American Institute of Mathematical Sciences

Advanced
June  2020, 14(3): 401-421. doi: 10.3934/ipi.2020019

$ \chi^2 $ test for total variation regularization parameter selection

J. Mead

Department of Mathematics, Boise State University, Boise, ID, USA

Received  March 2019 Revised  January 2020 Published  March 2020

Fund Project: This work was funded by National Science Foundation, DMS10431047

Total Variation (TV) is an effective method of removing noise in digital image processing while preserving edges. The scaling or regularization parameter in the TV process defines the amount of denoising, with a value of zero giving a result equivalent to the input signal. The discrepancy principle is a classical method for regularization parameter selection whereby data is fit to a specified tolerance. The tolerance is often identified based on the fact that the least squares data fit is known to follow a $ \chi^2 $ distribution. However, this approach fails when the number of parameters is greater than or equal to the number of data. Typically, heuristics are employed to identify the tolerance in the discrepancy principle and this leads to oversmoothing. In this work we identify a $ \chi^2 $ test for TV regularization parameter selection assuming the blurring matrix is full rank. In particular, we prove that the degrees of freedom in the TV regularized residual is the number of data and this is used to identify the appropriate tolerance. The importance of this work lies in the fact that the $ \chi^2 $ test introduced here for TV automates the choice of regularization parameter selection and can straightforwardly be incorporated into any TV algorithm. Results are given for three test images and compared to results using the discrepancy principle and MAP estimates.

Keywords: Regularization parameter, total variation, chi-squared test, discrepancy principle, regression.
Mathematics Subject Classification: 65F22, 62J07, 94A08.
Citation: J. Mead. $ \chi^2 $ test for total variation regularization parameter selection. Inverse Problems and Imaging, 2020, 14 (3) : 401-421. doi: 10.3934/ipi.2020019
References:
[1]

A. Ali and R. J. Tibshirani, The generalized lasso problem and uniqueness, Electron. J. Stat., 13 (2019), 2307-2347.  doi: 10.1214/19-EJS1569.

[2] R. C. Aster, B. Borchers and C. H. Thurber, Parameter Estimation and Inverse Problems, 2$^{nd}$ edition, Elsevier/Academic Press, Amsterdam, 2013.  doi: 10.1016/B978-0-12-385048-5.00001-X.
[3]

S. D. Babacan, R. Molina and A. K. Katsaggelos, Variational Bayesian blind deconvolution using a total variation prior, IEEE Trans. Image Process., 18 (2009), 12-26.  doi: 10.1109/TIP.2008.2007354.

[4]

T. Blu and F. Luisier, The SURE-LET approach to image denoising, IEEE Trans. Image Process., 16 (2007), 2778-2786.  doi: 10.1109/TIP.2007.906002.

[5]

G. Casella and R. L. Berger, Statistical Inference, 2$^{nd}$ edition Duxbury, 2001.

[6]

J. M. Bioucas-Dias, M. A. T. Figueiredo and J. P. Oliveira, Adaptive total variation image deconvolution: A majorization-minimization approach, 14$^{th }$ European Signal Processing Conference, (2006), 1–4.

[7]

S. H. Chan, R. Khoshabeh, K. B. Gibson, P. E. Gill and T. Q. Nguyen, An augmented Lagrangian method for total variation video restoration, IEEE Trans. Image Process., 20 (2011), 3097-3111.  doi: 10.1109/TIP.2011.2158229.

[8]

J. Dahl, P. C. Hansen, S. H. Jensen and T. L. Jensen, Algorithms and software for total variation image reconstruction via first-order methods, Numer. Algorithms, 53 (2010), 67-92.  doi: 10.1007/s11075-009-9310-3.

[9]

C.-A. Deledalle, S. Vaiter, J. Fadili and G. Peyré, Stein Unbiased GrAdient estimator of the Risk (SUGAR) for multiple parameter selection, SIAM J. Imaging. Sci., 7 (2014), 2448-2487.  doi: 10.1137/140968045.

[10]

J. C. De los Reyes and C.-B. Schönlieb, Image denoising: Learning the noise model via nonsmooth PDE-constrained optimization, Inverse Probl. Imaging, 7 (2013), 1183-1214.  doi: 10.3934/ipi.2013.7.1183.

[11]

N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J.-C. Olivo-Marin and J. Zerubia, Richardson-Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution, Microsc. Res. Tech., 69 (2006), 260-266.  doi: 10.1002/jemt.20294.

[12]

Y. Dong, M. Hintermüller and M. M. Rincon-Camacho, Automated regularization parameter selection in multi-scale total variation models for image restoration, J. Math. Imaging Vision, 40 (2011), 82-104.  doi: 10.1007/s10851-010-0248-9.

