Dual-grid parameter choice method with application to image deblurring

Markus Juvonen; Bjørn Jensen; Ilmari Pohjola; Yiqiu Dong; Samuli Siltanen

doi:10.3934/ammc.2025011

Article Contents

September 2025, Volume 5, 64-85. Doi: 10.3934/ammc.2025011

This volume Previous Article Image reconstruction in cone beam computed tomography using controlled gradient sparsity Next Article SDF: A self-supervised denoiser framework for X-Ray computed tomography

Dual-grid parameter choice method with application to image deblurring

1.
University of Helsinki, Finland
2.
Technical University of Denmark

^*Corresponding author: Markus Juvonen
^*Corresponding author: Markus Juvonen

Received: April 2025

Revised: September 2025

Published online: November 6, 2025

Juvonen and Siltanen are supported by Research Council of Finland [CoE decision 353097 and FAME Flagship decision 359182]. We thank Ensio Suonperä for his support in the implementation of the bilevel optimization method for parameter selection comparison.

Abstract / Introduction Full Text(HTML) Figure(20) / Table(3) Related Papers Cited by

Abstract

Variational regularization of ill-posed inverse problems is based on minimizing the sum of a data fidelity term and a regularization term. The balance between them is tuned using a positive regularization parameter, whose automatic choice remains an open question in general. A novel approach for parameter choice is introduced, based on the use of two slightly different computational models for the same inverse problem. Small parameter values should give two very different reconstructions due to amplification of noise. Large parameter values lead to two identical but trivial reconstructions. The optimal parameter is chosen between the extremes by matching image similarity of the two reconstructions with a pre-defined value. The efficacy of the new method is demonstrated by image deblurring using measured data and two different regularizers.

Keywords:

Mathematics Subject Classification: Primary: 65R32, 94A08; Secondary: 15A29.

Citation:

FullText(HTML)

Figure 1. Left: Simulated ground truth with no noise or blur. Middle: Blurred with radius 4 kernel and 4% added noise. Right: Blurred with radius 4 kernel and added 8% noise

Download: Full-size image PowerPoint slide

Figure 2. Left: "Ground truth", almost noiseless (ISO100). Middle: Slightly blurred, some noise (ISO1600). Right: Slightly blurred, more noise (ISO6400)

Download: Full-size image PowerPoint slide

Figure 3. "Ground truth" (ISO100) on the left. Slightly blurred and small noise (ISO6400) in the middle. Slightly blurred and more noise (ISO25600) on the right

Download: Full-size image PowerPoint slide

Figure 4. 10 of the 230 natural images used for the numerical test of the similarity hypothesis

Download: Full-size image PowerPoint slide

Figure 5. SSIM between images on the two grids. The blue graph indicates that for a regular photograph, a sub-pixel shift causes a significant change in the image as measured by SSIM. However, for images smoothed by Tikhonov or total variation denoising, the change caused by the shift is much weaker: their SSIM values are closer to 1

Download: Full-size image PowerPoint slide

Figure 6. Mean SSIM value between images on the two grids and standard deviation computed for all 230 images in Figure 5

Download: Full-size image PowerPoint slide

Figure 7. Dual-grid method for parameter choice for the case of simulated images shown in Figure 1. Plotted is the SSIM function between the two solutions SSIM$ ({{\mathbf g}}^{(\alpha)} , {{\mathbf f}}^{(\alpha)}) $ for two regularizers and two noise amplitudes. The SSIM threshold value is 0.985 in both cases. Top: Tikhonov regularization. The first regularization parameter after the selected threshold value is $ \alpha = 1.076 $ for the case with less noise (red curve) and $ \alpha = 1.467 $ for the higher noise case (blue curve). Bottom: Total variation regularization. The first regularization parameter after the selected threshold value is $ \alpha = 0.085 $ for the case with less noise (red curve) and $ \alpha = 0.158 $ for the higher noise case (blue curve)

Download: Full-size image PowerPoint slide

Figure 8. Geometric shapes images and reconstructions with regularization parameter values selected with dual-grid method as seen in Figure 7. First row: Ground truth, blur radius 4 + noise 4%, Tikhonov regularized, TV. Second row: Ground truth, blur radius 4 + noise 8%, Tikhonov regularized, TV regularized (Tikh threshold .985, TV threshold .985). PSNR values between the ground truth and the noisy images and reconstructions are visible under each image

