A
F amewo k
o
Digi al
Topology
E.
Dom’nguez
A.R.
F anc&
Dp o. Ing. EMc ica e In o mi ica
Facul ad de Ciencias.
U.
de Za agoza
E-50009
Za agoza
(SPAIN)
Dp o. Ing. Elk ica
e
In o m& ica
Facul ad de Ciencias.
U.
de Za agoza
E-50009
Za agoza
(SPAIN)
A.
Mkquez
Dp o. de Algeb a, Geome ia, Topologia
y
Compu acibn
Facul ad de Ma emL icas.
U.
de Se illa
Se illa
(SPAIN)
Abs mc
The
main
goal
o
his
pape
is
o show
he
unc ional a chi ec u e
o
a
amewo k
o
Dig-
i al
Topology.
This
a chi ec u e
has
ou
le els,
called
De ice, Logical, Concep ual and Con inuous
Le els.
In
each one
o
hem we
can
use
se e al mod-
els
acco ding
o
he
pa icula
p oblem. The models
in
he De ice Le el
ep esen
he
physical
p oblem
whe eas he models in
he
Con inuous Le el
a e
opological
spaces
which
allow
us
o
use
he well-
known
esul s
o
con inuous opology (ac ually, he
s onge
esul s
o
polyhed al opology). The o he
wo le els
a e
used
o And
a
digi al
solu ion. The
Logical Le el
is
close
o
he De ice Le el
and
i
is
used
o
p ocessing,
o
w i ing algo i hms
and
showing hei co ec ness. The Concep ual Le el
is
he nea es
o
he Con inuous Le el and i is used
o
ansla e
esul s
and
no ions
om
he Con inuous
Le el
o
he
Logical
Le el.
I. INTRODUCTION
The‘main pu pose o Digi al Topology is he s udy
o opological p ope ies o disc e e objec s which a e
go en digi izing con inuous objec s. Digi al Topology
plays a e y impo an ole in compu e ision, im-
age p ocessing and compu e g aphics. Bu a comple e
heo e ical ounda ion o a consis en heo y o digi al
spaces
is
s ill missing. Kong and Rosen eld gi e in
[5]
a e y good su ey abou his subjec .
Howe e , se e al heo ies ha e been de ised o he
analysis o he opological a ibu es o a digi al im-
age. Le us ecall, o example, Rosen eld’s combina o-
ial heo y
[9,lO,ll]
(gene alized by Kong and Roscoe
~~
Manusc ip ecei ed
July
1,
1993.
This
wo k
was
suppo ed
in
pa
by
he Dipu acih Gene al de A ag6n, he DGICYT and
he
Jun a
de Andalucia,
Spain.
in
[4])
and Khalimsky’s heo y based
in
a
pa icula
opological space
[3].
The common d awback o hese
heo ies is ha , in o de o include well-known esul s
o he Euclidean Topology, hey need
o
ew i e new
p oo s ins ead o exploi ing hose
coming
om con in-
uous opology. The eason
is
ha hese heo ies a e
a om he Euclidean plane (o space). Some o he
au ho s (Ko ale sky
[7],
Ankeney, Ri e
[l])
can use
esul s coming om Euclidean Topology bu hey ha e
p oblems in he image p ocessing because he models
a e a om he disc e e objec s which ep esen sc een
digi al images.
In his pape we in oduce
a
new poin
o
iew. We
p esen a amewo k, which ies o de ine a gene al
heo y o he de elopmen o Digi al Topology. To
do his, ou amewo k p oposes
a
mul ile el a chi ec-
u e whose main ea u e is he abili y o ansla ing
concep s, s a emen s, p oo s and algo i hms om con-
inuous opology wi hou ew i ing
a
pa allel heo y.
The
i s
le el ep esen s
a
compu e and he ollow-
ing le els consis o models mo e
and
mo e abs ac .
