Topological Persistence — Point Cloud Homology

Vietoris-Rips filtration: track birth/death of H₀ (components) and H₁ (loops) as ε grows

Point cloud + VR complex at ε

Persistence barcode (H₀=blue, H₁=red)

ε (filtration) 0.20

Point count 20

Shape

H₀ (components): —

H₁ (loops): —

ε: —

Persistent homology (Edelsbrunner-Letscher-Zomorodian 2002): track topological features across all scales. H₀ intervals: connected components (born at 0, die when merged). H₁ intervals: loops (born/die as simplicial complex forms/fills). Long bars = real topology.