TELKOM
NIKA
, Vol. 11, No. 9, September 20
13, pp.
5126
~51
3
2
ISSN: 2302-4
046
5126
Re
cei
v
ed
Jan
uary 29, 201
3
;
Revi
sed
Ju
n
e
4, 2013; Accepte
d
Ju
ne
17, 2013
The Blanket Fractal Dimensio
n based
on the Directed
Pattern Plate
Wang Yan*,
Cui Ze
y
a
n
Shen
ya
n
g
Lig
o
ng Un
ivisit
y, N
o
.6, nan
pin
g
cent
er Ro
ad, Hu
nna
n Ne
w
D
i
strict, Shen
yan
g
110
15
9
Lia
oni
ng, P.R.Chin
a, T
e
l: +
8
6
-
24-2
468
62
18, F
a
x:+
86-
24-2
4
686
22
0
*Corres
p
o
ndi
n
g
author, e-ma
i
l
:
w
y
2
2
2
8
@so
hu.com
,
515
04
34@
qq.com
A
b
st
r
a
ct
Bla
n
ket fracta
l di
me
nsio
n is
a kind of fr
actal
di
mens
io
n, w
h
ich is u
s
ually
use
d
in
image
process
i
ng, an
d
d
oesn'
t have
a
d
i
rect
io
n. T
h
i
s
pa
per w
i
l
l
pro
pose
to i
n
trod
u
c
e the
dir
e
ctio
n
of p
i
xel
into the
calcul
atio
n of the Bla
n
ket fra
c
tal
di
me
nsio
n,
and us
e it to detect the
ed
g
e
of imag
e. This al
gorith
m
w
ill
calcul
ate th
e
n
e
w
bla
n
ket frac
tal d
i
mens
ion
o
f
the i
m
age
at f
i
rst, and
the
n
d
e
tect the
ed
ge
w
i
th the
meth
o
d
of the ed
ge se
gmentati
on a
l
g
o
rith
m bas
ed
on the tra
d
itio
nal Bl
ank
et fracta
l di
me
nsi
on.
After this, it w
i
l
l
eli
m
i
nate a p
a
r
t
of pixels bas
e
d
on the IFS and Co
lla
ge
The
o
re
m. Experi
m
ents w
ill show
that this alg
o
rith
m
is abl
e to ov
er
come the
issu
e of do
ub
le-b
o
r
der
in t
he e
d
ge se
g
m
ent
ati
on a
l
gor
ith
m
b
a
se o
n
traditi
o
nal
Blank
et fractal di
me
nsio
n, and
extract the pr
ecision
edg
es w
i
th the different
directi
ons.
Ke
y
w
ords
: temp
let w
i
th dire
ctions, precis
io
n edg
e seg
m
e
n
tation, bl
ank
et fractal dimens
i
on, fractal
Copy
right
©
2013 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduction
Mandel
bo
rt used
self-simil
arity to descri
be
com
p
licated and inexa
c
t figure
s
, an
d pointed
out the fra
c
tal
geomet
ry. To co
ntra
st wit
h
the
Eu
clide
an ge
ometry,
fractal
geom
etry wa
s abl
e
to
descri
be th
e
natural
obj
ect
with
com
p
licated
cha
nge
s in a
better way. The fracta
l dimen
s
io
n
was
able to
u
s
e t
o
mea
s
u
r
e
a
nd a
nalyze th
e texture
and
the o
u
line
wi
th self-simil
arity [1]. Pentland
figured
out th
e blan
ket fra
c
tal dim
e
n
s
io
n ba
se
d
on
Mandel
bo
rt’s idea.
He g
o
t the informati
on
from the su
rf
ace of obj
ect,
and then det
ermi
n
ed the texture as
sm
ooth or ha
rsh
[2].
In these ye
a
r
s, a
s
the
speed
of co
m
put
ing imp
r
o
v
ed, the fra
c
tal theory d
e
v
eloped
widely. Many
edge extra
c
t
i
on algo
rithm
s
with t
he fra
c
tal theo
ry came out. Lili
Jian
g used t
h
e
differen
c
e
s
b
e
twee
n man
-
made o
b
ject
and natu
r
al b
a
ckgroun
d o
n
fractal di
m
ensi
on to det
ect
the edge of
man-m
ade
o
b
ject [3]. Mingqin Liu u
s
e
d
blancket fra
c
tal dimen
s
ion
to calculate
out
the dimen
s
io
n of pixel, and then detect
ed the edge
of
irreg
u
lar o
b
ject un
der t
he ba
ckgro
u
n
d
of
man-m
ade
o
b
ject.S.S [4, 5]. Chen
a
nd B.B Cha
udhu
ri poi
nted out the
edge
extra
c
tion
algorith
m
s
wit
h
the fra
c
tal t
heory [6, 7].
