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Applications of Computational Topology by Christopher Johnson

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Applications of Computational Topology by Christopher Johnson.

Masters Thesis, 2006

Abstract. Homology is a field of mathematics that classifies objects based on the number of n-dimensional holes (cuts, tunnels, voids, etc) they possess. The number of its real life applications is quickly growing which requires development of modern computational methods. In my thesis I will present results in two areas of research: Alpha Shapes and Cubical Homology. The Alpha Shapes method represents a point cloud as the union of balls centered at each point. Based on the mutual intersection of these balls a collection of simplices (i.e., points, lines, triangles, tetrahedra) called a simplicial complex is then built and its homology is computed. If the balls are allowed to grow, one can compute the so-called persistent homology, which gives a better idea of the shape of the space than homology alone. This method is particularly well suited for studying biological molecules as collections of balls. In particular, I will present algorithms and computer implementations of Alpha Shapes and Persistent Homology and apply them to the study of proteins. Cubical Homology is a recent approach that uses elementary squares or cubes, rather than the classical simplices, to build a complex. This gives us a natural way to study digital images, since they are stored as collections of pixels on a rectangular grid. This method also allows us to bypass the computationally intensive process of building a simplicial complex. Using persistent homology allows us to locate important features in the image and simplify it. In higher dimensional cases the method will provide topological information even when visualization of images is impossible.