Variational and differential models for shape modeling

The spread of new technologies led to a crucial role for the modeling of 3D objects, in particular for shape modeling, in a variety of applications, such as architecture, cultural heritage, industrial design, computer graphics, 3D radar scanning and others. Every task demands tailored surface processing of 3D geometric models, which defines the objects’ shape and features. We tackled certain surface processing tasks using differential and variational models. The quality of the numerical solution depends on the integrity of the given data, possibly suffering from damage or noise, and on the desired geometric properties to be preserved. Differential models rely on physics-inspired Partial Differential Equations (PDEs) to process surface data such as position, curvature and normal vectors. They are able to provide smooth, continuous representations of geometric structures and to exploit well-known physics equations. On the other hand, variational models compute the desired surface as the minimum of a suitable energy functional. They are built to encode initial surface features, through a data-fidelity term, and an a priori knowledge about the geometry of the desired result, through regularization or deformation terms. For the numerical solution of the proposed linear and nonlinear PDE models, we applied explicit, implicit or semi-implicit evolutive finite differences schemes. The numerical optimization methods used to solve the proposed variational models range from the gradient descent method on manifolds to the Alternate Direction Method of Multipliers. A fundamental role in both mathematical approaches is played by the shape descriptors, i.e. the type of representation used for geometric models, based on Euclidean coordinates or on intrinsic representations, like the Differential Coordinates. The proposed differential and variational models are applied to tackle challenging problems in shape analysis, such as removing noise from surfaces, filling in missing parts of surfaces, transferring textures between surfaces, and segmenting surfaces into meaningful regions.