Vietoris-Rips filtration, barcodes, and persistence diagrams
H₀ components: — | H₁ loops: —
Vietoris-Rips filtration: connect points within distance ε, then increase ε from 0 to ∞. Birth/death of connected components (H₀, blue) and loops (H₁, orange) are tracked. Long bars = real topology; short bars = noise. The persistence diagram (birth vs death) shows long-lived features far from the diagonal. Circle → one long H₁ bar at ε ≈ radius.