IAPRS, Vol. XXXIII, Ams e dam, 2000
APPLYING COMPUTER VISION TECHNIQUES TO TOPOGRAPHIC OBJECTS
Lau a Keyes, Adam Wins anley
Depa men o Compu e Science
Na ional Uni e si y o I eland Maynoo h
Co. Kilda e, I eland
[email p o ec ed], Adam[email p o ec ed]
Wo king G oup c#c#
KEY WORDS: shape analysis, shape desc ip ion, objec ecogni ion, Fou ie desc ip o s, momen in a ian s,
Bounda y chain-coding, scala desc ip o s.
ABSTRACT
Au oma ic s uc u ing ( ea u e coding and objec ecogni ion) o opog aphic da a, such as ha de i ed om ai su ey
o as e scanning la ge-scale pape maps, equi es he classi ica ion o objec s such as buildings, oads, i e s, ields
and ailways. The ecogni ion o objec s is la gely based on he ma ching o desc ip ions o shapes. Fou ie desc ip o s,
momen in a ian s, bounda y chain coding and scala desc ip o s a e widely used in image p ocessing and compu e
ision o desc ibe and classi y shapes. They ha e been de eloped o desc ibe shape i espec i e o posi ion, o ien a ion
and scale. The applicabili y o he abo e ou me hods o opog aphic shapes is desc ibed and hei use ulness
e alua ed.
1 INTRODUCTION
Au oma ic s uc u ing ( ea u e coding and objec ecogni ion) o opog aphic da a, such as ha de i ed om ai su ey
o as e scanning la ge-scale pape maps, equi es he classi ica ion o objec s such as buildings, oads, i e s, ields
and ailways. Shape and con ex a e he main a ibu es used by humans. Ou p ojec combines shape ecogni ion
echniques de eloped o compu e ision and con ex ual models de i ed om s a is ical language heo y o ecognise
objec s. This pape desc ibes he measu emen o shape o cha ac e ise ea u es ha will hen be used as inpu in o a
g aphical language model.
Much wo k has been done in compu e ision on he iden i ica ion and classi ica ion o objec s wi hin images.
Howe e , less p og ess has been made on au oma ing ea u e ex ac ion and seman ic cap u e in ec o g aphics. This is
pa ly because he low-le el g aphical con en o maps has o en been cap u ed manually (on digi ising ables e c.) and
he encoding o he seman ic con en has been seen as an ex ension o his. Howe e , he success ul au oma ion o
as e - ec o con e sion plus he la ge quan i y o new and a chi ed g aphical da a a ailable on pape makes he
au oma ion o ea u e ex ac ion desi able.
Fea u e ex ac ion and objec ecogni ion a e la ge esea ch a eas in he ield o image p ocessing and compu e ision.
Recogni ion is la gely based on he ma ching o desc ip ions o shapes. Nume ous shape desc ip ion echniques ha e
been de eloped in compu e ision, such as, bounda y chain coding, analysis o scala ea u es (dimension, a ea,
numbe o co ne s e c), Fou ie desc ip o s and momen in a ian s. These echniques a e well unde s ood when applied
o images and ha e been de eloped o desc ibe shapes i espec i e o posi ion, o ien a ion and scale. They can also be
easily applied o ec o g aphical shapes.
A desc ip ion o he abo e ou me hods o shape ecogni ion and hei applica ion o classi ying objec s on la ge-
scale maps is desc ibed he e. Unlike many applica ions whe e he shape ca ego ies a e e y speci ic ( o example
iden i ying a pa icula ai c a ype in a scene), he p oblem equi es he classi ica ion o a pa icula shape in o a
gene al class o simila objec shapes, o example, building, oad o s eam. A compa ison is made o he e ec i eness
o hese echniques in ecognising ea u es on la ge-scale opog aphic maps and plans.
