Tangent Estimation from Point Samples

Abstract

Let $\mathcal{M}$ be an m-dimensional smooth compact manifold embedded in $\mathbb {R}^d$, where m is a constant known to us. Suppose that a dense set of points are sampled from $\mathcal{M}$ according to a Poisson process with an unknown parameter. Let p be any sample point, let $\varrho$ be the local feature size at p, and let $\varrho \varepsilon$ be the distance from p to the $(n+1)$th nearest sample point for some n between $\left( {\begin{array}{c}m+1\\ 2\end{array}}\right) + 1$ and $\left( {\begin{array}{c}d+1\\ 2\end{array}}\right)$. Using the n sample points nearest to p, we can estimate the tangent space at p and it holds with probability $1 - O(n^{-1/3})$ that the angular error is $O(\varepsilon ^2)$. The running time is bounded by the time to compute the thin SVD of an $n \times \left( {\begin{array}{c}d+1\\ 2\end{array}}\right)$ matrix and the full SVD of an $n \times d$ matrix, which is usually $O(d^2n^2)$ in practice. We implemented the algorithm and experimentally verified its effectiveness on both noiseless and noisy data.