Finally, he las le el ep esen s he Euclidean Topol-
ogy. This akes us, om he disc e e wo ld (a compu e
sc een), and b ings us close o he con inuous one ( he
Euclidean plane o space).
In ou amewo k he e
is
no an uni e sal model
which can be used o sol ing all he p oblem in Digi-
al Topology. Ins ead, gi en
a
p oblem we mus choose
a sui able model in e e y le el. Acco ding o ou p o-
posal, wha
is
common
o
all he p oblems is he wo k-
ing me hodology and he mul ile el unc ional a chi ec-
u e.
To show ha ou heo y wo ks we p esen he solu-
ion o he well-known Digi al Jo dan Cu e P oblem
(as
many o he au ho s ha e done in o de o p o e he
65
Figu e
1:
Sc een model
Figu e
2:
A
digi al image
consis ency
o
hei heo ies).
So,
in he second sec ion
we choose he models
o
each le el in he a chi ec u e,
and hen, in he hi d sec ion, we gi e he p oo o he
esul ha is educed o gi ing he app op ia e no ions
and ansla ions. Finally, he o h sec ion is de o ed o
explain he unc ional a chi ec u e in
a
gene al con ex .
11.
THE MATHEMATICAL MODELS
We conside ha he sc een model
is
an in ini e ma-
ix
S
o
pixels wi h he shape showed in Fig
l
whe e
each pixel can ha e wo s a es ep esen ed by
a
do ed
o
black small squa e. In his con ex ,
a
digi al image
in he sc een model
S
is de ined by
a
se o black poin s
(see
a
digi al cu e in Fig
2).
The Digi al Jo dan Cu e P oblem consis s o p o -
ing ha
a
simple closed digi .al cu e, as ,lia
o
Fig
2,
di ides he sc een in wo connec ed componen s. E i-
den ly i is necessa y o de ine he meaning o “simple
closed digi al cu e” and “connec ed componen s.” Fo
his, we will s a by de ining he ma hema ical models
used in each le el
o
ou amewo k.
Since ou p oblems
has
a
,opological na u e, i is
na u al o conside
a
ans o ma ion om he sc een
model
S
o he g aph
E8
ep esen ed
in
he Fig
3.
Fo
he sake o simplici y we will suppose
ha
he e ex
Figu e
3:
The g aph
E8
Figu e
4:
The g aph
E:
se o
E8
is
Z2,
which is he se
o
all pai s
o
in ege
numbe s. The e ices o he g aph ep esen he pixels
and wo e ices a e adjacen i and only i hei co -
esponding pixels a e con iguous in he ob ious sense.
In o de o sol e ou p oblem, his ma hema ical model
ep esen s he Logical Le el.
In
i
he cu e becomes
he subg aph induced by he e ices co esponding o
black pixels.
Because
E8
is no plana , i is well-known ha wi hin
i we canno ep esen he ,opology o he Euclidean
plane (see Rosen eld
[9]).
To
sol e ha p oblem we
la en ou his g aph
in
a
na .u al way and we ge
he plana g aph
E;
ep esen ed
in
he Fig
4.
In
his
g aph he e a e wo di e en kinds o e ices. Some o
hem ep esen he pixels and he o he s, called middle
poin s, ep esen
a
deg ee o nea ness be ween he co -
esponding pixels;
in
ac
i
is he diagonal nea ness.
Obse e
ha
his g aph is
a
iangula ion
o
he Eu-
clidean plane. This makes up he Concep ual Le el
o
66
Figu e
5:
A digi al cu e
C
in
E8
Figu e
6:
A
digi al cu e
C'
in
ES
sol e ou p oblem.
On he o he hand, we de ine a digi al image
(o
digi al subspace) o one o hese g aphs
as
an induced
subg aph; ha is, a subg aph which con ains an edge
i and only i i con ains he wo e ices o he edge.
We ep esen he se o digi al images in
E8
and
E:
by
U(&)
and
U(&),
espec i ely.
