No abstract

We present anefficient algorithm for determining an aesthetically pleasing shape boundary connecting all the points in a given unorganized set of 2D points, with no other information than point coordinates. By posing shape construction as a minimisation problem which follows the Gestalt laws, our desired shape B min is non-intersecting, interpolates all points and minimizes a criterion related to these laws. The basis for our algorithm is an initial graph, an extension of the Euclidean minimum spanning tree but with no leaf nodes, called as the minimum boundary complex BC min . BC min and B min can be expressed similarly by parametrizing a topological constraint. A close approximation of BC min , termed BC 0 can be computed fast using a greedy algorithm. BC 0 is then transformed into a closed interpolating boundary B out in two steps to satisfy B min 's topological and minimization requirements. Computing B min exactly is an NP (Non-Polynomial)-hard problem, whereas B out is computed in linearithmic time. We present many examples showing considerable improvement over previous techniques, especially for shapes with sharp corners. Source code is available online.

Figure 1: Processing pipeline: An out-of-core hierarchical counting sort quickly generates chunks of suitable size which can then be indexed in parallel, and eventually merged into a single octree.

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.