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Revisiting Accurate Geometry for Morse-Smale Complexes
KTH, School of Electrical Engineering and Computer Science (EECS), Computer Science, Computational Science and Technology (CST).ORCID iD: 0009-0002-8825-0452
Indurad.
KTH, School of Electrical Engineering and Computer Science (EECS), Computer Science, Computational Science and Technology (CST).ORCID iD: 0000-0002-1498-9062
2024 (English)In: Proceedings - 2024 Topological Data Analysis and Visualization, TopoInVis 2024, Institute of Electrical and Electronics Engineers (IEEE) , 2024, p. 34-43Conference paper, Published paper (Refereed)
Abstract [en]

The Morse-Smale complex is a standard tool in visual data analysis. The classic definition is based on a continuous view of the gradient of a scalar function where its zeros are the critical points. These points are connected via gradient curves and surfaces emanating from saddle points, known as separatrices. In a discrete setting, the Morse-Smale complex is commonly extracted by constructing a combinatorial gradient assuming the steepest descent direction. Previous works have shown that this method results in a geometric embedding of the separatrices that can be fundamentally different from those in the continuous case. To achieve a similar embedding, different approaches for constructing a combinatorial gradient were proposed. In this paper, we show that these approaches generate a different topology, i.e., the connectivity between critical points changes. Additionally, we demonstrate that the steepest descent method can compute topologically and geometrically accurate Morse-Smale complexes when applied to certain types of grids. Based on these observations, we suggest a method to attain both geometric and topological accuracy for the Morse-Smale complex of data sampled on a uniform grid.

Place, publisher, year, edition, pages
Institute of Electrical and Electronics Engineers (IEEE) , 2024. p. 34-43
Keywords [en]
Accurate geometry, Discrete Morse theory, Morse-Smale complex, Topology
National Category
Computer graphics and computer vision
Identifiers
URN: urn:nbn:se:kth:diva-358219DOI: 10.1109/TopoInVis64104.2024.00008ISI: 001454374200004Scopus ID: 2-s2.0-85212879720OAI: oai:DiVA.org:kth-358219DiVA, id: diva2:1924853
Conference
2024 IEEE Topological Data Analysis and Visualization, TopoInVis 2024, St. Pete Beach, United States of America, Oct 13 2024 - Oct 14 2024
Note

Part of ISBN 9798331528447

QC 20250113

Available from: 2025-01-07 Created: 2025-01-07 Last updated: 2025-12-08Bibliographically approved

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Le Thanh, SonWeinkauf, Tino

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