Incremental Labelling of Voronoi Vertices for Shape Reconstruction

Abstract

We present an incremental Voronoi vertex labelling algorithm for approximating contours, medial axes and dominant points (high curvature points) from 2D point sets. Though there exist many number of algorithms for reconstructing curves, medial axes or dominant points, a unified framework capable of approximating all the three in one place from points is missing in the literature. Our algorithm estimates the normals at each sample point through poles (farthest Voronoi vertices of a sample point) and uses the estimated normals and the corresponding tangents to determine the spatial locations (inner or outer) of the Voronoi vertices with respect to the original curve. The vertex classification helps to construct a piece‐wise linear approximation to the object boundary. We provide a theoretical analysis of the algorithm for points non‐uniformly (ε‐sampling) sampled from simple, closed, concave and smooth curves. The proposed framework has been thoroughly evaluated for its usefulness using various test data. Results indicate that even sparsely and non‐uniformly sampled curves with outliers or collection of curves are faithfully reconstructed by the proposed algorithm.

BallMerge: High‐quality Fast Surface Reconstruction via Voronoi Balls

“…Handling sparse point sets [OMW16] was enabled via ρ‐ sampling. Similarly, outliers were addressed [PPT*19]. Additionally, non‐feature specific reconstruction [PM16], as well as self‐intersections and noise [PMM18] were covered, while optimal transport [dGCAD11] was shown to handle noise and outliers.…”

2D Points Curve Reconstruction Survey and Benchmark

“…In subsequent work, the authors [PPT ∗ 19] employed an incremental algorithm to classify Voronoi vertices into inner and outer with the help of normals estimated through Voronoi poles. Such a classification not only helps reconstructing the underlying curve but also aids in medial axis computation and dominant point detection.…”

“…Despite over three decades of tremendous research effort in the computational geometry, computer vision and graphics research communities, specific cases are still open in curve reconstruction, and there is no algorithm that would succeed on all types of problems. Recent research trends, however, address specific aspects of reconstruction such as improved sampling conditions [OMW16], reconstructing from fewer number of samples and curves with sharp corners [Ohr13], reconstruction from unstructured and noisy point clouds [OW18a], a unified framework for reconstruction [MPM15], incremental labeling techniques for curve extraction [PPT ∗ 19], and applications of curve reconstruction to hand‐drawn sketches [PM16].…”

Sampling Shape Contours Using Optimization over a Geometric Graph

“…For example, vertex detection using polygonal line approximation (PLA) [5]- [10] can be used to select detected vertices as sample points. Analogously, although dominant point (DP) detection [11]- [14] and high curvature point extraction [3], [15], [16] use different measures to calculate curvature, these methods are essentially the same because they can be used to determine curvature extremum points as sample points.…”