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Summer Research Fellowship Programme of India's Science Academies

Understanding Topological Methods for High Dimensional Data using Nerves, Mappers, Multiscale Mappers and Reeb Spaces

Maanvi Nunna

PES University, Bengaluru, 560085

Dr. Amit Chattopadhyay

International Institute of Information Technology, Bengaluru, 560100

Abstract

Topology is essentially the study of properties of space that are invariant even after certain transformations. Our aim is to better understand topology of higher dimensional and/or multivariate data by understanding the different work already done in this area. Initially the goal was to understand the working of the Marching Cubes algorithm, which is mainly used to reconstruct surfaces. The Marching Cubes algorithm is used for creating a triangular mesh from an implicit function representing an isosurface. The main principle is to iterate over a uniform grid of cubes and compute which of the vertices lie above or below the surface. The final mesh is the union of all the triangles. Once the images of every slice of the object has been masked (bit-wise mask for example), these can be used to reconstruct the original surface using the aforementioned algorithm. It is fast, accurate and works with arbitrarily shaped objects. The ambiguities posed in the marching cubes algorithm have been resolved by the Marching Tetrahedra algorithm. Next, the study is extended to the understanding of Mapper, which is essentially an algorithm or method used for topological analysis for shape and data analysis. A computational method is used to derive information about the higher dimensional data and reduce it to simplicial complexes. It is based on partial clustering achieved by a set of functions called filters. After deciding the filter functions which will decide the parameter space, the intervals and the percentage overlap between the intervals are parametrized. Then the inverse function is used on the covers of the parameter space, which results in clusters in the domain. A simplex is formed based on how many clusters intersect. All the simplices of the domain together form a simplicial complex or nerve of the domain. The simplicial complex resulting as the output from Mapper is analysed for the topological structures to understand and use them in practical applications.

Keywords: Marching cubes algorithm, isosurface, Mapper, parameter space, nerve, simplicial complex

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