.  Google Scholar

[27]

F. Natterer, The Mathematics of Computerized Tomography, John Wiley & Sons, Ltd., Chichester, 1986. doi: 10.1137/1.9780898719284.  Google Scholar

[28]

X. PanE. Y. Sidky and M. Vannier, Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?, Inverse Problems, 25 (2009), 36pp.  doi: 10.1088/0266-5611/25/12/123009.  Google Scholar

[29]

G. Peyré, Image processing with nonlocal spectral bases, Multiscale Model. Simul., 7 (2008), 703-730.  doi: 10.1137/07068881X.  Google Scholar

[30]

G. Peyré, S. Bougleux and L. Cohen, Non-local regularization of inverse problems, in ECCV 2008: Computer Vision, Lecture Notes in Computer Science, 5304, Springer, Berlin, Heidelberg, 2008, 57-68. doi: 10.1007/978-3-540-88690-7_5.  Google Scholar

[31]

E. T. Quinto, Singularities of the X-ray transform and limited data tomography in R2 and R3, SIAM J. Math. Anal., 24 (1993), 1215-1225.  doi: 10.1137/0524069.  Google Scholar

[32]

J. Radon, Uber die bestimmug von funktionen durch ihre integralwerte laengs geweisser mannigfaltigkeiten, Berichte Saechsishe Acad. Wissenschaft. Math. Phys., Klass, 69 (1917).   Google Scholar

[33]

L. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[34]

T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods in Banach Spaces, Radon Series on Computational and Applied Mathematics, 10, Walter de Gruyter GmbH & Co. KG, Berlin, 2012. doi: 10.1515/9783110255720.  Google Scholar

[35]

W. P. SegarsG. SturgeonS. MendoncaJ. Grimes and B. M. W. Tsui, 4D XCAT phantom for multimodality imaging research, Med. Phys., 37 (2010), 4902-4915.  doi: 10.1118/1.3480985.  Google Scholar

[36]

E. Y. Sidky and X. Pan, Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization, Phys. Med. Biol., 53 (2008).  doi: 10.1088/0031-9155/53/17/021.  Google Scholar

[37]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559.  doi: 10.1017/S0962492910000061.  Google Scholar

[38]

S. J. Vollmer, Dimension-independent MCMC sampling for inverse problems with non-gaussian priors, SIAM/ASA J. Uncertain. Quantif., 3 (2015), 535-561.  doi: 10.1137/130929904.  Google Scholar

[39]

S. J. Vollmer, Dimension-independent MCMC sampling for inverse problems with non-Gaussian priors, SIAM/ASA J. Uncertain. Quantif., 3 (2015), 535-561.  doi: 10.1137/130929904.  Google Scholar

[40]

G. Wang and H. Yu, The meaning of interior tomography, Phys. Med. Biol., 58 (2013), R161-186.  doi: 10.1088/0031-9155/58/16/R161.  Google Scholar

[41]

J. P. WardM. LeeJ. C. Ye and M. Unser, Interior tomography using 1D generalized total variation. Part Ⅰ: Mathematical foundation, SIAM J. Imaging Sci., 8 (2015), 226-247.  doi: 10.1137/140982428.  Google Scholar

[42]

J. YangH. YuM. Jiang and G. Wang, High-order total variation minimization for interior tomography, Inverse Problems, 26 (2010), 29pp.  doi: 10.1088/0266-5611/26/3/035013.  Google Scholar

[43]

Z. YaoZ. Hu and J. Li, A TV-Gaussian prior for infinite-dimensional Bayesian inverse problems and its numerical implementations, Inverse Problems, 32 (2016), 19pp.  doi: 10.1088/0266-5611/32/7/075006.  Google Scholar

[44]

X. ZhangM. BurgerX. Bresson and S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, SIAM J. Imaging Sci., 3 (2010), 253-276.  doi: 10.1137/090746379.  Google Scholar

[45]

X. Zhang and T. F. Chan, Wavelet inpainting by nonlocal toral variation, Inverse Probl. Imaging, 4 (2010), 191-210.  doi: 10.3934/ipi.2010.4.191.  Google Scholar

[46]

D. Zhou and B. Schölkopf, Regularization on discrete spaces, in Joint Pattern Recognition Symposium, Lecture Notes in Computer Science, 3663, Springer, Berlin, Heidelberg, 2005,361-368. doi: 10.1007/11550518_45.  Google Scholar

[47]

Q. ZhouW. LiuJ. Li and Y. M. Marzouk, An approximate empirical Bayesian method for large scale linear Gaussian inverse problems, Inverse Problems, 34 (2018), 18pp.  doi: 10.1088/1361-6420/aac287.  Google Scholar

Figure 1.  XCAT images: origin, ground truth and reference with different noise levels
Figure 2.  MAP results with sinogram noise level $ 5 $
Figure 3.  MAP results with sinogram noise level $ 20 $
Figure 4.  CM results with different sinogram data noise levels and references images. Upper: NLTG; Lower: TG
Figure 5.  The 95% confidence interval for different sinogram data noise level and references images. The range of the values is from 0 (black) to 100 (whitest). Upper: NLTG; Lower: TG
Table 1.  MAP results: PSNR and SSIM for different sinogram noise levels and reference images
Noise Ref. FBP TV TGV TG NLTV NLTG
5 $ u_\mathrm{ref}^1 $ 13.30/0.21 18.98/0.60 21.21/0.71 29.08/0.88 29.69/0.87 30.71/0.91
$ u_\mathrm{ref}^2 $ 28.22/0.87 28.42/0.84 28.88/0.86
20 $ u_\mathrm{ref}^1 $ 9.40/0.06 15.66/0.46 18.26/0.48 23.10/0.54 23.92/0.78 24.72/0.79
$ u_\mathrm{ref}^2 $ 22.51/0.49 23.13/0.75 23.63/0.74
Noise Ref. FBP TV TGV TG NLTV NLTG
5 $ u_\mathrm{ref}^1 $ 13.30/0.21 18.98/0.60 21.21/0.71 29.08/0.88 29.69/0.87 30.71/0.91
$ u_\mathrm{ref}^2 $ 28.22/0.87 28.42/0.84 28.88/0.86
20 $ u_\mathrm{ref}^1 $ 9.40/0.06 15.66/0.46 18.26/0.48 23.10/0.54 23.92/0.78 24.72/0.79
$ u_\mathrm{ref}^2 $ 22.51/0.49 23.13/0.75 23.63/0.74
Table 2.  CM results: SSIM and PSNR for different level of Sinogram noise and reference
PSNR SSIM
Noise Ref. NLTG TG NLTG TG
5 $ u_\mathrm{ref}^1 $ 27.73 21.44 0.80 0.46
$ u_\mathrm{ref}^2 $ 27.90 20.95 0.66 0.40
20 $ u_\mathrm{ref}^1 $ 25.97 19.58 0.62 0.37
$ u_\mathrm{ref}^2 $ 25.71 18.95 0.59 0.31
PSNR SSIM
Noise Ref. NLTG TG NLTG TG
5 $ u_\mathrm{ref}^1 $ 27.73 21.44 0.80 0.46
$ u_\mathrm{ref}^2 $ 27.90 20.95 0.66 0.40
20 $ u_\mathrm{ref}^1 $ 25.97 19.58 0.62 0.37
$ u_\mathrm{ref}^2 $ 25.71 18.95 0.59 0.31
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