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Parameter learning and fractional differential operators: Applications in regularized image denoising and decomposition problems
Department of Applied Mathematics, Mathematical Institute, University of Freiburg, Hermann-Herder-Str. 10, 79104 Freiburg i. Br., Germany |
In this paper, we focus on learning optimal parameters for PDE-based image denoising and decomposition models. First, we learn the regularization parameter and the differential operator for gray-scale image denoising using the fractional Laplacian in combination with a bilevel optimization problem. In our setting the fractional Laplacian allows the use of Fourier transform, which enables the optimization of the denoising operator. We prove stable and explainable results as an advantage in comparison to machine learning approaches. The numerical experiments correlate with our theoretical model settings and show a reduction of computing time in contrast to the Rudin-Osher-Fatemi model. Second, we introduce a new regularized image decomposition model with the fractional Laplacian and the Riesz potential. We provide an explicit formula for the unique solution and the numerical experiments illustrate the efficiency.
References:
[1] |
H. Antil and S. Bartels,
Spectral approximation of fractional PDEs in image processing and phase field modeling, Comput. Methods Appl. Math., 17 (2017), 661-678.
doi: 10.1515/cmam-2017-0039. |
[2] |
H. Antil, Z. W. Di and R. Khatri, Bilevel optimization, deep learning and fractional laplacian regularization with applications in tomography, Inverse Problems, 36 (2020), 064001.
doi: 10.1088/1361-6420/ab80d7. |
[3] |
H. Antil, E. Otárola and A. J. Salgado,
Optimization with respect to order in a fractional diffusion model: Analysis, approximation and algorithmic aspects, J. Sci. Comput., 77 (2018), 204-224.
doi: 10.1007/s10915-018-0703-0. |
[4] |
H. Antil and C. N. Rautenberg,
Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications, SIAM J. Math. Anal., 51 (2019), 2479-2503.
doi: 10.1137/18M1224970. |
[5] |
J.-F. Aujol, G. Gilboa, T. Chan and S. Osher,
Structure-texture image decomposition——modeling, algorithms, and parameter selection, International Journal of Computer Vision, 67 (2006), 111-136.
doi: 10.1007/s11263-006-4331-z. |
[6] |
J. Batson and L. Royer, {N}oise2{S}elf: Blind denoising by self-supervision, in Proceedings of the 36th International Conference on Machine Learning (eds. K. Chaudhuri and R. Salakhutdinov), vol. 97 of Proceedings of Machine Learning Research, PMLR, (2019), 524–533. |
[7] |
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511804441.![]() ![]() ![]() |
[8] |
F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, vol. 183 of Applied Mathematical Sciences, Springer, New York, 2013.
doi: 10.1007/978-1-4614-5975-0. |
[9] |
A. Bueno-Orovio, D. Kay and K. Burrage,
Fourier spectral methods for fractional-in-space reaction-diffusion equations, BIT, 54 (2014), 937-954.
doi: 10.1007/s10543-014-0484-2. |
[10] |
Y. Gousseau and J.-M. Morel,
Are natural images of bounded variation?, SIAM J. Math. Anal., 33 (2001), 634-648.
doi: 10.1137/S0036141000371150. |
[11] |
K. Kunish 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. |
[12] |
P. Liu and C.-B. Sch{ö}nlieb, Learning optimal orders of the underlying euclidean norm in total variation image denoising, arXiv preprint arXiv: 1903.11953. |
[13] |
Q. Liu, Z. Zhang and Z. Guo,
On a fractional reaction-diffusion system applied to image decomposition and restoration, Comput. Math. Appl., 78 (2019), 1739-1751.
doi: 10.1016/j.camwa.2019.05.030. |
[14] |
Y. Nesterov, Introductory Lectures on Convex Optimization, Springer US, 2004.
doi: 10.1007/978-1-4419-8853-9. |
[15] |
S. Osher, A. Solé and L. Vese,
Image decomposition and restoration using total variation minimization and the $H^{-1}$ norm, Multiscale Model. Simul., 1 (2003), 349-370.
doi: 10.1137/S1540345902416247. |
[16] |
G. Peyré,
The numerical tours of signal processing, Computing in Science & Engineering, 13 (2011), 94-97.
doi: 10.1109/MCSE.2011.71. |
[17] |
L. I. Rudin, S. 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. |
[18] |
J. Saranen and G. Vainikko, Periodic Integral and Pseudodifferential Equations with Numerical Approximation, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002.
