The purpose of this project is to study some mathematical models arising in Image Processing, Physics and Biology, whose main common feature is the presence of singularities, given by a system of evolving interfaces (networks of surfaces or lines), which persist during the evolution, raising several challenging problems both at theoretical and numerical level.
In the context of image processing, we are interested in the total variation flow, on which some important denoising models are based. Our goal is to understand the main features of this flow, as suggested by numerics: edge preservation, loss of contrast, staircasing effect, flat zones, regularization of corners.
A strongly related relevant model is the heat flow of certain convex, positively 1-homogeneous functionals describing the evolution of distributions of vortex filaments in the 3-d Ginzburg-Landau theory of superconductivity and Bose-Einstein condensation.
Our goal is to predict and observe some new phenomena with respect to the well-studied 2-d case: curvature equation and evolution law for the vortices, presence and evolution of free boundaries in the vorticity region.
We will also study variational models for hole surfaces reconstruction in a given image, based on functionals depending on curvatures (e.g. elastica, Willmore), and related geometric evolution models (e.g. mean curvature flow, 3-d crystalline flow) which are also relevant in Biology, for the description of the dynamics of wound healing and tissue repair.
We plan also to further apply Ginzburg-Landau based models in the reconstruction of 1-d elements of an image.
Our whole theoretical analysis will be accompanied by several numerical simulations.
The different aspects of the above problems require crucially the integration of the specific skills of the two partner teams and their leading experience in these topics, embracing Calculus of Variations, P.D.E., Geometric Measure Theory, Numerical Analysis and Scientifing Computing.