[13]

C. Dossal, M. Kachour, M. J. Fadili, G. Peyré and C. Chesneau, The degrees of freedom of the lasso for general design matrix, Statist. Sinica, 23 (2013), 809-828. 

[14]

B. Efron, T. Hastie, I. Johnstone and R. Tibshirani, Least angle regression, Ann. Statist., 32 (2004), 407-499.  doi: 10.1214/009053604000000067.

[15]

P. Getreuer, Rudin-Osher-Fatemi total variation denoising using Split Bregman, Image Processing On Line, 2 (2012), 74-95.  doi: 10.5201/ipol.2012.g-tvd.

[16]

M. L. Green, Statistics of Images, the TV Algorithm of Rudin-Osher-Fatemi for Image Denoising and an Improved Denoising Algorithm, UCLA CAM 02-55, 2002.

[17]

T. Goldstein and S. Osher, The split Bregman method for L$^1$-regularized problems, SIAM J. Imaging Sci., 2 (2009), 323-343.  doi: 10.1137/080725891.

[18]

P. Hall and D. M. Titterington, Common structure of techniques for choosing smoothing parameters in regression problems, J. Roy. Statist. Soc. Ser. B, 49 (1987), 184-198. 

[19]

P. C. Hansen, J. G. Nagy and D. P. O'Leary, Deblurring Images: Matrices, Spectra and Filtering, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006. doi: 10.1137/1.9780898718874.

[20]

P. C. Hansen, Regularization tools version 4.0 for Matlab 7.3, Numer. Algorithms, 46 (2007), 189-194.  doi: 10.1007/s11075-007-9136-9.

[21]

M. Hintermüller and C. N. Rautenberg, Optimal selection of the regularization function in a weighted total variation model. Part I: Modelling and theory, J. Math. Imaging Vision, 59 (2017), 498-514.  doi: 10.1007/s10851-017-0744-2.

[22]

M. Hintermüller, C. N. Rautenberg, T. Wu and A. Langer, Optimal selection of the regularization function in a weighted total variation model. Part II: Algorithm, its analysis and numerical tests, J. Math. Imaging Vision, 59 (2017), 515-533.  doi: 10.1007/s10851-017-0736-2.

[23]

J. Huang and D. Mumford, Statistics of natural images and models, Proceedings. 1999 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 1 (1999), 541-547. 

[24]

H. Liao, F. Li and M. K. Ng, Section of regularization parameter in total variation image restoration, J. Opt. Soc. Amer. A, 26 (2009), 2311-2320.  doi: 10.1364/JOSAA.26.002311.

[25]

S. Kotz, T. J. Kozubowski and K. Podgórski, The Laplace Distribution and Generalizations. A Revisit with Applications to Communications, Economics, Engineering, and Finance, Birkhäuser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0173-1.

[26]

K. Kunisch and T. Pock, A bilevel Optimization approach for parameter learning in variational models, SIAM J. Imaging Sci., 6 (2013), 938-983.  doi: 10.1137/120882706.

[27]

A. Langer, Automated parameter selection for total variation minimization in image restoration, J. Math. Imaging Vision, 57 (2017), 239-268.  doi: 10.1007/s10851-016-0676-2.

[28]

J. Lee and P. K. Kitanidis, Bayesian inversion with total variation prior for discrete geologic structure identification, Water Resour. Res., 49 (2013), 7658-7669. 

[29]

Y. Lin, B. Wohlberg and H. Guo, UPRE method for total variation parameter selection, Signal Processing, 90 (2010), 2546-2551. 

[30]

D. W. Marquardt, Generalized inverses, ridge regression, biased linear estimation, and nonlinear estimation, Technometrics, 12 (1970), 591-612. 

[31]

J. L. Mead, A priori weighting for parameter estimation, J. Inverse Ill-Posed Probl., 16 (2008), 175-193.  doi: 10.1515/JIIP.2008.011.

[32]

J. L. Mead and R. A. Renaut, A Newton root-finding algorithm for estimating the regularization parameter for solving ill-conditioned least squares problems, Inverse Problems, 25 (2009), 025002. doi: 10.1088/0266-5611/25/2/025002.

[33]

J. L. Mead and R. A. Renaut, Least squares problems with inequality constraints as quadratic constraints, Linear Algebra Appl., 432 (2010), 1936-1949.  doi: 10.1016/j.laa.2009.04.017.

[34]

J. L. Mead, Discontinuous parameter estimates with least squares estimators, Appl. Math. Comput., 219 (2013), 5210-5223.  doi: 10.1016/j.amc.2012.11.067.

