Pareto-Optimality of Cognitively Preferred Polygonal Hulls for Dot Patterns Antony Galton University of Exeter UK

The Problem To characterise the region occupied by a set of discrete point-like elements. Call the point-like elements ‘dots’, and the region their ‘footprint’. The footprint is a ‘higher-level’ entity: it is not the location of the dots as individuals, but of the aggregate or collective which has those dots as members.

Map Generalisation

Region Approximation

Location of Collectives

What’s wrong with the Convex Hull? The convex hull of a set of dots is the smallest convex region which contains all of the dots. It has well-known mathematical and computational properties. BUT it does not always give a highly representative footprint.

A ‘C’-shape and an ‘S’-shape

Their Convex Hulls

What we actually see

Some existing work Edelsbrunner et al (1983) Chaudhuri et al (1997) Garai and Chaudhuri (1999) Melkemi and Djebali (2000) Alani et al (2001) Arampatzis (2006) Galton and Duckham (2006) Moreira and Santos (2007) Duckham et al (2008?)

What is missing? A typical paper in this area –Proposes an algorithm for generating footprints for dot patterns –Explores its mathematical and computational characteristics –Examines its behaviour when applied to various dot patterns. What is missing is a principled way of evaluating that behaviour. ‘The shape produced by the algorithm is a good approximation to the perceived shape of the dot pattern’.

But what is ‘the perceived shape’ of the dot pattern? There is no unique solution. It is highly subjective. It is influenced by both the actual geometry of the dots and a multitude of subtle cognitive factors. Nobody seems to have investigated this.

Polygonal Hulls We shall restrict the investigation to shapes having the following properties: –It is a polygon whose vertices are members of the dot pattern –Any member of the dot pattern which is not a vertex lies in the interior of the polygon –The boundary of the polygon is a Jordan curve. A shape of this kind will be called a polygonal hull.

A dot pattern

Its Convex Hull

A very unlikely footprint

A more ‘reasonable’ footprint

What makes a good footprint? The convex hull can include large areas devoid of dots (e.g., perceived concavities) Of all the polygonal hulls, the convex hull has maximum area and minimum perimeter. The very jagged hull reduces the area but has a much longer perimeter. The ‘reasonable’ hull achieves a compromise between reducing the area and increasing the perimeter.

Area vs Perimeter AreaPerimeter Convex hull ‘Reasonable’ hull Jagged hull

Conflicting objectives A cognitively acceptable outline should –Not contain too much empty space –Not be too long and sinuous. To produce an optimal outline we should seek to simultaneously minimise both the area and the perimeter. But these are conflicting objectives, since the perimeter can only be minimised by maximising the area (convex hull).

Multi-objective Optimisation A polygonal hull with area A 1 and perimeter P 1 dominates a hull with area A 2 and perimeter P 2 so long as either A 1 < A 2 & P 1 < P 2 or A 1 < A 2 & P 1 < P 2. In seeking to minimise both area and perimeter we are looking for non- dominated hulls.

Pareto optimisation The non-dominated hulls form the Pareto set. When plotted in area-perimeter space (‘objective space’) these hulls lie along a line called the Pareto front. The Pareto front is the ‘south-western’ frontier of the set of points corresponding to all the hulls for a given dot pattern.

Example

HYPOTHESIS Our hypothesis is The points in area-perimeter space corresponding to polygonal hulls which best capture a perceived shape of a dot pattern lie on or close to the Pareto front.

Pilot Study A small pilot study was carried out to gain an initial estimation of the plausibility of the hypothesis. 8 dot patterns were presented to 13 subjects, who were asked to draw a polygonal outline which best captures the shape formed by each pattern of dots.

Dot Patterns Used in Pilot Study

Area-perimeter plots

Evaluating the Results For each outline drawn, the relative domination RD was computed. RD is the ratio of the number of hulls which dominate it to the maximum number of hulls which dominate any one hull for that dot pattern. By definition, 0 < RD < 1 Our hypothesis predicts that subjects should draw hulls with RD close to 0.

Results Pattern (# dots) HullsPareto Distinct responses Pareto responses Mean RD 1 (12) (12) (11) (12) (13) (11) (11) (11)

Pareto fronts, with selections

Summary of results 57 out of the 104 responses were Pareto optimal. The highest individual value for RD was The mean value for RD over all 104 responses was Therefore, on average, subjects hit the Pareto front with an error of 0.18%. A chi-squared test indicates statistical significance at the 0.1% level (in fact much better than this). The hypothesis is strongly supported by the results of the pilot study.

What Next? Many possible variations to explore: –Choice of dot patterns –Choice of experimental procedure –Application context –Other objective criteria –Evaluation of algorithms –Algorithm design –Extension to three dimensions

The immediate goal … … is to find ways of handling larger dot patterns. Computing the full set of polygonal hulls is computationally expensive, especially when a ‘brute force’ algorithm is used. Two plans of attack: –Look for a more efficient algorithm for computing the full hull-set –Estimate the distribution of the hulls in objective space by some form of sampling, e.g., using an evolutionary algorithm to home in on the Pareto front.

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