Three-Dimensional Features for Facial Gestures Simulation

Imageand range-based information provide a support for three-dimensional modeling of human faces. Their main objective is the generation of a bottom-up model given by a collection of facets which, after matching, can be applied to recognition, medicine or gestures simulation in multimedia applications. A three-dimensional model contains several layers including discrete clouds of points or superimposed piecewise linear (triangular or quadrangular meshes) or piecewise smooth models (usually obtained by smoothing PL-models, after smoothing edges). The availability of a piecewise smooth model allow to apply deformation methods for comparing different configurations by taking geometric features (appropriate points or meaningful curves) as control elements. Intrinsic geometric features are linked to curvature maps, and more specifically to parabolic curves which separate concave and convex regions, characterized by the local constancy of signature of the total curvature. Discrete models arising from points clouds and their superimposed PL-structures pose specific problems due to structural gaps between eventually discontinuous PL-structures and ideal PS-models, which concern to discrete inputs (including irregular distributions), incomplete character of information (requiring restoration relative to propagation models on surfaces bounding solids), automatic identification of intrinsic features relevant for surface segmentation (distinction between parabolic curves and apparent contours, appearing as critical locus for projections), and relations between different kinds of meshes and related transformations (arising from lifting quadrangular meshes in views), between others. All of them are meaningful for generating gestures from imageor range-based information. In this work we develop some tools which are focused towards: a) Estimation of topological invariants linked to local and global properties which can be read in terms of triangular or quadrangular meshes; b) Intrinsic character of features (curvature maps), and c) Detection and estimation of geometric features. Nevertheless the intrinsic character of geometric properties (linked to curvature maps), results obtained from appearances arising from meshes depend in a crucial way of the resolution chosen and how sampling procedure is performed. The three above problems are meaningful for estimating intrinsic data in terms of an approximate location of critical points (Morse Theory), interpretation of appearances in terms of intrinsic data (a discrete approach to the Geometry of Discriminant Locus for the projection map) and controlling propagation phenomena (Poisson models, in our case) in order to restore, estimate and modify facial gestures. We provide some illustrations of our results concerning to each above mentioned problem in terms of a discrete version of eventually Stratified Morse Theory (for including real or apparent singularities), an explicit description of curvature maps arising from local Monge form (local equation of surface at each point obtained from the closest points), Poisson-based propagation phenomena for volumetric restoration, grouping of vector references corresponding to principal curvatures, detection of parabolic curves and their application to generate gestures for scanned human true faces.