Vol. Xxxx (xxx) ACTA PHYSICA POLONICA A No. Xxx
Proceedings of the 3nd International Congress APMAS2013, April 24_28, 2013, Antalya, Turkey
2D
Cellular Automata w
ith a
n Image Pro
cessing Application
S. UGUZ
a
, U. SAHIN
b
, I. SIAP
c
AND H. AKIN
d
a
Department of Mathematics, Arts and Science Faculty, Harran University, Sanliurfa, 63120, Turkey
b
Multi Agent Biorobotic Laboratory, Rochester Institute of Technology, Rochester

NY USA
c
Depart
ment of Mathematics, Yildiz Technical University, 34210, Istanbul

Turkey
d
Department of Mathematics, Education Faculty, Zirve University, 27260, Gaziantep, Turkey
Abstract:
This paper investigates
the theor
etical aspects of two dimensional linear cellular
automata
with image
applications
.
We consider geometrical and visual aspects of patterns generated by
cellular automata
evolution. The present
work focuses
on the theory of two dimensional linear
cellular automata
with respect to uniform periodic and adia
batic
boundary
cellular
automata
(2D PB and AB) conditions. Multiple copies of any arbitrary image correspond to
cellular automata
find
so many
applications in real life situation e.g. textile design, DNA genetics research, etc.
Keywords:
Cellular automat
a, Self replicating patterns, Image Processing.
PACS:
02.10.Yn
;
07.05.Kf
;
02.10.Ox
.
1.
Introduction
Cellular
automata
(CAs
for
brevi
t
y)
i
n
tr
o
duced
b
y
Ulam and
v
on
Neumann [14]
in
the
early
1950’s
,
h
a
v
e
b
een
systematically
studied
b
y
Hedlund from
purely
mathematical
p
oi
n
t
of
view.
One
dimensional
CA
has
b
een
investigated
to
large
extend. H
ow
e
v
er,
little
interest
has
b
een
gi
v
en
to
t
w
o
dimensional
cel l ul ar
automata
(2DCA).
v
on
Neu
mann
[
14]
sh
ow
ed
that
a
cellular
automaton
can
b
e
uni
v
ersal.
D
ue
to
its complexi
t
y
,
v o n
Neumann
r ul e s
w
ere
ne
v
er
impleme
n
ted
on
a
computer.
In
the
b
eginning
of the
eig
h
ties, Stephen
W
olfram
[15]
has studied
i
n
m
u
c
h detail a
family
of
simple
one

dimensional
(
1D)
CA
rul
es
and
sh
ow
ed
that
e
v
en
these
simplest
r ul es
are capable of
e
m
ulating
c o mp l e x
behavior
.
Some
basic
and
precise
mathematical
m
o
dels
using
matrix
algebra
o
v
er
the
binary
field
whi
c
h
c
haracterize
the
b
eh
a
vior
of
2D
nearest neig
h
b
orh
o
o
d linear
CA
with
n
ull
and
p
eri
o
dic
b
oundary
conditions h
a
v
e
seen
in
the
literature
[7,
8].
CA
has
recei
v
ed
remarkable
atte
n
tion
in
the
last
few
decades [1,
2,
8].
Due to
its
structure
CA
has
given
the
op
p
ortuni
t
y
to
m
o
del
and
understand
ma
n
y
b
eh
a
viors
in
nature
easier.
Most
of
the
w
ork
for
CA
is
done
for
one
dimensional
case. The
set
of
pa
p
ers
[4,
8]
deal
s
with
the
b
eh
a
vior
of
the
uniform
2D
CA
o
v
er
binary
fields.
In
this
pa
p
er,
w
e
study
the
theory of 2

dimensional uniform
p
eri
o
dic
and
adiabatic
b
oundary
CA
(2D
PCA,
A
CA)
of
the
all
linear rules
(e.g.
v
on
Neumann, M
o
ore
neig
h
b
orh
oo
d and
the
others) and
applications
of
image
pr
o
cessing
for
self
replicating
patterns
(see
Figs.
1

