  Nowadays, big data of digital media (including images, videos and 3D
graphical models) are frequently modeled as low-dimensional manifold meshes
embedded in a high-dimensional feature space. In this paper, we summarized our
recent work on geodesic centroidal Voronoi tessellations(GCVTs), which are
intrinsic geometric structures on manifold meshes. We show that GCVT can find a
widely range of interesting applications in computer vision and graphics, due
to the efficiency of search, location and indexing inherent in these intrinsic
geometric structures. Then we present the challenging issues of how to build
the combinatorial structures of GCVTs and establish their time and space
complexities, including both theoretical and algorithmic results.
