Persistent Homology Gallery: Vietoris-Rips Barcodes of a Noisy Torus Point Cloud, Betti Numbers and Stability (3D)

Setting up the gallery...

Presets

Live readouts

Barcode (top: H1 loops, bottom: H0 pieces). Yellow dashed line = current scale.

Graphs

Persistence diagram: birth against death (points far above the diagonal are real)

Betti curves: pieces beta0 and loops beta1 against the scale

Stability: bottleneck distance of the barcode against the size of the perturbation

Numerical checks

running...

How to use

Pick a preset, then drag the scale epsilon slider (or let it grow). Each bead gets a ball of radius epsilon/2; pairs within epsilon are joined by a thread, and three mutually joined beads form a filled triangle. A loop that has been born and not yet filled is a one-dimensional hole; the gold and cyan tubes mark the longest ones. On the barcode a long bar is a real feature, a short grey one is sampling noise: raise Significance to hide more of it. Try Torus, too few points and Noisy torus to see when the answer is lost, and run the stability sweep to see that a small wobble of every point only moves the barcode by at most twice as much. Space pauses; the guided tour flies through the gallery.

The science