IAPRS, Vol. XXXIII, Ams e dam, 2000
2 SHAPE DESCRIPTION TECHNIQUES
The ecogni ion and desc ip ion o objec s plays a cen al ole in au oma ic shape analysis o compu e ision and i is
one o he mos amilia and undamen al p oblems in pa e n ecogni ion. Common examples a e he eading o
alphabe ic cha ac e s in ex and he au oma ic iden i ica ion o ai c a . Mos applica ions using Fou ie desc ip o s,
momen in a ian s, scala desc ip o s and bounda y chain coding o shape ecogni ion deal wi h he classi ica ion o
such de ini e shapes. To iden i y opog aphic objec s each o he echniques need o be ex ended o deal wi h gene al
ca ego ies o shapes, o example houses, pa cels and oads.
Figu e 1. Sec ion o a digi al map plan
The da a used o he expe imen s desc ibed in he ollowing sec ions was ex ac ed om ec o da a se s (NTF le el 2)
ep esen ing la ge-scale (1:1250) plans o he Isle o Man (Kelly and Hilde 1998), an example o which can be seen in
igu e1. A p e-p ocessing ope a ion was equi ed o ans o m he ec o da a om i s o iginal o m o a new o m
sui able o u he p ocessing. In his case he da a was p e-p ocessed o ex ac closed polygons om lines wi h he
same ea u e codes. A e ex ac ing he equi ed polygonal da a om he maps, an in e pola ion me hod was applied o
sample he shape bounda y a a ini e numbe (N) o equi-dis an poin s. These poin s a e hen s o ed in he app op ia e
o ma o p ocessing wi h each shape desc ip ion echnique.
2.1 Fou ie Desc ip o s
Fou ie ans o m heo y (Gonzalez and Win z 1977) has played a majo ole in image p ocessing o many yea s. I is a
commonly used ool in all ypes o signal p ocessing and is de ined bo h o one and wo-dimensional unc ions. In he
scope o his pape , he Fou ie ans o m echnique is used o shape desc ip ion in he o m o Fou ie desc ip o s. The
Fou ie desc ip o is a widely used all-pu pose shape desc ip ion and ecogni ion echnique (G anlund 1972, Wins anley
1998). The shape desc ip o s gene a ed om he Fou ie coe icien s nume ically desc ibe shapes and a e no malised o
make hem independen o ansla ion, scale and o a ion. These Fou ie desc ip o alues p oduced by he Fou ie
ans o ma ion o a gi en image ep esen he shape o he objec in he equency domain (Wallace and Win z 1980).
The lowe equency desc ip o s s o e he gene al in o ma ion o he shape and he highe equency he smalle de ails.
The e o e, he lowe equency componen s o he Fou ie desc ip o s de ine a ough shape o he o iginal objec
The Fou ie ans o m heo y can be applied in di e en ways o shape desc ip ion. One me hod wo ks on he change
in o ien a ion angle as he shape ou line is a e sed (Zahn and Roskies 1972), bu o he pu pose o his pape he
ollowing p ocedu e was implemen ed (Wood 1986). The bounda y o he image is ea ed as lying in he complex
plane. So he ow and column co-o dina es o each poin on he bounda y can be exp essed as a complex numbe , x + jy
whe e j is sq (-1). T acing once a ound he bounda y in he coun e -clockwise di ec ion a a cons an speed yields a
sequence o complex numbe s, ha is, a one-dimensional unc ion o e ime. In o de o ep esen a e sal a a cons an
IAPRS, Vol. XXXIII, Ams e dam, 2000
speed i is necessa y o in e pola e equi-dis an poin s a ound he bounda y. T a e sing he bounda y mo e han once
esul s in a pe iodic unc ion. The Fou ie ans o m o a con inuous unc ion o a a iable x is gi en by he equa ion:
 




