Geometry-driven Collapses for Converting a Čech Complex into a Triangulation of a Nicely Triangulable Shape

Abstract

Given a set of points that sample a shape, the Rips complex of the points is often used to provide an approximation of the shape easily-computed. It has been proved that the Rips complex captures the homotopy type of the shape, assuming that the vertices of the complex meet some mild sampling conditions. Unfortunately, the Rips complex is generally high-dimensional. To remedy this problem, it is tempting to simplify it through a sequence of collapses. Ideally, we would like to end up with a triangulation of the shape. Experiments suggest that, as we simplify the complex by iteratively collapsing faces, it should indeed be possible to avoid entering a dead end such as the famous Bing’s house with two rooms. This paper provides a theoretical justification for this empirical observation. We demonstrate that the Rips complex of a point cloud (for a well-chosen scale parameter) can always be turned into a simplicial complex homeomorphic to the shape by a sequence of collapses, assuming that the shape is nicely triangulable and well-sampled (two concepts we will explain in the paper). To establish our result, we rely on a recent work which gives conditions under which the Rips complex can be converted into a Čech complex by a sequence of collapses. We proceed in two phases. Starting from the Čech complex, we first produce a sequence of collapses that arrives to the Čech complex, restricted by the shape. We then apply a sequence of collapses that transforms the result into the nerve of some covering of the shape. Along the way, we establish results which are of independent interest. First, we show that the reach of a shape cannot decrease when intersected with a (possibly infinite) collection of balls, assuming the balls are small enough. Under the same hypotheses, we show that the restriction of a shape with respect to an intersection of balls is either empty or contractible. We also provide conditions under which the nerve of a family of compact sets undergoes collapses as the compact sets evolve over time. We believe conditions are general enough to be useful in other contexts as well.

Appendix: \(C^{1,1}\) diffeomorphisms preserve nicely triangulable manifolds

The goal of this section is to prove Theorem 6.

Proof of Theorem 6

Let $x \in M$. Since M is a compact $C^{1,1} k$-manifold embedded in $\mathbb {R}^d$, there exists a k-dimensional affine space $T_M(x) \subset \mathbb {R}^d$ tangent to M at x. Let $\pi _x : M \rightarrow T_M(x)$ be the orthogonal projection onto the tangent space $T_M(x)$ and let $\pi _{\Phi (x)}' : M' \rightarrow T_{M'}(\Phi (x))$ the orthogonal projection onto $T_{M'}(\Phi (x))$. Since M and $M'$ are compact, we can find two constants K and $K'$ independent of x such that:

$$\begin{aligned} \forall y \in M, \quad&\Vert y-\pi _x(y)\Vert < K \Vert y-x\Vert ^2, \end{aligned}$$

(1)

$$\begin{aligned} \forall y \in M', \quad&\Vert y-\pi _{\Phi (x)}'(y)\Vert < K' \Vert y-\Phi (x)\Vert ^2. \end{aligned}$$

(2)

Given $t_0 > 0$, we consider the open set $U_x = M \cap {\text {B}}^\circ (x, t_0)$ and adjust $t_0$ in such a way that

• 1. The restriction $\pi _x : U_x \rightarrow \pi _x(U_x)$ is a homeomorphism for all $x \in M;$

• 2. The restriction $\pi _{\Phi (x)}' : \Phi (U_x) \rightarrow \pi _{\Phi (x)}'(\Phi (U_x))$ is also a homeomorphism for all $x \in M$.

For sake of conciseness, we only sketch a justification for the existence of such a $t_0>0$. The local property (i.e. the existence of $t_0>0$ for a given $x \in M$) follows easily from the definition of embedded $C^{1,1} k$-manifolds. Indeed, the assumption of a regular embedding entails that $\pi _x$ has full rank derivative at x and the inverse function theorem can be applied to get the local property. In order to get a uniform $t_0>0$ (the requested global property) one can establish first the following strengthening of the local property: For any $x \in M$, there is $t_x>0$ such that for any $y \in M \cap B^\circ (x, t_x)$, the restriction of $\pi _y$ to $M \cap B^\circ (x, t_x)$ is a $C^1$ homeomorphism. Compactness of M can then be used in the usual manner to get a uniform $t_0$.

