| Spaces arise naturally in the process of solving problems in many fields of inquiry. This thesis focuses on capturing and understanding the topological properties of spaces. As such, the contributions of this thesis belong to the emerging field of computational topology. This thesis advocates a combinatorial approach, utilizing geometry-encoded filtrations as the input for the algorithms. The approach includes using simplicial homology for capturing connectivity.; Using this approach, the main contributions of this thesis are: (1) a new measure of importance for topological attributes called persistence, (2) extension of Morse complexes for piece-wise linear 2-manifolds, (3) and extension of the linking number invariant for simplicial complexes.; The thesis also gives algorithms for computing the theoretically defined measures or structures in each case. Using persistence, we may distinguish between topological noise and features of a space. This differentiation enables us to simplify a space topologically. To denoise two-dimensional density functions, we first construct Morse complexes over their underlying space. Applying persistence, we create a hierarchy of progressively coarser Morse complexes. The thesis describes implementations of the algorithms and presents experimental evidence of their feasibility on a variety of data. |