dxeu uF uxj

2
(1)
When dealing wi h disc e e images he Disc e e Fou ie T ans o m (DFT) is used. So equa ion (1) ans o ms o:
 
N
N
x
xj
eu
N
uF

2
1
0
1










(2)
The a iable x is complex, so by using he expansion e[-j A] = cos (A) – j. sin (A) whe e N is he numbe o equally
spaced samples, equa ion (2) becomes:
    










1
0
sin.cos.)
1N
x
AxjAxjyx
N
uF
(3)
omplex numbe s, ob ained by he a e sal o he objec con ou , gi es he Fou ie
esc ip o alues o ha shape.
g he wo coe icien s wi h la ges magni ude and se ing hei phase angle
qual o ze o (Keyes and Wins anley 1999).
o poin s N de ining he shape o be a powe o wo. In
e case o his p ojec i was decided o use 512 sample poin s.
ie Spec um builds a new lis and disposes o he Fou ie ans o m lis . The
sul is 510 Fou ie desc ip o e ms.
whe e A = 2

u/x.
The DFT o he sequence o c
d
The Fou ie desc ip o alues can be no malised o make hem independen o ansla ion, scale and o a ion o he
o iginal shape. Simply, ansla ion o he shape by a complex quan i y ha ing x and y componen s, co esponds o
adding a cons an x + jy o each poin ep esen ing he bounda y. Scaling a shape is achie ed by mul iplying all co-
o dina e alues by a cons an ac o . The DFT esul s in all membe s o he co esponding Fou ie se ies being
mul iplied by he same ac o . So by di iding each coe icien by he same membe , no malisa ion o size is achie ed.
Ro a ion no malisa ion is achie ed by indin
e
To apply he Fou ie desc ip o echnique o he da a se ex ac ed om he Isle o Man map, he poin s a e s o ed as a
se ies o complex numbe s and hen p ocessed using he Fou ie ans o m esul ing in ano he complex se ies also o
leng h N. I he o mula o he disc e e Fou ie ans o m we e di ec ly applied each e m would equi e N i e a ions o
sum. As he e a e N e ms o be calcula ed, he compu a ion ime would be p opo ional o N2. So he algo i hm chosen
o compu e he Fou ie desc ip o s was he Fas Fou ie T ans o m (FFT) o which he compu a ion ime is
p opo ional o NlogN. The FFT algo i hm equi es he numbe
h
The FFT algo i hm is applied o hese 512 coe icien s. The lis is no malised o ansla ion, o a ion and scale. This
esul s in he i s wo e ms always ha ing he alues 0 and 1.0 espec i ely which makes hem edundan o
classi ica ion. Calcula ion o he Fou
e
Gi en wo se s o Fou ie desc ip o s, how do we measu e hei deg ee o simila i y? An app op ia e classi ica ion is
necessa y i unknown shapes a e o be compa ed o a lib a y o known shapes. I wo shapes, A and B, p oduce a se o
IAPRS, Vol. XXXIII, Ams e dam, 2000
alues ep esen ed by a(i) and b(i) hen he dis ance be ween hem can be gi en as c(i) = a(i) – b(i). I a(i) and b(i) a e
iden ical hen c(i) will be ze o. I hey a e di e en hen he magni udes o he coe icien s in c(i) will gi e a easonable
measu e o he di e ence. I p o es mo e con enien o ha e one alue o ep esen his a he han he se o alues ha
make up c(i). The easies way is o ea c(i) as a ec o in a mul i-dimensional space, in which case i s leng h, which
ep esen s he dis ance be ween he planes, is gi en by he squa e oo o he sum o he squa es o he elemen s o c(i).
2.2 Momen In a ian s
Ghee, 1977). These momen
a ian alues a e in a ian wi h espec o ansla ion, scale and o a ion o he shape.
he no malised cen al momen s a se o se en
ian momen s can be compu ed which a e independen o o a ion.
lemen a ion hey a e compu ed in he disc e e o m. Gi en a unc ion (x,y), hese egula momen s a e
e ined by:
(4)
e o he m en is p + q) he e p and q a e bo h
(5)
al momen s. The co-
dina e he cen e o i y o he image a e cal ula ed using equa ion (5) :
(6)
The cen al momen s can hen be de ined in hei disc e e ep esen a ion as:
dxdyyx yx qp
pq ),(


Momen in a ian s ha e been equen ly used as ea u es o image p ocessing, emo e sensing, shape ecogni ion and
classi ica ion. Momen s can p o ide cha ac e is ics o an objec ha uniquely ep esen i s shape. In a ian shape
ecogni ion is pe o med by classi ica ion in he mul idimensional momen in a ian ea u e space. Se e al echniques
ha e been de eloped ha de i e in a ian ea u es om momen s o objec ecogni ion and ep esen a ion. These
echniques a e dis inguished by hei momen de ini ion, such as he ype o da a exploi ed and he me hod o de i ing
in a ian alues om he image momen s. I was Hu ( Hu, 1962), ha i s se ou he ma hema ical ounda ion o wo-
dimensional momen in a ian s and demons a ed hei applica ions o shape ecogni ion. They we e i s applied o
ai c a shapes and we e shown o be quick and eliable (Dudani, B eeding and Mc
in
Hu de ines se en o hese shape desc ip o alues compu ed om cen al momen s h ough o de h ee ha a e
independen o objec ansla ion, scale and o ien a ion. T ansla ion in a iance is achie ed by compu ing momen s ha
a e no malised wi h espec o he cen e o g a i y so ha he cen e o mass o he dis ibu ion is a he o igin (cen al
momen s). Size in a ian momen s a e de i ed om algeb aic in a ian s bu hese can be shown o be he esul o a
simple size no malisa ion. F om he second and hi d o de alues o
in a
T adi ionally, momen in a ian s a e compu ed based on he in o ma ion p o ided by bo h he shape bounda y and i s
in e io egion (Hu 1962). The momen s used o cons uc he momen in a ian s a e de ined in he con inuous bu o
p ac ical imp
d
Mpq is he wo-dimensional momen o he unc ion (x,y). The o d om ( w
na u al numbe s. Fo implemen a ion in digi al om his becomes:
),( yx yx
pq 