[35]

J. L. Mead and C. C. Hammerquist, $\chi^2$ tests for the choice of the regularization parameter in nonlinear inverse problems, SIAM J. Matrix Anal. Appl., 34 (2013), 1213-1230.  doi: 10.1137/12088447X.

[36]

V. A. Morozov, Methods for Solving Incorrectly Posed Problems, Translated from the Russian by A. B. Aries, translation edited by Z. Nashed, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5280-1.

[37]

M. K. Ng, P. Weiss and X. Yuan, Solving Constrained total-variation image restoration and reconstruction problems via alternating direction methods, SIAM J. Sci. Comput., 32 (2010), 2710-2736.  doi: 10.1137/090774823.

[38]

J. P. Oliveira, J. M. Bioucas-Dias and M. A. T. Figueiredo, Adaptive total variation image deblurring: A majorization-minimization approach, Signal Processing, 89 (2009), 1683-1693. 

[39]

L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms. Experimental mathematics: Computational issues in nonlinear science, Phys. D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.

[40]

A. Srivastava, A. B. Lee, E. P. Simoncelli and S.-C. Zhu, On advances in statistical modeling of natural images, J. Math. Imaging Vision, 18 (2003), 17-33.  doi: 10.1023/A:1021889010444.

[41]

T. Teuber, G. Steidl and R. H. Chan, Minimization and parameter estimation for seminorm regularization models with $I$-divergence constraints, Inverse Problems, 29 (2013), 035007, 28 pp. doi: 10.1088/0266-5611/29/3/035007.

[42]

R. J. Tibshirani and J. Taylor, The solution path of the generalized lasso, Ann. Statist., 39 (2011), 1335-1371.  doi: 10.1214/11-AOS878.

[43]

R. J. Tibshirani and J. Taylor, Degrees of freedom in lasso problems, Ann. Statist., 40 (2012), 1198-1232.  doi: 10.1214/12-AOS1003.

[44]

D. M. Titterington, Choosing the regularization parameter in image restoration, IMS Lecture Notes Monogr. Ser., 20 (1991), 392-402.  doi: 10.1214/lnms/1215460514.

[45]

G. Wahba, Bayesian "confidence intervals" for the cross-validated smoothing spline, J. Roy. Statist. Soc. Ser. B, 45 (1983), 133-150.  doi: 10.1111/j.2517-6161.1983.tb01239.x.

[46]

Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Imaging Sci., 1 (2008), 248-272.  doi: 10.1137/080724265.

[47]

Y.-W. Wen and R. H. Chan, Parameter selection for total-variation-based image restoration using discrepancy principle, IEEE Trans. Image Process., 21 (2012), 1770-1781.  doi: 10.1109/TIP.2011.2181401.

show all references

References:
[1]

A. Ali and R. J. Tibshirani, The generalized lasso problem and uniqueness, Electron. J. Stat., 13 (2019), 2307-2347.  doi: 10.1214/19-EJS1569.

[2] R. C. Aster, B. Borchers and C. H. Thurber, Parameter Estimation and Inverse Problems, 2$^{nd}$ edition, Elsevier/Academic Press, Amsterdam, 2013.  doi: 10.1016/B978-0-12-385048-5.00001-X.
[3]

S. D. Babacan, R. Molina and A. K. Katsaggelos, Variational Bayesian blind deconvolution using a total variation prior, IEEE Trans. Image Process., 18 (2009), 12-26.  doi: 10.1109/TIP.2008.2007354.

[4]

T. Blu and F. Luisier, The SURE-LET approach to image denoising, IEEE Trans. Image Process., 16 (2007), 2778-2786.  doi: 10.1109/TIP.2007.906002.

[5]

G. Casella and R. L. Berger, Statistical Inference, 2$^{nd}$ edition Duxbury, 2001.

[6]

J. M. Bioucas-Dias, M. A. T. Figueiredo and J. P. Oliveira, Adaptive total variation image deconvolution: A majorization-minimization approach, 14$^{th }$ European Signal Processing Conference, (2006), 1–4.

[7]

S. H. Chan, R. Khoshabeh, K. B. Gibson, P. E. Gill and T. Q. Nguyen, An augmented Lagrangian method for total variation video restoration, IEEE Trans. Image Process., 20 (2011), 3097-3111.  doi: 10.1109/TIP.2011.2158229.

[8]

J. Dahl, P. C. Hansen, S. H. Jensen and T. L. Jensen, Algorithms and software for total variation image reconstruction via first-order methods, Numer. Algorithms, 53 (2010), 67-92.  doi: 10.1007/s11075-009-9310-3.