Download: Full-size image PowerPoint slide

Figure 9. Dual-grid method for parameter choice for the case of the playing card images shown in Figure 2. Plotted is the SSIM function between the two solutions SSIM$ ({{\mathbf g}}^{(\alpha)} , {{\mathbf f}}^{(\alpha)}) $ for two regularizers and two noise amplitudes. Top: Tikhonov regularization. The first regularization parameter after the selected SSIM threshold value of 0.985 is $ \alpha = 0.742 $ for the case with less noise (red curve) and $ \alpha = 0.616 $ for the higher noise case (blue curve). Bottom: Total variation regularization. The first regularization parameter after the selected SSIM threshold value of 0.97 is $ \alpha = 0.168 $ for the case with less noise (red curve) and $ \alpha = 0.085 $ for the higher noise case (blue curve)

Download: Full-size image PowerPoint slide

Figure 10. Queen playing card and reconstructions with regularization parameter values selected with dual-grid method as seen in Figure 9. First row: "Ground truth" sharp image (ISO100), blur+little noise(ISO1600), Tikhonov, TV. Second row: "Ground truth" sharp image (ISO100), blur+more noise(ISO6400), Tikhonov, TV. (Tikh threshold .985, TV threshold .97)

Download: Full-size image PowerPoint slide

Figure 11. Queen playing card images from Figure 10 cropped for a more detailed view. First row: "Ground truth" sharp image (ISO100), blur+little noise(ISO1600), Tikhonov, TV. Second row: "Ground truth" sharp image (ISO100), blur+more noise(ISO6400), Tikhonov, TV. (Tikh threshold .985, TV threshold .97)

Download: Full-size image PowerPoint slide

Figure 12. Dual-grid method for parameter choice for the case of the books images shown in Figure 3. Plotted is the SSIM function between the two solutions SSIM$ ({{\mathbf g}}^{(\alpha)} , {{\mathbf f}}^{(\alpha)}) $ for two regularizers and two noise amplitudes. Top: Tikhonov regularization. The first regularization parameter after the selected threshold value of 0.985 is $ \alpha = 0.742 $ for the case with less noise (red curve) and $ \alpha = 0.84 $ for the higher noise case (blue curve). Bottom: Total variation regularization. The first regularization parameter after the selected threshold value of 0.955 is $ \alpha = 0.376 $ for the case with less noise (red curve) and $ \alpha = 0.312 $ for the higher noise case (blue curve)

Download: Full-size image PowerPoint slide

Figure 13. Books images and reconstructions with regularization parameter values selected with dual-grid method as seen in Figure 12. First row: "Ground truth" sharp image (ISO100), Blur+little noise, Tikhonov, TV. Second row: "Ground truth" sharp image (ISO100), blur+more noise, Tikhonov, TV (Tikh threshold .985, TV threshold .955)

Download: Full-size image PowerPoint slide

Figure 14. Books images from Figure 13 cropped for more detailed look. First row: "Ground truth" sharp image (ISO100), blur+little noise, Tikhonov, TV. Second row: "Ground truth" sharp image (ISO100), blur+more noise, Tikhonov, TV (Tikh. threshold .985, TV threshold .955)

Download: Full-size image PowerPoint slide

Figure 15. Discrepancy principle for parameter choice for the case of simulated images shown in Figure 1. Plotted is the function $ \Psi(\alpha) $ defined in (15) for two regularizers and two noise amplitudes. Top: Tikhonov regularization. The closest regularization parameter to the estimated noise level $ \delta $ is $ \alpha = 0.352 $ for the case with less noise (red curve) and $ \alpha = 0.427 $ for the higher noise case (blue curve). Bottom: Total variation regularization. The closest regularization parameter to the estimated noise level $ \delta $ is $ \alpha = 0.821 $ for the case with less noise (red curve) and $ \alpha = 1.894 $ for the higher noise case (blue curve)

Download: Full-size image PowerPoint slide

Figure 16. Comparison of optimal reconstructions according to dual-grid method and discrepancy principle for the simulated image with noise level 4%. Left: Ground truth. Upper row: Dual-grid results. Bottom row: Discrepancy principle results. PSNR values compared to the ground truth image are under each reconstruction