 qp
To no malise o ansla ion in he image plane, he image cen oids a e used o de ine he cen
o s o g a c and a e gi en by
00

10

x

y01
00




 qp
pq yyxx )()(

IAPRS, Vol. XXXIII, Ams e dam, 2000
(7)
The momen s a e u he no malised o he e ec s o change o scale using he ollowing o mula:




00
pq
pq 
(8)
Whe e he no malisa ion ac o :  = (p + q / 2) +1. F om he no malised cen al momen s a se o se en alues can be
calcula ed and a e de ined by:
1 = 20 + 02
2 = (20 - 02)2 + 4211
3 = (30 - 312)2 + (03 - 321)2
4 = (30 + 12)2 + (03 + 21)2
5 = (330 - 312)(30 + 12)[(30 + 12)2
–3(21 + 03)2] + (321 - 03)(21 + 03)
 [3(30 + 12)2 – (21 + 03)2]
6 = (20 - 02)[(30 + 12)2 – (21 + 03)2]
+ 411(30 + 12)(21 + 03)
7 = (321 - 03)(30 + 12)[(30 + 12)2
- 3(21 + 03)2] + (312 - 30)(21 + 03)
 [3(30 + 12)2 – (21 + 30)2] (9)
These se en in a ian momen s, I, 1  I  7, se ou by Hu, we e addi ionally shown o be independen o o a ion.
Howe e hey a e compu ed o e he shape bounda y and i s in e io egion.
2.2.1 New momen s
Fo he pu pose o his pape he momen in a ian s a e compu ed using he shape bounda y only and a e p o en o be
in a ian unde objec ansla ion, scale and o a ion (Chau -Chin Chen 1993). Then, using he same no a ion o
con enience, he momen de ini ion in equa ion (1) can be exp essed as:
(10)


C
qp
pq dsyx
Fo p, q = 0,1,2,3, whe e c is he line in eg al along he cu e C and ds =

((dx)2 + (dy)2). The cen al momen s can be
simila ly de ined as:

C
qp
pq dsyyxx )()(


IAPRS, Vol. XXXIII, Ams e dam, 2000
(11)
Gi en ha he cen oids a e as in he egula me hod:
00
01


y
00
10


x
(12)
Fo a digi al image, hen equa ion (11) becomes



C
qp
pq yyxx
),(
)()(

(13)
Thus he cen al momen s a e in a ian o ansla ion. These new cen al momen s can also be no malised such ha hey
a e scaling in a ian .
(14)