8
)
.
We
prese
n
t
some
illustrati
v
e
examples
and
fi
gures
to
explain
the
meth
o
d
in
details.
Using
the
rule
matrices
obtained
in
this
w
ork,
the
prese
n
t
pa
p
er
co
n
tributes
further
to
the
algebraic
structure
of
these
CA
and
relates its
applications
studied
b
y
differe
n
t
authors
previously
(i.e. [5,
15]).
The
li
near
co
m
bination
of
the
neig
h
b
oring
cells
on
whi
c
h
ea
c
h
cell
v
alue
is
de
p
ende
n
t
is
called
the
rule
n
u
m
b
er of
the
2D
CA
o
v
er
the
field
Z
2
.
Regarding
the
neig
h
b
orh
oo
d
of
the
extreme cells,
there exist
four
differe
n
t
approa
c
hes.
•
A
n
ull
b
oundary
(NB)
CA
is
the
one
whi
c
h
the
extreme
cells
are
connected
to
0

state.
•
A
p
eri
o
dic
b
oundary
(PB
)
CA
is
the
one
whi
c
h
the
extreme
cells
are
adjace
n
t
to
ea
c
h
other.
•
A
adiabatic
b
oundary
(AB)
CA
is
duplicating
the
v
alue
of
the
cell
in
an
extra
virtual
nei ghbor
.
2.
Rule
Matrices
w
ith Primary Rules
1
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
,
0 0 0 0 0 0
0 0 0 0 0 1
0 0 0 0 0 0
T
2
0 0 0 0 0 0
1 0 0 0 0 0
0 1 0 0 0 0
,
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
T
Lemm
a
1. [7]
The
r
ep
r
esentation
of
the
next
state
of
a
l
l
primary
rules
(1
,
2
,
4
,
8
,
16
,
32
,
64
,
128
and
256)
under
the
nu
l
l
b
oundary
c
ondition
can
b
e
given
by
using
the
auxili
ary
matrices
T
1
and
T
2
defin
e
d
a
b
ove
in
the
fo
l
lowing
way:
R
u
l
e
1
N
:
[
X
t
+1
]
=
[
X
t
]
R
u
l
e
2
N
:
[
X
t
+1
]
=
[
X
t
][
T
2
]
R
u
l
e
4
N
:
[
X
t
+1
]
=
[
T
1
][
X
t
][
T
2
]
R
u
l
e
8
N
:
[
X
t
+1
]
=
[
T
1
][
X
t
]
R
u
l
e
16
N
:
[
X
t
+1
]
=
[
T
1
][
X
t
][
T
1
]
R
u
l
e
32
N
:
[
X
t
+1
]
=
[
X
t
][
T
1
]
R
u
l
e
64
N
:
[
X
t
+1
]
=
[
T
2
][
X
t
][
T
1
]
R
u
l
e
128
N
:
[
X
t
+1
]
=
[
T
2
][
X
t
]
R
u
l
e
256
N
:
[
X
t
+1
]
=
[
T
2
][
X
t
][
T
2
]
.
Rule
matrices
under
p
eri
o
dic
b
oundary
1
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
,
0 0 0 0 0 0
0 0 0 0 0 1
1 0 0 0 0 0
p
T
2
0 0 0 0 0 1
1 0 0 0 0 0
0 1 0 0 0 0
,
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
p
T
Theorem
2.
(
Peri
o
dic
case
)
The
Matrix
for
any
p
eri
o
dic
b
oundary
CA
rule
(PB)
c
an
b
e
r
ep
r
esent
e
d
as
( ),
p p p
p p p
p p p
PB mn mn
p p p
p p p
p p p
A B O O O D
C A B O O O
O C A B O O O
T
O O O C A B O
O O O C A B
E O O O C A
whe
r
e
A
p
,
B
p
,
C
p
,
D
p
,
E
p
a
r
e
one
of
the
fo
l
lowing
matri
c
es of
the
o
r
der
of
n
×
n
0
,
I
,