Download: Full-size image PowerPoint slide

Figure 17. Comparison of optimal reconstructions according to dual-grid method and discrepancy principle for the simulated image with noise level 8%. Left: Ground truth. Upper row dual-grid. Lower row discrepancy principle. PSNR values compared to the ground truth image are under each reconstruction

Download: Full-size image PowerPoint slide

Figure 18. Discrepancy principle for parameter choice for the case of the playing card images shown in Figure 2. Plotted is the function $ \Psi(\alpha) $ defined in (15) for two regularizers and two noise amplitudes. Top: Tikhonov regularization. We only get a parameter value for the higher noise case. The closest value to the intersection with the estimated noise level is $ \alpha = 0.091 $ for the higher noise case (blue curve). Bottom: Total variation regularization. The discrepancy principle fails to give us parameter values for the estimated noise levels

Download: Full-size image PowerPoint slide

Figure 19. Discrepancy principle for parameter choice for the case of the books images shown in Figure 3. Plotted is the function $ \Psi(\alpha) $ defined in (15) for two regularizers and two noise amplitudes. Top: Tikhonov regularization. The closest regularization parameter to the estimated noise level $ \delta $ is $ \alpha = 0.062 $ for the case with less noise (red curve) and $ \alpha = 0.134 $ for the higher noise case (blue curve). Bottom: total variation regularization. We fail to find a parameter value for the case with less noise (red curve). The closest regularization parameter to the estimated noise level $ \delta $ is $ \alpha = 0.005 $ for the higher noise case (blue curve)

Download: Full-size image PowerPoint slide

Figure 20. Box plots of $ \alpha $ across 15 of the 230 natural images for each SSIM threshold $ \{0.95, 0.96, 0.97, 0.98, 0.99\} $. Boxes show the interquartile range (IQR = $ [Q_1, Q_3] $), the horizontal line is the median, whiskers extend to the most extreme values within $ 1.5\times\mathrm{IQR} $. The vertical axis is labeled as powers of ten, but boxes are computed on $ \log_{10}(\alpha) $, so equal vertical distances correspond to multiplicative factors in $ \alpha $. Medians increase monotonically with the threshold (0.0129, 0.0239, 0.0444, 0.2086, 0.8402). The IQR also grows (0.0116, 0.0327, 0.1258, 0.3485, 0.8534), indicating substantial image-to-image variability at fixed thresholds

Download: Full-size image PowerPoint slide

Table 1. Optimal Tikhonov regularization parameter values $ \alpha $ for all image examples according to the discrepancy principle and the proposed dual-grid method

Image test	Tikh. discrep.$ \alpha $	Tikh. Dual-grid $ \alpha $
Simulated 4%	0.352	1.076
Simulated 8%	0.427	1.467
Queen low	-	0.742
Queen high	0.091	0.616
Books low	0.062	0.742
Books high	0.134	0.840

| Show Table

DownLoad: CSV

Table 2. Optimal TV parameter values $ \alpha $ for all image examples. The discrepancy principle, the proposed dual-grid method and the FIFB bilevel optimization algorithm

Image test	TV discrep.$ \alpha $	TV Dual-grid $ \alpha $	TV Bilevel (FIFB) $ \alpha $
Simulated 4%	0.894	0.085	0.064
Simulated 8%	2.263	0.158	0.123
Queen low	-	0.168	0.085
Queen high	-	0.085	0.048
Books low	-	0.376	0.440
Books high	0.005	0.312	0.559

| Show Table

DownLoad: CSV

Table 3. TV deblurring parameter $ \alpha $ per image for each SSIM threshold using the dual-grid method. Values rounded to three significant digits for readability

Image #	SSIM threshold
Image #	0.95	0.96	0.97	0.98	0.99
Im1	0.00508	0.00808	0.015	0.0824	0.387
Im2	0.00593	0.00944	0.015	0.0605	0.452
Im3	0.015	0.0239	0.038	0.0824	0.209
Im4	0.00593	0.00944	0.0175	0.0605	0.284
Im5	0.0205	0.0518	0.153	0.452	1.34
Im6	0.0326	0.0824	0.209	0.528	1.82
Im7	0.0129	0.0326	0.131	0.528	2.13
Im8	0.0110	0.0205	0.0444	0.244	1.15
Im9	0.0175	0.0326	0.0706	0.209	0.720
Im10	0.00319	0.00508	0.00944	0.0279	0.0962
Im11	0.00808	0.0150	0.0380	0.153	0.840
Im12	0.0824	0.179	0.284	0.616	1.34
Im13	0.00373	0.00593	0.0129	0.0518	0.387
Im14	0.0129	0.0279	0.0824	0.284	0.981
Im15	0.0175	0.0706	0.153	0.387	0.981

