What is the Region Occupied by a Set of Points?

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What is the Region Occupied by a Set of Points?

• Antony Galton
• University of Exeter, UK
• Matt Duckham
• University of Melbourne, Australia

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The General Problem

• To assign a region to a set of points, in order
  to represent the location or configuration of the
  points as an aggregate, abstracting away from the
  individual points themselves.

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Example Generalisation
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Example Generalisation
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Example Clustering
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Example Clustering
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Evaluation Criteria
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Are outliers allowed?
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Must the points lie in the interior?
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Can the region be topologically non-regular?
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Can the region be disconnected?
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Can the boundary be curved?
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Can the boundary be non-Jordan?
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How much empty space is allowed?
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Questions about method

• How easily can the method be generalised to three
  (or more) dimensions?
• What is the computational complexity of the
  algorithm?

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Other criteria

• Perceptual
• Cognitive
• Aesthetic
• We do not consider these!

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Why not use the Convex Hull?
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The C shape is lost!
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A non-convex region is better
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Another Example
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Convex hull is connected
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Non-convex shows two islands
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Edelsbrunners a-shape

• H. Edelsprunner, D. Kirkpatrick and R. Seidel,
  On the Shape of a Set of Points in the Plane,
  IEEE Transactions on Information Theory, 1983.

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A -Shape

• M. Melkemi and M. Djebali, Computing the shape
  of a planar points set, Pattern Recognition,
  2000.

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DSAM Method

• H. Alani, C. B. Jones and D. Tudhope,Voronoi-base
  d region approximation for geographical
  information retrieval with gazeteers, IJGIS, 2001

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The Swinging Arm Method

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A set of points
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Their convex hull
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The swinging arm
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Non-convex hull r 2
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Non-convex hull r 3
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Non-convex hull r 4
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Non-convex hull r 5
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Non-convex hull r 6
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Non-convex hull r 6(Anticlockwise)
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Non-convex hull r 7
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Non-convex hull r 7(anticlockwise)
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Non-convex hull r 8
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Convex Hull (r17.117)
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Properties of footprints obtained by the swinging
arm method

• No outliers
• Points on the boundary
• May be topologically non-regular
• May be disconnected
• Always polygonal (possibly degenerate)
• May have large empty spaces
• May have non-Jordan boundary

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Properties of the swinging arm method

• Does not generalise straightforwardly to 3D (must
  use a swinging flap).
• Complexity could be as high as O(n3).
• Essentially the same results can be obtained by
  the close pairs method (see paper).

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Delaunay triangulation methods

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Characteristic hull 0.98 l 1.00
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Characteristic hull 0.91 l lt 0.98
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Characteristic hull 0.78 l lt 0.91
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Characteristic hull 0.64 l lt 0.78
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Characteristic hull 0.63 l lt 0.64
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Characteristic hull 0.61 l lt 0.63
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Characteristic hull 0.56 l lt 0.61
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Characteristic hull 0.51 l lt 0.56
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Characteristic hull 0.40 l lt 0.51
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Characteristic hull 0.39 l lt 0.40
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Characteristic hull 0.34 l lt 0.39
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Characteristic hull 0.28 l lt 0.34
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Characteristic hull 0.25 l lt 0.28
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Characteristic hull 0.23 l lt 0.25
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Characteristic hull 0.22 l lt 0.23
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Characteristic hull 0.00 l lt 0.22
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Properties of footprints obtained by the
Characteristic Hull method

• No outliers
• Points on the boundary
• May not be topologically non-regular
• May not be disconnected
• Always polygonal
• May have large empty spaces
• May not have non-Jordan boundary

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Properties of footprints obtained by the
Characteristic Hull method

• Complexity is reported as O(n log n), but relies
  on regularity constraints
• See Duckham, Kulik, Galton, Worboys (in prep).
  Draft at http//www.duckham.org

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General properties of Delaunay methods

• DT constrains solution space substantially more
  than SA and CP methods
• Lower bound of O(n log n) on DT methods
• Extensions to three dimensions may be problematic

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Discussion

• Correct footprint is necessarily application
  specific, but some general properties can be
  identified
• Axiomatic definition of a hull operator does not
  accord well with these shapes
• Footprint formation and clustering are often
  conflated in methods