[9]

C.-A. Deledalle, S. Vaiter, J. Fadili and G. Peyré, Stein Unbiased GrAdient estimator of the Risk (SUGAR) for multiple parameter selection, SIAM J. Imaging. Sci., 7 (2014), 2448-2487.  doi: 10.1137/140968045.

[10]

J. C. De los Reyes and C.-B. Schönlieb, Image denoising: Learning the noise model via nonsmooth PDE-constrained optimization, Inverse Probl. Imaging, 7 (2013), 1183-1214.  doi: 10.3934/ipi.2013.7.1183.

[11]

N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J.-C. Olivo-Marin and J. Zerubia, Richardson-Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution, Microsc. Res. Tech., 69 (2006), 260-266.  doi: 10.1002/jemt.20294.

[12]

Y. Dong, M. Hintermüller and M. M. Rincon-Camacho, Automated regularization parameter selection in multi-scale total variation models for image restoration, J. Math. Imaging Vision, 40 (2011), 82-104.  doi: 10.1007/s10851-010-0248-9.

[13]

C. Dossal, M. Kachour, M. J. Fadili, G. Peyré and C. Chesneau, The degrees of freedom of the lasso for general design matrix, Statist. Sinica, 23 (2013), 809-828. 

[14]

B. Efron, T. Hastie, I. Johnstone and R. Tibshirani, Least angle regression, Ann. Statist., 32 (2004), 407-499.  doi: 10.1214/009053604000000067.

[15]

P. Getreuer, Rudin-Osher-Fatemi total variation denoising using Split Bregman, Image Processing On Line, 2 (2012), 74-95.  doi: 10.5201/ipol.2012.g-tvd.

[16]

M. L. Green, Statistics of Images, the TV Algorithm of Rudin-Osher-Fatemi for Image Denoising and an Improved Denoising Algorithm, UCLA CAM 02-55, 2002.

[17]

T. Goldstein and S. Osher, The split Bregman method for L$^1$-regularized problems, SIAM J. Imaging Sci., 2 (2009), 323-343.  doi: 10.1137/080725891.

[18]

P. Hall and D. M. Titterington, Common structure of techniques for choosing smoothing parameters in regression problems, J. Roy. Statist. Soc. Ser. B, 49 (1987), 184-198. 

[19]

P. C. Hansen, J. G. Nagy and D. P. O'Leary, Deblurring Images: Matrices, Spectra and Filtering, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006. doi: 10.1137/1.9780898718874.

[20]

P. C. Hansen, Regularization tools version 4.0 for Matlab 7.3, Numer. Algorithms, 46 (2007), 189-194.  doi: 10.1007/s11075-007-9136-9.

[21]

M. Hintermüller and C. N. Rautenberg, Optimal selection of the regularization function in a weighted total variation model. Part I: Modelling and theory, J. Math. Imaging Vision, 59 (2017), 498-514.  doi: 10.1007/s10851-017-0744-2.

[22]

M. Hintermüller, C. N. Rautenberg, T. Wu and A. Langer, Optimal selection of the regularization function in a weighted total variation model. Part II: Algorithm, its analysis and numerical tests, J. Math. Imaging Vision, 59 (2017), 515-533.  doi: 10.1007/s10851-017-0736-2.

[23]

J. Huang and D. Mumford, Statistics of natural images and models, Proceedings. 1999 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 1 (1999), 541-547. 

[24]

H. Liao, F. Li and M. K. Ng, Section of regularization parameter in total variation image restoration, J. Opt. Soc. Amer. A, 26 (2009), 2311-2320.  doi: 10.1364/JOSAA.26.002311.

[25]

S. Kotz, T. J. Kozubowski and K. Podgórski, The Laplace Distribution and Generalizations. A Revisit with Applications to Communications, Economics, Engineering, and Finance, Birkhäuser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0173-1.

[26]

K. Kunisch and T. Pock, A bilevel Optimization approach for parameter learning in variational models, SIAM J. Imaging Sci., 6 (2013), 938-983.  doi: 10.1137/120882706.

[27]

A. Langer, Automated parameter selection for total variation minimization in image restoration, J. Math. Imaging Vision, 57 (2017), 239-268.  doi: 10.1007/s10851-016-0676-2.

[28]

J. Lee and P. K. Kitanidis, Bayesian inversion with total variation prior for discrete geologic structure identification, Water Resour. Res., 49 (2013), 7658-7669. 

[29]

Y. Lin, B. Wohlberg and H. Guo, UPRE method for total variation parameter selection, Signal Processing, 90 (2010), 2546-2551. 

[30]

D. W. Marquardt, Generalized inverses, ridge regression, biased linear estimation, and nonlinear estimation, Technometrics, 12 (1970), 591-612. 