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	S. W. Anzengruber and R. Ramlau, Morozov's discrepancy principle for tikhonov-type functionals with nonlinear operators, Inverse Problems, 26 (2009), 025001, 17 pp. doi: 10.1088/0266-5611/26/2/025001.
[2]	S. Arridge, P. Maass, O. Öktem and C.-B. Schönlieb, Solving inverse problems using data-driven models, Acta Numerica, 28 (2019), 1-174. doi: 10.1017/S0962492919000059.
[3]	A. Bakushinskii, Remarks on choosing a regularization parameter using the quasi-optimality and ratio criterion, USSR Computational Mathematics and Mathematical Physics, 24 (1984), 181-182. doi: 10.1016/0041-5553(84)90253-2.
[4]	F. Bauer and M. Reiß, Regularization independent of the noise level: An analysis of quasi-optimality, Inverse Problems, 24 (2008), 055009, 16 pp. doi: 10.1088/0266-5611/24/5/055009.
[5]	A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145. doi: 10.1007/s10851-010-0251-1.
[6]	K. Chen, E. Loli Piccolomini and F. Zama, An automatic regularization parameter selection algorithm in the total variation model for image deblurring, Numerical Algorithms, 67 (2014), 73-92. doi: 10.1007/s11075-013-9775-y.
[7]	C. Clason, B. Jin and K. Kunisch, A duality-based splitting method for $\ell^{1}$-tv image restoration with automatic regularization parameter choice, SIAM Journal on Scientific Computing, 32 (2010), 1484-1505. doi: 10.1137/090768217.
[8]	E. Davoli, R. Ferreira, I. Fonseca and J. A. Iglesias, Dyadic partition-based training schemes for tv/tgv denoising, Journal of Mathematical Imaging and Vision, 66 (2024), 1070-1108. doi: 10.1007/s10851-024-01213-x.
[9]	J. C. De los Reyes, C.-B. Schönlieb and T. Valkonen, Bilevel parameter learning for higher-order total variation regularisation models, Journal of Mathematical Imaging and Vision, 57 (2017), 1-25. doi: 10.1007/s10851-016-0662-8.
[10]	Y. Dong, M. Hintermüller and M. M. Rincon-Camacho, Automated regularization parameter selection in multi-scale total variation models for image restoration, Journal of Mathematical Imaging and Vision, 40 (2011), 82-104. doi: 10.1007/s10851-010-0248-9.
[11]	H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, volume 375. Springer Science & Business Media, 1996.
[12]	K. Hamalainen, A. Kallonen, V. Kolehmainen, M. Lassas, K. Niinimaki and S. Siltanen, Sparse tomography, SIAM Journal on Scientific Computing, 35 (2013), B644-B665. doi: 10.1137/120876277.
[13]	P. C. Hansen, Analysis of discrete ill-posed problems by means of the l-curve, SIAM Review, 34 (1992), 561-580. doi: 10.1137/1034115.
[14]	P. C. Hansen, J. G. Nagy and D. P. O'leary, Deblurring Images: Matrices, Spectra, and Filtering, SIAM, 2006. doi: 10.1137/1.9780898718874.
[15]	B. Harrach, T. Jahn and R. Potthast, Beyond the bakushinkii veto: Regularising linear inverse problems without knowing the noise distribution, Numerische Mathematik, 145 (2020), 581-603. doi: 10.1007/s00211-020-01122-2.
[16]	S. Kindermann, L. D. Mutimbu and E. Resmerita, A numerical study of heuristic parameter choice rules for total variation regularization, Journal of Inverse and Ill-Posed Problems, 22 (2014), 63-94. doi: 10.1515/jip-2012-0074.
[17]	K. Kunisch and T. Pock, A bilevel optimization approach for parameter learning in variational models, SIAM Journal on Imaging Sciences, 6 (2013), 938-983. doi: 10.1137/120882706.
[18]	M. A. Lukas, Strong robust generalized cross-validation for choosing the regularization parameter, Inverse Problems, 24 (2008), 034006, 16 pp. doi: 10.1088/0266-5611/24/3/034006.
