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

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.