[31]

J. L. Mead, A priori weighting for parameter estimation, J. Inverse Ill-Posed Probl., 16 (2008), 175-193.  doi: 10.1515/JIIP.2008.011.

[32]

J. L. Mead and R. A. Renaut, A Newton root-finding algorithm for estimating the regularization parameter for solving ill-conditioned least squares problems, Inverse Problems, 25 (2009), 025002. doi: 10.1088/0266-5611/25/2/025002.

[33]

J. L. Mead and R. A. Renaut, Least squares problems with inequality constraints as quadratic constraints, Linear Algebra Appl., 432 (2010), 1936-1949.  doi: 10.1016/j.laa.2009.04.017.

[34]

J. L. Mead, Discontinuous parameter estimates with least squares estimators, Appl. Math. Comput., 219 (2013), 5210-5223.  doi: 10.1016/j.amc.2012.11.067.

[35]

J. L. Mead and C. C. Hammerquist, $\chi^2$ tests for the choice of the regularization parameter in nonlinear inverse problems, SIAM J. Matrix Anal. Appl., 34 (2013), 1213-1230.  doi: 10.1137/12088447X.

[36]

V. A. Morozov, Methods for Solving Incorrectly Posed Problems, Translated from the Russian by A. B. Aries, translation edited by Z. Nashed, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5280-1.

[37]

M. K. Ng, P. Weiss and X. Yuan, Solving Constrained total-variation image restoration and reconstruction problems via alternating direction methods, SIAM J. Sci. Comput., 32 (2010), 2710-2736.  doi: 10.1137/090774823.

[38]

J. P. Oliveira, J. M. Bioucas-Dias and M. A. T. Figueiredo, Adaptive total variation image deblurring: A majorization-minimization approach, Signal Processing, 89 (2009), 1683-1693. 

[39]

L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms. Experimental mathematics: Computational issues in nonlinear science, Phys. D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.

[40]

A. Srivastava, A. B. Lee, E. P. Simoncelli and S.-C. Zhu, On advances in statistical modeling of natural images, J. Math. Imaging Vision, 18 (2003), 17-33.  doi: 10.1023/A:1021889010444.

[41]

T. Teuber, G. Steidl and R. H. Chan, Minimization and parameter estimation for seminorm regularization models with $I$-divergence constraints, Inverse Problems, 29 (2013), 035007, 28 pp. doi: 10.1088/0266-5611/29/3/035007.

[42]

R. J. Tibshirani and J. Taylor, The solution path of the generalized lasso, Ann. Statist., 39 (2011), 1335-1371.  doi: 10.1214/11-AOS878.

[43]

R. J. Tibshirani and J. Taylor, Degrees of freedom in lasso problems, Ann. Statist., 40 (2012), 1198-1232.  doi: 10.1214/12-AOS1003.

[44]

D. M. Titterington, Choosing the regularization parameter in image restoration, IMS Lecture Notes Monogr. Ser., 20 (1991), 392-402.  doi: 10.1214/lnms/1215460514.

[45]

G. Wahba, Bayesian "confidence intervals" for the cross-validated smoothing spline, J. Roy. Statist. Soc. Ser. B, 45 (1983), 133-150.  doi: 10.1111/j.2517-6161.1983.tb01239.x.

[46]

Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Imaging Sci., 1 (2008), 248-272.  doi: 10.1137/080724265.

[47]

Y.-W. Wen and R. H. Chan, Parameter selection for total-variation-based image restoration using discrepancy principle, IEEE Trans. Image Process., 21 (2012), 1770-1781.  doi: 10.1109/TIP.2011.2181401.

Figure 1.  Example of Theorem 3
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Gaussian filter with variance 9. BSNR 20 (Left), BSNR 30 (Middle), BSNR 40 (Right)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Uniform 15 x 15 filter. BSNR 20 (Left), BSNR 30 (Middle), BSNR 40 (Right)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Uniform 15 x 15 filter. BSNR 20 (Left), BSNR 30 (Middle), BSNR 40 (Right)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Left vertical axis: $ \chi^2 $ value (- -) and target value $ mn\sigma^2 $ ($ \cdot \cdot $). Right vertical axis: ISNR (–). Gaussian filter with variance 9
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Left vertical axis: $ \chi^2 $ value (- -) and target value $ mn\sigma^2 $ ($ \cdot \cdot $). Right vertical axis: ISNR (–). Uniform filter 15 x 15
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  ISNR: Gaussian blur with variance 9
BSNR MAP estimate Discrepancy $ \chi^2 $ test Maximum
Camerman $ (m=n=256) $
40 3.8419 5.5893 5.6180 6.2762
30 2.1214 3.4184 3.4507 4.2052
20 1.4033 2.1465 2.1675 2.7272
MRI $ (m=n=256) $
40 5.2304 6.2520 6.3032 7.0209
30 2.5435 4.2183 4.2656 5.0079
20 1.4242 2.5236 2.5483 3.2903
Mountain $ (m=480, n=640) $
40 2.0275 3.0206 3.0242 3.2735
30 0.9674 1.8473 1.8526 2.1171
20 0.5623 1.0222 1.0265 1.3306
Table Options