doi: 10.1007/978-3-662-04796-5. |
[19] |
J. Sprekels and E. Valdinoci,
A new type of identification problems: Optimizing the fractional order in a nonlocal evolution equation, SIAM J. Control Optim., 55 (2017), 70-93.
doi: 10.1137/16M105575X. |
[20] |
D. Ulyanov, A. Vedaldi and V. Lempitsky,
Deep image prior, International Journal of Computer Vision volume, 128 (2020), 1867-1888.
doi: 10.1007/s11263-020-01303-4. |
show all references
References:
[1] |
H. Antil and S. Bartels,
Spectral approximation of fractional PDEs in image processing and phase field modeling, Comput. Methods Appl. Math., 17 (2017), 661-678.
doi: 10.1515/cmam-2017-0039. |
[2] |
H. Antil, Z. W. Di and R. Khatri, Bilevel optimization, deep learning and fractional laplacian regularization with applications in tomography, Inverse Problems, 36 (2020), 064001.
doi: 10.1088/1361-6420/ab80d7. |
[3] |
H. Antil, E. Otárola and A. J. Salgado,
Optimization with respect to order in a fractional diffusion model: Analysis, approximation and algorithmic aspects, J. Sci. Comput., 77 (2018), 204-224.
doi: 10.1007/s10915-018-0703-0. |
[4] |
H. Antil and C. N. Rautenberg,
Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications, SIAM J. Math. Anal., 51 (2019), 2479-2503.
doi: 10.1137/18M1224970. |
[5] |
J.-F. Aujol, G. Gilboa, T. Chan and S. Osher,
Structure-texture image decomposition——modeling, algorithms, and parameter selection, International Journal of Computer Vision, 67 (2006), 111-136.
doi: 10.1007/s11263-006-4331-z. |
[6] |
J. Batson and L. Royer, {N}oise2{S}elf: Blind denoising by self-supervision, in Proceedings of the 36th International Conference on Machine Learning (eds. K. Chaudhuri and R. Salakhutdinov), vol. 97 of Proceedings of Machine Learning Research, PMLR, (2019), 524–533. |
[7] |
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511804441.![]() ![]() ![]() |
[8] |
F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, vol. 183 of Applied Mathematical Sciences, Springer, New York, 2013.
doi: 10.1007/978-1-4614-5975-0. |
[9] |
A. Bueno-Orovio, D. Kay and K. Burrage,
Fourier spectral methods for fractional-in-space reaction-diffusion equations, BIT, 54 (2014), 937-954.
doi: 10.1007/s10543-014-0484-2. |
[10] |
Y. Gousseau and J.-M. Morel,
Are natural images of bounded variation?, SIAM J. Math. Anal., 33 (2001), 634-648.
doi: 10.1137/S0036141000371150. |
[11] |
K. Kunish 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. |
[12] |
P. Liu and C.-B. Sch{ö}nlieb, Learning optimal orders of the underlying euclidean norm in total variation image denoising, arXiv preprint arXiv: 1903.11953. |
[13] |
Q. Liu, Z. Zhang and Z. Guo,
On a fractional reaction-diffusion system applied to image decomposition and restoration, Comput. Math. Appl., 78 (2019), 1739-1751.
doi: 10.1016/j.camwa.2019.05.030. |
[14] |
Y. Nesterov, Introductory Lectures on Convex Optimization, Springer US, 2004.
doi: 10.1007/978-1-4419-8853-9. |
[15] |
S. Osher, A. Solé and L. Vese,
Image decomposition and restoration using total variation minimization and the $H^{-1}$ norm, Multiscale Model. Simul., 1 (2003), 349-370.
doi: 10.1137/S1540345902416247. |
[16] |
G. Peyré,
The numerical tours of signal processing, Computing in Science & Engineering, 13 (2011), 94-97.
doi: 10.1109/MCSE.2011.71. |
[17] |
L. I. Rudin, S. 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. |
[18] |
J. Saranen and G. Vainikko, Periodic Integral and Pseudodifferential Equations with Numerical Approximation, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002.
doi: 10.1007/978-3-662-04796-5. |
[19] |
J. Sprekels and E. Valdinoci,
A new type of identification problems: Optimizing the fractional order in a nonlocal evolution equation, SIAM J. Control Optim., 55 (2017), 70-93.
doi: 10.1137/16M105575X. |
[20] |
D. Ulyanov, A. Vedaldi and V. Lempitsky,
Deep image prior, International Journal of Computer Vision volume, 128 (2020), 1867-1888.
doi: 10.1007/s11263-020-01303-4. |




















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