T
1
p
,
T
2
p
,
I
+
T
1
p
,
I
+
T
2
p
,
T
1
p
+
T
2
p
and
I
+
T
1
p
+
T
2
p
.
Lemma
3.
The
next
state
of
a
l
l
primary
rules
(1,
2,
4,
8,
16,
32,
64,
128,
256)
of
2D
p
eri
o
dic
c
e
l
lular
automaton
ov e r
Z
2
c
an
b
e
r
ep
r
esent
e
d
as
fo
l
lows:
R
u
l
e
1
P
:
[
X
t
+1
]
=
[
X
t
]
R
u
l
e
2
P
:
[
X
t
+1
]
=
[
X
t
][
T
2
p
]
R
u
l
e
4
P
:
[
X
t
+1
]
=
[
T
1
p
][
X
t
][
T
2
p
]
R
u
l
e
8
P
:
[
X
t
+1
]
=
[
T
1
p
][
X
t
]
R
u
l
e
16
P
:
[
X
t
+1
]
=
[
T
1
p
][
X
t
][
T
1
p
]
R
u
l
e
32
P
:
[
X
t
+1
]
=
[
X
t
][
T
1
p
]
R
u
l
e
64
P
:
[
X
t
+1
]
=
[
T
2
p
][
X
t
][
T
1
p
]
R
u
l
e
128
P
:
[
X
t
+1
]
=
[
T
2
p
][
X
t
]
R
u
l
e
256
P
:
[
X
t
+1
]
=
[
T
2
p
][
X
t
][
T
2
p
]
.
Rule
matrices
under
adiabatic
b
oundary
1
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
,
0 0 0 0 0 0
0 0 0 0 0 1
0 0 0 0 0 1
a
T
2
1 0 0 0 0 0
1 0 0 0 0 0
0 1 0 0 0 0
,
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
a
T
The
auxiliary
ma
trices
T
1
a
and
T
2
a
for
the
adiabatic
b
oundary
case
are
defined
as
foll
o
ws;
hence
w
e
get
the
foll
o
wing
general
rule
matrix
result
for
the
adiabatic
case
as
a
theorem.
Theorem
4.
(
A
dia
b
atic
c
ase)
The
rule
matrix
f or
any
adia
b
atic
b
oundary
CA
rule
(AB)
c
an
b
e
r
ep
r
e
sent
e
d
as
( ),
a a
a a a
a a a
AB mn mn
a a a
a a a
a a
A B O O O O
C A B O O O
O C A B O O O
T
O O O C A B O
O O O C A B
O O O O C A
whe
r
e
A
a
,
B
a
,
C
a
a
r
e
one
of
the
fo
l
lowing
matri
c
es of
the
o
r
der
of
n
×
n:
0
,
I
,
T
1
a
,
T
2
a
,
I
+
T
1
a
,
I
+
T
2
a
,
T
1
a
+
T
2
a
and
I
+
T
1
a
+
T
2
a
.
Lemma
5.
The
next
state
of
a
l
l
primary rules
(1,
2,
4,
8,
16,
32,
64,
128,
256)
of
2D
adia
b
atic
c
e
l
lular
automaton
wi t h
Z
2
c
an
b
e
r
ep
r
esent
e
d as
fo
l
lows:
R
u
l
e
1
AB
:
[
X
t
+1
]
=
[
X
t
]
,
R
u
l
e
2
AB
:
[
X
t
+1
]
=
[
X
t
][
T
1
a
]
t
,
R
u
l
e
4
AB
:
[
X
t
+1
]
=
[
T
1
a
][
X
t
][
T
1
a
]
t
R
u
l
e
8
AB
:
[
X
t
+1
]
=
[
T
1
a
][
X
t
]
,
R
u
l
e
16
AB
:
[
X
t
+1
]
=
[
T
1
a
][
X
t
][
T
2
a
]
t
,
R
u
l
e
32
AB
:
[
X
t
+1
]
=
[
X
t
][
T
2
a
]
t
R
u
l
e
64
AB
:
[
X
t
+1
]
=
[
T
2
a
][
X
t
][
T
2
a
]
t
,
R
u
l
e
128
AB
:
[
X
t
+1
]
=
[
T
2
a
][
X
t
]
,
R
u
l
e
256
AB
:
[
X
t
+1
]
=
[
T
2
a
][
X
t
][
T
1
a
]
t
3.
Application o
f
Im
a
ge Processing
Self
replicating
pat t er n
generation
i s
one
of
the
most
i
n
teresting
topic
and
research
area
in
nonlinear
science. A
motif
is
considered
as
a
basic
sub