Otherwise, th
ere
we
re e
d
g
e
extra
c
tion a
l
gorithm
s b
a
sed
on the Bro
w
n
Rand
om Fiel
d Model, the scale fra
c
tal, and multi-fractal [8-12].
2. The Classi
cal Blanke
t
Fractal dimension
2.1.
The Formula
tion of th
e Bl
anke
t Frac
ta
l Dimension
The blan
ket fractal di
men
s
ion assum
e
s
the im
age a
s
its height ha
s the direct p
r
oportio
n
with the hill
of grey level. The grey lev
e
l is
covered by the surfac
e
with the di
stance of
. The
distan
ce
of u
p
and d
o
wn
is the volum
e
of the ima
ge. Ch
angin
g
the amo
u
n
t
of
is able to
cal
c
ulate
out
different vol
u
me. The
dif
f
eren
ce
of the volume
s of
the adja
c
e
n
t
is ma
rked a
s
()
A
.Asso
ciated u
pper
su
rface is
()
u
,the lower surface is
()
b
,the formul
ation is
belo
w
：
11
|(
,
)
(
,
)
1
|
(
,
)
m
a
x
{
(
,)
1
,
m
a
x
(
,)
}
mn
i
j
ui
j
u
i
j
u
m
n
(1)
11
|(
,
)
(
,
)
1
|
(
,
)
m
i
n
{
(
,)
1
,
m
i
x
(
,)
}
mn
i
j
bi
j
b
i
j
b
m
n
(2)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
The Blan
ket Fra
c
tal Dim
ensio
n ba
sed
on the Direct
ed Pattern Plate (Wang Ya
n)
5127
In the formula
t
ion
00
(,
)
(
,
)
(,
)
ui
j
b
i
j
I
i
j
,
(,
)
Ii
j
is the grey level of pixel
(,
)
ij
.
The form
ulation of the volume of image i
s
belo
w
:
)
,
(
)
,
(
,
j
i
b
j
i
u
V
j
i
(3)
The form
ulation of the image’s
surfa
c
e
area:
2
)
(
1
V
V
A
(4)
Peleg u
s
e
d
Mandlel
bort’
s idea to fig
u
re out the
im
age’
s surfa
c
e
are
a
of the
blan
ket
fractal dim
e
si
on:
D
F
A
2
)
(
(5)
Above F i
s
a
co
nsta
nt val
ue
whi
c
h i
s
related to
the
image, n
o
t re
lated to
. D is the
image’
s blan
ket fractal dim
ensi
on. Getting the loga
rithm from (5
):
)
log(
)
2
(
)
log(
))
(
log(
D
F
A
(6)
C
o
un
te
r
po
in
t
(lo
g
(
1
),
l
o
g
(
1
)
)
A
，
(log(
),
log
(
))
A
，
(log(
1
)
,
l
og
(
1
))
A
pro
c
e
s
ses the lin
e
bridg
e
an
d g
e
t linear
slop
e
h
=
2
D
. After that, it is able to get the im
age’
s blan
ket
fractal
dimen
s
ion D=2-
h
.
2.2. Analy
z
in
g the Blan
ke
t Frac
tal Dim
e
nsion
The blan
ket fractal dim
e
n
s
ion use
s
the image’
s grey l
e
vel to calcul
ate the volume, and
finally transf
e
r the im
ag
e’s volum
e
t
o
the ima
g
e
’
s bla
n
ket fractal di
men
s
ion. The
r
efo
r
e
,
analyzi
ng the
pro
c
ed
ure o
f
calculating
the image’
s
volume is be
neficial to th
e analy
z
ing t
h
e
pro
c
ed
ure of cal
c
ulatin
g the dire
cted bla
n
ket fra
c
tal di
mensi
on.
As
the
u
above,
the up
per surface
cal
c
ulate
s
fro
m
the
iteration
. D
u
r
i
ng
th
is
pr
oc
e
d
u
r
e
,
letting the se
ekin
g point
(,
)
ui
j
plus 1 a
nd cho
o
sin
g
the big
gest poi
nt am
ong the five p
o
ints of its
4-conn
ecte
d amount as
th
e
amo
unt of t
he iteration
(,
)
ij
. When
=
1
, as th
e
F
i
gu
r
e
be
lo
w
,
the
maximum is f
r
om the conv
olution of the pattern plate
and the pixel.
Figure 1.
