Abstract-We propose a novel geometric framework for analyzing 3D faces, with the specific goals of comparing, matching, and averaging their shapes. Here we represent facial surfaces by radial curves emanating from the nose tips and use elastic shape analysis of these curves to develop a Riemannian framework for analyzing shapes of full facial surfaces. This representation, along with the elastic Riemannian metric, seems natural for measuring facial deformations and is robust to challenges such as large facial expressions (especially those with open mouths), large pose variations, missing parts, and partial occlusions due to glasses, hair, etc. This framework is shown to be promising from both -empirical and theoretical -perspectives. In terms of the empirical evaluation, our results match or improve the state-of-the-art methods on three prominent databases: FRGCv2, GavabDB, and Bosphorus, each posing a different type of challenge. From a theoretical perspective, this framework allows for formal statistical inferences, such as the estimation of missing facial parts using PCA on tangent spaces and computing average shapes.

a b s t r a c tIn this paper we address the problem of modeling and analyzing human motion by focusing on 3D body skeletons. Particularly, our intent is to represent skeletal motion in a geometric and efficient way, leading to an accurate action-recognition system. Here an action is represented by a dynamical system whose observability matrix is characterized as an element of a Grassmann manifold. To formulate our learning algorithm, we propose two distinct ideas: (1) in the first one we perform classification using a Truncated Wrapped Gaussian model, one for each class in its own tangent space. (2) In the second one we propose a novel learning algorithm that uses a vector representation formed by concatenating local coordinates in tangent spaces associated with different classes and training a linear SVM. We evaluate our approaches on three public 3D action datasets: MSR-action 3D, UT-kinect and UCF-kinect datasets; these datasets represent different kinds of challenges and together help provide an exhaustive evaluation. The results show that our approaches either match or exceed state-of-the-art performance reaching 91.21% on MSR-action 3D, 97.91% on UCF-kinect, and 88.5% on UT-kinect. Finally, we evaluate the latency, i.e. the ability to recognize an action before its termination, of our approach and demonstrate improvements relative to other published approaches.