Learning the Geometric Structure of Manifolds with Singularities Using the Tensor Voting Graph

Abstract

We present a general framework that addresses manifolds with singularities and multiple intersecting manifolds, which is also robust against a large number of outliers. We suggest a hybrid local–global method that leverages the algorithmic capabilities of the tensor voting framework and, unlike tensor voting, is capable of reliably inferring the global structure of complex manifolds by using a unique graph construction, called the tensor voting graph (TVG). Moreover, we propose to explicitly and directly resolve the ambiguities near the intersections with a novel algorithm, which uses the TVG and the positions of the points near the manifold intersections. Experimental results in estimating geodesic distances and clustering demonstrate that our framework outperforms the state of the art, especially on geometric complex settings such as when the tangent spaces at the intersections points are not orthogonal and in the presence of a large amount of outliers.

Appendix: Theoretical Analysis

In this section we prove that given two sub-manifolds that are intersecting, and then under certain conditions, the maximal principal angle of between the local tangent spaces is smaller for points which belong to the same manifold in the local intersection area. Thus, given three points which are sufficiently close to the local intersection area, where only two points belong to the same manifold, the maximal principal angle will be smaller between the pair which belongs to the same manifold. This result serves as a motivation for our ambiguity resolution algorithm, which allows us to untangle the manifolds in the intersection area. The proof is given for the case of $K=2$ intersecting manifolds for simplicity, and can be extended to $K>2$ .

Preliminary Definitions Let $\mathbb {R}^{N}$ be the ambient space. Let $M_{d}(\varvec{k})$ denote the class of connected, $C^{2}$ and compact d dimensional manifolds without boundary embedded in $\mathbb {R}^{N}$, with reach at least $\frac{1}{\varvec{k}}$, which is a notion used to quantify smoothness[8].

Formally, the reach of $M\subset \mathbb {R}^{N}$ is the supremum over $r>0$ such that, for each $\mathbf {y}\in B(\mathbf {x},r)$, where

$$\begin{aligned} B(\mathbf {x},r)=\left\{ \mathbf {y}\in \mathbb {R}^{N}: ||\mathbf {x}- \mathbf {y}||<r \right\} \end{aligned}$$

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there is a unique point in M nearest $\mathbf {x}$. For sub-manifolds without boundaries, the reach coincides with the condition number $1 \backslash \tau$ [8].

Given two connected, $C^{2}$ and compact d dimensional manifolds $M_{1}, M_{2} \in M_{d}(\varvec{k})$. Define $X_{J}$ as the set of points where $M_{1}, M_{2}$ intersect

$$\begin{aligned} X_{J}= \left\{ \mathbf {x}\in M_{1}\cap M_{2} | \mathbf {x}=\mathbf {s}_{\mathbf {1}}=\mathbf {s}_{\mathbf {2}}, \mathbf {s}_{\mathbf {1}} \in M_{1}, \mathbf {s}_{\mathbf {2}} \in M_{2} \right\} \end{aligned}$$

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Let $\theta _J$ to be the set of all principal angles corresponding to the maximal principal angle $\theta _{\max }$ at J:

$$\begin{aligned} \theta _J(M_{1},M_{2})=\left\{ \theta _{\max }\left( T_{M_1}(\mathbf {s}_\mathbf{1}),T_{M_{2}}(\mathbf {s}_{\mathbf {2}}) \right) |\mathbf {s}_{\mathbf {1}},\mathbf {s}_{\mathbf {2}}\in X_{J} \right\} \end{aligned}$$

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and also let

$$\begin{aligned} \theta _{J}^{\inf }= \underset{(\mathbf {s}_{\mathbf {1}},\mathbf {s}_{\mathbf {2}}) }{\inf } \left\{ \theta _{\max }(\mathbf {s}_{\mathbf {1}},\mathbf {s}_{\mathbf {2}}) \right| \theta _{\max } (\mathbf {s}_{\mathbf {1}}, \mathbf {s}_{\mathbf {2}}) \in \theta _J \} \end{aligned}$$

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to be the infimum obtained over all possible maximal principal angles at the intersections points.