The
Pattern Plate for Cal
c
ulatin
g the
U
p
pe
r
Su
r
f
ace
w
h
en
1
Figure 2. The
Pattern Plate for Cal
c
ulatin
g the
Upp
e
r Surfa
c
e whe
n
1
Whe
n
=5, a
s
the figure, the
maximum is
from
the co
nvolution of the pattern plate
and
the pixel.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 9, September 201
3: 512
6 – 5132
5128
As the analyzing ab
ove, durin
g the proce
dur
e of calcul
ating the
image’s vol
u
me by
blan
ket fracta
l dimen
s
io
n,
becau
se
of u
s
ing
the
pixel
of 4
-
conn
ect
ed
regio
n
, its volume
ha
s t
he
same
weig
ht for each direction. Thi
s
kind of bl
an
ket fractal dim
ensi
on is a fractal dim
e
n
s
ion
without directi
on.
2.3. The Res
u
lt of Clas
sical Blanke
t
Fractal
Edge
Segmenta
tion Algorith
m
and Analy
z
ing
Usi
ng typical
lena g
r
ey le
vel image a
s
sam
p
le,
3. The Fig
u
re
is a re
sult of u
s
ing
cla
ssi
cal bl
an
ket fra
c
tal di
mensi
on to proce
s
s the ed
ge se
gmentat
ion algo
rithm.
Figure 3. The
Result of usi
ng Cla
s
sical
Blanket F
r
act
a
l Dimen
s
io
n to Proce
s
s the
Eedge Segm
entation Algo
rithm
Figure 4. Wh
en
3, the Reg
i
ons of Adja
cent
Pixel under th
e Effect are Covered
From
analy
z
i
ng the
Figu
re
3, it is cl
ear t
o
see
th
at in t
he result of u
s
ing
cl
assi
cal
blan
ket
fractal dim
e
n
s
ion to p
r
o
c
e
ss th
e edg
e segmentatio
n
algorith
m
, the pheno
men
o
n
of double e
d
ge
is mo
re
re
ma
rka
b
le. It is b
e
ca
use the
weights of pixe
l und
er th
e ef
fect from
ea
ch directio
n a
r
e
same
(a
s the
Figure 2
)
，
a
nd the m
a
jor
regio
n
s of adj
ace
n
t pixel u
nder the effe
ct are same,
(as
Figure 4).
Wh
en
=3
，，
the re
gion of adj
acent two pixel
A
and B und
er the effe
ct has 2
5
poi
nts.
The re
gion of
A under the effect is the red triangl
e. The regi
on of B under the e
ffect is the black
circle. Th
e ye
llow b
a
ckg
r
o
und i
s
the
re
gion fo
r both
of A and B u
nder the effe
ct and it
ha
s
18
points.
As the an
alyzing above, d
u
r
ing the p
r
o
c
edure of
cla
s
sical blan
ket f
r
actal
dimen
s
ion, the
major regio
n
of the adjace
n
t pixel under the effect
for both two poi
nts are
covered. If the region
of the
regi
stration of
adj
a
c
ent
pixel i
n
clude
s e
dge,
the
re
sults of
adja
c
e
n
t pix
e
l A a
n
d
B a
r
e
clo
s
ed. It is easily hap
pen
ed the phe
no
menon of do
u
b
le edg
e in such
situation.
3. Directed Blanket Fr
actal Dimensio
n
3.1. The For
mula of Dire
cted
Blanke
t Fractal
Dimension
Putting the di
rectio
n of pix
e
l into the p
r
oc
e
dure of
calcul
ating the
uppe
r a
nd l
o
we
r is
able to
dire
ct
the directio
n
to the bla
n
ket frac
tal
dim
ensi
on. From
the directe
d
blan
ket fra
c
t
a
l
dimen
s
ion,
e
x
amining
wh
e
t
her the
ed
ge
is in
the
adja
c
ent
regi
on of
pixel in
one
dire
ction i
s
a
b
le
to examine the edge in diff
erent di
re
ctio
ns.
The Fig
u
re 5
belo
w
sho
w
s
the 12
situati
ons
of expan
ding the
cal
c
ulation of u
p
p
e
r a
n
d
lowe
r su
rfa
c
e
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
The Blan
ket Fra
c
tal Dim
ensio
n ba
sed
on the Direct
ed Pattern Plate (Wang Ya
n)
5129
Figure 5.The
12 Situations
of Expanding
the Ca
l
c
ulatio
n of Uppe
r an
d Lower Surf
ace
Usi
ng the D7 as the examp
l
e, t
he calcula
t
ion of upper
surfa
c
e i
s
:
11
1
(
,
)
m
a
x
{
(
,
)
1,
(
-
1,
+1
)
(
+1,
+
1
)
}
ui
j
u
i
j
u
i
j
u
i
j
，
(7)
Putting the di
rectio
n into t
he calculatio
n of
up
per
an
d lower
su
rfa
c
e, a
s
the fo
rmulation
(1), the result will be direct
ed blanket fractal dimension.
