On the Shape of a Set of Points

Radke, John

doi:10.1016/b978-0-444-70467-2.50014-x

Cited by 17 publications

(8 citation statements)

References 19 publications

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“…In [6] and [12] a distinction is made between the "internal" and "external" shapes of a finite point set. It is stated that "the external shape of a point set is exhibited by identifying the essential extreme points of the set and, among these, joining essential neighbors" and that "the internal shape of a point set is exhibited by identifying essential points of the set and, among these, joining essential neighbors."…”

Section: Relation To Other Workmentioning

confidence: 99%

“…It is stated that "the external shape of a point set is exhibited by identifying the essential extreme points of the set and, among these, joining essential neighbors" and that "the internal shape of a point set is exhibited by identifying essential points of the set and, among these, joining essential neighbors." In [6] and [12] the external shape is described by using the a-shape, where the internal shape is described by using a so-called fl-shape. This fl-shape is based on neighborhoods, with parameter/3 regulating the size of the neighborhood.…”

Section: Relation To Other Workmentioning

confidence: 99%

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Shape of an arbitrary finite point set in IR2

Worring

Smeulders

1994

J Math Imaging Vis

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Abstract. We study the shape of a finite point set in Ill 2, where the points are not bound to a regular grid like Z 2. The shape of a connected point set in IR 2 is captured by its boundary. For a finite point set the boundary is a directed graph that connects points identified as boundary points. We argue that to serve as a proper boundary definition the directed graph should regulate scale, be minimal, have an increasing interior and be consistent with the boundary definition of connected objects. We propose to use the directed variant of the a-shape as defined by Edelsbrunner et al (1983), which we call the a-graph. The a-graph is based on a generalization of the convex hull. The computational aspects of the a-graph have been extensively studied, but little attention has been paid to the potential use of the a-graph as a shape descriptor or a boundary definition. In this paper we prove that the oz-graph satisfies the aforementioned criteria. We also prove a relation between the a-graph and the opening scale space from mathematical morphology. In fact, the a-hull provides a generalization of this scale space.

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Section: Relation To Other Workmentioning

confidence: 99%

Section: Relation To Other Workmentioning

confidence: 99%

Shape of an arbitrary finite point set in IR2

Worring

Smeulders

1994

J Math Imaging Vis

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“…A classical way for associating a "shape" to a given distribution of points on the plane is to connect pairs of points that are deemed close by some proximity measure, computing in this way a graph, called a proximity graph, associated to the set of points. Many different measures of proximity have been defined (each giving rise to different types of proximity graphs) and among them the proximity regions described above play a central role [18], [21], [23], [13]. If, for example, one wishes to give a set of points the "shape" of a tree, it is necessary to determine which proximity regions will induce on the points such a shape.…”

Section: Applicationsmentioning

confidence: 99%

Proximity constraints and representable trees (extended abstract)

et al. 1995

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“…The Gabriel, modified Gabriel, relative neighborhood and relatively closes~ drawings studied in [26], [25], [3], [5], [35] are all examples of members of a family of drawings called fl-drawings. In 1985, Kirkpatrick and Radke [23,30] R(x, y, 0o). The regions defined above are referred to as lune-based/7-regions.…”

Section: A Delaunay Drawingmentioning

confidence: 99%

Proximity drawability: A survey extended abstract

1995

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Abstract. Increasing attention has been given recently to drawings of graphs in which edges connect vertices based on some notion of proximity. Among such drawings are Gabriel, relative neighborhood, Delaunay, sphere of influence, and minimum spanning drawings. This paper attempts to survey the work that has been done to date on proximity drawings, along with some of the problems which remain open in this area. Proximity DrawingsIn 1969, Gabriel and Sokal [15] presented a method for associating a graph to a set of geographic data points P by connecting points z, y E P with an edge if and only if the closed disk having the segment ~ as diameter contained no other point of P. This graph, now called the Gabriel graph of P, is just one example of what have come to be called proximity graphs. Loosely speaking, a proximity graph is a graph constructed from a set P of points in some metric space by connecting pairs of points which are deemed to be "sufficiently" close together. A set P can give rise to a variety of different proximity graphs depending upon the definition of closeness used. Early work in this area was concerned for the most part with the problems of determining notions of proximity which might best capture the "internal structure" of a set of points and, having done so, of efficiently computing the proximity graph of a given set of points. For a survey of such results, see Jaromczyk and Toussaint [21].More recently, increasing attention has been given to the proximity drawing problem: given a graph G and a definition of proximity, determine whether a set P of points exists such that the proximity graph of P is the given graph, and if so, compute such a set. Clearly the set P, if it exists, gives rise to a straight-line

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On the Shape of a Set of Points

Cited by 17 publications

References 19 publications

Shape of an arbitrary finite point set in IR2

Shape of an arbitrary finite point set in IR2

Proximity constraints and representable trees (extended abstract)

Proximity drawability: A survey extended abstract

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