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Connected Components on Lie Groups and Applications to Multi-Orientation Image Analysis.

Nicky J van den Berg1, Olga Mula2, Leanne Vis1

  • 1Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands.

Journal of Mathematical Imaging and Vision
|March 30, 2026
PubMed
Summary
This summary is machine-generated.

A new algorithm identifies connected components in compact sets on Lie groups using a distance threshold (δ). This method effectively distinguishes crossing structures and groups aligned components, outperforming standard algorithms for applications like retinal vascular tree analysis.

Keywords:
Connected componentsImage analysisLie groupsMedical image analysisMorphological operatorsVessel tree identification

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Area of Science:

  • Computational geometry
  • Differential geometry
  • Image analysis

Background:

  • Connected components analysis is crucial for understanding complex structures in various scientific fields.
  • Standard algorithms often struggle with overlapping or crossing structures, particularly in non-Euclidean spaces.
  • Lie groups offer a powerful framework for analyzing data with inherent symmetries, such as rotations and translations.

Purpose of the Study:

  • To develop and analyze a novel algorithm for finding δ-connected components of compact sets in Lie groups.
  • To demonstrate the algorithm's ability to group structures based on proximity and alignment.
  • To apply the algorithm to identify vascular structures in retinal images, differentiating crossings and grouping aligned segments.

Main Methods:

  • An iterative algorithm involving morphological dilations with Hamilton-Jacobi-Bellman kernels on a Lie group G.
  • Utilizing δ-thickened sets and persistence diagrams to determine the optimal δ value.
  • Employing affinity matrices for grouping δ-connected components based on local properties.
  • Applying an orientation score transform to retinal images, mapping them to the Lie group SE(2).

Main Results:

  • The algorithm converges in a finite number of steps.
  • The method successfully identifies δ-connected components that distinguish between crossing and non-crossing structures.
  • Application to retinal images reveals efficient identification of vascular tree branches, including overlapping ones.
  • The approach offers advantages over standard connected component algorithms in R² for complex structural analysis.

Conclusions:

  • The developed algorithm provides an effective method for analyzing connected components in Lie groups.
  • This approach offers superior performance in identifying and grouping complex structures compared to traditional methods.
  • The algorithm shows significant potential for applications in medical image analysis and other fields involving geometric data.