Boundary Estimation from Point Clouds: Algorithms, Guarantees and Applications

Abstract

We investigate identifying the boundary of a domain from sample points in the domain. We introduce new estimators for the normal vector to the boundary, distance of a point to the boundary, and a test for whether a point lies within a boundary strip. The estimators can be efficiently computed and are more accurate than the ones present in the literature. We provide rigorous error estimates for the estimators. Furthermore we use the detected boundary points to solve boundary-value problems for PDE on point clouds. We prove error estimates for the Laplace and eikonal equations on point clouds. Finally we provide a range of numerical experiments illustrating the performance of our boundary estimators, applications to PDE on point clouds, and tests on image data sets.

Appendices

Appendix A. Proof of Lemma 3.1

The following lemma will be useful in proving Lemma 3.1.

Lemma A.1

(Covering with spherical segments) Let $r \le 1$ and $0<a<b\le r$. For $u\in \mathbb {S}^{d-1}$ and $0< a < b\le r$ define the spherical sector by

$$\begin{aligned}S_{a,b}^u = \{x\in B(0,r) \, : \, a \le x\cdot u\le b\}.\end{aligned}$$

Suppose $\Sigma \subset \mathbb {S}^{d-1}$ is a finite set satisfying the following property:

$$\begin{aligned} \text { for all } u\in \mathbb {S}^{d-1} \text { there exists } v\in \Sigma \text { such that } |u-v|\le \delta . \end{aligned}$$

(A.1)

Then, for any $u\in \mathbb {S}^{d-1}$ we can find $v\in \Sigma$ such that

$$\begin{aligned}S^{v}_{a+\delta b,b-\delta b}\subset S^u_{a,b}.\end{aligned}$$

Proof

Let $u\in \mathbb {S}^{d-1}$ and fix a $v\in \Sigma$ satisfying (A.1). Suppose that $x\in S_{a+\delta b,b-\delta b}^v$. Then we have

$$\begin{aligned}a+\delta b \le x\cdot v \le b-\delta b.\end{aligned}$$

We have

$$\begin{aligned}|x\cdot v - x\cdot u| =| x\cdot (v-u)| \le |x| |u-v| \le \delta |x| \le \delta b,\end{aligned}$$

since $|x|\le b-\delta \le b$. Therefore

$$\begin{aligned}x \cdot u \le b-\delta b + \delta b=b \ \ \text {and} \ \ x\cdot u \ge a + \delta b - \delta b=a .\end{aligned}$$

Therefore $x\in S_{a,b}^u$, which shows that for each $u\in \mathbb {S}^{d-1}$ there exists $v\in \Sigma$ such that

$$\begin{aligned}S_{a,b}^u \supset S_{a+\delta b,b-\delta b}^v.\end{aligned}$$

Hence, the event that $S_{a,b}^u$ is empty for some $u\in \mathbb {S}^{d-1}$ is contained in the event that $S_{a+\delta b,b-\delta b}^v$ is empty for some $v\in \Sigma$—a finite collection of events.

$\square$

Remark A.2

($\varepsilon$-nets and upper bound on $|\Sigma |$) Recall that an $\varepsilon$-net of $\mathbb {S}^{d-1}$ is the set of points in $\mathbb {S}^{d-1}$ such that the pairwise distance is at least $\varepsilon$. Then we define a maximal $\varepsilon$-net of the sphere to be an $\varepsilon$-net such that no point on $\mathbb {S}^{d-1}$ can be added while preserving the lower bound for the pairwise distance.

Then, observe that any maximal $\varepsilon$-net of the unit sphere satisfies the condition of Lemma A.1. If $\Sigma _\varepsilon =\{x^1,\cdots ,x^{N_\varepsilon }\}$ is a maximal $\varepsilon$-net of $\mathbb {S}^{d-1}$, then for each $x\in \mathbb {S}^{d-1}$ there exists $x^i\in \Sigma _{\varepsilon }$ such that $|x-x^i|\le \varepsilon$. To see this, suppose $|x^*- x^i|>\varepsilon$ for all $i=1,\cdots ,N_\varepsilon$. Then

$$\begin{aligned}B(x^*,\varepsilon /2)\cap B(x^i,\varepsilon /2) =\emptyset \text { for all } x^i\in \Sigma _\varepsilon .\end{aligned}$$

Thus $\Sigma _{\varepsilon }\cap \{x^*\}$ should also be an $\varepsilon$-net, which contradicts the maximality of $\Sigma _\varepsilon$.

