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

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[13]

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

[14]

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

[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.

[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.

[17]

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

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[27]

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

[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.

[29]

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

[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.

[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.

[32]

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

[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.

[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.

[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.

[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.

[37]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[13]

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

[14]

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

[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.

[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.

[17]

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

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[27]

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

[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.

[29]

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

[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.

[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.

[32]

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

[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.

[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.

[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.

[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.

[37]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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
[1]

Xiaoqun Zhang, Tony F. Chan. Wavelet inpainting by nonlocal total variation. Inverse Problems and Imaging, 2010, 4 (1) : 191-210. doi: 10.3934/ipi.2010.4.191

[2]

Víctor Almeida, Jorge J. Betancor. Variation and oscillation for harmonic operators in the inverse Gaussian setting. Communications on Pure and Applied Analysis, 2022, 21 (2) : 419-470. doi: 10.3934/cpaa.2021183

[3]

Adriana González, Laurent Jacques, Christophe De Vleeschouwer, Philippe Antoine. Compressive optical deflectometric tomography: A constrained total-variation minimization approach. Inverse Problems and Imaging, 2014, 8 (2) : 421-457. doi: 10.3934/ipi.2014.8.421

[4]

Masoumeh Dashti, Stephen Harris, Andrew Stuart. Besov priors for Bayesian inverse problems. Inverse Problems and Imaging, 2012, 6 (2) : 183-200. doi: 10.3934/ipi.2012.6.183

[5]

Haijuan Hu, Jacques Froment, Baoyan Wang, Xiequan Fan. Spatial-Frequency domain nonlocal total variation for image denoising. Inverse Problems and Imaging, 2020, 14 (6) : 1157-1184. doi: 10.3934/ipi.2020059

[6]

Guillaume Bal, Ian Langmore, Youssef Marzouk. Bayesian inverse problems with Monte Carlo forward models. Inverse Problems and Imaging, 2013, 7 (1) : 81-105. doi: 10.3934/ipi.2013.7.81

[7]

Raymond H. Chan, Haixia Liang, Suhua Wei, Mila Nikolova, Xue-Cheng Tai. High-order total variation regularization approach for axially symmetric object tomography from a single radiograph. Inverse Problems and Imaging, 2015, 9 (1) : 55-77. doi: 10.3934/ipi.2015.9.55

[8]

Johnathan M. Bardsley. Gaussian Markov random field priors for inverse problems. Inverse Problems and Imaging, 2013, 7 (2) : 397-416. doi: 10.3934/ipi.2013.7.397

[9]

Tan Bui-Thanh, Omar Ghattas. A scalable algorithm for MAP estimators in Bayesian inverse problems with Besov priors. Inverse Problems and Imaging, 2015, 9 (1) : 27-53. doi: 10.3934/ipi.2015.9.27

[10]

Kui Lin, Shuai Lu, Peter Mathé. Oracle-type posterior contraction rates in Bayesian inverse problems. Inverse Problems and Imaging, 2015, 9 (3) : 895-915. doi: 10.3934/ipi.2015.9.895

[11]

Andrew J. Majda, Michal Branicki. Lessons in uncertainty quantification for turbulent dynamical systems. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3133-3221. doi: 10.3934/dcds.2012.32.3133

[12]

Jing Li, Panos Stinis. Mori-Zwanzig reduced models for uncertainty quantification. Journal of Computational Dynamics, 2019, 6 (1) : 39-68. doi: 10.3934/jcd.2019002

[13]

H. T. Banks, Robert Baraldi, Karissa Cross, Kevin Flores, Christina McChesney, Laura Poag, Emma Thorpe. Uncertainty quantification in modeling HIV viral mechanics. Mathematical Biosciences & Engineering, 2015, 12 (5) : 937-964. doi: 10.3934/mbe.2015.12.937

[14]

Alex Capaldi, Samuel Behrend, Benjamin Berman, Jason Smith, Justin Wright, Alun L. Lloyd. Parameter estimation and uncertainty quantification for an epidemic model. Mathematical Biosciences & Engineering, 2012, 9 (3) : 553-576. doi: 10.3934/mbe.2012.9.553

[15]

Ryan Bennink, Ajay Jasra, Kody J. H. Law, Pavel Lougovski. Estimation and uncertainty quantification for the output from quantum simulators. Foundations of Data Science, 2019, 1 (2) : 157-176. doi: 10.3934/fods.2019007

[16]

Richard Archibald, Feng Bao, Yanzhao Cao, He Zhang. A backward SDE method for uncertainty quantification in deep learning. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022062

[17]

Michael Herty, Elisa Iacomini. Uncertainty quantification in hierarchical vehicular flow models. Kinetic and Related Models, 2022, 15 (2) : 239-256. doi: 10.3934/krm.2022006

[18]

You-Wei Wen, Raymond Honfu Chan. Using generalized cross validation to select regularization parameter for total variation regularization problems. Inverse Problems and Imaging, 2018, 12 (5) : 1103-1120. doi: 10.3934/ipi.2018046

[19]

Barbara Brandolini, Francesco Chiacchio, Cristina Trombetti. Hardy type inequalities and Gaussian measure. Communications on Pure and Applied Analysis, 2007, 6 (2) : 411-428. doi: 10.3934/cpaa.2007.6.411

[20]

Junxiong Jia, Jigen Peng, Jinghuai Gao. Posterior contraction for empirical bayesian approach to inverse problems under non-diagonal assumption. Inverse Problems and Imaging, 2021, 15 (2) : 201-228. doi: 10.3934/ipi.2020061

2020 Impact Factor: 1.639

Metrics

  • PDF downloads (542)
  • HTML views (414)
  • Cited by (1)

[Back to Top]