Download as excel

BSNR MAP estimate Discrepancy $ \chi^2 $ test Maximum
Camerman $ (m=n=256) $
40 3.8419 5.5893 5.6180 6.2762
30 2.1214 3.4184 3.4507 4.2052
20 1.4033 2.1465 2.1675 2.7272
MRI $ (m=n=256) $
40 5.2304 6.2520 6.3032 7.0209
30 2.5435 4.2183 4.2656 5.0079
20 1.4242 2.5236 2.5483 3.2903
Mountain $ (m=480, n=640) $
40 2.0275 3.0206 3.0242 3.2735
30 0.9674 1.8473 1.8526 2.1171
20 0.5623 1.0222 1.0265 1.3306
Table 2.  ISNR: Uniform 15 x 15
BSNR MAP estimate Discrepancy $ \chi^2 $ test Maximum
Camerman $ (m=n=256) $
40 5.0019 7.0914 7.1123 7.6329
30 2.8671 5.3398 5.3556 5.6426
20 1.8228 3.6031 3.6241 4.0441
MRI $ (m=n=256) $
40 5.6696 8.1718 8.2201 9.2978
30 3.2113 5.9225 5.9510 6.5944
20 1.7017 3.8260 3.8474 4.5641
Mountain $ (m=480, n=640) $
40 2.8357 4.0904 4.0938 4.3440
30 1.6074 2.9915 2.9945 3.1432
20 0.9594 2.1004 2.1049 2.2803
Table Options

Download as excel

BSNR MAP estimate Discrepancy $ \chi^2 $ test Maximum
Camerman $ (m=n=256) $
40 5.0019 7.0914 7.1123 7.6329
30 2.8671 5.3398 5.3556 5.6426
20 1.8228 3.6031 3.6241 4.0441
MRI $ (m=n=256) $
40 5.6696 8.1718 8.2201 9.2978
30 3.2113 5.9225 5.9510 6.5944
20 1.7017 3.8260 3.8474 4.5641
Mountain $ (m=480, n=640) $
40 2.8357 4.0904 4.0938 4.3440
30 1.6074 2.9915 2.9945 3.1432
20 0.9594 2.1004 2.1049 2.2803
Table 3.  PSNR: Gaussian blur with variance 9
BSNR MAP estimate Discrepancy $ \chi^2 $ test Maximum ISNR
Camerman $ (m=n=256) $
40.0 25.3775 27.1479 27.1772 27.8249
30.0 23.6516 24.9772 25.0077 25.6775
20.0 22.6682 23.3500 23.3718 24.0393
MRI $ (m=n=256) $
40.0 27.7080 28.8271 28.8750 29.5010
30.0 24.9866 26.7687 26.8176 27.5807
20.0 23.6173 24.6765 24.7007 25.4519
Mountain $ (m=480, n=640) $
40.0 19.0529 20.0588 20.0624 20.3040
30.0 17.9755 18.8666 18.8720 19.1485
20.0 17.4244 17.8787 17.8830 18.1806
Table Options

Download as excel

BSNR MAP estimate Discrepancy $ \chi^2 $ test Maximum ISNR
Camerman $ (m=n=256) $
40.0 25.3775 27.1479 27.1772 27.8249
30.0 23.6516 24.9772 25.0077 25.6775
20.0 22.6682 23.3500 23.3718 24.0393
MRI $ (m=n=256) $
40.0 27.7080 28.8271 28.8750 29.5010
30.0 24.9866 26.7687 26.8176 27.5807
20.0 23.6173 24.6765 24.7007 25.4519
Mountain $ (m=480, n=640) $
40.0 19.0529 20.0588 20.0624 20.3040
30.0 17.9755 18.8666 18.8720 19.1485
20.0 17.4244 17.8787 17.8830 18.1806
Table 4.  PSNR: Uniform 15 x 15. The value in parentheses is the percent relative difference from the PSNR obtained with the $ \lambda $ that achieves maximum ISNR
BSNR MAP estimate Discrepancy $ \chi^2 $ test Maximum ISNR
Camerman $ (m=n=256) $
40.0 24.1719 (9.84) 26.3170 (1.84) 26.3371 (1.77) 26.8111
30.0 22.0192 (11.10) 24.4882 (1.14) 24.5037 (1.08) 24.7709
20.0 20.8288 (9.70) 22.5266 (2.34) 22.5493 (2.24) 23.0660
MRI $ (m=n=256) $
40.0 25.0072 (12.82) 27.5739 (3.87) 27.6225 (3.70) 28.6832
30.0 22.5294 (13.29) 25.1393 (3.24) 25.1710 (3.12) 25.9823
20.0 20.9540 (11.69) 23.1799 (2.31) 23.1974 (2.23) 23.7269
Mountain $ (m=480, n=640) $
40.0 18.3899 (7.56) 19.6504 (1.22) 19.6538 (1.20) 19.8931
30.0 17.1524 (8.24) 18.5660 (0.68) 18.5688 (0.66) 18.6924
20.0 16.4198 (7.48) 17.5595 (1.06) 17.5633 (1.04) 17.7470
Table Options