[19]	A. Meaney, M. A. Brix, M. T. Nieminen and S. Siltanen, Image reconstruction in cone beam computed tomography using controlled gradient sparsity, arXiv preprint, arXiv: 2412.07465, 2024.
[20]	A. Neubauer, The convergence of a new heuristic parameter selection criterion for general regularization methods, Inverse Problems, 24 (2008), 055005, 10 pp. doi: 10.1088/0266-5611/24/5/055005.
[21]	K. Niinimaki, M. Lassas, K. Hamalainen, A. Kallonen, V. Kolehmainen, E. Niemi and S. Siltanen, Multiresolution parameter choice method for total variation regularized tomography, SIAM Journal on Imaging Sciences, 9 (2016), 938-974. doi: 10.1137/15M1034076.
[22]	S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, Multiscale Modeling & Simulation, 4 (2005), 460-489. doi: 10.1137/040605412.
[23]	M. Pragliola, L. Calatroni, A. Lanza and F. Sgallari, Admm-based residual whiteness principle for automatic parameter selection in single image super-resolution problems, Journal of Mathematical Imaging and Vision, 65 (2023), 99-123. doi: 10.1007/s10851-022-01110-1.
[24]	Z. Purisha, S. S. Karhula, J. H. Ketola, J. Rimpeläinen, M. T. Nieminen, S. Saarakkala, H. Kröger and S. Siltanen, An automatic regularization method: An application for 3-d x-ray micro-ct reconstruction using sparse data, IEEE Transactions on Medical Imaging, 38 (2018), 417-425. doi: 10.1109/TMI.2018.2865646.
[25]	Z. Purisha, J. Rimpeläinen, T. Bubba and S. Siltanen, Controlled wavelet domain sparsity for x-ray tomography, Measurement Science and Technology, 29 (2017), 014002. doi: 10.1088/1361-6501/aa9260.
[26]	L. Rudin and S. Osher, Total variation based image restoration with free local constraints, In Proceedings of 1st International Conference on Image Processing, volume 1, 1994, 31-35. doi: 10.1109/ICIP.1994.413269.
[27]	E. Suonperä, Codes for "linearly convergent bilevel optimization with single-step inner methods", May 2023. doi: 10.5281/zenodo.7974062.
[28]	E. Suonperä and T. Valkonen, Linearly convergent bilevel optimization with single-step inner methods, Computational Optimization and Applications, 87 (2024), 571-610. doi: 10.1007/s10589-023-00527-7.
[29]	A. N. Tikhonov, On the solution of ill-posed problems and the method of regularization, Dokl. Akad. Nauk SSSR, 151 (1963), 501-504.
[30]	A. Toma, B. Sixou and F. Peyrin, Iterative choice of the optimal regularization parameter in tv image restoration, Inverse Probl. Imaging, 9 (2015), 1171-1191. doi: 10.3934/ipi.2015.9.1171.
[31]	C. R. Vogel and M. E. Oman, Fast, robust total variation-based reconstruction of noisy, blurred images, IEEE Transactions on Image Processing, 7 (1998), 813-824. doi: 10.1109/83.679423.
[32]	Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM Journal on Imaging Sciences, 1 (2008), 248-272. doi: 10.1137/080724265.
[33]	Z. Wang, A. Bovik, H. Sheikh and E. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Transactions on Image Processing, 13 (2004), 600-612. doi: 10.1109/TIP.2003.819861.
[34]	Y.-W. Wen and R. H. Chan, Parameter selection for total-variation-based image restoration using discrepancy principle, IEEE Transactions on Image Processing, 21 (2011), 1770-1781. doi: 10.1109/TIP.2011.2181401.
[35]	Y.-W. Wen and R. H. Chan, Using generalized cross validation to select regularization parameter for total variation regularization problems, Inverse Problems and Imaging, 12 (2018), 1103-1120. doi: 10.3934/ipi.2018046.