Persistent Homology — Filtration & Persistence Diagram

Grow balls of radius ε around random points. Watch connected components (H₀) merge and loops (H₁) form and fill. The persistence diagram records each feature's birth–death pair — long bars = robust topology.

ε (radius)

Point count

H₀ birth H₁ birth dead

Persistence diagram
Each point = (birth ε, death ε) of a topological feature.

Points far from the diagonal persist long — true topology.
Points near diagonal = noise.

H₀: connected components (blue)
H₁: 1-cycles / loops (orange)

Stability theorem (Cohen-Steiner 2007):
d_bottleneck ≤ ‖f − g‖∞

Drag the ε slider to animate the filtration in real time.