138 research outputs found
Applications of Computational Homology
Homology is a field of topology that classifies objects based on the number of n- dimensional holes (cuts, tunnels, voids, etc.) they possess. The number of its real life ap- plications is quickly growing, which requires development of modern computational meth- ods. In my thesis, I will present methods of calculation, algorithms, and implementations of simplicial homology, alpha shapes, and persistent homology.
The Alpha Shapes method represents a point cloud as the union of balls centered at each point, and based on these balls, a complex can be built and homology computed. If the balls are allowed to grow, one can compute the persistent homology, which gives a better understanding of the shape of the object represented by the point cloud by eliminating noise.
These methods are particularly well suited for studying biological molecules. I will test the hypothesis that persistent homology can describe some important features of a protein\u27s shape
Algebraic Topology
The chapter provides an introduction to the basic concepts of Algebraic
Topology with an emphasis on motivation from applications in the physical
sciences. It finishes with a brief review of computational work in algebraic
topology, including persistent homology.Comment: This manuscript will be published as Chapter 5 in Wiley's textbook
\emph{Mathematical Tools for Physicists}, 2nd edition, edited by Michael
Grinfeld from the University of Strathclyd
Computing Persistent Homology within Coq/SSReflect
Persistent homology is one of the most active branches of Computational
Algebraic Topology with applications in several contexts such as optical
character recognition or analysis of point cloud data. In this paper, we report
on the formal development of certified programs to compute persistent Betti
numbers, an instrumental tool of persistent homology, using the Coq proof
assistant together with the SSReflect extension. To this aim it has been
necessary to formalize the underlying mathematical theory of these algorithms.
This is another example showing that interactive theorem provers have reached a
point where they are mature enough to tackle the formalization of nontrivial
mathematical theories
Advanced homology computation of digital volumes via cell complexes
Given a 3D binary voxel-based digital object V, an algorithm for computing homological information for V via a polyhedral cell complex is designed. By homological information we understand not only Betti numbers, representative cycles of homology classes and homological classification of cycles but also the computation of homology numbers related additional algebraic structures defined on homology (coproduct in homology, product in cohomology, (co)homology operations,...). The algorithm is mainly based on the following facts: a) a local 3D-polyhedrization of any 2×2×2 configuration of mutually 26-adjacent black voxels providing a coherent cell complex at global level; b) a description of the homology of a digital volume as an algebraic-gradient vector field on the cell complex (see Discrete Morse Theory [5], AT-model method [7,5]). Saving this vector field, we go further obtaining homological information at no extra time processing cost
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