On curve reconstruction in Riemannian Manifolds: ordering motion frames

Abstract

In this work we generalize the computational geometric curve reconstruction approach to curves embedded in Riemannian manifolds. We prove that the minimum spanning tree, given a sufficiently dense sample, correctly reconstructs smooth arcs which can be used to reconstruct simple closed curves in Riemannian manifolds. The proof is based on the behavior of a curve segment inside a tubular neighborhood of the curve. To take care of the local topological structure of the underlying manifold, a tubular neighborhood is constructed using the injectivity radius of the underlying Riemannian manifold. We also present examples of successfully reconstructed curves and apply curve reconstruction to ordering motion frames. To give a specific example, think of a graphic game designer designing a game. To design a path of an object and the way the object moves along that path he must first create a sequence of orientations and displacements in the space. A typical method of animation is to begin with the first frame and the last frame. The graphic designer will create in between frames iteratively. For the movements along the path, he may create intermediate frames in an order which best suits his imagination. Now he provides these frames to an interpolator. At this stage he is also required to provide an ordering of the frames to the interpolator. Results presented in this work provide a way to automate the process of ordering the frames created by a graphic designer. In this work we present a uniform sampling criterion, an ordering algorithm and an interpolation scheme that reconstructs an approximation to the original motion. In addition an attempt has been made here to generalize the computational geometric curve reconstruction approach to curved spaces (Riemannian Manifolds). This problem is at the junction of Computational Geometry and Differential Geometry