pattern.
P
attern
ge
neration is
the
pr
o
cess
of
transforming
copies
of
the
motif
a
b
out
the
arr
a
y
(1D),
plane
(2D)
or
space
(3D)
in
order
to
create
the
whole
re
p
eating
pattern
with
no
ov
erlaps
and blank
[14,
15].
These
patterns
h
a
v
e some
mathematical
pro
p
erties
wh i c h
ma
k
e
gene
rating
al gor i t hm
p
ossible.
A
cellular
automaton
is
a
g
o
o
d
candidate
algorithmic
approa
c
h
used
for
pattern
generation.
Figure
1.
An
application
of
Rule
8
wit
n
ull
(NB),
p
eri
o
dic
(PB)
and
adiabatic
(AB)
b
oundary
res
p
ecti
v
ely
after
32
iterations
of
the
firs
t
image.
Figure
2.
An
application
of
Rule
65
after
32
iterations
of
the
first
image.
Figure
3.
An
application
of
Rule
82
after
32
iterations
of
the
first
image.
Figure
4.
Application
of
Rule
112
after
32
iterations
of
the
first
image.
Creating
algo
rithmic
approa
c
h
for
generating
self
replicating
patt erns
of
digital
images
(motif
as
in
first
image)
is
im
p
orta
n
t
and
sometimes
difficult
task.
Mea
n
while
ma
n
y
resear
c
hers
face
with
ma
n
y
c
hallenges
in
building
and
de
v
eloping
tiling
algorithms
such
as
pr
o
v
iding
simple
and
applicable
algorithm
to
descri
b
e
high
complex
patterns
m
o
del.
Gr
o
wth
from
simple
motif
in
2D
CAs
can
pr
o
duce
self
replicating
pat t er ns
with
complicated
boundar i es
(
n
ull,
p
eri
o
dic,
adiabatic
and
reflexi
v
e),
c
haracterized
b
y
a
v
arie
t
y
of
gr
o
wth
dimensions. The
approa
c
h
gi
v
en
here
leads
to
an
accurate
algorithm
for
generating
di f f er ent
patterns.
In
this
pa
p
er we
use
the
CAs
with
all
the
nearest
neighborhoods
to
generate
self
replicate
patterns
of
digital
images.
F
or
applying
2D
n
ull,
p
eri
o
dic
and
adiabatic
CA
linear
rules
in
image
pr
o
cessing,
w
e
ta
k
e
a
binary
matrix
o f
size
(100
×
100)
due
to
computational
limitations.
W
e
map
ea
c
h
element
of
the
matrix
to
a
unique
pixel
on
the
screen
(writing
new
M
A
TLAB
codes
)
and
we
color
a
pixel
white
for 0
,
black
for
1 for
the
matrix
el e ment s
.
Then
w
e
ta
k
e
another
i ma ge
(as
a
motif
)
whose
size
is
less
than (30
×
30)
fo
r
whi
c
h
patterns
are
to
b
e
generated
and
put
it
in
the
ce
n
ter
of
the
binary
matrix.
This
is
the
w
a
y
,
h
o
w
the
image
is
dr
a
wn
within
an
area
of
(100
×
100)
pixels. It
is
obser
v
ed
from
the
figures
that
the
self
replicating
patterns
can
b
e
generated
only
when
n
u
m
b
er
of
re
p
etition
is
2
n
where
(n=4).
A
neig
h
b
orh
o
o
d
funct i on
that
s
p
ecifies
whi
c
h
of
the
cells
adjace
n
t
cells
affects
its
state
al so
determ
ines
h
o
w
ma
n
y
copi es
will
b
e
obtained
from
the
self

