@InProceedings{lethanh24,
  author    = {Le Thanh, Son and Ankele, Michael and Weinkauf, Tino},
  booktitle = {IEEE Topological Data Analysis and Visualization (TopoInVis)},
  title     = {Revisiting Accurate Geometry for Morse-Smale Complexes},
  year      = {2024},
  address   = {Tampa, FL, USA},
  month     = oct,
  pages     = {34-43},
  abstract  = {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.},
  day       = {14},
  doi       = {10.1109/TopoInVis64104.2024.00008},
  keywords  = {Geometry;Visualization;Accuracy;Data analysis;Shape;Feature extraction;Sampling methods;Vectors;Topology;Standards;Discrete Morse theory;Morse-Smale complex;Topology;Accurate geometry},
  url = {http://tinoweinkauf.net/publications/abslethanh24.html},
}