Tangent Estimation from Point Samples

Let M be an m-dimensional smooth compact manifold embedded in R, where m is a constant known to us. Suppose that a dense set of points are sampled from M according to a Poisson process with an unknown parameter. Let p be any sample point, let % be the local feature size at p, and let %ε be the distance from p to the (n + 1)th nearest sample point for some n between ( m+1 2 ) + 1 and ( d+1 2 ) . 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(ε). The running time is bounded by the time to compute the thin SVD of an n× ( d+1 2 ) matrix and the full SVD of an n× d matrix, which is usually O(dn) in practice. We implemented the algorithm and experimentally verified its effectiveness on both noiseless and noisy data. ∗Research supported by the Research Grant Council, Hong Kong, China (project no. 612109). Part of the work was done while Chiu was at HKUST. †Dept. of Computer Science and Engineering, HKUST, Hong Kong. ‡National Institute of Informatics (NII), Tokyo, Japan. §JST, ERATO, Kawarabayashi Large Graph Project.