🍩 Database of Original & Non-Theoretical Uses of Topology

(found 3 matches in 0.00088s)

  1. Topology of Frame Field Meshing (2020)

    Piotr Beben
    Abstract In the past decade frame fields have emerged as a promising approach for generating hexahedral meshes for CFD and CAE applications. One important problem asks for construction of a boundary aligned frame field with prescribed singularity constraints that correspond to a valid hexahedral mesh. We give a necessary and sufficient condition in terms of solutions to a system of monomial equations whose variables are in the binary octahedral group. Along the way we look at frame field design from an algebraic topological perspective, proving various results, some known, some new.

  2. Persistence Diagrams for Exploring the Shape Variability of Abdominal Aortic Aneurysms (2024)

    Dario Arnaldo Domanin, Matteo Pegoraro, Santi Trimarchi, Maurizio Domanin, Piercesare Secchi
    Abstract Abdominal aortic aneurysm consists of a permanent dilation in the abdominal portion of the aorta and, along with its associated pathologies like calcifications and intraluminal thrombi, is one of the most important pathologies of the circulatory system. The shape of the aorta is among the primary drivers for these health issues, with particular reference to all the characteristics which affect the hemodynamics. Starting from the computed tomography angiography of a patient, we propose to summarize such information using tools derived from Topological Data Analysis, obtaining persistence diagrams which describe the irregularities of the lumen of the aorta. We showcase the effectiveness of such shape-related descriptors with a series of supervised and unsupervised case studies.

  3. A Stable Multi-Scale Kernel for Topological Machine Learning (2015)

    Jan Reininghaus, Stefan Huber, Ulrich Bauer, Roland Kwitt
    Abstract Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes.