On the shape of a set of points in the plane

Abstract

A generalization of the convex hull of a finite set of points in Akl and Toussaint [ 11, for instance, discuss the relevance the plane is introduced and analyzed. This generalization leads to a family of straight-line graphs, "o-shapes," which seem to capture the intuitive of the convex hull problem to pattern recognition. By notions of "fine shape" and "crude shape" of point sets. It is shown that identifying and ordering the extreme points of a point set, a-shapes are subgraphs of the closest point or furthest point Delaunay the convex hull serves to characterize, at least in a rough triangulation. Relying on this result an optimal O(n log n) algorithm that way, the "shape" of such a set. Jarvis [9] presents several constructs o-shapes is developed. algorithms based on the so-called nearest neighbor logic that compute what he calls the "shape" of a finite set of I. INTR~DUOTION points. The "shape," in Jarvis' terminology, is a notion T HE efficient construction of convex hulls for finite sets of points in the plane is one of the most exhaustively examined problems in the rather young field often referred to as "computational geometry." Part of the motivation is theoretical in nature. It seems to stem from the fact that the convex hull problem, like sorting, is easy to formulate and visualize. Furthermore, the convex hull problem, again like sorting, plays an important role as a component of a large number of more complex problems. Nevertheless, much of the work on convex hulls is motivated by direct applications in some of the more practical branches of computer science.

A high‐resolution approach for the spatiotemporal analysis of forest canopy space using terrestrial laser scanning data

“…A common approach is to compute a 2D or 3D α‐shape around a set of crown points to measure crown projection area and crown volume (CV) (e.g., Bayer, Seifert, & Pretzsch, 2013; Fernández‐Sarría et al., 2013; Olsoy, Glenn, Clark, & Derryberry, 2014) as well as more complex metrics such as crown sinuosity and roughness (Martin‐Ducup, Schneider, & Fournier, 2016). α‐shapes are a generalization of the convex hull concept (Edelsbrunner, Kirkpatrick, & Seidel, 1983), and by adjusting the value of α, even small details and concave surface structures in crown morphology can be captured. The rationale for using α‐shapes is often to simplify the point cloud and the amount of data without losing information on occupied crown space (Rutzinger, Pratihast, Elberink, Sander, & Vosselman, 2011).…”

Storage and Manipulation of Vague Spatial Objects Using Existing GIS Functionality

“…The α-shapes are a generalization of convex hulls. The convex hull of a point set S may be defined as the intersection of all closed half-planes that contain all points of S. This notion is generalized to α-hulls in [8]. For positive (yet sufficiently small) α, the α-hull of S is the intersection of all closed discs with radius 1/α that contain all points of S. Large α produce a curved hull of which the boundary consists of parts of circles (with radius 1/α) that pass through extreme points of S. As α approaches zero, the curved hull converges to the convex hull.…”

“…A generalized disc of radius 1/α is a disc of radius 1/α for α > 0; it is the complement of a disc of radius −1/α for α < 0, and a half-plane for α = 0. For a point set S and a real value α, the α-hull of S is the intersection of all closed generalized discs of radius 1/α that contain all points of S. Two other concepts, α-extreme and α-neighbour, are used in [8] to define the α-shape concept. A point p of a set S is termed an α-extreme in S if there exists a closed generalized disc of radius 1/α such that p lies on its boundary, and it contains all points of S. Two α-extremes p and q are said to be α-neighbours if there exists a closed generalized disc of radius 1/α with both points on its boundary, and which contains all points of S. An α-shape of S is then the straight line graph whose vertices are the α-extremes and whose edges connect the α-neighbours [8].…”

“…4(c) for four points. The set of α-extremes becomes larger when α decreases: the set of α 1 -extreme points of a point set S is a subset of α 2 -extreme points of S if α 1 > α 2 [8]. We are interested in shapes finer than the convex hull, therefore we work only with negative α values.…”

“…The α-shape of a point set S is a subset of the Delaunay triangulation of S [8]. It is built by first constructing the Delaunay triangulation of S, then testing for Delaunay edges whether they are in the α-shape.…”