Bounded Approximations of Geodesics for Triangular Manifolds with Partially Missing Data

In this paper we present an algorithm to compute approximate geodesic distances on a triangular manifold S containing n vertices with partially missing data. The proposed method computes an approximation of the geodesic distance between two vertices pi and p j on S and provides a maximum relative error bound of the approximation. The error bound is shown to be worst-case optimal. The algorithm approximates the geodesic distance without trying to reconstruct the missing data by embedding the surface in a low dimensional space via multi-dimensional scaling (MDS). We derive a new method to add an object to the embedding computed via least-squares MDS.