TY - JOUR AB - For a locally finite set in R2, the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in R2 is locally finite, coarsely dense, and generic with probability one. For such a set, the distributions of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-1 Delaunay mosaics given by Miles (Math. Biosci. 6, 85–127 (1970)). AU - Edelsbrunner, Herbert AU - Garber, Alexey AU - Ghafari, Mohadese AU - Heiss, Teresa AU - Saghafian, Morteza ID - 14345 JF - Discrete and Computational Geometry SN - 0179-5376 TI - On angles in higher order Brillouin tessellations and related tilings in the plane ER - TY - CHAP AB - To compute the persistent homology of a grayscale digital image one needs to build a simplicial or cubical complex from it. For cubical complexes, the two commonly used constructions (corresponding to direct and indirect digital adjacencies) can give different results for the same image. The two constructions are almost dual to each other, and we use this relationship to extend and modify the cubical complexes to become dual filtered cell complexes. We derive a general relationship between the persistent homology of two dual filtered cell complexes, and also establish how various modifications to a filtered complex change the persistence diagram. Applying these results to images, we derive a method to transform the persistence diagram computed using one type of cubical complex into a persistence diagram for the other construction. This means software for computing persistent homology from images can now be easily adapted to produce results for either of the two cubical complex constructions without additional low-level code implementation. AU - Bleile, Bea AU - Garin, Adélie AU - Heiss, Teresa AU - Maggs, Kelly AU - Robins, Vanessa ED - Gasparovic, Ellen ED - Robins, Vanessa ED - Turner, Katharine ID - 11440 SN - 9783030955182 T2 - Research in Computational Topology 2 TI - The persistent homology of dual digital image constructions VL - 30 ER - TY - CONF AB - Digital images enable quantitative analysis of material properties at micro and macro length scales, but choosing an appropriate resolution when acquiring the image is challenging. A high resolution means longer image acquisition and larger data requirements for a given sample, but if the resolution is too low, significant information may be lost. This paper studies the impact of changes in resolution on persistent homology, a tool from topological data analysis that provides a signature of structure in an image across all length scales. Given prior information about a function, the geometry of an object, or its density distribution at a given resolution, we provide methods to select the coarsest resolution yielding results within an acceptable tolerance. We present numerical case studies for an illustrative synthetic example and samples from porous materials where the theoretical bounds are unknown. AU - Heiss, Teresa AU - Tymochko, Sarah AU - Story, Brittany AU - Garin, Adélie AU - Bui, Hoa AU - Bleile, Bea AU - Robins, Vanessa ID - 10828 SN - 9781665439022 T2 - 2021 IEEE International Conference on Big Data TI - The impact of changes in resolution on the persistent homology of images ER - TY - JOUR AU - Adams, Henry AU - Kourimska, Hana AU - Heiss, Teresa AU - Percival, Sarah AU - Ziegelmeier, Lori ID - 10071 IS - 9 JF - Notices of the American Mathematical Society SN - 0002-9920 TI - How to tutorial-a-thon VL - 68 ER - TY - CONF AB - Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functionsthat facilitates the efficient search for new materials and material properties. We prove invarianceunder isometries, continuity, and completeness in the generic case, which are necessary featuresfor the reliable comparison of crystals. The proof of continuity integrates methods from discretegeometry and lattice theory, while the proof of generic completeness combines techniques fromgeometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and relatedinclusion-exclusion formulae. We have implemented the algorithm and describe its application tocrystal structure prediction. AU - Edelsbrunner, Herbert AU - Heiss, Teresa AU - Kurlin , Vitaliy AU - Smith, Philip AU - Wintraecken, Mathijs ID - 9345 SN - 1868-8969 T2 - 37th International Symposium on Computational Geometry (SoCG 2021) TI - The density fingerprint of a periodic point set VL - 189 ER - TY - CONF AB - We present an efficient algorithm to compute Euler characteristic curves of gray scale images of arbitrary dimension. In various applications the Euler characteristic curve is used as a descriptor of an image. Our algorithm is the first streaming algorithm for Euler characteristic curves. The usage of streaming removes the necessity to store the entire image in RAM. Experiments show that our implementation handles terabyte scale images on commodity hardware. Due to lock-free parallelism, it scales well with the number of processor cores. Additionally, we put the concept of the Euler characteristic curve in the wider context of computational topology. In particular, we explain the connection with persistence diagrams. AU - Heiss, Teresa AU - Wagner, Hubert ED - Felsberg, Michael ED - Heyden, Anders ED - Krüger, Norbert ID - 833 SN - 03029743 TI - Streaming algorithm for Euler characteristic curves of multidimensional images VL - 10424 ER -