replicating
p r o c e s s
.
In the
t wo
dimensional
a nd
eight
neig
h
b
orh
o
o
ds
case,
this
should
b
e
at most
eig
h
t
copies
of
the
original
image
itself. This
situation
brings
Figure
5.
An
application
of
Rule
189
af
ter
16
iterations
of
the
first
image
Figure
6.
Application
of
Rule
201
after
16
iterations
of
the
first
image.
Figure
7.
Application
of
Rule
261
after
32
iterations
of
the
first
image.
Figure
8.
Application
of
Rule
345
after 16
iterations
of
the
f
irst
image
also
some
the
limitation
ov
er
the
matrix
size
of the
images
to
b
e
replicated.
The
matrix
size
of the
original
images
should
l
ow
er
30
p
erce
n
t
of
the
displ
a
y
matrix in
all
directions.
If
the
first
image exceeds 30
p
erce
n
tage
of
the length of
r
o
w
or
column
of
the
displ
a
y
mat r i x
,
self
replication
pat t er n
when
the
iteration
n
u
m
b
er
t
rea
c
hes
to
16
d
o
es
not
o
ccur.
Also
b
eh
a
viors
for
differe
n
t
b
oundaries
pr
o
duce
differe
n
t
sha
p
es when
t
=
16. Hence
w
e
h
a
v
e
a
classification device
and
tables
up
to
self
r
eplicating
pat t er n
n
u
m
b
er
and
for
the
case
seed
image
less
than
30
p
erce
n
tage
of
the
displ
a
y
matrix
(see
Figs.
1

8), these
will
b
e
prese
n
ted
in
the
next
studies
.
4.
Conclusion
In
this pa
p
er
w
e
discuss
the theory
2

dimensional, uniform
p
eri
o
dic
and
adiabatic
b
oundary
CAs
of linear rules
and
applications
of
image
pr
o
cessing. It is
seen
that
CAs
theory can
b
e
applied
successfully
in
self
replicating
patterns
of
image
pr
o
cessing.
The
some
c
haracterization
and
applications
on
a
2D
finite
CA
b
y
using
matrix
a l g e b r a
built
on
Z
3
are planned
to
next
studies.
H
o
w
e
v
er
after making use
of
the
matrix
represe
n
tation
of 2D
CA,
it
will
b
e
pr
o
vided an
algorithm
to
obtain the
n
u
m
b
er
of
Garden of
Eden
configurations for
the
2D
CA
defined
b
y
some
rules.
Acknowledgements:
This work
is supported by "The Scientific and Technological Research Council of Turkey" (TÜB
İTAK)
(Project Number: 110T713).
References
[1]
Akın
H
.
,
On
the
di r ect i onal
e
n
tro
p
y
of
Z
2
actions
generated
b
y
additive
cellular
automata,
Appl.
Math.
Computation
170
(1)
(2005)
339

346.
DOI:
10.1016/j.amc.2004.11.032
[2]
Akın H., Siap
I.,
On
cellular
automata
ov
er Galois
rings,
Infor
.
Pr
o
ces
.
Letters,
103
(1)
(2007)
24

27.
DOI:
10.1016/j.ipl.2007.02.002
[3]
Akın H.,
Siap
I., Uguz
S.,
Structure
of
2

dimensional
hexagonal
cellular
automata
,
AIP
Conf
.
Pr
o
ceed
.
,
V
olume
1309,
(2010)
p.
16