pq
pq 
00
Whe e he no malisa ion ac o is:  = p + q + 1. The se en momen in a ian alues can hen be calcula ed as be o e
using he esul s ob ained om he compu a ion o equa ion’s (10) o (14) abo e.
Using he same da a se s as in he Fou ie desc ip o me hod desc ibed ea lie , he momen s echnique is applied.
Howe e , o momen s he poin s ex ac ed om he map a e s o ed no as complex numbe s bu ep esen he x and y
co-o dina es o he polygonal shape. These poin s a e p ocessed by a momen ans o ma ion on he ou line o he
shape, which p oduces se en momen in a ian alues ha a e no malised wi h espec o ansla ion, scale and o a ion
using he o mulae abo e. The esul ing se o alues can be used o disc imina e be ween he shapes. Classi ica ion is
ca ied ou using he same me hod desc ibed in sec ion 2.1 o Fou ie desc ip o s.
2.3 Bounda y chain coding and scala desc ip o s
Bounda y chain-coding and scala desc ip o s a e also used as shape desc ip ion echniques o his expe imen . Chain-
code ea s he poin s o a cu e and in pa icula a egion bounda y, di ec ly and in a s ic ly local ashion. The basis o
he echnique is, essen ially, o s a wi h a poin ha is belie ed o be on he bounda y (some local edge poin ) and o
ex end he bounda y by adding a neighbou ing poin in he con ou di ec ion ( ha is, he di ec ion which is no mal o
he g adien di ec ion). This p ocess is ei e a ed, s a ing a his new bounda y pixel. This bounda y chain code
echnique encodes piecewise linea cu es as a sequence o di ec ed s aigh -line segmen s called links. The e a e eigh
possible di ec ions o a link be ween a poin and i s neighbou . These eigh di ec ions a e numbe ed ‘0’ h ough ‘7’ and
mo e coun e -clockwise, as shown in igu e 3a. Each o hese can be conside ed as an angula di ec ion, in mul iples o
45 deg ees, which a e mo ed o go om one pixel o he nex . The absolu e co-o dina es (x, y) o he i s bounda y
pixel oge he wi h he chain code, ep esen a comple e desc ip ion o he disc e e egion bounda y.
IAPRS, Vol. XXXIII, Ams e dam, 2000
(a) (b)
Figu e 3 (a) shows he eigh neighbou s numbe ed ‘0’ h ough ‘7’, coun e clockwise. (b), indica es he di ec ion o
each link ( he di ec ion ha can be a elled be ween a poin and i s neighbou ).
BCC:31111757555
Figu e 4 A Bounda y Chain Code (BCC) ep esen a ion o a simple shape.
As acing a ound he bounda y con inues i builds a bounda y chain code ep esen a ion o he con ou by eco ding all
he bounda y pixels isi ed. Figu e 4 shows a bounda y ep esen a ion o a simple shape, which is based upon he wo k
o F eeman (F eeman 1961). I is assumed ha he bounda y pixels co espond o he pixels exhibi ing he local
maximum g adien magni ude and ha he g adien di ec ion indica es, app oxima ely a leas , he neighbou ing
bounda y pixel. Mo e o mally, he bounda y di ec ion D o an edge poin is gi en by he edge g adien G as D = G + 2
o he o wa d di ec ion and as D = G –2 o he e e se di ec ion (+90° and -90° espec i ely). By acing he
bounda y, candida es o inclusion in he bounda y a e gi en by di ec ions, D, D + 1, and D – 1. These candida es a e,
he pixel di ec ly ahead o he cu en pixel and one pixel ei he side o i , o example i D = 1 (bounda y di ec ion is
45°) hen pixels 0, 1 and 2 a e chosen as candida es. The po en ial o each o hese candida es o be a bounda y poin is
e alua ed. I he pixel has been isi ed p e iously o i he pixel o e laps he image bounda y, hen i is assumed o ha e
a nega i e po en ial. The candida e wi h he highes posi i e po en ial is selec ed as he nex bounda y poin om which
o con inue he ace. This poin is implici ly included in he lis o bounda y poin s by upda ing he BCC.
This BCC echnique ou lined abo e is dealing speci ically wi h digi ised, edge de ec ed pixel images. Howe e , o he
pu pose o his pape his o m o shape desc ip ion needs o be adap ed o deal wi h he ec o map da a desc ibed in
he beginning o sec ion 2, which will be seen in la e in he pape . Gi en he BCC ep esen a ion o a co pus o shapes
in wha way can hey be classi ied. While he BCC is a use ul me hod o he ep esen a ion o shapes, ecogni ion is
