The Chordal Distance Transform of Geometric Loops and its Persistent Homology

Abstract

We present an isometry and parametrisation invariant of embeddings of $S^1$ into Euclidean space. We do so by representing the distance between pairs of points on the embedded circle as a function on a Möbius band, the two-point finite subset space of $S^1$. We call this function the chordal distance transform of the embedding. We show that the sublevel set persistent homology of the chordal distance transform satisfies the desired isometry and parametrisation invariance, and is a continuous transform with respect to the Whitney topology on the space of circle embeddings and the bottleneck distance in the space of persistence diagrams. We then considered the generic behaviour of the chordal distance transform for $C^2$ and finite piecewise linear embeddings. In the $C^2$-case, we show that non-boundary critical points of the chordal distance transform are finite and non-degenerate on an open and dense subset of circle embeddings. Consequently, its persistent homology is pointwise finite dimensional for generic $C^2$-embeddings. In the finite piecewise linear case, we also find piecewise-continuous analogues of non-degenerate critical points, and give generic conditions for the homological critical points of the chordal distance transform to be non-degenerate. In order to gain a geometric interpretation of the chordal distance transform and its persistent homology, we give a geometric characterisation of the $C^2$ and finite piecewise linear non-degenerate critical points. Finally, we consider how the chordal distance transform can be generalised to capture geometric features involving $n\geq 2$ points on an embedded shape, as a function on the $n$-point finite subset space.

Figures (9)

• Figure 1: The chordal distance transform transforms a geometric loop represented by an embedding $\gamma$ into a function $\mathcal{T}_\gamma$ on the Möbius band. This gives an isometry invariant representation of our geometric loop. The persistent homology of the sublevel set filtration of $\mathcal{T}_\gamma$ yields a summary of the geometry of the loop that is invariant with respect to reparametrisations of $\gamma$.
• Figure 2: We derive the chordal distance transform of a circle embedding $\gamma$ from the distance matrix of pairs of points, as a function $d: S^1 \times S^1 \to \mathop{\mathrm{\mathbb{R}}}\nolimits$. Since the distance matrix is symmetric with respect to permutation $d(z_1,z_2) = d(z_2,z_1)$ (visually represented by reflecting across the diagonal), we remove the redundant information introduced by symmetry by only considering the distance evaluated on pairs of points on the circle, modulo permutation $\{z_1, z_2\}$. This gives us a function $\mathsf{T}_\gamma$ on the two-point subset space $\mathbf{F}_{2}(S^1)$ of $S^1$, which is homeomorphic to the Möbius band $\mathcal{M}$. We call this function the chordal distance transform of $\gamma$, as each $\{z_1, z_2\}$ in $\mathbf{F}_{2}(S^1)$ corresponds to a chord on the circle, and $\mathsf{T}_\gamma(\{z_1, z_2\})$ encodes the length of each chord on the embedded loop. On the left, we illustrate the distance matrix of an ellipse $\gamma$ with major axis = 2, minor axis = 1. On the right, we illustrate the chordal distance trasnform $\mathsf{T}_\gamma: \mathbf{F}_{2}(S^1) \to \mathop{\mathrm{\mathbb{R}}}\nolimits$ as derived from $d$.
• Figure 3: Heat map representations of the chordal distance transform of a circle (\ref{['fig:dist3']}) and an ellipse (\ref{['fig:dist4']}), as a function on the Möbius band. The ellipse here has major axis $a=8$ and minor axis $b=1$.
• Figure 4: Here we see $\mathrm{UConf}_{3}((0,1))$ as cut out by the planes, $x=y$, $x=z$ and $y=z$, where we consider $x<y<z$. This is homeomorphic to the open 3-simplex as described in \ref{['ex:confint']}.
• Figure 5: A cut and gluing argument showing that $\mathbf{F}_{2}(S^1) = (S^1 \times S^1) / S_2$ is the Möbius band. Expressing the torus as a square with opposite sides identified, the $S_2$ action 'folds' the torus across the dashed diagonal. The resultant quotient space is the triangle with the indicated pair of edges identified with the orientation given in the diagram. To see that this space is homeomorphic to a Möbius band $\mathcal{M}$, we cut along the opposite diagonal giving us two triangles and another identification. We then identify the vertical and horizontal sides of the original triangle to obtain the square on the right, with a pair of opposing edges identified with opposite orientation.

Theorems & Definitions (117)

• Definition 1.1
• Remark 1.2
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Example 2.4
• Example 2.5
• Remark 2.6
• Definition 2.7
• Definition 2.8