Department of Computer Science, State University of New York at Stony Brook
Dissertation for Doctor of Philosophy in Computer Science, August 2005
Even though there has been significant growth in solid and volumetric representations, rapid advances in manufacturing technology, imaging technology, and material science in recent years pose new challenges which have not been fully resolved by the current representation techniques.
In this dissertation, we address the challenges by proposing a novel framework for solid model representation based on subdivision process. We call our approach Multiresolution Solid Objects, or MSO. Within the MSO framework, we represent volumetric objects by a series of new solid subdivision schemes. Subdivision solids share many benefits of subdivision surfaces, yet they have the complexity and the mathematical challenges that are unique to the high dimensional cases.
This dissertation follows our development of MSO framework. We begin with the discussion of general polyhedral mesh structures in 3D space and develop a new structured mesh in 3D based on simplicial complexes, called Octet-truss. The dissertation continues with the derivation of new solid subdivision schemes over the regular meshes, and the generalization to arbitrary tetrahedral and hexahedral meshes. We present (i) a box-spline based approximate solid scheme; (ii) an interpolatory solid scheme over arbitrary complex meshes; (iii) an interpolatory solid scheme over arbitrary hexahedral meshes; and finally (iv) a unified subdivision schemes for multidimensional objects based on box-splines. Each scheme is derived from volumetric splines, a weighted perturbation of linear interpolation, and the Lagrange interpolating polynomials. The derivation is followed by the proving of the convergence and smoothness of the schemes using well-established mathematical techniques.In rare cases, we provide the empirical data which highly suggest the convergence and the smoothness of the schemes.
In addition to the theoretical contributions, this dissertation presents numerous practical implementations to emphasize the benefits of our framework. These include arbitrary shape design, heterogeneous material modeling, free-form design, non-manifold object representation, boundary and feature representation and free-form deformation. Finally, we describe few new applications including high quality partitioning and fitting, and feature preserving volume filtering, which lead toward a wide-range of applications in the future.
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