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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
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.
Example: Generalisation
Example: Generalisation
Example: Clustering
Example: Clustering
Evaluation Criteria
Are outliers allowed?
Must the points lie in the interior?
Can the region be topologically non-regular?
Can the region be disconnected?
Can the boundary be curved?
Can the boundary be non-Jordan?
How much ‘empty space’ is allowed?
Questions about method
• How easily can the method be generalised to three (or more) dimensions?
• What is the computational complexity of the algorithm?
Other criteria
• Perceptual
• Cognitive
• Aesthetic
• …
We do not consider these!
Why not use the Convex Hull?
The ‘C’ shape is lost!
A non-convex region is better
Another Example
Convex hull is connected
Non-convex shows two ‘islands’
Edelsbrunner’s -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.
A -Shape
• M. Melkemi and M. Djebali, ‘Computing the shape of a planar points set’, Pattern Recognition, 2000.
DSAM Method• H. Alani, C. B. Jones and D. Tudhope,‘Voronoi-
based region approximation for geographical information retrieval with gazeteers’, IJGIS, 2001
The Swinging Arm Method
A set of points …
Their convex hull …
The swinging arm
Non-convex hull: r = 2
Non-convex hull: r = 3
Non-convex hull: r = 4
Non-convex hull: r = 5
Non-convex hull: r = 6
Non-convex hull: r = 6(Anticlockwise)
Non-convex hull: r = 7
Non-convex hull: r = 7(anticlockwise)
Non-convex hull: r = 8
Convex Hull (r=17.117…)
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
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).
Delaunay triangulation methods
Characteristic hull: 0.98 ≤ l ≤ 1.00
Characteristic hull: 0.91 ≤ l < 0.98
Characteristic hull: 0.78 ≤ l < 0.91
Characteristic hull: 0.64 ≤ l < 0.78
Characteristic hull: 0.63 ≤ l < 0.64
Characteristic hull: 0.61 ≤ l < 0.63
Characteristic hull: 0.56 ≤ l < 0.61
Characteristic hull: 0.51 ≤ l < 0.56
Characteristic hull: 0.40 ≤ l < 0.51
Characteristic hull: 0.39 ≤ l < 0.40
Characteristic hull: 0.34 ≤ l < 0.39
Characteristic hull: 0.28 ≤ l < 0.34
Characteristic hull: 0.25 ≤ l < 0.28
Characteristic hull: 0.23 ≤ l < 0.25
Characteristic hull: 0.22 ≤ l < 0.23
Characteristic hull: 0.00 ≤ l < 0.22
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
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
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
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
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