LAUSR

We propose a novel framework to learn the spatiotemporal variability in longitudinal 3D shape datasets composed of objects that deform over time. This problem is challenging since deforming objects, called 4D surfaces, come with arbitrary spatial parameterizations and evolve at different speeds. Thus, they need to be spatiotemporally registered onto each other. We treat 3D surfaces as a point in a shape space equipped with an elastic metric that measures the amount of bending and stretching the surfaces undergo as they deform. A 4D surface then becomes a trajectory in this space and thus, their statistical analysis becomes the problem of analyzing trajectories embedded in a nonlinear Riemannian manifold. Our core contribution is the mapping of the surfaces to the space of Square-Root Normal Fields where the L2 metric is equivalent to the partial elastic metric in the space of surfaces. By solving the spatial registration in this space, analyzing 4D surfaces becomes the problem of analyzing trajectories embedded in a Euclidean space. We develop the building blocks that enable such analysis. These include the spatiotemporal registration of and computation of geodesics between arbitrarily parameterized 4D surfaces, computation of statistical summaries of 4D surfaces, and the synthesis of 4D surfaces.