Multiresolution in geometric modelling : subdivision mark points and ternary subdivision

• Main Author: Hassan, Mohamed Fathih
• Published: University of Cambridge 2005
• Subjects: 510
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.603847

The term <i>multiresolution </i>captures the concept of representing a single mathematical model in several levels of detail or accuracy. The 3D geometric models we have in mind are organized sets of point data, typically in the form of triangle meshes, such as those used to represent terrain models or free-form surfaces in computer graphics. There are two obvious approaches to creating a multiresolution model. The first is to have a low resolution model from which we create a higher resolution one. For this we can use recursive subdivision and remeshing techniques. Conversely, we may have a high resolution model and desire a lower one. For this we can use discrete fairing and mesh decimation. This thesis focuses on recursive subdivision. We describe two new methods for analysing subdivision schemes. The first of which, mark point analysis, extends the established spectral analysis of the subdivision matrix at a vertex to the other mark points of the scheme. This provides additional necessary conditions for smoothness that we can impose on a scheme. We have used this method to help us derive two new families of subdivision schemes, both in the univariate and bivariate settings. The second method extends the established univariate theory to obtain sufficient conditions for smoothness from the binary case to the ternary case, and we posit how this would extend to any arity. This not only extends the method for proving the continuity class, but also provides an estimate for the Hölder exponent. We also present four new subdivision schemes. First, we present two new families of univariate interpolating subdivision schemes. These are then extended into the bivariate setting to give two new families of interpolating schemes on the triangular mesh. We establish their major properties using the smoothness analyses presented earlier and other techniques. Both the methods and the schemes have opened up interesting avenues for further research.