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

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dc.contributor.advisor Professor Hamid Krim, Committee Chair en_US Aouada, Djamila en_US 2010-04-02T18:54:56Z 2010-04-02T18:54:56Z 2009-04-27 en_US
dc.identifier.other etd-03232009-204551 en_US
dc.description.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. en_US
dc.rights I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dis sertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to NC State University or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report. en_US
dc.subject Manifold theory en_US
dc.subject 3D shape comparison en_US
dc.subject Object classification en_US
dc.subject Differential geometry en_US
dc.title Geometric, Statistical, and Topological Modeling of Intrinsic Data Manifolds: Application to 3D Shapes. en_US PhD en_US dissertation en_US Electrical Engineering en_US

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