Download as excel

BSNR MAP estimate Discrepancy $ \chi^2 $ test Maximum ISNR
Camerman $ (m=n=256) $
40.0 24.1719 (9.84) 26.3170 (1.84) 26.3371 (1.77) 26.8111
30.0 22.0192 (11.10) 24.4882 (1.14) 24.5037 (1.08) 24.7709
20.0 20.8288 (9.70) 22.5266 (2.34) 22.5493 (2.24) 23.0660
MRI $ (m=n=256) $
40.0 25.0072 (12.82) 27.5739 (3.87) 27.6225 (3.70) 28.6832
30.0 22.5294 (13.29) 25.1393 (3.24) 25.1710 (3.12) 25.9823
20.0 20.9540 (11.69) 23.1799 (2.31) 23.1974 (2.23) 23.7269
Mountain $ (m=480, n=640) $
40.0 18.3899 (7.56) 19.6504 (1.22) 19.6538 (1.20) 19.8931
30.0 17.1524 (8.24) 18.5660 (0.68) 18.5688 (0.66) 18.6924
20.0 16.4198 (7.48) 17.5595 (1.06) 17.5633 (1.04) 17.7470
Table 5.  SSIM: Gaussian blur with variance 9
BSNR MAP estimate Discrepancy $ \chi^2 $ test Maximum ISNR
Camerman $ (m=n=256) $
40.0 0.7956 0.8402 0.8408 0.8379
30.0 0.7393 0.7809 0.7818 0.7855
20.0 0.7024 0.7271 0.7278 0.7327
MRI $ (m=n=256) $
40.0 0.8496 0.8717 0.8725 0.8767
30.0 0.7846 0.8279 0.8288 0.8332
20.0 0.7246 0.7686 0.7693 0.7607
Mountain $ (m=480, n=640) $
40.0 0.5301 0.6197 0.6201 0.6343
30.0 0.4440 0.5156 0.5160 0.5377
20.0 0.3868 0.4348 0.4352 0.4526
Table Options

Download as excel

BSNR MAP estimate Discrepancy $ \chi^2 $ test Maximum ISNR
Camerman $ (m=n=256) $
40.0 0.7956 0.8402 0.8408 0.8379
30.0 0.7393 0.7809 0.7818 0.7855
20.0 0.7024 0.7271 0.7278 0.7327
MRI $ (m=n=256) $
40.0 0.8496 0.8717 0.8725 0.8767
30.0 0.7846 0.8279 0.8288 0.8332
20.0 0.7246 0.7686 0.7693 0.7607
Mountain $ (m=480, n=640) $
40.0 0.5301 0.6197 0.6201 0.6343
30.0 0.4440 0.5156 0.5160 0.5377
20.0 0.3868 0.4348 0.4352 0.4526
Table 6.  SSIM: Uniform 15 x 15
BSNR MAP estimate Discrepancy $ \chi^2 $ test Maximum ISNR
Camerman $ (m=n=256) $
40.0 0.7571 0.8281 0.8285 0.8158
30.0 0.6766 0.7660 0.7662 0.7628
20.0 0.6300 0.6931 0.6938 0.6984
MRI $ (m=n=256) $
40.0 0.7683 0.8341 0.8349 0.8270
30.0 0.6732 0.7694 0.7702 0.7553
20.0 0.6037 0.6934 0.6938 0.6706
Mountain $ (m=480, n=640) $
40.0 0.4609 0.5834 0.5837 0.5965
30.0 0.3602 0.4801 0.4804 0.4906
20.0 0.3060 0.3901 0.3904 0.4003
Table Options

