| We give a provably correct algorithm to reconstruct a $k$-dimensional manifold embedded in $d$-dimensional Euclidean space. Input to our algorithm is a point sample coming from an unknown manifold. Our approach is based on two main ideas~: the notion of tangential Delaunay complex, and the technique of sliver removal by weighting the sample points. Differently from previous methods, we do not construct any subdivision of the embedding $d$-dimensional space. As a result, the running time of our algorithm depends only linearly on the extrinsic dimension $d$ while it depends quadratically on the size of the input sample, and exponentially on the intrinsic dimension $k$. To the best of our knowledge, this is the first certified algorithm for manifold reconstruction whose complexity depends linearly on the ambient dimension. We also prove that for a dense enough sample the output of our algorithm is isotopic to the manifold and a close geometric approximation of the manifold. |
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