Now, let $\Sigma _\delta$ be any $\delta$-net – i.e. $\varepsilon$-net with $\varepsilon =\delta$. Then $\{B(v^i,\delta /2):\,v^i\in \Sigma _\delta \}$ is a collection of disjoint balls, all contained in $B(0,1+\delta /2)\setminus B(0,1-\delta /2)$. Thus, base on a simple volumetric argument, we can deduce

$$\begin{aligned} |\Sigma _\delta | \le 2d\left( 1+\frac{2}{\delta }\right) ^{d-1}, \end{aligned}$$

(A.2)

Proof of Lemma 3.1

• (1) Let $\{v_i\}_{i=1}^M=\Sigma \subset \mathbb {S}^{d-1}$ be a maximal $\delta$-net. By Lemma A.1 and Remark A.2, for any $u\in \mathbb {S}^{d-1}$ we can find $v_k\in \Sigma$ such that

  $$\begin{aligned}S_{a+b\delta ,b-b\delta }^{v_k}\subset S_{a,b}^u.\end{aligned}$$

  This means that if all of $S_{a+b\delta ,b-b\delta }^{v_i}$ are nonempty, all of $S_{a,b}^u$ is nonempty for $u\in \mathbb {S}^{d-1}$ hence

  $$\begin{aligned}{{\hat{d}}}_r^1(x^0)\ge a.\end{aligned}$$

  Without loss of generality, assume $x^0\in \mathbb {R}^d$ is the origin, and let $\alpha =d_\Omega (x^0)\wedge \frac{ r }{2}$. Denote by $K_{a,b}^u\subset S_{a,b}^u$ the cone of maximal height sharing the base with $S_{a,b}^u$. Note that $b\le \alpha$ implies $K_{a,b}^u\subset {{\overline{B}}}(x_0, r )\cap \Omega$. On the other hand, we need $a\ge (1-\lambda )\alpha -t$ to deduce the desired lower bound on ${{\hat{d}}}_ r ^1$. Thus choose

  $$\begin{aligned}a=(1-\lambda )\alpha -t,\,b=\alpha .\end{aligned}$$

  Further, we need the height of $S_{a+b\delta ,b-b\delta }^{v_i}$ to scale like t, in order to lower bound the volume. Thus we need

  $$\begin{aligned} b-b\delta -(a+b\delta )=(1-2\delta )b-\alpha =(1-2\delta )\alpha -(1-\lambda )\alpha -t =(\lambda -2\delta )\alpha +t. \end{aligned}$$

  As we are interested in $t\lesssim r ^2\ll \alpha \sim \varepsilon$, we need $\lambda -2\delta \ge 0$, hence

  $$\begin{aligned}\delta \le \frac{\lambda }{2}.\end{aligned}$$

• (2) Following the discussion in the previous step, let $\Sigma =\{v^1,\cdots ,v^{N_\lambda }\}$ be a maximal $\frac{\lambda }{2}$-net of $\mathbb {S}^{d-1}$, and write

  $$\begin{aligned}S^i = S^{v^i}_{a+b\lambda /2,b-b\lambda /2} \text { where } a=(1-\lambda ) \alpha ,\,b=\alpha ,\text { and }. \end{aligned}$$

  Thus, to show (3.2) holds with probability at least $1-n^{-\gamma }$, it suffices to show

  $$\begin{aligned}\mathbb {P}(\text { No point in } S^i)\le (1-\rho _{\min }|S^i\cap \Omega |)^{n}\le N_\lambda ^{-1}n^{-\gamma } \text { for all } i=1,\cdots ,N_\lambda .\end{aligned}$$

• (3) We first compute the lower bound for $|S^i\cap \Omega |$. Temporarily write $a'=a+b\lambda /2,\,b'=b-b\lambda /2$. Let $K_{a',b'}^i$ be the cone of height $b'-a'=t$ sharing the base of $S^i$. Note that $K_{a',b'}^i\subset S^i\cap \Omega$ and its base has radius $\sqrt{r^2-(a')^2}= r \sqrt{1-(a'/ r )^2}$. As the $|K_{a',b'}^i|$ is independent of i, we may drop the superscript and deduce

  $$\begin{aligned}&|S^i\cap \Omega |\ge |K_{a',b'}|=\int _0^t \omega _{d-1}\left( r \sqrt{1-(a'/ r )^2}\frac{s}{t}\right) ^{d-1}\\ {}&\quad ds =\frac{1}{d}\omega _{d-1}t r ^{d-1}(1-(a'/ r )^2)^{\frac{d-1}{2}}. \end{aligned}$$

  As $a'\le b\le \alpha \le r /2$, we have $(1-(a'/r)^2)^{(d-1)/2}\ge 2^{-(d-1)/2}$. Hence, for each $i=1,\cdots ,N_\lambda$

  $$\begin{aligned}\mathbb {P}(\text { No point in } S^i)\le (1-\rho _{\min }|K_{a',b'}|)^{n}\le \left( 1-\frac{\rho _{\min }}{d2^{(d-1)/2}}t r ^{d-1}\right) ^{n}.\end{aligned}$$