To prove our main claim, we need the following lemma, which was proved in [8]. This results give a bound on the maximal principal angle between the local tangent space on a smooth manifold which is much more tight than the one obtained in [28], and therefore is the one we use.

Lemma 1

[8]. For $M \subset M_{d}(\varvec{k})$ and any $\mathbf {s}, \mathbf {s^{\prime }} \in M$

$$\begin{aligned} \theta _{\max }\left( T_{M}(\mathbf {s}),T_{M}(\mathbf {s^{\prime }}) \right) < 2 \text{ asin }\left( \min \left\{ \frac{\varvec{k}}{2} \left\| \mathbf {s^{\prime }}-\mathbf {s} \right\| ,1\right\} \right) \end{aligned}$$

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Lemma 2

Suppose $M_{1}, M_{2} \subset M_{d}({\varvec{k}})$, and assume that $\theta _{J}(\mathbf {s}_{\mathbf {1}},\mathbf {s}_{\mathbf {2}})>0$, $\forall \mathbf {s}_{\mathbf {1}}, \mathbf {s}_{\mathbf {2}} \in X_{J}$, and that the intersection set has a strictly positive reach. Given three points $\mathbf {y}, \mathbf {t}, \mathbf {z}$, and assume that, without loss of generality, $\mathbf {y}, \mathbf {t} \in M_{1}, \mathbf {z} \in M_{2}$. Then, for $r>0$ which obeys $r=\text{ min }\left\{ \text{ asin }(\theta _{J}^{\inf }), \frac{(\theta _{J}^{\inf })}{4\varvec{k}},1\right\}$, we have that for all $\mathbf {y} , \mathbf {t} \in M_{1}, \mathbf {z} \in M_{2}$, where $\mathbf {y}, \mathbf {z}, \mathbf {t} \in B(\mathbf {x},r)$, the following inequality is satisfied:

$$\begin{aligned} \theta _{\max }\left( T_{M_1}(\mathbf {t}),T_{M_1}(\mathbf {y}) \right) < \theta _{\max }\left( T_{M_1}(\mathbf {y}),T_{M_2}(\mathbf {z}) \right) \end{aligned}$$

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Proof

We first prove that $\text{ inf }_{\theta \in \theta _J}\theta =C1>0$: We claim that there exists $C1>0$ such that for all $\mathbf {x} \in X_{J}$, where $\mathbf {x}=\mathbf {s^{\prime }}=\mathbf {t^{\prime }}$ , $\mathbf {s^{\prime }} \in M_{1}, \mathbf {t^{\prime }} \in M_{2}$, then for all $\mathbf {s}\in M_{1}, \mathbf {t}\in M_{2},$ where $\mathbf {s}, \mathbf {t} \in B(\mathbf {x},\epsilon )$, we have that $\theta _{\max } \left( T_{M_{1}}(\mathbf {s}), T_{M_{2}} (\mathbf {t}) \right) >0$. Assume in contrast that there exists $\mathbf {x} \in X_{J}$ , $\mathbf {x}=s^{\prime }=t^{\prime }$ such that for all $\epsilon >0$, $\mathbf {s}\in M_{1}, \mathbf {t}\in M_{2}, \mathbf {s}, \mathbf {t} \in B(\mathbf {x},\epsilon )$ such that $\theta _{\max } \left( T_{M_{1}}(\mathbf {s}), T_{M_{2}} (\mathbf {t}) \right) =0$. Thus, by this assumption, there exists sequences $\left\{ \mathbf {s}_{\mathbf{n}} \right\} _{n=1}^{\infty }, \left\{ \mathbf {t}_{{ \mathbf n}} \right\} _{n=1}^{\infty }$ such that $\theta _{\max }\left( T_{M_{1}}(\mathbf {s}_{n}),T_{M_{2}}(\mathbf { t}_{n}) \right) =0$, $\forall n$. Since $M_{2}$ is compact, there exists a converging sub-sequence $\left\{ \mathbf {t}_{nk} \right\} _{k=1}^{\infty }\subset M_{2}$ such that $\left\{ \mathbf {t}_{nk} \right\} _{nk=1}^{\infty } \rightarrow \mathbf {t^{\prime }} \in M_{2}$. On the other hand, we have that $\theta _{\max }\left( T_{M_1}(\mathbf {s}_{n}),T_{M_2}(\mathbf {t^{\prime }}) ) \right) =0$ $\forall n$. Since $M_{1}$ is compact, $\left\{ \mathbf { s}_{n} \right\} _{n=1}^{\infty }$ has a converging sub-sequence $\left\{ \mathbf {s}_{nk} \right\} _{k=1}^{\infty } \rightarrow \mathbf {s^{\prime }} \in M_{1}$. Thus we have that $\theta _{\max }\left( T_{M_1}(\mathbf {s}_{nk}),T_{M_2}(\mathbf {t^{\prime }})) \right) =0$ and $\theta _{\max }\left( T_{M_1}(\mathbf {s^{\prime }}),T_{M_2}(\mathbf {t^{\prime }}) ) \right) >0$. Letting $\mathbf {v}_{\mathbf {s}_{nk}}, \mathbf {v}_{\mathbf {t}_{nk}}$ denote the vectors corresponding to the maximal principal angles of the local tangent spaces $T_{M_1}(\mathbf {s}_{nk})$ and $T_{M_{2}}(\mathbf {t}_{nk})$, we have that

