🍩 Database of Original & Non-Theoretical Uses of Topology
(found 3 matches in 0.000843s)
A Multi-Parameter Persistence Framework for Mathematical Morphology (2021)Yu-Min Chung, Sarah Day, Chuan-Shen Hu
AbstractThe field of mathematical morphology offers well-studied techniques for image processing. In this work, we view morphological operations through the lens of persistent homology, a tool at the heart of the field of topological data analysis. We demonstrate that morphological operations naturally form a multiparameter filtration and that persistent homology can then be used to extract information about both topology and geometry in the images as well as to automate methods for optimizing the study and rendering of structure in images. For illustration, we apply this framework to analyze noisy binary, grayscale, and color images.
Euler Characteristic Surfaces (2021)Gabriele Beltramo, Rayna Andreeva, Ylenia Giarratano, Miguel O. Bernabeu, Rik Sarkar, Primoz Skraba
AbstractWe study the use of the Euler characteristic for multiparameter topological data analysis. Euler characteristic is a classical, well-understood topological invariant that has appeared in numerous applications, including in the context of random fields. The goal of this paper is to present the extension of using the Euler characteristic in higher-dimensional parameter spaces. While topological data analysis of higher-dimensional parameter spaces using stronger invariants such as homology continues to be the subject of intense research, Euler characteristic is more manageable theoretically and computationally, and this analysis can be seen as an important intermediary step in multi-parameter topological data analysis. We show the usefulness of the techniques using artificially generated examples, and a real-world application of detecting diabetic retinopathy in retinal images.