Geometric, Statistical, and Topological Modeling of Intrinsic Data Manifolds: Application to 3D Shapes.

Abstract

The increasing size and complexity of data often invokes the extraction of information from their reduced representations while preserving their inherent structure. In this thesis, we explore the statistical, geometric and topological intrinsic information contained in high dimensional data. We focus on applications related to 3-dimensional objects, and model their 2-dimensional surfaces using compact curved-skeletal models that we refer to as “squigraphs†. These models are multi-level representations that superpose global topological and local geometric 3D shape descriptors. Squigraphs are subsequently used for classification, and ensure a high discrimination between in-class 3-dimensional shapes. The extraction of squigraphs starts by sampling the surface of an object for a resulting set of curves. This may be accomplished by defining an appropriate intrinsic characteristic function on the surface itself, referred to as a Morse function; which we use in a two-phase approach. To ensure the invariance of the final representation to isometric transforms, we choose the Morse function to be an intrinsic global geodesic function. The first phase is a coarse representation through a reduced topological Reeb graph. We use it for a meaningful decomposition of shapes into primitives. At 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 corresponding to this evolution of curves into R3, and obtain a simple space curve. We further define a Riemannian metric to quantitatively compare the geometry of shapes. We point the flexibility of our techniques for other applications, namely, face recognition, behavioral modeling, and sensor network data analysis. While all these applications face the same curse of dimensionality, we show that they may be formalized under similar geometrical settings.