The collection of pairs $\{(U_x,\pi _x)\}_{x \in M}$ forms an atlas in the $C^{1,1}$ structure of M. Similarly, the collection of pairs $\{(\Phi (U_x),\pi _{\Phi (x)}')\}_{x \in M}$ forms an atlas in the $C^{1,1}$ structure of $M'$. Let ${\mathcal {T}}_M(x)$ be the linear space associated to $T_M(x)$ and denote by $D\Phi _x$ the derivative of $\Phi$ at x, seen as a linear map between ${\mathcal {T}}_M(x)$ and ${\mathcal {T}}_{M'}(\Phi (x))$. Since M is compact, there is a constant $K_\Phi$ independent of x such that

$$\begin{aligned} \forall y \in M, \quad \Vert \Phi (y)-\Phi (x)-D\Phi _x(\pi _x(y)-x)| < K_\Phi \Vert y-x\Vert ^2, \end{aligned}$$

(3)

and two positive numbers $\kappa _2 \ge \kappa _1 > 0$, again independent of x by compactness of M, such that

$$\begin{aligned} \forall u \in {\mathcal {T}}_M(x), \quad \kappa _1 \Vert u\Vert \le \Vert D\Phi _x (u) \Vert \le \kappa _2 \Vert u\Vert . \end{aligned}$$

(4)

Consider the affine function $\hat{\Phi }_x : T_M(x) \rightarrow T_{M'}(\Phi (x))$ defined by $\hat{\Phi }_x (y) = \Phi (x) + D\Phi _x (y-x)$. Combining Eqs. (1), (2) and (3), we can find a constant $L_\Phi$ independent of x such that for all $t < t_0$ and all compact sets $A \subset M \cap B(x,t)$:

$$\begin{aligned} d_H(\hat{\Phi }_x \circ \pi _x(A) , \pi _{\Phi (x)}' \circ \Phi (A) ) < L_\Phi t^2. \end{aligned}$$

(5)

Now, assume that M is nicely triangulable and let us prove that $M'$ is also nicely triangulable. By definition, we can find $\rho _0 > 0$ and $\eta _0>0$ such that, for all $0 < \rho < \rho _0$, there is a $(\rho ,\eta _0\rho )$-nice triangulation T of M with respect to some $(h, \mathcal {C})$. Suppose $\mathcal {C}= \{C_v \mid v \in V\}$ and consider the covering $\mathcal {C}' = \{ \Phi (C_v) \mid v \in V\}$, the homeomorphism $h' = \Phi \circ h : |T| \rightarrow M'$, the real numbers $\rho '=2\kappa _2\rho$ and $\eta '_0 = \frac{\kappa _1 \eta _0 - 5 L_\Phi \rho }{2\kappa _2}$. Let us prove that by choosing $\rho$ small enough, T is a $(\rho ',\eta _0'\rho ')$-nice triangulation of $M'$ with respect to $(\mathcal {C}',h')$. In other words, we need to check that conditions (ii) and (iii) of Definition 2 are satisfied for $\mathcal {C}= \mathcal {C}', h=h', \rho =\rho '$ and $\delta = \eta _0'\rho '$. Take $v \in V$ and set $x=h(v), C = C_v, S = h({\text {St}}_{T}(v))$.

(ii) By definition of T, we have $x \in C \subset {\text {B}}^\circ (x, \rho )$. Taking the image of this relation under $\Phi$ and choosing $\rho >0$ small enough, we get that $\Phi (x) \in \Phi (C) \subset \Phi ({\text {B}}^\circ (x, \rho )) \subset {\text {B}}^\circ (\Phi (x), \rho ')$. The last inclusion is obtained by combining Eqs. (1), (3) and (4).

