Browsing by Subject "simplicial homology"
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(2021)The topological data analysis studies the shape of a space at multiple scales. Its main tool is persistent homology, which is based on other homology theory, usually simplicial homology. Simplicial homology applies to finite data in real space, and thus it is mainly used in applications. This thesis aims to introduce the theories behind persistent homology and its application, image completion algorithm. Persistent homology is motivated by the question of which scale is the most essential to study data shape. A filtration contains all scales we want to explore, and thus it is an essential tool of persistent homology. The thesis focuses on forming a filtaration from a Delaunay triangulation and its subcomplexes, alpha-complexes. We will found that these provide sufficient tools to consider homology classes birth and deaths, but they are not particularly easy to use in practice. This observation motivates to define a regional complement of the dual alpha graph. We found that its components' and essential homology classes' birth and death times correspond. The algorithm utilize this observation to complete images. The results are good and mainly as could be expected. We discuss that algorithm has potential since it does need any training or other input parameters than data. However, future studies are needed to imply it, for example, in three-dimensional data.
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(2019)We study a subcategory of topological spaces called polyhedrons. In particular, the work focuses on simplicial complexes out of which polyhedrons are constructed. With simplicial complexes we can calculate the homology groups of polyhedrons. These are computationally easier to handle compared to singular homology groups. We start by introducing simplicial complexes and simplicial maps. We show how polyhedrons and simplicial complexes are related. Simplicial maps are certain maps between simplicial complexes. These can be transformed to piecewise linear maps between polyhedrons. We prove the simplicial approximation theorem which states that for any continuous function between polyhedrons we can find a piecewise linear map which is homotopic to the continuous function. In section 4 we study simplicial homology groups. We prove that on polyhedrons the simplicial homology groups coincide with singular homology groups. Next we give an algorithm for calculating the homology groups from matrix presentations of boundary homomorphisms. Also examples of these calculations are given for some polyhedrons. In the last section, we assign an integer called the Lefschetz number for continuous maps from a polyhedron to itself. It is calculated using the induced map between homology groups of the polyhedron. With the help of Hopf trace theorem we can also calculate the Lefschetz number using the induced map between chain complexes of the polyhedron. We prove the Lefschetz fixed-point theorem which states that if the Lefschetz number is not zero, then the continuous function has a fixed-point.
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