A Framework of Calculus on Facial Surfaces

Facial surfaces play an important role in different applications such as computer graphics and biometric. A few works have been proposed to study the space of facial surfaces. In this paper, we represent a facial surface as a path on the space of closed curves in R³, called facial curves, and we study its differential geometry. A new Riemannian metric is then proposed to construct a geodesic path between two given facial surfaces. We first construct a geodesic path between arbitrary two facial surfaces and we define and compute the Karcher mean of several facial surfaces in this Riemannian framework. Many experimental examples are presented to demonstrate our approach.