no mally based upon o he desc ip o s de i ed om he BCC. Fo example, he momen shape desc ip o s discussed
p e iously can be gene a ed om he bounda y poin s gi en by he BCC, which is u ilised o his p ojec .
IAPRS, Vol. XXXIII, Ams e dam, 2000
Scala desc ip o s a e based on scala ea u es de i ed om he bounda y o an objec . They use nume ous aspec s o
he objec o pe o ming shape ecogni ion. Simple examples o such ea u es include:
 he pe ime e leng h;
 he a ea o he shape;
 he a io o he a ea o a shape o he squa e o he leng h o i s pe ime e (A/P2);
 he numbe o nodes;
 he numbe o co ne s;
Recogni ion and classi ica ion o he esul ing scala alues, o desc ibe a shape, can be e alua ed h ough he dis ance
be ween he ec o s in he n-dimensional space in he same way as he classi ica ion me hod desc ibed in sec ion 2 o
he Fou ie desc ip o and momen in a ian s echniques.
RESULTS
In his sec ion a sample o he esul s p oduced by he applica ion o he Fou ie desc ip o , momen in a ian s and
scala desc ip o echniques a e p esen ed o e alua e and compa e hei use ulness in shape disc imina ion o gene al
opog aphic ea u es. E alua ion o he BCC me hod is no p esen ed in his wo k as his is s ill an on-going expe imen .
Figu e 5 plo s he a e age alues, ob ained o i e ca ego ies o objec s om he sample maps (using he Fou ie
desc ip o me hod in his example). This shows ha in o de o classi y shapes wi h any deg ee o ce ain y, he
a ia ion wi hin classes mus be less han ha be ween classes.
0.03 0.04 0.05 0.06
0.1
0.2
0
0.02
0.04
0.06
FD(2)
FD(3)
FD
(4)
Building
Pa cel
Road
Railway
S eam
Figu e 5. A e age alues o i e sample classes
To compa e he h ee shape ecogni ion echniques used, se e al sample shapes om he map (buildings and pa cels)
we e used as es images. Figu es 6a, 6b, 7 and 8 show plo s o he Fou ie desc ip o , momen in a ian s and scala
desc ip o echniques espec i ely o a small ep esen a i e sample o buildings and land pa cels. Each plo shows he
deg ee o which he wo se s o objec s clus e in h ee-dimensional space. As can be seen in Figu e 6 (a), hese wo se s
a e no dis inc . The e idence he e o e indica es ha no malised FD’s a e no e y good o use in shape desc ip ion
whe e he da a se s a e o a e y gene al shape. No e, ha due o no malisa ion he i s wo e ms, FD(0) = 0 and
FD(1)= 1 a e edundan in compa ison. Howe e , because he polygons a e o a known scale he expe imen was
conduc ed using a Fou ie desc ip o echnique ha is no no malised o scale, ha is, FD(1)  1 and he e o e can be
used as a compa ison. The esul ing clus e s ( igu e 6 (b)) a e much mo e signi ican indica ing ha using ou ie
desc ip o s wi hou scale no malisa ion gi es an imp o emen in objec disc imina ion.
IAPRS, Vol. XXXIII, Ams e dam, 2000
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
0
0.05
0.1
0.15
0.2
FD(2)
FD(3)
FD(4)
0
2000
4000
6000
0
2000
4000
6000
0
200
400
600
800
FD(1)
FD(2)
FD(3)
Figu e 6, (a): Clus e ing o he polygon shapes in h ee-dimensional space o he ea u es FD(2), FD(3) and FD(4), (b):
Clus e ing o he polygon shapes in h ee-dimensional space o he ea u es FD(1), FD(2) and FD(3), no no malised o
scale.
Buildings Land Pa cels
Numbe o polygons analysed 537 1095
Mean FD alues FD(2) = 0.0422
FD(3) =0.0795
FD(4) = 0.0416
FD(2) =0.0489
FD(3) = 0.0672
FD(4) = 0.0279
Va iance in FD’s (2) FD(2) =0.0073
FD(3) =0.0067
FD(4) = 0.0049
FD(2) =0.0088
FD(3) = 0.0030
FD(4) = 0.0016
Repea abili y (3) FD(2) =0.2562
FD(3) = 0.2457
FD(4) = 0.2100
FD(2) =0.2814
FD(3) = 0.1644
FD(4) = 0.1200
Dis ance be ween means o buildings and pa cels FD(2) =0.0067
FD(3) = 0.0123
FD(4) = 0.0137
Table 1: Compa ison o epea abili y wi hin ea u e classes and dis ance be ween classes o Fou ie desc ip o s.
Table 1 shows he measu emen s o he sample o building and land pa cels. The epea abili y o he measu emen s o
he class, ep esen ed as 3 imes he s anda d de ia ion, is sizeably la ge han he dis ance be ween he mean alues o
he wo classes.
0
0.05
0.1
0.15
0.2
0.25
0
0.05
0.1
0.15
0.2
0
2
4
6
8
x 10
-4
IM(1)
IM(2)
IM(3)
Figu e 7. Clus e ing o he polygon shapes in h ee dimensional space o he ea u e IM(1), IM(2) and IM(3).