(iii) Let us choose a positive real number $\rho < \min { \{ \rho _0, \frac{t_0}{2}\}}$ small enough to ensure that $\eta _0' > 0$ and let us prove that $M' \cap [ {\text {Conv}}(C)]^{\oplus \eta _0'\rho '} \subset S$. By choice of T as a $(\rho ,\eta _0\rho )$-nice triangulation of M with respect to $(h,\mathcal {C})$, we have that $M \cap {\text {Conv}}(C)^{\oplus \eta _0 \rho } \subset S$. Furthermore, $C \subset B(x,\rho )$ and $S \subset B(x,2\rho )$. Thus, by choosing $\rho < \frac{t_0}{2}$, we have $S \subset U_x$ and

$$\begin{aligned} U_x \cap {\text {Conv}}(C)^{\oplus \eta _0 \rho } \subset S. \end{aligned}$$

Taking the image by the homeomorphism $\pi _x : U_x \rightarrow \pi _x(U_x)$ on both sides and using $\pi _x(A \cap B) = \pi _x(A) \cap \pi _x(B)$ we get

$$\begin{aligned} \pi _x({\text {Conv}}(C)^{\oplus \eta _0 \rho }) \subset \pi _x(S). \end{aligned}$$

Let $B_k(0,r)$ denote the k-dimensional ball of ${\mathcal {T}}_M(x)$ centered at the origin with radius r. Writing $A\oplus B = \{ a + b \mid a \in A, b \in B\}$ for the Minkowski sum of A and B, it is not too difficult to prove that $\pi _x(A^{\oplus \delta }) = \pi _x(A) \oplus B_k(0,\delta )$. It follows that

$$\begin{aligned} \pi _x({\text {Conv}}(C)) \oplus B_k(0,\eta _0\rho ) \subset \pi _x(S). \end{aligned}$$

Taking the image under $\hat{\Phi }_x$ on both sides we get

$$\begin{aligned} \hat{\Phi }_x \circ \pi _x({\text {Conv}}(C)) \oplus D \Phi _x B_k(0,\eta _0\rho ) \subset \hat{\Phi }_x \circ \pi _x(S)). \end{aligned}$$

Let $B'_k(0,r)$ denote the k-dimensional ball of ${\mathcal {T}}_{M'}(\Phi (x))$ centered at the origin with radius r. Using Eq. (4) we get that $B_k'(0,\kappa _1\eta _0\rho ) \subset D \Phi _x B_k(0,\eta _0\rho )$. Since $\hat{\Phi }_x$ and $\pi _x$ are both affine, so is the composition and therefore $\hat{\Phi }_x \circ \pi _x({\text {Conv}}(C)) = {\text {Conv}}(\hat{\Phi }_x \circ \pi _x(C))$. It follows that

$$\begin{aligned} {\text {Conv}}(\hat{\Phi }_x \circ \pi _x(C)) \oplus B_k'(0,\kappa _1 \eta _0 \rho ) \subset \hat{\Phi }_x \circ \pi _x(S). \end{aligned}$$

Recalling that $C \subset B(x,\rho )$ and $S \subset B(x,2\rho )$ and combining the above inclusion with Eq. (5) we obtain

$$\begin{aligned} {\text {Conv}}(\pi '_x \circ \Phi (C)) \oplus B_k(0, \kappa _1 \eta _0 \rho - 5 L_\Phi \rho ^2) \subset \pi '_x \circ \Phi (S). \end{aligned}$$

Interchanging ${\text {Conv}}$ and $\pi _x'$, noting that $\eta '_0\rho ' = \kappa _1 \eta _0 \rho - 5 L_\Phi \rho ^2$ and using $\pi '_x(A^{\oplus \delta }) = \pi '_x(A) \oplus B_k(0,\delta )$ we get

$$\begin{aligned} \pi '_x({\text {Conv}}(\Phi (C))^{ \oplus \eta _0' \rho '}) \subset \pi '_x \circ \Phi (S). \end{aligned}$$

Since $\pi '_x : \Phi (U_x) \rightarrow \pi _{\Phi (x)}'(\Phi (U_x))$ is homeomorphic, we thus obtain $M' \cap {\text {Conv}}(\Phi (C))^{\oplus \eta _0' \rho } \subset \Phi (S)$ as desired. $\square$