Journal Articles
Interpretable Model Learning in Variational Imaging: A Bilevel Optimization Approach, The IMA Journal of Applied Mathematics, hxad024 (2023)
Juan Carlos De los Reyes, David Villacís
In this paper, we investigate the use of bilevel optimization for model learning in variational imaging problems. Bilevel learning is an alternative approach to traditional deep learning methods, that leads to fully interpretable models. However, it requires a detailed analytical insight into the underlying mathematical model. We focus on the bilevel learning problem for total variation models with spatially- and patch-dependent parameters. Our study encompasses the directional differentiability of the solution mapping, the derivation of optimality conditions, and the characterization of the Bouligand subdifferential of the solution operator. We also propose a two-phase trust-region algorithm for solving the problem and present numerical tests using the CelebA dataset.
Optimality Conditions for Bilevel Imaging Learning Problems with Total Variation Regularization, SIAM Journal on Imaging Sciences Vol. 15, Iss. 4 (2022)
Juan Carlos De los Reyes, David Villacís
We address the problem of optimal scale-dependent parameter learning in total variation image denoising. Such problems are formulated as bilevel optimization instances with total variation denoising problems as lower-level constraints. For the bilevel problem, we are able to derive M-stationarity conditions, after characterizing the corresponding Mordukhovich generalized normal cone and verifying suitable constraint qualification conditions. We also derive B-stationarity conditions, after investigating the Lipschitz continuity and directional differentiability of the lower-level solution operator. A characterization of the Bouligand subdifferential of the solution mapping, by means of a properly defined linear system, is provided as well. Based on this characterization, we propose a two-phase nonsmooth trust-region algorithm for the numerical solution of the bilevel problem and test it computationally for two particular experimental settings.
Bilevel Optimization Methods in Imaging, Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging (2021)
Juan Carlos De los Reyes, David Villacís
Optimization techniques have been widely used for image restoration tasks, as many imaging problems may be formulated as minimization ones with the recovered image as the target minimizer. Recently, novel optimization ideas also entered the scene in combination with machine learning approaches, to improve the reconstruction of images by optimally choosing different parameters/functions of interest in the models. This chapter provides a review of the latest developments concerning the latter, with special emphasis on bilevel optimization techniques and their use for learning local and nonlocal image restoration models in a supervised manner. Moreover, the use of related optimization ideas within the development of neural networks in imaging will be briefly discussed.
First Order Methods for High Resolution Image Denoising, Latin American Journal of Computing Vol. 4, Iss. 3 (2017)
David Villacís
In this paper we are interested in comparing theperformance of some of the most relevant first order non-smoothoptimization methods applied to the Rudin, Osher and Fatemi (ROF) Image Denoising Model and a Primal-Dual Chambolle-Pock Image Denoising Model. Because of the properties of theresulting numerical schemes it is possible to handle these computationspixelwise, allowing implementations based on parallelparadigms which are helpful in the context of high resolution imaging.
Preprints
Bilevel Imaging Learning Problems as Mathematical Programs with Complementarity Constraints,
Juan Carlos De los Reyes, David Villacís
We investigate a family of bilevel imaging learning problems where the lower-level instance corresponds to a convex variational model involving first-and second-order nonsmooth regularizers. By using geometric properties of the primal-dual reformulation of the lower-level problem and introducing suitable changes of variables, we are able to reformulate the original bilevel problems as Mathematical Programs with Complementarity Constraints (MPCC). For the latter, we prove tight constraint qualification conditions (MPCC-MFCQ and partial MPCC-LICQ) and derive Mordukovich (M-) and Strong (S-) stationarity conditions. The S-stationarity system for the MPCC turns also into S-stationarity conditions for the original formulation. Second-order sufficient optimality conditions are derived as well. The proposed reformulation may be extended to problems in function spaces, leading to MPCC’s with additional constraints on the gradient of the state. Finally, we report on some numerical results obtained by using the proposed MPCC reformulations together with available large-scale nonlinear programming solvers.
Open Datasets
Photographic dataset - Playing Cards
David Villacis, Santeri Kaupinmäki, Samuli Siltanen, Teemu Helenius
This is a photographic dataset collected for testing image processing algorithms. The idea is to have images that can exploit the properties of total variation, therefore a set of playing cards was distributed on the scene. The dataset is made available at www.fips.fi/photographic_dataset2.php