  The expression on the right is less than $N_\lambda ^{-1}n^{-\gamma }$ if

  $$\begin{aligned}n\log \left( 1-\frac{\rho _{\min }}{d2^{(d-1)/2}}t r ^{d-1}\right) \le -\gamma \log n-\log N_\lambda ,\end{aligned}$$

  or equivalently

  $$\begin{aligned}t r ^{d-1}\ge \frac{d2^{(d-1)/2}(1-e^{-\frac{\gamma \log n+\log N_\lambda }{n}})}{\rho _{\min }\omega _{d-1}}.\end{aligned}$$

  As $1-e^{-x}\le x$, it suffices for $t, r$ to satisfy

  $$\begin{aligned}t r ^{d-1}\ge \frac{d2^{(d-1)/2}}{\rho _{\min }\omega _{d-1}}\left( \frac{\gamma \log n+\log N_\lambda }{n}\right) .\end{aligned}$$

• (4) We claim that $\log N_\lambda \le \gamma (d-1)\log n$. By setting $\delta =\frac{\lambda }{2}$ in (A.2), we know

  $$\begin{aligned}N\le 2d\left( 1+\frac{4}{\lambda }\right) ^{d-1}=2d\left( \frac{\lambda +4}{\lambda }\right) ^{d-1}.\end{aligned}$$

  By hypothesis $n\ge d\vee \frac{\lambda +4}{\lambda }$ and $\gamma >2$, we see

  $$\begin{aligned}n^{\gamma (d-1)}\ge n^{d-1} n^{d-1}\ge 2d \left( \frac{\lambda +4}{\lambda }\right) ^{d-1}\ge N_\lambda .\end{aligned}$$

  Thus $\gamma \log n +\log N\le d\gamma \log n$, and it suffices for $t, r$ to satisfy

  $$\begin{aligned}t r ^{d-1}\ge \frac{d^2 2^{(d-1)/2}\gamma }{\rho _{\min }\omega _{d-1}}\left( \frac{\log n}{n}\right) .\end{aligned}$$

  This completes the proof $\square$

Appendix B. Proof of Proposition 6.3

Proof

The proof is split into several steps.

1. Let $y\in \partial \Omega$ satisfy $d_\Omega (x_*)=|x_*-y|$. Let $z\in \partial B(x^0,\varepsilon )$ be along the line from $x_*$ to y. Then we have

$$\begin{aligned}d_{\Omega }(z) \le d_\Omega (x_*) - |x_*-z|\end{aligned}$$

and so by the property defining $x_*$ we have $x_*=z$; that is $x_*\in \partial B(x^0,\varepsilon )$. Since $d_\Omega$ is 1-Lipschitz, we have $d_\Omega (x_*) \ge d_\Omega (x^0)-\varepsilon$. By a similar argument as above, we have $d_\Omega (x_*) \le d_\Omega (x^0) - \varepsilon$, and so

$$\begin{aligned}d_\Omega (x_*) = d_\Omega (x^0) - \varepsilon .\end{aligned}$$

Now, note that the function

$$\begin{aligned}g(r) = d_\Omega (x_* + r p)\end{aligned}$$

is 1-Lipschitz and satisfies $g(\varepsilon ) = d_\Omega (x^0) = g(0) + \varepsilon$. It follows that $g(l) = g(0) + r$ for $0\le r\le \varepsilon$, and so

$$\begin{aligned} d_\Omega (x_* + r p) = d_\Omega (x_*) + r \ \ \text { for }0 \le r \le \varepsilon . \end{aligned}$$

(B.1)

2. Since $d_\Omega - \frac{1}{R}|x-x_*|^2$ is a concave function, there exists $q\in \mathbb {R}^n$ such that

$$\begin{aligned}d_\Omega (x) - d_\Omega (x_*) \le q\cdot (x-x_*)+ \frac{1}{R}|x-x_*|^2. \end{aligned}$$

for all $x\in \Omega$. By (B.1) we have

$$\begin{aligned}r = d_\Omega (x_* + rp) - d_\Omega (x_*) \le r q\cdot p + \frac{r^2}{R}\end{aligned}$$

for $0 \le r \le \varepsilon$. Therefore

$$\begin{aligned}q\cdot p \ge 1 - \frac{r}{R}.\end{aligned}$$

Sending $r\rightarrow 0^+$ we find that $p\cdot q \ge 1$.

