Surface- and volume-based techniques for shape modeling and analysis

Extending a shape-driven map to the interior of the input shape and to the surrounding volume is a difficult problem since it typically relies on the integration of shape-based and volumetric information, together with smoothness conditions, interpolating constraints, preservation of feature values at both a local and global level. In this context, this course revises the main out-of-sample approximation schemes for both 3D shapes and d-dimensional data, and provides a unified discussion on the integration of surface- and volume-based shape information. Then, it describes the application of shape-based and volumetric techniques to shape modeling and analysis through the definition of volumetric shape descriptors; shape processing through volumetric parameterization and polycube splines; feature-driven approximation through kernels and radial basis functions. We also discuss the Hamilton's Ricci flow, which is a powerful tool to compute the conformal structure of the shapes and to design Riemannian metrics of manifolds by prescribed curvatures and shape descriptors using conformal welding. We conclude the presentation by discussing applications to shape analysis and medicine, open problems, and future perspectives.