26.
DOI:
10.1063/1.3525111
[4]
Chatto
p
d
h
ya
y
P
.,
Choud
h
ury
P
.
P
.,
Dihidar
K.,
Characteriz
ation
of
a
particular
h
ybrid
transformation
of
t
w
o

dimensional
cellular
automata,
Comput.
Mat
h.
App.
38,
(1999),
p.
207

216
DOI:
10.1016/S0898

1221(99)00227

8
[5]
Choud
h
ur
y
,
P
.
P
.,
Sah
o
o, S.,
Hassan,
S.
S.,
Basu,
S.,
Ghosh,
D.,
Kar,
D.,
Ghosh,
Ab.,
Ghosh,
Av.,
Ghosh
A.K.,
Classifi
cation
of
cellular
automata
rules
based
on
their
pro
p
erties,
I
n
t.
J.
o
f
Comp.
Cogn.
8,
(2010), p.
50

54.
[6]
Chou
H.H., Reggia
J.
A.,
Emergence
of
self

replicating
structures
in
a
cellular
automata
space,
P
h
ysica
D:
110,
(1997),
252

276.
doi.org/10.101
6/S0167

2789(97)00132

2
[7]
Choud
h
ury
P
.
P
.,
N
ay
ak
B.
K.,
Sah
o
o
S.,
Rath
S.
P
.,
Theory
and
a
p p l i c a t i o n s
of
t
w
o
dimensional,
null

b
oundar
y
,
nine

neig
h
b
orh
oo
d,
cellular
automata
linear
rules
,
Tech. Report
No.
ASD/2005/4,
13
M
a
y
2005
,
arXiv:0804.2346
.
[8]
Dihidar
K
.,
Choud
h
ury
P
.
P
.,
Matrix
algebraic
for
m
ulae
concerning
some
exceptional
rules
of
t
w
o
dimensional
cellular
automata,
Inf.
Sci.
165
(2004)
91

101.
doi.org/10.1016/j.ins.2003.09.024
[9]
Gr
a
vner J.,
Gliner
G.,
P
elfrey
M.
,
Replication
in
one

dimensional
cellular
automata,
P
h
ysica
D:
240,
(18),
(2011),
1460

1474.
doi.org/10.1016/j.physd.2011.06.015
[10]
Siap
I.,
Akın H.,
Uguz
S.,
Structure
and
re
v
ersibili
t
y
of
2D
hexagonal
cellular
automata,
Comput
.
Math
.
Applications,
62, (2011
)
4161

4169.
DOI:
10.1016/j.camwa.2011.09.066
[11]
Siap
I., Akın H.
and
Sah
F.,
Characterization
of
t
w
o
dimensional
cellular
automata
ov
er
ter
nary
fields,
J
.
of
the
F
rank
.
Inst
.
,
348
(2011),
1258

1275.
DOI:
10.1016/j.jfranklin.2010.02.002
[12]
Siap I.,
Akın
H.
and Sah F.,
Garden
of
Eden
configurations
for
2

D
cellular
automaton
with
rule 2460N,
Infor
.
Sciences,
180
18
(2010)
3562

3571.
DOI:
10.1016
/j.jfranklin.2010.02.002
[13]
Uguz
S.,
Akın
H.,
Siap
I.,
Re
v
ersibili
t
y
algorithms
for
3

state
hexagonal
cellular
automata
with
p
eri
o
dic
boundaries
,
I
n
tern
.
J
.
Bifur
.
and
Chaos
,
23
, (2013)
1350101

1

15,
DOI: 10.1142/S0218127413501010
[14]
v
on
Neumann
J.,
The
theory
of
self

repr
o
ducing
automata, (Edited
b
y
A.
W.
Burks),
Univ. of
Illinois
Press,
Urbana,
(1966).
[15]
W
olfram
S.,
Rev. M
o
d.
P
h
ys.
55
(
3) (19
83)
601

644.
doi/10.1103/RevModPhys.55.601
.
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