3. We now claim that $|q|\le 1$, which combined with $p\cdot q \ge 1$ from part 2 implies that $p=q$ and completes the proof. To see this, since $B(x^0,\varepsilon )\subset \Omega$, we have $B(x_*,r)\subset \Omega$ for $r>0$ sufficiently small. Now, the dynamic programming principle gives

$$\begin{aligned}0 = \min _{x\in B(x_*,r)}\left\{ d_\Omega (x) - d_\Omega (x_*)+|x-x_*| \right\} \le \min _{x\in B(x_*,r)}\left\{ q\cdot (x-x_*)+|x-x_*| \right\} + \frac{r^2}{R}.\end{aligned}$$

Setting $x-x_*=-|x-x_*|q/|q|$ we have

$$\begin{aligned}0 \le \min _{x\in B(x_*,r)}\left\{ |x-x_*|(1-|q|) \right\} + \frac{r^2}{R} = -r(|q|-1)_+ + \frac{r^2}{R}.\end{aligned}$$

Sending $r\rightarrow 0^+$ we obtain $|q|\le 1$, which completes the proof. $\square$

Appendix C. Concentration Inequalities

For reference, we state the Chernoff bounds, Hoeffding inequality, and the Bernstein inequality, which are concentration of measure inequalities used to control the variance of our normal and distance estimators. We refer the reader to [14] for a general reference on concentration inequalties. Proofs of the exact inequalities below can also be found in [18, Chapter 5].

Theorem C.1

(Chernoff bounds) Let $X_1,X_2\dots ,X_n$ be a sequence of i.i.d. Bernoulli random variables with parameter $p\in [0,1]$ (i.e., $\mathbb {P}(X_i=1)=p$ and $\mathbb {P}(X_i=0)=1-p$). Then for any $\varepsilon >0$ we have

$$\begin{aligned} \mathbb {P}\left( \sum _{i=1}^n X_i \ge (1+\varepsilon )np\right) \le \exp \left( -\frac{np\,\varepsilon ^2}{2(1+\tfrac{1}{3} \varepsilon )} \right) , \end{aligned}$$

(C.1)

and for any $0 \le \varepsilon < 1$ we have

$$\begin{aligned} \mathbb {P}\left( \sum _{i=1}^n X_i \le (1-\varepsilon )np\right) \le \exp \left( -\frac{1}{2}np\,\varepsilon ^2 \right) , \end{aligned}$$

(C.2)

Theorem C.2

(Hoeffding inequality) Let $X_1,X_2\dots ,X_n$ be a sequence of i.i.d. real-valued random variables with finite expectation $\mu =\mathbb {E}[X_i]$, and write $S_n=\frac{1}{n}\sum _{i=1}^n X_i$. Assume there exists $b>0$ such that $|\mathcal {X}-\mu |\le b$ almost surely. Then for any $t>0$ we have

$$\begin{aligned} \mathbb {P}(S_n-\mu \ge t)\le \exp \left( -\frac{nt^2}{2b^2}\right) . \end{aligned}$$

(C.3)

Theorem C.3

(Bernstein Inequality) Let $X_1,X_2\dots ,X_n$ be a sequence of i.i.d. real-valued random variables with finite expectation $\mu =\mathbb {E}[X_i]$ and variance $\sigma ^2=\text {Var}(X_i)$, and write $S_n=\frac{1}{n}\sum _{i=1}^n X_i$. Assume there exists $b>0$ such that $|\mathcal {X}-\mu |\le b$ almost surely. Then for any $t>0$ we have

$$\begin{aligned} \mathbb {P}(S_n-\mu \ge t)\le \exp \left( -\frac{nt^2}{2(\sigma ^2 + \tfrac{1}{3} bt)} \right) . \end{aligned}$$

(C.4)

Appendix D. List of Constants

We list the explicit constants that appear in Sects. 2 and 3. Below $\omega _d$ is the volume of unit ball in d dimensions, and $\gamma >2$ is a parameter of choice related to the error rate in the following way: $\mathbb {P}(\text { Boundary test fails })=O(n^{-\gamma })$.

$$\begin{aligned} C_{x}&= 2\omega _{d-1}+\frac{LR\omega _{d}}{\rho _{min}}, \\ C_{y}&= \frac{\omega _{d-1}}{2(d+1)}, \\ C_ r&= \frac{1}{R}\max \left[ \left( \frac{3\gamma \rho _{\max }d^2\omega _{d}R^2}{{C_{x}}^2 \rho _{\min }^2 }\right) ^{\frac{1}{d+2}},\left( \frac{4\gamma C_yd^2 2^{(d-1)/2}}{13\rho _{\min }\omega _{d-1}C_{x}}\right) ^{\frac{1}{d+1}}\right] . \end{aligned}$$