Now we ha e
a
na u al ans o ma ion
A
:
O(
Es)
-
U(E;)
de ined
as
ollows: Gi en a digi al image
C
in
Ea,
n(C)
=
c'
is he subg aph induced by he e ices
in
C
and he middle e ices de ined by wo diagonal
e ices in
C
(see he Fig
5
and
6).
In a na u al way we
ha e a ans o ma ion
A*
:
o(E,')
-
o(E8).
Gi en a
digi al image
C'
in
E,'
n*(C*)
=
C
is he subg aph
in-
duced by he e ices in
C'
ha a e no middle e ices.
Also
we ha e a ans o ma ion
j
:
(?(E,)
-
,C(R2)
in-
duced by he embedding o
E,
in
he Euclidean plane
R2,
whe e
C(R2)
is he se
o
polygona subspaces.
In his
way.
we ha e he a chi ec u e ep esen ed by
he ollowing diag am
whe e
O(S)
is
he
se
o digi al images in he sc een
model
S
and
i
is
he
1-1
ans o ma ion be ween
O(E8)
and
O(S).
67
III. THE DIGITAL JORDAN CURVE THEOREM
In he logical and concep ual le els,
a
simple digi al
cu e
is
he subg aph induced by
a
sequence o e ices
{PO,.
.
.
,pn}
SO
ha
Pi
is
adjacen
o
pj
i
and only
i
li
-
jl
5
1.
The cu e
is
called closed i , in addi ion,
po
=
pn.
Then a digi al objec in he
sc een
model
S
is
called a simple closed digi al cu e
i
i s
image by
i-'
is
his ype o cu e in
Ea.
These de ini ions ag ee wi h.
hose usually adop ed
in
he li e a u e (see
[5]).
In his way,
a
simple closed
digi al
cu e
C
in he
sc een model
S
is, by de ini ion, ans o med h ough
i
in such a cu e in
Ea,
deno ed
also
by
C.
I
is
ob ious
ha he image by
17
o
a
simple closed digi al cu e
C
in
E8
is a simple closed digi al cu e
C'
in
E;.
And,
also, he image by he embedding
j
o
C
is
a polygonal
Jo dan Cu e
C'
in
R2
(see Fig
5,
6).
In
his way, we
ha e ansla ed ou ini ial Jo dan Cu e P oblem o
he Sc een Model o analogous p oblems
in
Ea,
E,'
and
R2.
Now hen,
i
is
well-known ha he solu ion o his
p oblem in
R2
is
he Polygonal Jo dan Cu e Theo em.
So
ha , ou goal is o ansla e his heo em, by mean
o he ans o ma ions
j
and
R',
o
E8
in o de o ind
a solu ion o ou p oblem in his model.
In he li e a u e, he e exis s se e al equi alen
s a emen s o he Polygonal Jo dan Cu e Theo em.
He e, we conside one
o
hem ha is app op ia ed o
ind an algo i hm sol ing his p oblem and o p o e i s
co ec ness. The base o his s a emen is he no ion
o ans e sal in e sec ion be ween a hal -line, which
is
pa allel o he axis
OX,
and a polygonal cu e.
Le
D
be a polygonal cu e in
R2
whose se o
e ices
{PO,.
.
.
,pn}
is coun e -clockwise o de ed. Le
,
=
{(z,y);
y
=
y,
and
z
2
zq}
be
a
hal -line, whe e
q
=
(
z,
y,).
The e exis s a
ans e sal in e sec ion
be-
ween
D
and
q
i
one o he ollowing si ua ions occu s:
(a)
,
in e sec s he edge de ined by he e ices
pi
and
p1+i
in only a poin
P
B
{pi,Pi+1}-
(b)
he e exis s
i
E
(0,.
.
.
,
n}
such ha o some
k
2
0
2-
Pi-1
>
Yq
and
Yi+k+l
<
Yq
o
Pi-1
<
Yq
and
Yi+k+1
>
Yq
3.
x,
2
xq
o e e y
i
5
j
L
i
+
k.
