Curve Reconstruction with Many Fewer Samples

Abstract

We consider the problem of sampling points from a collection of smooth curves in the plane, such that the Crust family of proximity‐based reconstruction algorithms can rebuild the curves. Reconstruction requires a dense sampling of local features, i.e., parts of the curve that are close in Euclidean distance but far apart geodesically. We show that ε < 0.47‐sampling is sufficient for our proposed HNN‐Crust variant, improving upon the state‐of‐the‐art requirement of ε < ‐sampling. Thus we may reconstruct curves with many fewer samples. We also present a new sampling scheme that reduces the required density even further than ε < 0.47‐sampling. We achieve this by better controlling the spacing between geodesically consecutive points. Our novel sampling condition is based on the reach, the minimum local feature size along intervals between samples. This is mathematically closer to the reconstruction density requirements, particularly near sharp‐angled features. We prove lower and upper bounds on reach ρ‐sampling density in terms of lfs ε‐sampling and demonstrate that we typically reduce the required number of samples for reconstruction by more than half.

FitConnect: Connecting Noisy 2D Samples by Fitted Neighbourhoods

“…A similar method [WYZ*14] fails to reconstruct curves from moderately sparse point sets as well, as can be seen in their figure 28. Our algorithm handles such sparse point sets without noise well since it behaves like HNN‐Crust , which it extends seamlessly, as can be seen in the results [OMW16].…”

“…Reconstruction from Noise‐free Samples Ohrhallinger et al . [OMW16] give a detailed overview of the evolution of these reconstruction algorithms which are often based on sampling assumptions. Starting by requiring uniform sampling density [EKS83, KR85, FMG94, Att97, DT14, DT15, Ste08, ST09], Amenta et al .…”

“…Starting by requiring uniform sampling density [EKS83, KR85, FMG94, Att97, DT14, DT15, Ste08, ST09], Amenta et al . [ABE98] introduced the ε‐sampling condition based on the lfs which spurred further development [Gol99, DK99, Alt01, Len06, PM16], extending it to handle open curves [DMR99], sharp corners [DW02, FR01] and modifying the sampling condition to get tighter bounds [OM13, OMW16].…”

“…We first revisit the algorithm HNN‐Crust [OMW16], which reconstructs curves by interpolating samples (assumed to be noise‐free). Then we describe its properties and show how we can extend it seamlessly to samples contaminated with noise.…”

“… HNN‐Crust reconstruction of an edge‐pair for a sample s (image courtesy of [OMW16]). Edge e 0 connects s to its nearest neighbour n 0 .…”