Analysis and application of subdivision surfaces

Abstract

Subdivision surfaces are a convenient representation for modeling objects of arbitrary topological type. In this dissertation, we investigate the analysis of a piecewise smooth subdivision scheme, and we apply the scheme to reconstruct objects from non-uniformly sampled data points.Defined as the limit of repeated refinement of a mesh of 3D control points, subdivision surfaces require analysis to establish convergence to a well-defined, tangent plane smooth $(G\sp1)$ surface. Recent research has focused on analyzing smooth surface schemes in which the rules are symmetrical about each vertex and edge. However, a scheme for creating surfaces with sharp features has rules that do not exhibit this symmetry. In this dissertation, we extend the use of eigenanalysis and characteristic maps to analyze a piecewise smooth subdivision scheme that generalizes quartic triangular B-spline surfaces.Subdivision surfaces are suitable for optimized surface fitting and have been used in the reconstruction of objects from 3D data. Previous methods have created accurate representations of objects from dense and uniform data samples. As a practical low cost alternative, we present an algorithm for creating a subdivision surface from data sampled uniformly along closed curves and non-uniformly within the regions they enclose.