🍩 Database of Original & NonTheoretical Uses of Topology
(found 4 matches in 0.001731s)


The Extended Persistent Homology Transform of Manifolds With Boundary (2022)
Katharine Turner, Vanessa Robins, James MorganAbstract
The Extended Persistent Homology Transform (XPHT) is a topological transform which takes as input a shape embedded in Euclidean space, and to each unit vector assigns the extended persistence module of the height function over that shape with respect to that direction. We can define a distance between two shapes by integrating over the sphere the distance between their respective extended persistence modules. By using extended persistence we get finite distances between shapes even when they have different Betti numbers. We use Morse theory to show that the extended persistence of a height function over a manifold with boundary can be deduced from the extended persistence for that height function restricted to the boundary, alongside labels on the critical points as positive or negative critical. We study the application of the XPHT to binary images; outlining an algorithm for efficient calculation of the XPHT exploiting relationships between the PHT of the boundary curves to the extended persistence of the foreground. 
Persistent Homology on Grassmann Manifolds for Analysis of Hyperspectral Movies (2016)
Sofya Chepushtanova, Michael Kirby, Chris Peterson, Lori ZiegelmeierAbstract
The existence of characteristic structure, or shape, in complex data sets has been recognized as increasingly important for mathematical data analysis. This realization has motivated the development of new tools such as persistent homology for exploring topological invariants, or features, in large data sets. In this paper, we apply persistent homology to the characterization of gas plumes in time dependent sequences of hyperspectral cubes, i.e. the analysis of 4way arrays. We investigate hyperspectral movies of LongWavelength Infrared data monitoring an experimental release of chemical simulant into the air. Our approach models regions of interest within the hyperspectral data cubes as points on the real Grassmann manifold Gk,ï źn whose points parameterize the kdimensional subspaces of \$\$\mathbb \R\\textasciicircumn\$\$Rn, contrasting our approach with the more standard framework in Euclidean space. An advantage of this approach is that it allows a sequence of time slices in a hyperspectral movie to be collapsed to a sequence of points in such a way that some of the key structure within and between the slices is encoded by the points on the Grassmann manifold. This motivates the search for topological features, associated with the evolution of the frames of a hyperspectral movie, within the corresponding points on the Grassmann manifold. The proposed mathematical model affords the processing of large data sets while retaining valuable discriminatory information. In this paper, we discuss how embedding our data in the Grassmann manifold, together with topological data analysis, captures dynamical events that occur as the chemical plume is released and evolves. 
Manifold Learning for Coherent Design Interpolation Based on Geometrical and Topological Descriptors (2023)
D. Muñoz, O. Allix, F. Chinesta, J. J. Ródenas, E. NadalAbstract
In the context of intellectual property in the manufacturing industry, knowhow is referred to practical knowledge on how to accomplish a specific task. This knowhow is often difficult to be synthesised in a set of rules or steps as it remains in the intuition and expertise of engineers, designers, and other professionals. Today, a new research line in this concern spotup thanks to the explosion of Artificial Intelligence and Machine Learning algorithms and its alliance with Computational Mechanics and Optimisation tools. However, a key aspect with industrial design is the scarcity of available data, making it problematic to rely on deeplearning approaches. Assuming that the existing designs live in a manifold, in this paper, we propose a synergistic use of existing Machine Learning tools to infer a reduced manifold from the existing limited set of designs and, then, to use it to interpolate between the individuals, working as a generator basis, to create new and coherent designs. For this, a key aspect is to be able to properly interpolate in the reduced manifold, which requires a proper clustering of the individuals. From our experience, due to the scarcity of data, adding topological descriptors to geometrical ones considerably improves the quality of the clustering. Thus, a distance, mixing topology and geometry is proposed. This distance is used both, for the clustering and for the interpolation. For the interpolation, relying on optimal transport appear to be mandatory. Examples of growing complexity are proposed to illustrate the goodness of the method.