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