Metodi variazionali e PDE per l’elaborazione delle immagini

Corso di Dottorato "Metodi variazionali e PDE per l’elaborazione delle immagini"- Cicli di seminari In this course we will present some classical and recent approaches for some problems in image reconstruction (denoising, deblurring, inpainting, shadow-removal…) formulated in terms of appropriate minimisation problems in infinite-dimensional functional spaces. We will further draw connections between these minimisation problems and parabolic Partial Differential Equations (PDEs) based on non-linear diffusion and possibly combined with transport terms. For the practical implementation of the models above, we will review standard finite difference stencils discussing their extensions to anisotropic diffusion and diffusion-transport problems. The course will be complemented by some practical MATLAB classes where simple exemplar problems will be solved by means of some reference iterative algorithms. Classical examples of imaging problems (denoising, deblurring, inpainting, segmentation..). Formulation as ill-posed inverse problems. Variational regularisation methods: regularisation term VS data fitting. Statistical interpretation: MAP estimation (2h) Sobolev spaces, standard methods in calculus of variations: a review. Total variation, the space of functions of bounded variations (2h) Second-order parabolic PDEs for image processing: heat equation, mean-curvature flow. Applications to image processing: linear VS non-linear PDEs. Regularisation of non-smoothness: lagged diffusivity. Anisotropic diffusion and diffusion-transport problems. (4h) Finite differences stencils for PDE-based imaging models. (2h) Numerical implementation and simulations in MATLAB for PDE-based models for image reconstruction (deblurring, inpainting, face fusion). (5h) Gli orari e le aule saranno specificati alla pagina web del dottorato ed inviati di volta in volta secondo il calendario 9/5: 2h (Teoria), mattina - 10/5: 2h (Teoria), mattina - 13/5: 2h (Teoria), mattina 14/5: 2h (Laboratorio), mattina 15/5: 2h (Teoria), mattina + 1h (Laboratorio), pomeriggio 16:5: 2h (Teoria), mattina - 17/5: 2h (Laboratorio), mattina