Construction and analysis of subdivision schemes from a linear algebra perspective.

Subdivision schemes are efficient tools for generating smooth curves and surfaces as limit of an iterative algorithm based on simple refinement rules starting from few control points defining a polyline or a mesh. Aim of this thesis is to give a complete framework regarding the tools used for the analysis of subdivision schemes and to exploit them to construct new subdivision schemes. We focus our attention on some linear algebra structures that allow to give an exhaustive characterization on the analysis of convergence and smoothness of the limit curves and surfaces produced. Moreover, we propose general sufficient conditions to check the convergence of non-stationary subdivision schemes on arbitrary manifold topology meshes, exploiting the eigenproperties of a block-circulant matrix. These linear algebra tools are fundamental for the construction and analysis of subdivision schemes on arbitrary manifold topology meshes. The use of this kind of meshes is extremely important: regular meshes do not allow us to design the complex models used in computer aided design as well as in biomedical imaging segmentation. Moreover, non-stationary subdivision schemes allow us to design particular shapes such as ellipsoids and tori, thanks to their capability of generating exponential polynomials. In the univariate setting, to work out necessary and sufficient conditions for the Cr continuity of a subdivision scheme, we should exploit the joint spectral radius of a set of matrices.