Squigraphs for Fine and Compact Modeling of 3-D Shapes

We propose to superpose global topological and local geometric 3-D shape descriptors in order to define one compact and discriminative representation for a 3-D object. While a number of available 3-D shape modeling techniques yield satisfactory object classification rates, there is still a need for a refined and efficient identification/recognition of objects among the same class. In this paper, we use Morse theory in a two-phase approach. To ensure the invariance of the final representation to isometric transforms, we choose the Morse function to be a simple and intrinsic global geodesic function defined on the surface of a 3-D object. The first phase is a coarse representation through a reduced topological Reeb graph. We use it for a meaningful decomposition of shapes into primitives. During the second phase, we add detailed geometric information by tracking the evolution of Morse function´s level curves along each primitive. We then embed the manifold of these curves into ??³, and obtain a single curve. By combining phase one and two, we build new graphs rich in topological and geometric information that we refer to as squigraphs. Our experiments show that squigraphs are more general than existing techniques. They achieve similar classification rates to those achieved by classical shape descriptors. Their performance, however, becomes clearly superior when finer classification and identification operations are targeted. Indeed, while other techniques see their performances dropping, squigraphs maintain a performance rate of the order of 97%.