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Revisiting Accurate Geometry for Morse-Smale Complexes

Son Le Thanh, Michael Ankele, Tino Weinkauf

TL;DR

Problem: discrete Morse-Smale complexes computed by steepest-descent on uniform grids fail to geometrically approximate the continuous embedding, while probabilistic gradient choices improve geometry but alter topology. Approach: analyze how gradient construction and data sampling affect both geometry and topology, identify fixed minima/maxima and variable saddles, and propose converting a uniform grid into a richer triangle mesh via mid-edge and interior points with a per-cell Delaunay triangulation and a small $\\epsilon$-simplification. Findings: topology can diverge between methods across 2D/3D data, but triangle-grid enrichment yields geometric fidelity close to the continuous case and preserves the overall topology of steepest-descent, demonstrated on synthetic functions and Hurricane Isabel data. Significance: provides a practical, scalable route to faithful Morse-Smale analysis on data sampled on uniform grids, improving visualization and topological data analysis outcomes.

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.

Revisiting Accurate Geometry for Morse-Smale Complexes

TL;DR

Problem: discrete Morse-Smale complexes computed by steepest-descent on uniform grids fail to geometrically approximate the continuous embedding, while probabilistic gradient choices improve geometry but alter topology. Approach: analyze how gradient construction and data sampling affect both geometry and topology, identify fixed minima/maxima and variable saddles, and propose converting a uniform grid into a richer triangle mesh via mid-edge and interior points with a per-cell Delaunay triangulation and a small -simplification. Findings: topology can diverge between methods across 2D/3D data, but triangle-grid enrichment yields geometric fidelity close to the continuous case and preserves the overall topology of steepest-descent, demonstrated on synthetic functions and Hurricane Isabel data. Significance: provides a practical, scalable route to faithful Morse-Smale analysis on data sampled on uniform grids, improving visualization and topological data analysis outcomes.

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.
Paper Structure (15 sections, 6 theorems, 7 equations, 11 figures, 1 table)

This paper contains 15 sections, 6 theorems, 7 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Introduction
  3. Background
  4. Morse function and Morse-Smale complex
  5. Discrete Morse theory
  6. Topological Simplification
  7. Accurate Geometry for Morse-Smale Complexes
  8. Topological Inconsistencies
  9. Identical Parts
  10. Different Parts
  11. Summary
  12. Effect of Sampling Methods
  13. Suggestion for Uniform Grids
  14. Increased computational effort
  15. Conclusion

Key Result

Lemma 1

The homotopy expansion of the $1$-skeleton of each $R({\mathbf v})$ covers all of its $0$-simplices.

Figures (11)

  • Figure 1: The function from \ref{['formula-AnaFunc1']} has been sampled on different types of grids and the Morse-Smale complex has been extracted using different methods. The goal of accurate geometry is to achieve a geometric embedding that is similar to the one from continuous topology. While it is well-known that the steepest descent method is not able to achieve this on uniform grids, we show that it can achieve accurate geometry on certain grid types and provide a suggestion of how to convert uniform grids accordingly.
  • Figure 2: Given the same input data from \ref{['formula-Matrix1']}, the two different methods for computing the discrete gradient field can result in different positions for saddle points as indicated by the red circles. The white spheres, white lines, and squares represent the $0$-cells, $1$-cells, and $2$-cells. The blue and red segments illustrate the pairings between the $0$- and $1$-cells as well as the $1$- and $2$-cells, respectively.
  • Figure 3: Given the same input data from \ref{['formula-Matrix1']}, the two different methods for computing the discrete gradient field can result in a different connectivity of the separatrices. The separatrices connecting to the same critical points are colored black. The separatrices connecting to different minima are highlighted in yellow. The red separatrix only presents for the steepest descent method, and is missing from the output of the probabilistic method.
  • Figure 4: Morse-Smale complexes of the simple 3D data set from \ref{['formula-Tensor 1']} computed using different methods. Many $1$-saddle points (green) have different positions (marked by the orange circles). This also leads to a different connectivity of the separatrices (marked by the red circle).
  • Figure 5: The data set from \ref{['formula-Matrix1']} has been bilinearly interpolated on a $40\times 40$ uniform grid. The Morse-Smale complexes have been extracted using the different methods and we observe connectivity changes for some separatrices (highlighted in red and yellow).
  • ...and 6 more figures

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • Corollary 1
  • Corollary 2