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: 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(n 3 ). 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

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