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$ \chi^2 $ test for total variation regularization parameter selection
Department of Mathematics, Boise State University, Boise, ID, USA |
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.
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. |






BSNR | MAP estimate | Discrepancy | Maximum | |
Camerman |
||||
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 |
||||
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 |
||||
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 |
BSNR | MAP estimate | Discrepancy | Maximum | |
Camerman |
||||
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 |
||||
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 |
||||
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 |
BSNR | MAP estimate | Discrepancy | Maximum | |
Camerman |
||||
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 |
||||
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 |
||||
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 |
BSNR | MAP estimate | Discrepancy | Maximum | |
Camerman |
||||
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 |
||||
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 |
||||
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 |
BSNR | MAP estimate | Discrepancy | Maximum ISNR | |
Camerman |
||||
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 |
||||
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 |
||||
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 |
BSNR | MAP estimate | Discrepancy | Maximum ISNR | |
Camerman |
||||
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 |
||||
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 |
||||
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 |
BSNR | MAP estimate | Discrepancy | Maximum ISNR | |
Camerman |
||||
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 |
||||
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 |
||||
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 |
BSNR | MAP estimate | Discrepancy | Maximum ISNR | |
Camerman |
||||
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 |
||||
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 |
||||
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 |
BSNR | MAP estimate | Discrepancy | Maximum ISNR | |
Camerman |
||||
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 |
||||
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 |
||||
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 |
BSNR | MAP estimate | Discrepancy | Maximum ISNR | |
Camerman |
||||
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 |
||||
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 |
||||
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 |
BSNR | MAP estimate | Discrepancy | Maximum ISNR | |
Camerman |
||||
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 |
||||
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 |
||||
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 |
BSNR | MAP estimate | Discrepancy | Maximum ISNR | |
Camerman |
||||
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 |
||||
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 |
||||
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 |
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