4D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Data

We propose a novel framework to learn the spatiotemporal variability in longitudinal 3D shape data sets, which contain observations of subjects that evolve and deform over time. This problem is challenging since surfaces come with arbitrary spatial and temporal parameterizations. Thus, they need to be spatially registered and temporally aligned onto each other. We solve this spatiotemporal registration problem using a Riemannian approach. We treat a 3D surface as a point in a shape space equipped with an elastic metric that measures the amount of bending and stretching that the surfaces undergo. A 4D surface can then be seen as a trajectory in this space. With this formulation, the statistical analysis of 4D surfaces becomes the problem of analyzing trajectories embedded in a nonlinear Riemannian manifold. However, computing spatiotemporal registration and statistics on nonlinear spaces relies on complex nonlinear optimizations. Our core contribution is the mapping of the surfaces to the space of Square-Root Normal Fields (SRNF) where the L2 metric is equivalent to the partial elastic metric in the space of surfaces. By solving the spatial registration in the SRNF space, analyzing 4D surfaces becomes the problem of analyzing trajectories embedded in the SRNF space, which is Euclidean. Here, we develop the building blocks that enable such analysis. These include the spatiotemporal registration of arbitrarily parameterized 4D surfaces even in the presence of large elastic deformations and large variations in their execution rates, the computation of geodesics between 4D surfaces, the computation of statistical summaries, such as means and modes of variation, and the synthesis of random 4D surfaces. We demonstrate the performance of the proposed framework using 4D facial surfaces and 4D human body shapes.

KShapeNet: Riemannian network on Kendall shape space for Skeleton based Action Recognition

Deep Learning architectures, albeit successful in most computer vision tasks, were designed for data with an underlying Euclidean structure, which is not usually fulfilled since pre-processed data may lie on a non-linear space. In this paper, we propose a geometry aware deep learning approach for skeleton-based action recognition. Skeleton sequences are first modeled as trajectories on Kendall's shape space and then mapped to the linear tangent space. The resulting structured data are then fed to a deep learning architecture, which includes a layer that optimizes over rigid and non rigid transformations of the 3D skeletons, followed by a CNN-LSTM network. The assessment on two large scale skeleton datasets, namely NTU-RGB+D and NTU-RGB+D 120, has proven that proposed approach outperforms existing geometric deep learning methods and is competitive with respect to recently published approaches.

Aggregated Pairwise Classification of Statistical Shapes

The classification of shapes is of great interest in diverse areas ranging from medical imaging to computer vision and beyond. While many statistical frameworks have been developed for the classification problem, most are strongly tied to early formulations of the problem - with an object to be classified described as a vector in a relatively low-dimensional Euclidean space. Statistical shape data have two main properties that suggest a need for a novel approach: (i) shapes are inherently infinite dimensional with strong dependence among the positions of nearby points, and (ii) shape space is not Euclidean, but is fundamentally curved. To accommodate these features of the data, we work with the square-root velocity function of the curves to provide a useful formal description of the shape, pass to tangent spaces of the manifold of shapes at different projection points which effectively separate shapes for pairwise classification in the training data, and use principal components within these tangent spaces to reduce dimensionality. We illustrate the impact of the projection point and choice of subspace on the misclassification rate with a novel method of combining pairwise classifiers.