Wi h his de ini ion we can s a e he nex well-known
esul .
Polygonal Jo dan Cu e Theo em.
Le
D
be
a simple closed polygonal cu e in
R2,
hen
R2
D
has wo connec ed componen s (one o hem bounded
and he o he one unbounded). Mo eo e , a poin
q
E
R2
D
belongs o he bounded componen i and only
i
#( ,
4
D)
E
1
(mod
2),
whe e
#( q
,+,
D)
ep esen s
he numbe o ans e sal in e sec ions be ween
D
and
q.
I
is
impo an o poin ou ha he esul we wan
o ansla e o
E8
is applied o polygonal cu es
c'
coming om a digi al cu e
C
in
E8
and hal -lines
p ,
whe e
p'
belongs o
Z2.
In his way, i is easy o obse e
ha he ans e sal in e sec ions be ween one o such
a hal -line
pl
and
one o such
a
cu e
C'
ne e occu s
in case (a) o he de ini ion abo e. Tha is, his kind o
in e sec ion always has a e ex
o
he cu e.
So
ha ,
we can ansla e he no ion o ans e sal in e sec ion
o
E8
and
E,'
in such a way ha
#@p'
m
C')
=
#(.P'
1
C')
=
#(.p
m
C)
On he o he hand, in he Concep ual Le el, ep-
esen ed by
E;,
we can conside he na u al no ion o
connec ion induced by he g aph s uc u e. This no ion
coincides wi h he no ion o connec ion induced by he
opology o he Euclidean plane h ough he embed-
ding
j.
So
we can conside he connec ed componen s
o
E:
c'.
Since
E;
is
a
iangula ion o
R2
i is no
di icul o p o e ha he numbe
b
connec ed compo-
nen s o
E,'
C'
and
R2
C'
ag ee; e en mo e, each
componen
I<*
o
E;
c'
is he ini e iangula ion o
a
componen
IC'
o
R2
C'
in he ollowing sense:
1.
Gi en
Ii'
he e
is
one and only one componen
I "
o
R2
C'
such ha
I<*
c
IC'.
2.
I{*
is
induced by he e ices o
E,'
in
I;'.
These p ope ies show us ha he componen s o
E,'
c'
ep esen he componen s o
R'
C'.
Now we can ansla e .he Jo dan Cu e Theo em
o
E,'
in he ollowing
way.
Jo dan Cu e Theo eiu
in
E;.
Le
C'
be
a
simple closed digi al cu e
in
E;,
hen
E;
C"
has
wo connec ed componen s (one
o
hem bounded and
he o he one unbounded). Mo eo e ,
a
poin
y'
E
E:
C'
belongs o he bounded componen
i
and
only
i
#( p.
mC)
E
1
(mod
2).
Figu e
7:
Componen s o
E,'
C'
68
Figu e
8:
Top-componen s
o
E8
c
Finally, we need o conside an app op ia e no ion o
componen in
&.
Le
c
be a simple closed digi al cu e
in
Ea.
we call
a
op-componen
o
E8
c
o he image
by
i '
o a connec ed componen o
E;
c'.
Obse e
ha , in gene al, a op-componen o
E8
c
does no
coincide wi h
a
connec ed componen o
ES
C
(see
Fig
7,
8).
In his way we ha e p o ed
Jo dan Cu e Theo em
in
Ea.
Le
C
be
a
simple
closed digi al cu e in
Ea.
hen
E0
C
has wo con-
nec ed op-componen s (one o hem bounded and he
o he one unbounded). Mo eo e ,
a
poin
p
E
E8
C
belongs o he bouiided op-componen
i
and only
i
#(
l,
C)
E
1
(mod
2).
Obse e ha ile p e ious p oo no oul~ p o es he
gi en p oblem
bu
also
allows
o ansla e he well-
known algo i hm o P epa a a
[8]
(coming om
Com-
pu a ional Geome y)
o
sol e he digi al cu e inclu-
sion p obleni.
IV. THE FUNCTIONAL ARCHITECTURE
The p e ious sec ions con ain a pa icula ins ance
o he me hodology p oposed in ou amewo k. In his
sec ion, we will p esen he gene al unc ional a chi ec-
u e o his amewo k. Ou amewo k has ou le -
els, called De ice, Logical, Concep ual and Con inuous
Le els.
In he De ice Le el we ep esen he objec s in a
compu e sc een ( ypically a digi al image). This le el
has a e y small deg ee o abs ac ion and we only ep-
esen he physical aspec s o he objec s.
A
second le el o abs ac ion is ob ained in he Log-
ical Le el. We conside in i he aspec s o p oximi y o
he objec s
so,
we can s udy some p ope ies o opo-
logical na u e. The main unc ion o his le el is o
be he suppo o w i ing he algo i hms and o p o e
hei co ec ness.
In gene al, he le el abo e is a om he ma hema -
ical model
in
which we ha e
a
solu ion o ou p oblem.
So
we need he Concep ual Le el
as
an in e ace be-
ween he le el abo e and he Con inuous Le el. To
ealize his in e ace i is necessa y o ansla e: (1) Ob-
jec s and p ope ies om he Logical Le el o he Con-
cep ual Le el and ice e sa;
(2)
Objec s and p ope ies
om he Concep ual Le el o he Con inuous Le el;
(3)
P ope ies o objec s in he Con inuous Le el o p op-
e ies o objec s in he Concep ual Le el.
Finally, he Con inuous Le el is used o ind
a
con-
inuous solu ion. Obse e ha , ac ually, he objec s
and
concep s ob ained oni lie Logical Le el a e in-
side lie Polyhed al Topology a he han he Con in-
uous Topology and
so
we can use he mo e powe ul
ools o his ield. Tlie objec s o
ou
physical p oblem
ha e been ansla ed by consecu i e abs ac ions om
lie De ice Le el. Now we mus ind
a
con inuous
so-
lu ion
in
his le el by using he well-known esul s o
Polyhed al Topology and we aiisla e
i a
o he Logical
Le el ac oss he Concep ual Le el.
When we ha e
a
coiic e e p oblem and
a
pa icula
sc een model we
mis ,
choose speci ic models
in
each
le el
and
unc ions
which
can suppo he unc .ioliali y
liab we ha e desc ibed. Speci ically, suppose ha hese
chosen niodels
a e
U,
L,
C
a id
S
o he De ice, Log-
ical, Concep ual and Con inuous Le el, espec i ely.
Le
C?(D),
(?(I,),
O(C)
and
O(S)
be he se s o lie ob-
jec s (i.e., subs uc u es in
sonie
ma ~hema ical sense)
o
liese models.
So
we lia e 4he ollowing unc ional
a chi ec u e
We ep esen ou physical'objec s
in
he mode!
D
and we ansla e i o he model
L
by he unc ion
i.
I we ha e
in
L
enough knowledge o sol e he p oblem
we do no need o use he es o he models; when
we
ha e he solu ion, we in e p e
i
in
D
by he unc ion
i.
O he wise, we ansla e he objec s
o
he model
C.
I
we can ind a solu ion in
i
we
ansla e
i
o
he model
L
by he unc ion
T*.
Bu
i
e en in
C
we canno ind
a
solu ion, we ansla e he objec s o he model
S
whe e
we can apply all o he qui e powe ul ools and eml s
o Polyhed al Topology and, i we ind
a
solu ion, we
ansla e i o
L
by he unc ions
j
and
T*.
F om a heo e ical poin o iew, he pa icula
s uc u es ha we need in he le els depend
on
he
p oblem we wan o sol e. Bu , in gene al, he e
is
a
basic s uc u e o a wide ange o p oblems. Fo
exam-
ple, he basic s uc u e o he plana Digi al Topology
is he one shown in he pa ag aphs abo e.
In addi ion, his amewo k can be used o sol e
p oblems which ha e no been p oposed up
ill
now
in Digi al Topology. An impo an example
is
he digi-
al Shoen lies heo em which s a es ha
a
simple closed
digi al Jo dan cu e su ounds
a
digi al disk. The solu-
ion o his p oblem
is
well-known in Plana Euclidean
Topology. Thus, ou mul ile el me hodology can be
applied (choosing sui able models) o ob ain he co e-
sponding digi al e sion.
Mo eo e , his amewo k also wo ks in highe di-
mensions.
As
an example, he p oo gi en o he digi al
Jo dan cu e heo em can easily be adap ed o sol e he
co esponding 3-dimensional p oblem (compa e his
so-
lu ion wi h
[GI,
whe e Koppe man e
al.
ew i e a new
p oo o his esul ).
V. FINAL REMARKS
In his pape , we ha e de eloped a gene al ame-
wo k ha allows o use e y powe ul ools and e-
sul s ( hose o Polyhed al Topology) in Digi al Topol-
ogy. Tlie models used in he a chi ec u e depend on
he Sc een Model. Fo example, i ou Sc een Model
is ep esen ed by he Fig 9(a), he g aph used
as
Logi-
cal Model is he one in he Fig 9(b) (called hexagonal
g aph).
In
his case, each pai o cells has he same con-
nec i i y deg ee and he g aph is plana ,
so
we choose
lie same g aph o ep esen he Concep ual Le el.
Ob-
se e ha , in his case, his model e i ies he Jo dan
Cu e Theo em. The p oo is he same
as
in he case
o he g aph o he &adjacencies.
The e a e models which do no e i y he Jo dan
Cu e Theo em. An example o his
is
he g aph
E4,
ep esen ed by lie Fig 10(b), which is he logical model
o
lie sc een model ep esen ed by he Fig 10(a). This
g aph
is
plana ,
so
we mus conside he same g aph
69

Figu e 9: (a) Sc een model; (b) The g aph E6
(4 (b)
Figu e 10: (a) Sc een model; (b) The g aph
E4
in he Concep ual Le el. Thus he op-connec ion is
equi alen o he 4connec ion. Now, he e ices o
a minimal cycle on his g aph de ine a closed simple
digi al cu e bu i s complemen is connec ed.
Ve y equen ly, some au ho s ha e shown a p oo
o he Digi al Jo dan Cu e Theo em in o de o p o e
he consis ency
o
hei heo ies.
Fo
his eason we
also ha e chosen i o ou amewo k. In he li e a u e
he e a e se e al p oo s o his heo em using di e -
en echniques. The i s au ho who ga e a p oo
was
Rosen eld who p esen ed wo e sions in a se ies o pa-
pe s ([9,10,12]). One is aking an 8-cu e (i.e.,
a
cu e
in he g aph
E8)
and p o ing ha i s complemen has
wo 4-connec ed componen s (i.e., connec ed by a cs in
he g aph
E4
o he 4adjacencies). The o he is aking
a 4cu e (i.e., a cu e in he g aph
E4)
and p o ing
ha i s complemen has wo $-connec ed componen s
(i.e., connec ed by a cs in
E*).
I is easy o obse e ha he heo em p esen ed
ill
his pape includes bo h e sions. This is a di ec conse-
quence o he ollowing p ope y. Gi en
a
closed digi al
cu e
c
in
E8,
hen:
2. I
C
is
an
4cu e, he op-componen s o
E8
C
coincide wi h he 8-connec ed componen s.
-
ACKNOWLEDGEMENTS
We would like o hank Julio Rubio o his e y use ul
commen s on an ea lie d a o his pape .
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Rosen eld, Connec i i y
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1.
I
c
is an 8-cu e, he op-componen s o
E8
C'
coincide wi h he 4-connec ed componen s.
70