$$\begin{aligned} \left| \left\langle \mathbf {v}_{\mathbf {s}_{nk}},\mathbf {v}_{\mathbf {t^{\prime }}} \right\rangle \right| =1 , \left| \left\langle \mathbf {v}_{\mathbf {s^{\prime }}}, \mathbf {v}_{t^{\prime }} \right\rangle \right| \ne 1, \end{aligned}$$

and $\mathbf {v}_{\mathbf {s}_{nk}} \rightarrow \mathbf {v}_{\mathbf {s^{\prime }}} \forall k$, which is a contradiction to the compactness of $M_{1}$.

We use the results above to conclude the proof. Let $r=\text{ min }\left\{ \text{ asin }(\theta _{J}^{\inf }), \frac{(\theta _{J}^{\inf })}{4\varvec{k}},1\right\}$. Without loss of generality, choose $\mathbf {s}, \mathbf {s^{\prime }} \in M_{1}$ and $\mathbf {t} \in M_{2}$ where $\mathbf {s}, \mathbf {s^{\prime }} , \mathbf {t} \in B(\mathbf {x},r)$ with $\mathbf {x} \in X_{J}$. Using Lemma 1 and applying the arc-sine function which is increasing—we have that:

$$\begin{aligned} \theta _{\max }\left( T_{M_1}(\mathbf {s}),T_{M_1}(\mathbf {s^{\prime }}) \right) \le 2 \text{ asin }\left( \min \left\{ \frac{\varvec{k}}{2} \left\| \mathbf {s^{\prime }}-\mathbf {s} \right\| ,1\right\} \right) \nonumber \\ \le 2\text{ asin }\left( \varvec{k} \frac{r}{2} \right) \le 2\text{ asin }(\theta _{J}^{\inf }) \le 2(\theta _{J}^{\inf }/4)=\theta _{J}^{\inf }/2 \end{aligned}$$

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On the other hand, we have that $\theta _{\max }\left( T_{M_{1}}(\mathbf {s}),T_{M_{2}}((\mathbf {t}) \right) > \theta _{J}^{\inf }/2$ for each $\mathbf {s} \in M_{1}, \mathbf {t} \in M_{2}$, and thus we have that $\theta _{\max }\left( T_{M_1}(\mathbf {s}),T_{M_1}(\mathbf {s^{\prime }}) \right) < \theta _{\max }\left( T_{M_1}(\mathbf {s}),T_{M_2}(\mathbf {t}) \right) .$ $\square$