@article{14345, abstract = {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)).}, author = {Edelsbrunner, Herbert and Garber, Alexey and Ghafari, Mohadese and Heiss, Teresa and Saghafian, Morteza}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, publisher = {Springer Nature}, title = {{On angles in higher order Brillouin tessellations and related tilings in the plane}}, doi = {10.1007/s00454-023-00566-1}, year = {2023}, } @inproceedings{10828, abstract = {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.}, author = {Heiss, Teresa and Tymochko, Sarah and Story, Brittany and Garin, Adélie and Bui, Hoa and Bleile, Bea and Robins, Vanessa}, booktitle = {2021 IEEE International Conference on Big Data}, isbn = {9781665439022}, location = {Orlando, FL, United States; Virtuell}, pages = {3824--3834}, publisher = {IEEE}, title = {{The impact of changes in resolution on the persistent homology of images}}, doi = {10.1109/BigData52589.2021.9671483}, year = {2022}, } @inbook{11440, abstract = {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.}, author = {Bleile, Bea and Garin, Adélie and Heiss, Teresa and Maggs, Kelly and Robins, Vanessa}, booktitle = {Research in Computational Topology 2}, editor = {Gasparovic, Ellen and Robins, Vanessa and Turner, Katharine}, isbn = {9783030955182}, pages = {1--26}, publisher = {Springer Nature}, title = {{The persistent homology of dual digital image constructions}}, doi = {10.1007/978-3-030-95519-9_1}, volume = {30}, year = {2022}, } @inproceedings{9345, abstract = {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.}, author = {Edelsbrunner, Herbert and Heiss, Teresa and Kurlin , Vitaliy and Smith, Philip and Wintraecken, Mathijs}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, issn = {1868-8969}, location = {Virtual}, pages = {32:1--32:16}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{The density fingerprint of a periodic point set}}, doi = {10.4230/LIPIcs.SoCG.2021.32}, volume = {189}, year = {2021}, } @article{10071, author = {Adams, Henry and Kourimska, Hana and Heiss, Teresa and Percival, Sarah and Ziegelmeier, Lori}, issn = {1088-9477}, journal = {Notices of the American Mathematical Society}, number = {9}, pages = {1511--1514}, publisher = {American Mathematical Society}, title = {{How to tutorial-a-thon}}, doi = {10.1090/noti2349}, volume = {68}, year = {2021}, } @inproceedings{833, abstract = {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.}, author = {Heiss, Teresa and Wagner, Hubert}, editor = {Felsberg, Michael and Heyden, Anders and Krüger, Norbert}, issn = {03029743}, location = {Ystad, Sweden}, pages = {397 -- 409}, publisher = {Springer}, title = {{Streaming algorithm for Euler characteristic curves of multidimensional images}}, doi = {10.1007/978-3-319-64689-3_32}, volume = {10424}, year = {2017}, }