Download as excel

BSNR MAP estimate Discrepancy $ \chi^2 $ test Maximum ISNR
Camerman $ (m=n=256) $
40.0 0.7571 0.8281 0.8285 0.8158
30.0 0.6766 0.7660 0.7662 0.7628
20.0 0.6300 0.6931 0.6938 0.6984
MRI $ (m=n=256) $
40.0 0.7683 0.8341 0.8349 0.8270
30.0 0.6732 0.7694 0.7702 0.7553
20.0 0.6037 0.6934 0.6938 0.6706
Mountain $ (m=480, n=640) $
40.0 0.4609 0.5834 0.5837 0.5965
30.0 0.3602 0.4801 0.4804 0.4906
20.0 0.3060 0.3901 0.3904 0.4003
[1]

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
[2]

Vinicius Albani, Adriano De Cezaro, Jorge P. Zubelli. On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancy-based strategy. Inverse Problems and Imaging, 2016, 10 (1) : 1-25. doi: 10.3934/ipi.2016.10.1
[3]

Juan Carlos De los Reyes, Estefanía Loayza-Romero. Total generalized variation regularization in data assimilation for Burgers' equation. Inverse Problems and Imaging, 2019, 13 (4) : 755-786. doi: 10.3934/ipi.2019035
[4]

Yuyuan Ouyang, Yunmei Chen, Ying Wu. Total variation and wavelet regularization of orientation distribution functions in diffusion MRI. Inverse Problems and Imaging, 2013, 7 (2) : 565-583. doi: 10.3934/ipi.2013.7.565
[5]

Liang Ding, Weimin Han. Morozov's discrepancy principle for $ \alpha\ell_1-\beta\ell_2 $ sparsity regularization. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022035
[6]

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
[7]

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
[8]

Rinaldo M. Colombo, Francesca Monti. Solutions with large total variation to nonconservative hyperbolic systems. Communications on Pure and Applied Analysis, 2010, 9 (1) : 47-60. doi: 10.3934/cpaa.2010.9.47
[9]

Vinicius Albani, Adriano De Cezaro. A connection between uniqueness of minimizers in Tikhonov-type regularization and Morozov-like discrepancy principles. Inverse Problems and Imaging, 2019, 13 (1) : 211-229. doi: 10.3934/ipi.2019012
[10]

Lican Kang, Yanming Lai, Yanyan Liu, Yuan Luo, Jing Zhang. High-dimensional linear regression with hard thresholding regularization: Theory and algorithm. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022034
[11]

Matteo Negri. Crack propagation by a regularization of the principle of local symmetry. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : 147-165. doi: 10.3934/dcdss.2013.6.147
[12]

Xianchao Xiu, Ying Yang, Wanquan Liu, Lingchen Kong, Meijuan Shang. An improved total variation regularized RPCA for moving object detection with dynamic background. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1685-1698. doi: 10.3934/jimo.2019024
[13]

Yunho Kim, Paul M. Thompson, Luminita A. Vese. HARDI data denoising using vectorial total variation and logarithmic barrier. Inverse Problems and Imaging, 2010, 4 (2) : 273-310. doi: 10.3934/ipi.2010.4.273
[14]

Mujibur Rahman Chowdhury, Jun Zhang, Jing Qin, Yifei Lou. Poisson image denoising based on fractional-order total variation. Inverse Problems and Imaging, 2020, 14 (1) : 77-96. doi: 10.3934/ipi.2019064
[15]

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
[16]

Sören Bartels, Marijo Milicevic. Iterative finite element solution of a constrained total variation regularized model problem. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1207-1232. doi: 10.3934/dcdss.2017066
[17]

Wei Wang, Ling Pi, Michael K. Ng. Saturation-Value Total Variation model for chromatic aberration correction. Inverse Problems and Imaging, 2020, 14 (4) : 733-755. doi: 10.3934/ipi.2020034
[18]

Yunhai Xiao, Junfeng Yang, Xiaoming Yuan. Alternating algorithms for total variation image reconstruction from random projections. Inverse Problems and Imaging, 2012, 6 (3) : 547-563. doi: 10.3934/ipi.2012.6.547
[19]

Juan C. Moreno, V. B. Surya Prasath, João C. Neves. Color image processing by vectorial total variation with gradient channels coupling. Inverse Problems and Imaging, 2016, 10 (2) : 461-497. doi: 10.3934/ipi.2016008
[20]

Liyan Ma, Lionel Moisan, Jian Yu, Tieyong Zeng. A stable method solving the total variation dictionary model with $L^\infty$ constraints. Inverse Problems and Imaging, 2014, 8 (2) : 507-535. doi: 10.3934/ipi.2014.8.507

2021 Impact Factor: 1.483

Tools

Metrics

  • PDF downloads (379)
  • HTML views (142)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy