Designing Tissue-like P Sys ems o Image
Segmen a ion on Pa allel A chi ec u es
Ja ie Ca ne o1, Daniel D´ıaz-Pe nil1, Miguel A. Gu i´e ez-Na anjo2
1Compu a ional Algeb aic Topology and Applied Ma hema ics Resea ch G oup
Depa men o Applied Ma hema ics I
Uni e si y o Se illa
A da. Reina Me cedes s/n, 41012, Se illa, Spain
[email p o ec ed], [email p o ec ed]
2Resea ch G oup on Na u al Compu ing
Depa men o Compu e Science and A i icial In elligence
Uni e si y o Se illa
A da. Reina Me cedes s/n, 41012, Se illa, Spain
[email p o ec ed]
Summa y. P oblems associa ed wi h he ea men o digi al images ha e se e al in-
e es ing ea u es om a bio-inspi ed poin o iew. One o hem is ha hey can be
sui able o pa allel p ocessing, since he same sequen ial algo i hm is usually applied in
di e en egions o he image. In his pape we epo a wo k-in-p og ess o a ha dwa e
implemen a ion in Field P og ammable Ga e A ays (FPGAs) o a amily o issue-like
P sys ems which sol es he segmen a ion p oblem in digi al images.
1 In oduc ion
Memb ane Compu ing is a compu a ional pa adigm inspi ed in he unc ioning o
li ing cells and issues. One o i s cha ac e is ic ea u es is he use o pa allelism as
a compu a ion ool. In many o he models, he de ices pe o m he compu a ion
by applying pa alleliza ion in a double sense: on he one hand, se e al ules can be
applied simul aneously in each memb ane; on he o he hand, all he memb anes
pe o m he compu a ion a he same ime.
In spi e o ecen e o s [15], i seems ha in he nex u u e he e will no be
an implemen a ion o P sys ems in i o o in i o. All he possible app oaches o
he heo e ical model lean on he cu en compu e a chi ec u es.
In his line, many e o s ha e been made o ob aining a simula ion o he
P sys em beha io wi h cu en compu e s [13, 16]. Mos o hese simula o s a e
hough o unning on one-p ocesso compu e s. These sequen ial machines only
pe o m one ac ion pe ime uni and he pa allelism o he memb ane compu ing
de ices is los . This bo le-neck p oduces a se ious disc epancy be ween he heo-
44 J. Ca ne o e al.
e ical e iciency o he P sys ems and he ealis ic esou ces needed o pe o ming
a compu a ion.
In he las yea s, acco ding wi h he de elopmen o new pa allel a chi ec u es,
new a emp s ha e been made o app oaching he compu a ion o P sys ems by
pe o ming se e al ac ions in he same s ep. This does no mean a eal implemen-
a ion o he P sys em, bu i can be conside ed as a new s ep owa d a mo e
ealis ic simula ion.
The i s pa allel and dis ibu ed simula o s we e p esen ed in 2003. In [12],
a pa allel implemen a ion o ansi ion P sys ems was p esen ed. The p og am
was designed o a clus e o 64 dual p ocesso nodes and i was implemen ed
and es ed on a Linux clus e a he Na ional Uni e si y o Singapo e. In [32],
a pu ely dis ibu i e simula o o P sys ems was p esen ed. I was implemen ed
using Ja a’s Remo e Me hods In oca ion o connec a numbe o compu e s ha
in e change da a. The class o P sys ems ha he simula o can accep is a subse
o he NOP2(coo, a ) amily o sys ems, which ha e he compu a ional powe o
Tu ing machines.
Also in 2003, Pe eska and Teusche [29] p esen ed a pa allel ha dwa e im-
plemen a ion o a special class o memb ane sys ems. The implemen a ion was
based on a uni e sal memb ane ha dwa e componen ha allows e icien ly un P
sys em on a econ igu able ha dwa e known as Field P og ammable Ga e A ays
(FPGAs) [35]. Recen ly, a new esea ch line has a isen due o a no el de ice a -
chi ec u e called CUDAT M , (Compu e Uni ied De ice A chi ec u e) [39]. I is a
gene al pu pose pa allel compu ing a chi ec u e ha allows he pa allel compu e
engine in NVIDIA G aphic P ocesso Uni s (GPUs) o sol e many complex com-
pu a ional p oblems in a mo e e icien way han on a CPU [5, 6, 7]. Following he
esea ch line s a ed in [29], Van Nguyen e al. ha e p oposed he use ha dwa e
implemen a ion o memb ane compu ing applica ions [22, 23, 24, 25] based on
econ igu able compu ing echnology called Recon ig-P.
In his pape , we also explo e he possibili ies o he Field P og ammable Ga e
A ays (FPGAs) o building a ha dwa e implemen a ion o P sys ems. The P
sys em model chosen o he implemen a ion has been issue-like P sys ems and
as a case s udy we conside he segmen a ion p oblem in 2D images.
Segmen a ion in compu e ision (see [31]), e e s o he p ocess o pa i ioning
a digi al image in o mul iple segmen s (se s o pixels). The goal o segmen a ion
is o simpli y and/o change he ep esen a ion o an image in o some hing ha
is mo e meaning ul and easie o analyze. Image segmen a ion is ypically used o
loca e objec s and bounda ies (lines, cu es, e c.) in images. Mo e p ecisely, image
segmen a ion is he p ocess o assigning a label o e e y pixel in an image such
ha pixels wi h he same label sha e ce ain isual cha ac e is ics. Technically,
he p ocess consis s on assigning a label o each pixel, in such way ha pixels
wi h he same label o m a meaning ul egion. The e exis di e en echniques o
segmen an image. Some echniques a e clus e ing me hods [1, 36], his og am-based
me hods [34], Wa e shed ans o ma ion me hods [33], image py amids me hods
Designing Tissue-like P Sys ems o Pa allel A chi ec u es 45
[18] o g aph pa i ioning me hods [37, 38]. Some o he p ac ical applica ions o
image segmen a ion a e medical imaging [36] o ace ecogni ion [17].
Segmen a ion in Digi al Image y has se e al ea u es which make i sui able
o echniques inspi ed by na u e. One o hem is ha i can be pa alleled and
locally sol ed. Rega dless how la ge is he pic u e, he segmen a ion p ocess can
be pe o med in pa allel in di e en local a eas o i . Ano he in e es ing ea u e
is ha he basic necessa y in o ma ion can be easily encoded by bio-inspi ed ep-
esen a ions.
In he li e a u e, one can ind se e al a emp s o b idging p oblems om
Digi al Image y wi h Na u al Compu ing as he wo ks by K.G. Sub amanian e
al. [8, 9] o he wo k by Chao and Nakayama whe e Na u al Compu ing and Al-
geb aic Topology a e linked by using Neu al Ne wo ks [10] (ex ended Kohonen
mapping). In his pape , we will use an in o ma ion encoding and echniques bo -
owed om Memb ane Compu ing. This pape is a new s ep in he esea ch s a ed
a [4], whe e he au ho s p esen an implemen a ion o a memb ane solu ion o a
segmen a ion p oblem using ha dwa e p og amming. In his pape , we p esen a
di e en amily o issue-like P sys ems o sol e he p oblem and epo he ha d-
wa e implemen a ion. In wha ollows we assume he eade is al eady amilia
wi h he basic no ions and he e minology unde lying P sys ems3.
The pape is o ganized as ollows: i s ly, we p esen ou bio-inspi ed o mal
amewo k. Nex , we p esen a amily o issue-like P sys ems designed o ob ain
an edge-based segmen a ion o a 2D digi al image. Then, gene al conside a ions
abou designing ha dwa e P sys ems a e s udied, ocusing on he segmen a ion
p oblem. The pape inishes wi h some conclusions and u u e wo k.
2 Fo mal F amewo k: Tissue-like P Sys ems
Tissue-like P sys ems we e p esen ed by Ma ´ın–Vide e al. in [21]. They ha e
wo biological inspi a ions (see [20]): in e cellula communica ion and coope a ion
be ween neu ons. The common ma hema ical model o hese wo mechanisms is
a ne wo k o p ocesso s dealing wi h symbols and communica ing hese symbols
along channels speci ied in ad ance.
The main ea u es o his model, om he compu a ional poin o iew, a e
ha cells do no ha e pola iza ion and he memb ane s uc u e is a gene al g aph.
Fo mally, a issue-like P sys em wi h inpu o deg ee q≥1 is a uple
Π= (Γ, Σ, E, w1, . . . , wq,R, iΠ, oΠ),
whe e
1. Γis a ini e alphabe , whose symbols will be called objec s;
2. Σ(⊂Γ) is he inpu alphabe ;
3We e e o [26] o basic in o ma ion in his a ea, o [28] o a comp ehensi e p esen-
a ion and he web si e [40] o he up- o-da e in o ma ion.
46 J. Ca ne o e al.
3. E ⊆ Γ( he objec s in he en i onmen );
4. w1, . . . , wqa e s ings o e Γ ep esen ing he mul ise s o objec s associa ed
wi h he cells a he ini ial con igu a ion;
5. Ris a ini e se o communica ion ules o he ollowing o m:
(i, u/ , j)
o i, j ∈ {0,1,2, . . . , q}, i 6=j,u, ∈Γ∗;
6. iΠ∈ {1,2, . . . , q}is he inpu cell;
7. oΠ∈ {0,1,2, . . . , q}is he ou pu cells
A issue-like P sys em o deg ee q≥1 can be seen as a se o qcells (each one
consis ing o an elemen a y memb ane) labelled by 1,2, . . . , q. We will use 0 o
e e o he label o he en i onmen , iΠdeno es he inpu egion and oΠdeno es
he ou pu egion (which can be he egion inside a cell o he en i onmen ).
The s ings w1, . . . , wqdesc ibe he mul ise s o objec s placed in he qcells
o he P sys em. We in e p e ha E ⊆ Γis he se o objec s placed in he
en i onmen , each one o hem a ailable in an a bi a y la ge amoun o copies.
The communica ion ule (i, u/ , j) can be applied o e wo cells labelled by i
and jsuch ha uis con ained in cell iand is con ained in cell j. The applica ion
o his ule means ha he objec s o he mul ise s ep esen ed by uand a e
in e changed be ween he wo cells. No e ha i ei he i= 0 o j= 0 hen he
objec s a e in e changed be ween a cell and he en i onmen .
Rules a e used as usual in he amewo k o memb ane compu ing, ha is, in a
maximally pa allel way (a uni e sal clock is conside ed). In one s ep, each objec
in a memb ane can only be used o one ule (non-de e minis ically chosen when
he e a e se e al possibili ies), bu any objec which can pa icipa e in a ule o
any o m mus do i , i.e., in each s ep we apply a maximal se o ules.
Acon igu a ion is an ins an aneous desc ip ion o he P sys em Π. Gi en a
con igu a ion, we can pe o m a compu a ion s ep and ob ain a new con igu a ion
by applying he ules in a pa allel manne as i is shown abo e. A sequence o
compu a ion s eps is called a compu a ion. A con igu a ion is hal ing when no
ules can be applied o i . Then, a compu a ion hal s when he P sys em eaches
a hal ing con igu a ion.
3 Segmen ing Digi al Images
Apoin se is simply a opological space consis ing o a collec ion o objec s called
poin s and a opology which p o ides o such no ions as nea ness o wo poin s,
he connec i i y o a subse o he poin se , he neighbo hood o a poin , bounda y
poin s, and cu es and a cs.
The mos common poin se s occu ing in image p ocessing a e disc e e subse s
o N-dimensional Euclidean space Rnwi h n= 1,2 o 3 oge he wi h he disc e e
opology. The e is no es ic ion on he shape o he disc e e subse s o Rnused
in applica ions o image algeb a o sol e ision p oblems.
Designing Tissue-like P Sys ems o Pa allel A chi ec u es 47
Fo a poin se Xin Z, a neighbo hood unc ion om Xin Z, is a unc ion
N:X→2Z. Fo each poin x∈X,N(x)⊆Z. The se N(x) is called a
neighbo hood o x.
The e a e wo neighbo hood unc ion on subse s o Z2which a e o pa icula
impo ance in image p ocessing, he on Neumann neighbo hood and he Moo e
neighbo hood. The i s one N:X→2Z2is de ined by N(x) = {y:y=
(x1±j, x2)o y = (x1, x2±k), j, k ∈ {0,1}}, whe e x= (x1, x2)∈X⊂Z2.
While he Moo e neighbo hood M:X→2Z2is de ined by M(x) = {y:y=
(x1±j, x2±k), j, k ∈ {0,1}}, whe e x= (x1, x2)∈X⊂Z2. The on Neumann and
Moo e neighbo hood a e also called he ou neighbo hood (4-adjacency) and eigh
neighbo hood (8-adjacency), espec i ely. In his pape , we wo k wi h 4-adjacency.
The poin se s wi h he usual ope a ions has an algeb a s uc u e (see [30]).
An Z- alued image on Xis any elemen o ZX. Gi en an Z- alued image
I∈ZX, i.e. I:X→Z, hen Zis called he se o possible ange alues o I
and X he spa ial domain o I. The g aph o an image is also e e ed o as he
da a s uc u e ep esen a ion o he image. Gi en he da a s uc u e ep esen a ion
I={(x, I(x)) : x∈X}, hen an elemen (x, I(x)) is called a pic u e elemen o
pixel. The i s coo dina e xo a pixel is called he pixel loca ion o image poin ,
and he second coo dina e I(x) is called he pixel alue o Ia loca ion x.
Fo example, Xcould be a subse o Z2whe e x= (i, j) deno es spa ial loca ion,
and Zcould be a subse o N,N3, e c. So, gi en an image I∈ZZ2, a pixel o I
is he o m ((i, j), I(x)), which will be deno ed by I(x)ij. We call he se o colo s
o alphabe o colo s o he image se o he unc ion Iwi h domain Xand he
image poin o each pixel is called associa ed colo . We can conside an o de in
his se . In his pape , we deno e Zas CI. Usually, we conside in digi al image a
p ede ined alphabe o colo s C. We de ine h=|C| as he size (numbe o colo s) o
C. In his pape , we wo k wi h images in g ey scale, hen C={0, . . . , 255}, whe e
0 codi y he black colo and 255 he whi e colo .
By echnical easons, we use below di e en ways o codi y a same pixel. Fo ex-
ample, i we ake he pixel ((i, j), a) we could codi y wi h he ollowing exp essions:
aij,Aij ,a0
ij,aij , (a, l)ij wi h l∈N, e c.
A egion could be de ined by a subse o he domain o Iwhose poin s a e all
mapped o he same (o simila ) pixel alue by I. So, we can conside he egion
Rias he se {x∈X:I(x) = i}bu his kind o egions has no o be connec ed.
We p e e o conside a egion as a maximal connec ed subse o a se like Ri.
We say wo egions 1, 2a e adjacen when a less a pai o pixel x1∈ 1and
x2∈ 2a e adjacen . We say x1and x2a e bo de pixels. I I(x1)< I(x2) we say
x1is an edge pixel. The se o connec ed edge pixels wi h he same pixel alue is
called a bounda y be ween wo egions.
F om a gene al poin o iew, segmen a ion e e s o he p ocess o pa i ioning
a digi al image in o mul iple egions. Th esholding is a me hod o image segmen a-
ion whose basic aim is o ob ain a bina y image om a colou one. The idea is o
spli he se o pixels in o wo se s (black and whi e) depending on i s b igh and a
ixed alued, he h eshold. I he b igh o he pixel is g ea e han he h eshold,

48 J. Ca ne o e al.
hen he pixel is labelled as objec . O he wise, i is labelled as backg ound. A e
labelling, a new bina y image is c ea ed by colou ing each pixel whi e o black,
depending on he label.
The basic h esholding me hod can be gene alized in a na u al way. Ins ead
o ge ing a bina y image by labelling he o iginal se o pixels by {0,1}, we can
conside a la ge se o labels, {1, . . . , k}so we ob ain a inal image wi h kle els.
Ano he na u al gene aliza ion is o eplace he colou in o ma ion by ano he
scale on he ea u es o he pixel (b igh , in ensi y, g ay scale, e c.).
Edge de ec ion is an impo an ope a ion in a la ge numbe o image p ocessing
applica ions, such as image segmen a ion, cha ac e ecogni ion and scene analysis.
In his pape we wo k wi h he i s one, he edge-based segmen a ion o 2D
digi al images p oblem (2D-ES p oblem), which is desc ibed as ollows: Gi en a
digi al 2D image wi h pixels o (possibly) di e en colo s, ob ain he bounda ies o
egions in ha image.
In o de o p o ide a loga i hmic- ime uni o m solu ion o ou p oblem, we
design a amily o issue-like P sys ems, Π. Gi en an image Io size n2, we ake
he P sys em Π(n, k) o he amily o wo k wi h I. The inpu da a (image I) is
codi ied by a se o objec s a0
ij, wi h a∈ C and 1 ≤i, j ≤nand kis e e ed o
he numbe o p ocessing cells. So, when we wo k wi h a pa allel a chi ec u e we
do no ha e o know p e iously an exac numbe o p ocesso s o wo k. Then, we
in oduce he pa ame e k o sol e his p oblem.
The unc ioning o a P sys em o he amily consis s o he ollowing s ages:
•Fi s o all, he P sys em gene a es 8 auxilia y copies o he inpu da a. Then,
we ha e 9 codi ica ions o he inpu image, bu one o hem is dis inguished
o he es . So, we can wo k wi h each pixel wi hou aking in o accoun wha
happens wi h he es o he image.
•Second, he P sys em applies a basic noise il e in o de o elimina e some
pickle noise ha could a ec he segmen a ion p ocess. The P sys em will
apply he la gely used a e age il e because o i s simplici y and good esul s.
Fo each pixel, he p ocess consis s o calcula ing he a e age a e age o i s
adjacen pixels. I he dis ance be ween he pixel and i s a e age is g ea e
han a h eshold ρ, he pixel will be conside ed as noise and i will be eplaced
by i s a e age colou .
•Nex , he P sys em pe o ms a h esholding o he image o sol e he p oblem
o deg ada ion o colou s o pixels in he bounda y o adjacen egions wi h
di e en colou s.
•Once his p ocess is inished, he P sys em applies a ansla ion o ules de ined
in [11] ob aining an edge-based segmen a ion o he image ook o he p e ious
s age.
The amily Π={Π(n, k) : n, k ∈N}o issue-like P sys ems o deg ee k+ 1 is
de ined as ollows:
Fo each n, k ∈N,
Π(n, k) = (Γ, Σ, E, w1, . . . , wk+1,R, iΠ, oΠ),
Designing Tissue-like P Sys ems o Pa allel A chi ec u es 49
de ined as ollows:
•Γ=Σ∪ {aij, a00
ij,¯aij, Aij , A0
ij, A00
ij, Aij , Aij,(a, 1)ij,(a, 2)ij,(a, 3)ij : 1 ≤i, j ≤
n, a ∈ C} is he wo king alphabe ;
• he inpu alphabe is Σ={a0
ij : 1 ≤i, j ≤n, a ∈ C, I(i, j) = a};
• he en i onmen alphabe is E=Γ Σ;
• he mul ise s o he cells a e w1={{ν3
ij, ν3
ji :i= 0, n + 1,0≤j≤n+ 1}},
w2=···=wk+1 =Tdn2/ke, espec i ely. We call o he las kcells as p ocessing
cells;
•Ris he ollowing se o communica ion ules:
1. (1, a0
ij/a8
ijAij ,0)
o 1 ≤i, j ≤n.
These ules a e used o gene a e new elemen s, so he P sys em can wo k in
pa allel wi h each pixel and o ge wha happen wi h he es o he image.
The P sys em i s uses hese elemen s o wo k wi h he noise o ou image.
2. 
1,
ci−1j−1di−1jei−1j+1
bij−1Aij ij+1
li+1j−1hi+1jgi+1j+1
/ T, 

o
– 1 ≤i, j ≤n,
–a, b, c, d, e, , g, h, l ∈ C ∪ {ν}.
This ype o ules a e used o ansla e each objec Aij and one copy o hei
neighbou s (objec s) o a p ocessing cell. We a e su e ha all he pixels no
go o he same cell, because ou P sys em has n2o n2+1 objec s T sp ead
o e p ocessing cells, each one wi h a simila numbe o copies o T.
3. 
 ,
ci−1j−1di−1jei−1j+1
bij−1Aij ij+1
li+1j−1hi+1jgi+1j+1
/ α0
ij,0

o
– 1 ≤i, j ≤n,
–a, b, c, d, e, , g, h, l ∈ C ∪ {ν},
– We ake µas he numbe o pixels wi h colo s in Cand ν= 0. Then,
a (a)=(b+c+d+e+ +g+h+i)/µ,
–αis he nea es colou in C o he a e age colou a (a) wi h |α−a (a)|>
ρ, wi h ρ∈R.
4. 
 ,
ci−1j−1di−1jei−1j+1
bij−1Aij ij+1
ii+1j−1hi+1jgi+1j+1
/ a0
ij,0

o
– 1 ≤i, j ≤n,
–a, b, c, d, e, , g, h, i ∈ C ∪ {ν},
50 J. Ca ne o e al.
– We ake µas he numbe o pixels wi h colo s in Cand ν= 0. Then,
a (a)=(b+c+d+e+ +g+h+i)/µ,
–|a−a (a)| ≤ ρ, whe e ρ1∈R.
This se o ules is used o de ec he noise and co ec i wi h he a e age
colou o i s adjacen pixels. We ind he e a local h esholding (wi h espec
o he colo s) wi h p ede ined h eshold ρ. In ac , we a e simula ing one
o he mo e ypical algo i hms o emo e noise. The P sys em changes he
no a ion o he objec s which a e codi ying pixels and hey adop he o m
a0
ij, wi h a∈ C.
5. ( , b0
ij/A0
ij,0)
o
– 1 ≤i, j ≤n,
–τ= (|C|/ρ2), l= 0,1,2, . . . , ρ2,
– I b∈ C hen a∈ C (a < b ≤a+ (τ−1) and a=τ·l) o (b=a=τ·l),
– I b=ν hen A=ν.
These ules a e used o disc e ize he colo s di iding he se o colo s in ρ2
subse s o leng h ν. We ind he e a gene al h esholding (wi h espec o
he colo s) wi h p ede ined h eshold ν.
6. ( , A0
ij/T, 1)
o
– 0 ≤i, j ≤n+ 1, 2 ≤ ≤k+ 1,
–a∈ C.
This se o ules a e used o send ou ans o med image o he cell 1. Now,
he objec s A0
ij encode he pixels o ou image.
7. (1, A0
ij/A00
ijAij a8
ij,0)
o
– 0 ≤i, j ≤n+ 1,
–a∈ C ∪ {ν}.
The P sys em uses hese ules o gene a e enough copies o ou image o
pe o m he segmen a ion p ocess in he cells 2, . . . , k and k+1. The objec s
A00
ij a e used in he second pa o he segmen a ion. The es o he objec s
a e used in he i s pa o he segmen a ion.
8. 
 ,
ci−1j−1di−1jei−1j+1
bij−1Aij ij+1
ii+1j−1hi+1jgi+1j+1
/ T, 

o
– 1 ≤i, j ≤n,
– and a, b, c, d, e, , g, h, i ∈ C ∪ {ν}.
These ules a e de ined o send new objec s o he p ocessing cells o do
he i s pa o he segmen a ion. We look o edge pixels.
Designing Tissue-like P Sys ems o Pa allel A chi ec u es 51
9. ( , Aijbkl/Aijbkl,0),
o
– 1 ≤i, j, k, l ≤n, (i, j),(k, l) adjacen pixels,
–a, b ∈ C and a < b.
These ules a e used o ma k edge pixels. In ac , he he P sys em b ings
om he en i onmen an objec o he o m Aij o each edge pixel. Ou
p oblem is he edge pixels no always a e adjacen . So, we do no ha e an
only one se o connec ed edge pixel o ming a bounda y. Then, we should
add he necessa y pixel o connec all he edge pixels o a bounda y.
10. ( , Aij/T, 1)
o
– 1 ≤i, j ≤n,
–a∈ C.
These ules send he edge pixels o he cell 1.
11. (1, Aij/(a, 1)2
ij,0)
o
– 0 ≤i, j ≤n+ 1,
–a∈ C ∪ {ν}.
The P sys em uses hese ules o gene a e wo copies o ou edge pixels o
pe o m he second pa o he segmen a ion in p ocessing cells.
12. (1, A00
ij/(a, 2)2
ij,0)
o
– 0 ≤i, j ≤n+ 1,
–a∈ C ∪ {ν}.
The P sys em uses hese ules o gene a e enough copies o ou image o
pe o m he second pa o he segmen a ion in p ocessing cells.
13. µ1,(a, 1)i−1j−1(a, 2)i−1j
(b, 2)ij−1(a, 1)ij / T, ¶µ1,(b, 2)i−1j−1(a, 1)i−1j
(a, 1)ij−1(a, 2)ij / T, ¶
µ1,(a, 1)i−1j−1(b, 2)i−1j
(a, 2)ij−1(a, 1)ij / T, ¶µ1,(a, 2)i−1j−1(a, 1)i−1j
(a, 1)ij−1(b, 2)ij / T, ¶
o
– 1 ≤i, j ≤n,
–a, b ∈ C.
These ules a e de ined o send new objec s o he p ocessing cells o do
he second pa o he segmen a ion. We look o new edge pixels.
14. µ1,(a, 1)i−1j−1(a, 2)i−1j
(b, 2)ij−1(a, 1)ij /(a, 3)i−1j−1(a, 3)i−1j
(b, 2)ij−1(a, 3)ij , ¶
µ1,(b, 2)i−1j−1(a, 1)i−1j
(a, 1)ij−1(a, 2)ij /(b, 2)i−1j−1(a, 3)i−1j
(a, 3)ij−1(a, 3)ij , ¶
58 J. Ca ne o e al.
Fig. 8. 16x16 colo image segmen a ion
5 Conclusions and Fu u e Wo ks
P oblems associa ed wi h he ea men o Digi al Images ha e se e al in e es ing
ea u es om a bio-inspi ed poin o iew. One o hem is ha hey can be sui able
o pa allel p ocessing, since he same sequen ial algo i hm is usually applied in
di e en egions o he image.
In his pape , we s udy he ad an ages and d awbacks o conside ing a ha d-
wa e implemen a ion o issue-like P sys ems sol ing he segmen a ion p oblem on
a ha dwa e p og amming ool (FPGA). The heo e ical s udy has been made ia
he language p og amming VHDL [2] and cu en ly we a e in he p ocess o he
eal ha dwa e implemen a ion.
In addi ion, al hough he segmen a ion example showed he e is a synch onous
issue-like P sys em, we wan in he nex u u e o wo k wi h asynch onous issue-
like P sys ems in o de o op imize pe o mance.
Many ques ions emains open as u u e wo k. One o hem is he ea men
o he noise in images wi h Memb ane Compu ing echniques, o he pa alleliza-
ion and au oma iza ion o he choice o he h eshold by a i icial in elligence
echniques.
Acknowledgemen s
DDP and MAGN acknowledge he suppo o he p ojec s TIN-2009-13192 o he
Minis e io de Ciencia e Inno aci´on o Spain and he suppo o he P ojec o
Excellence o he Jun a de Andaluc´ıa, g an P08-TIC-04200. JC acknowledges he
suppo o he p ojec MTM2009-12716 o he Minis e io espa˜nol de Educaci´on
y Ciencia, he p ojec PO6-TIC-02268 o Excellence o Jun a de Andaluc´ıa, and
he Compu a ional Topology and Applied Ma hema ics PAICYT esea ch g oup
FQM-296.
Re e ences
1. Abdala, D.D., Jiang, X.: Fibe segmen a ion using cons ained clus e ing. In: Zhang,
D., Sonka, M. (eds.) ICMB. Lec u e No es in Compu e Science, ol. 6165, pp. 1–10.
Sp inge (2010)

Designing Tissue-like P Sys ems o Pa allel A chi ec u es 59
2. Ashenden, P.J.: The Designe ’s Guide o VHDL. Mo gan Kau mann Publishe s Inc.,
San F ancisco, CA, USA, 2nd edn. (2001)
3. Bo ego-Rope o, R., D´ıaz-Pe nil, D., P´e ez-Jim´enez, M.J.: Tissue simula o : A g aph-
ical ool o issue P sys ems. In: Vaszil, G. (ed.) P oceedings o he In e na ional
Wo kshop Au oma a o Cellula and Molecula Compu ing. pp. 23–34. MTA SZ-
TAKI, Budapes , Hunga y (Augus 2007), sa elli e o he 16 h In e na ional Sympo-
sium on Fundamen als o Compu a ional Theo y
4. Ca ne o, J., D´ıaz-Pe nil, D., Molina-Ab il, H., Real, P.: Image segmen a ion inspi ed
by cellula models using ha dwa e p og amming. Image-A 1(3), 143–150 (2010)
5. Cecilia, J.M., Ga c´ıa, J.M., Gue e o, G.D., Ma ´ınez-del-Amo , M.A., P´e ez-
Hu ado, I., P´e ez-Jim´enez, M.J.: Implemen ing P sys ems pa allelism by means
o GPUs. In: P˘aun e al. [27], pp. 227–241
6. Cecilia, J.M., Ga c´ıa, J.M., Gue e o, G.D., Ma ´ınez-del-Amo , M.A., P´e ez-
Hu ado, I., P´e ez-Jim´enez, M.J.: Simula ing a P sys em based e icien solu ion
o SAT by using GPUs. Jou nal o Logic and Algeb aic P og amming 79(6), 317–325
(2010)
7. Cecilia, J.M., Ga c´ıa, J.M., Gue e o, G.D., Ma ´ınez-de-Amo , M.A., P´e ez-Hu ado,
I., P´e ez-Jim´enez, M.J.: Simula ion o P sys ems wi h ac i e memb anes on CUDA.
B ie ings in Bioin o ma ics 11(3), 313–322 (2010)
8. Ce e chi, R., G ama o ici, R., Jonoska, N., Sub amanian, K.G.: Tissue-like P sys ems
wi h ac i e memb anes o pic u e gene a ion. Fundamen a In o ma icae 56(4), 311–
328 (2003)
9. Ce e chi, R., Mu yam, M., P˘aun, Gh., Sub amanian, K.G.: A ay- ew i ing P sys-
ems. Na u al Compu ing 2(3), 229–249 (2003)
10. Chao, J., Nakayama, J.: Cubical singula simplex model o 3D objec s and as com-
pu a ion o homology g oups. In: 13 h In e na ional Con e ence on Pa e n Recog-
ni ion (ICPR’96). ol. IV, pp. 190–194. IEEE Compu e Socie y, IEEE Compu e
Socie y, Los Alami os, CA, USA (1996)
11. Ch is inal, H.A., D´ıaz-Pe nil, D., Real, P.: Segmen a ion in 2D and 3D image using
issue-like P sys em. In: Bay o-Co ochano, E., Eklundh, J.O. (eds.) CIARP. Lec u e
No es in Compu e Science, ol. 5856, pp. 169–176. Sp inge (2009)
12. Ciobanu, G., Wenyuan, G.: P sys ems unning on a clus e o compu e s. In: Ma ´ın-
Vide e al. [19], pp. 123–139
13. D´ıaz-Pe nil, D., G aciani, C., Gu i´e ez-Na anjo, M.A., P´e ez-Hu ado, I., Ma io J.
P´e ez-Jim´enez, M.: So wa e o P sys ems. In: P˘aun e al. [28], pp. 437–454
14. D´ıaz-Pe nil, D., Gu i´e ez-Na anjo, M.A., Molina-Ab il, H., Real, P.: A bio-inspi ed
so wa e o segmen ing digi al images. In: Naga , A.K., Thambu aj, R., Li, K.,
Tang, Z., Li, R. (eds.) P oceedings o he 2010 IEEE Fi h In e na ional Con e ence
on Bio-Inspi ed Compu ing: Theo ies and Applica ions BIC-TA. ol. 2, pp. 1377 –
1381. IEEE Compu e Socie y (2010)
15. Ge shoni, R., Keinan, E., P˘aun, Gh., Pi an, R., Ra ne , T., Shoshani, S.: Resea ch
opics a ising om he (planned) P sys ems implemen a ion expe imen in Tech-
nion. In: D´ıaz-Pe nil, D., G aciani, C., Gu i´e ez-Na anjo, M.A., P˘aun, Gh., P´e ez-
Hu ado, I., Riscos-N´u˜nez, A. (eds.) Six h B ains o ming Week on Memb ane Com-
pu ing. pp. 183–192. F´enix Edi o a, Se illa, Spain (2008)
16. Gu i´e ez-Na anjo, M.A., P´e ez-Jim´enez, M.J., Riscos-N´u˜nez, A.: A ailable mem-
b ane compu ing so wa e. In: Ciobanu, G., P´e ez-Jim´enez, M.J., P˘aun, Gh. (eds.)
Applica ions o Memb ane Compu ing, pp. 411–436. Na u al Compu ing Se ies,
Sp inge (2006)
60 J. Ca ne o e al.
17. Kim, S.H., Kim, H.G., Tchah, K.H.: Objec o ien ed ace de ec ion using colou ans-
o ma ion and ange segmen a ion. Elec onics Le e s, IEEE 34, 979–980 (1998)
18. K opa sch, W.G., Haxhimusa, Y., Ion, A.: Mul i esolu ion image segmen a ions in
g aph py amids. In: Kandel, A., Bunke, H., Las , M. (eds.) Applied G aph Theo y
in Compu e Vision and Pa e n Recogni ion, S udies in Compu a ional In elligence,
ol. 52, pp. 3–41. Sp inge (2007)
19. Ma ´ın-Vide, C., Mau i, G., P˘aun, Gh., Rozenbe g, G., Salomaa, A. (eds.): Mem-
b ane Compu ing, In e na ional Wo kshop, WMC 2003, Ta agona, Spain, July 17-
22, 2003, Re ised Pape s, Lec u e No es in Compu e Science, ol. 2933. Sp inge
(2004)
20. Ma ´ın-Vide, C., P˘aun, Gh., Pazos, J., Rod ´ıguez-Pa ´on, A.: Tissue P sys ems. The-
o e ical Compu e Science 296(2), 295–326 (2003)
21. Ma ´ın-Vide, C., Pazos, J., P˘aun, Gh., Rod ´ıguez-Pa ´on, A.: A new class o symbolic
abs ac neu al ne s: Tissue P sys ems. In: Iba a, O.H., Zhang, L. (eds.) COCOON.
Lec u e No es in Compu e Science, ol. 2387, pp. 290–299. Sp inge (2002)
22. Nguyen, V., Kea ney, D., Gioiosa, G.: Balancing pe o mance, lexibili y, and scal-
abili y in a pa allel compu ing pla o m o memb ane compu ing applica ions. In:
Ele he akis, G., Ke alas, P., P˘aun, Gh., Rozenbe g, G., Salomaa, A. (eds.) Wo k-
shop on Memb ane Compu ing. Lec u e No es in Compu e Science, ol. 4860, pp.
385–413. Sp inge (2007)
23. Nguyen, V., Kea ney, D., Gioiosa, G.: An algo i hm o non-de e minis ic objec
dis ibu ion in p sys ems and i s implemen a ion in ha dwa e. In: Co ne, D.W.,
F isco, P., P˘aun, Gh., Rozenbe g, G., Salomaa, A. (eds.) Wo kshop on Memb ane
Compu ing. Lec u e No es in Compu e Science, ol. 5391, pp. 325–354. Sp inge
(2008)
24. Nguyen, V., Kea ney, D., Gioiosa, G.: A egion-o ien ed ha dwa e implemen a ion
o memb ane compu ing applica ions. In: P˘aun e al. [27], pp. 385–409
25. Nguyen, V., Kea ney, D., Gioiosa, G.: An ex ensible, main ainable and elegan ap-
p oach o ha dwa e sou ce code gene a ion in econ ig-P. Jou nal o Logic and Alge-
b aic P og amming 79(6), 383–396 (2010)
26. P˘aun, Gh.: Memb ane Compu ing. An In oduc ion. Sp inge -Ve lag, Be lin, Ge -
many (2002)
27. P˘aun, Gh., P´e ez-Jim´enez, M.J., Riscos-N´u˜nez, A., Rozenbe g, G., Salomaa, A.
(eds.): Memb ane Compu ing, 10 h In e na ional Wo kshop, WMC 2009, Cu ea de
A ges, Romania, Augus 24-27, 2009. Re ised Selec ed and In i ed Pape s, Lec u e
No es in Compu e Science, ol. 5957. Sp inge (2010)
28. P˘aun, Gh., Rozenbe g, G., Salomaa, A. (eds.): The Ox o d Handbook o Memb ane
Compu ing. Ox o d Uni e si y P ess (2010)
29. Pe eska, B., Teusche , C.: A econ igu able ha dwa e memb ane sys em. In: Ma ´ın-
Vide e al. [19], pp. 269–285
30. Ri e , G.X., Wilson, J.N., Da idson, J.L.: Image algeb a: An o e iew. Compu e
Vision, G aphics, and Image P ocessing 49(3), 297–331 (1990)
31. Shapi o, L.G., S ockman, G.C.: Compu e Vision. P en ice Hall PTR, Uppe Saddle
Ri e , NJ, USA (2001)
32. Sy opoulos, A., Mama as, L., Allilomes, P.C., So i iades, K.T.: A dis ibu ed simu-
la ion o ansi ion P sys ems. In: Ma ´ın-Vide e al. [19], pp. 357–368
33. Ta abalka, Y., Chanusso , J., Benedik sson, J.A.: Segmen a ion and classi ica ion o
hype spec al images using Wa e shed ans o ma ion. Pa e n Recogni ion 43(7),
2367–2379 (2010)
Designing Tissue-like P Sys ems o Pa allel A chi ec u es 61
34. Tobias, O.J., Sea a, R.: Image segmen a ion by his og am h esholding using uzzy
se s. IEEE T ansac ions on Image P ocessing 11(12), 1457–1465 (2002)
35. T imbe ge , S.M.: Field-P og ammable Ga e A ay Technology. Kluwe Academic
Publishe s, No well, MA, USA (1994)
36. Wang, D., Lu, H., Zhang, J., Liang, J.Z.: A knowledge-based uzzy clus e ing me hod
wi h adap a ion penal y o bone segmen a ion o c images. In: P oceedings o he
2005 IEEE Enginee ing in Medicine and Biology 27 h Annual Con e ence. pp. 6488–
6491 (2005)
37. Yazid, H., A o , H.: Image segmen a ion using wa e shed ans o ma ion o acial
exp ession ecogni ion. In: IFMBE P oceedings, 4 h Kuala Lumpu In e na ional
Con e ence on Biomedical Enginee ing. pp. 575–578 (2008)
38. Yuan, X., Si u, N., Zou idakis, G.: A na ow band g aph pa i ioning me hod o
skin lesion segmen a ion. Pa e n Recogni ion 42(6), 1017–1028 (2009)
39. NVIDIA Co po a ion. NVIDIA CUDAT M P og amming Guide.
h p://www.n idia.com/objec /cuda home new.h ml
40. P sys em web page. h p://ppage.psys ems.eu