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What is the Region Occupied bya Set of Points?

Antony GaltonUniversity of Exeter, UK 

Matt Duckham

University of Melbourne, Australia

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The Geneal Poblem

To assign a region to a set of

points, in order to represent thelocation or configuration of the

points as an aggregate,

abstracting away from theindividual points themselves.

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!"ample# Genealisation

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!"ample# Genealisation

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!"ample# $lusteing

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!"ample# $lusteing

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!%aluation $iteia

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Ae outlies allo&ed?

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Must the points lie in the inteio?

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$an the egion be topologically non'

egula?

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$an the egion be disconnected?

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$an the bounday be cu%ed?

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$an the bounday be non'(odan?

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)o& much *empty space+ is

allo&ed?

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,uestions about method

- )o& easily can the method be

genealised to thee .o moe/dimensions?

-What is the computationalcomple"ity of the algoithm?

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Why not use the $on%e" )ull?

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The *$+ shape is lost1

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A non'con%e" egion is bette 

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Anothe !"ample

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$on%e" hull is connected

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 2on'con%e" sho&s t&o *islands+

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!delsbunne+s α'shape

- )3 !delspunne4 D3

5ikpatick and R3

Seidel4 *On the Shapeof a Set of Points in

the Plane+4 IEEE

Transactions on

 Information Theory467893

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A 'Shape

- M3 Melkemi and

M3 D:ebali4

*$omputing theshape of a plana

 points set+4

 Pattern

 Recognition4;

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DSAM Method

- )3 Alani4 $3 =3 (ones and D3 Tudhope4*>oonoi' based egion appo"imation fo geogaphical

infomation etie%al &ith gaetees+4 @(G@S4 ;

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The S&inging Am Method

 

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A set of points 0

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Thei con%e" hull 0

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The s&inging am

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 2on'con%e" hull# r ;

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 2on'con%e" hull# r 9

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 2on'con%e" hull# r B

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 2on'con%e" hull# r   C

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 2on'con%e" hull# r

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 2on'con%e" hull# r

.Anticlock&ise/

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 2on'con%e" hull# r E

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 2on'con%e" hull# r E

.anticlock&ise/

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 2on'con%e" hull# r 8

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$on%e" )ull .r=6E366E0/

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Popeties of the s&inging am

method- Does not genealise staightfo&adly to 9D

.must use a *s&inging flap+/3

- $omple"ity could be as high as O.n9/3

- !ssentially the same esults can be obtained

 by the *close pais+ method .see pape/3

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Delaunay tiangulation methods

 

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$haacteistic hull#

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$haacteistic hull#

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$haacteistic hull#

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$haacteistic hull#

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$haacteistic hull#

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$haacteistic hull#

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$haacteistic hull#

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$haacteistic hull#

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$haacteistic hull#

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$haacteistic hull#

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$haacteistic hull#

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$haacteistic hull#

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$haacteistic hull#

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$haacteistic hull#

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$haacteistic hull#

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Popeties of footpints obtained by

the $haacteistic )ull method-  2o outlies

- Points on the bounday

- May not be topologically non'egula 

- May not be disconnected

- Al&ays polygonal

- May ha%e lage empty spaces

- May not ha%e non'(odan bounday

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Popeties of footpints obtained by

the $haacteistic )ull method- $omple"ity is epoted as O.n log n/4 but

elies on egulaity constaints

- See Duckham4 5ulik4 Galton4 Woboys .in

 pep/3 Daft at http#&&&3duckham3og

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Geneal popeties of Delaunay

methods- DT constains solution space substantially

moe than SA and $P methods

- Ho&e bound of O.n log n/ on DT methods

- !"tensions to thee dimensions may be

 poblematic

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Discussion

- I$oectJ footpint is necessaily

application specific4 but some geneal

 popeties can be identified- A"iomatic definition of a hull opeato does

not accod &ell &ith these shapes

- Kootpint fomation and clusteing ae oftenconflated in methods