Functional 3D shape maps
Finding a map between two 3D shapes is a fundamental problem in computer graphics with applications in morphing, geometry synthesis and animation. The standard approach to this problem is to search a space of possible maps built on the basis of correspondences between parts that are similar. The EU-funded GEN-MAPS-3D-SHAPES (Generalized maps for the analysis of 3D shapes and shape collections) project focused on analysing and visualising a given map. Researchers were motivated by the lack of metrics for map evaluation that highlight areas where maps fail to meet certain quality criteria. Researchers proposed to address this challenge in a unified way by means of functional representation of the map and performing spectral analysis on this representation. The new method provides detailed multi-scale information about the distortion introduced by the map. In this new approach, the functional map representation puts in correspondence real-valued functions rather than points on the shapes. The GEN-MAPS-3D-SHAPES team demonstrated that such functional maps provide a powerful tool for visualising maps between shapes as well as collections of such maps. Their next step was to define a functional operator describing differences between two shapes as comparable objects instead of merely distances between points. This made it easier to compute shape differences and variability in an entire shape collection. Use of the operator to align two shape collections provided an intrinsic way of computing differences between shapes as well as shape analogies and to parametrise the intrinsic variability in the collections. To align the collections, researchers solved a set of linear equations. Before completion of the GEN-MAPS-3D-SHAPES project, researchers extended the toolbox of functional operators to tangent vector fields on surfaces. Exploring the new operator properties and designing efficient methods for their computation led to a broad range of applications beyond shape analysis. Functional vector field representations found applications to viscous fluid flow simulations, capturing intricate fingering effects. The simulations proved to be robust and efficient since they were based on novel formulations in terms of mass transport.