3.2. Analy
z
in
g the Dire
cte
d
Blanke
t Fr
actal Dimen
s
ion
Defini
tion:
In
the
pro
c
e
dure of
cal
c
ulatin
g the
bl
an
ket
fractal
dime
n
s
ion, th
e
con
v
olution
template whi
c
h u
s
e
s
the i
t
eration o
r
th
e pre
-
calculat
ed pixel with
different wei
ghts is
dire
ct
ed
pattern plate.
Usi
ng th
e
sa
me
way in
pa
rt 1 to
an
alyze the
affecte
d
region
in
di
recte
d
blan
ke
t fractal
dimen
s
ion
pi
xel, usin
g th
e num
be
r 7 i
n
Figu
re
5,
whe
n
=5, get
ting the m
a
ximum fro
m
th
e
pattern plate
and pixel con
v
olution, as F
i
gure 6.
Figure 6. Wh
en
=
5
, getting the Maximum
from the Pattern Plate an
d
Pixel Convolution
Figure 7. The
Affected Reg
i
on of Blanket
Fractal Dimension Pixel
As the
analyzing of affe
cte
d
re
gion
of pi
xel in
Figu
re
6, the Fig
u
re
7 sh
ows th
e
affected
regio
n
of bla
n
ket fra
c
tal d
i
mensi
on pix
e
l A whe
n
3
. In Figu
re 7,
yellow, re
d, and b
r
o
w
n
rep
r
e
s
ent the
pixel in different x-coordin
a
tes u
nde
r th
e effect fro
m
A. It is clea
r to se
e that in t
he
whol
e figure,
the affected region of A is
the regi
on inn
e
r the da
she
d
triangle.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 9, September 201
3: 512
6 – 5132
5130
4. The Classi
cal Blanke
t
Fractal edge
Segmenta
tion Algorith
m
w
i
th
Direc
t
ed Patter
n Plate
The ite
r
ation
functio
n
fa
mily is a
n
i
m
porta
nt the
o
ry of fractal
geom
etry b
e
ca
use it
explain
s
the
gene
ration
type a
nd
cont
rols th
e im
a
g
i
ne g
ene
ratio
n
. The
coll
ag
e theo
ry expl
ains
the level of self-simil
arity of a focus poi
n
t
in
fractal ge
ometry and th
e corre
s
p
ondi
ng fixed point
.
This a
r
ticl
e u
s
e
s
affine tra
n
sformation
as it
eration f
unctio
n
famil
y
, uses th
e
stand
ard
deviation
of
the pixel
an
d its
adja
c
e
n
t fra
c
tal di
mensi
on
after affine
tran
sformation
a
s
the
evaluation for the pixel similarity.
4.1. Based o
n
the Ro
ta
tio
n
of Pixel
In the digital
image, the
r
e
are o
n
ly 8 sit
uat
ion
s
ba
se
d on the
rota
tion of pixel: identity
transfo
rm, ce
ntral vertical
line doubl
e over, cent
ral h
o
rizontal line
double over,
main diagon
al
doubl
e over, back-diag
ona
l double ove
r
, clockwi
s
e rotation
o
90
, c
l
ock
w
is
e rotation
o
180
, and
clo
c
kwi
s
e rot
a
tion
o
270
. The
Fi
gure
8
1
L
-
8
L
rep
r
e
s
ent
s the
8
situations. Am
ong th
en, the
0
L
is
the origin
al image.
Figure 8. 8 Situations b
a
sed on the Rotation of Pixel
4.2. The Cla
ssical Blan
k
e
t Fra
c
tal E
dge Se
gmen
tation Algori
t
hm
w
i
th Directed
Patte
r
n
Plate
First of all, u
s
ing di
re
cted
blanket fract
a
l dimen
s
ion
and the ide
a
of cla
ssi
cal
blanket
fractal ed
ge
segm
entation
algorithm to cal
c
ulate
out
each pixel of blan
ket fracta
l dimensi
on for
examining
th
e imagi
ne e
d
ge, an
d then
bases on th
e
di
fferent pixel
s
in th
e e
dge,
the pixel f
r
a
c
ta
l
in adj
acent
region
with
sa
me di
re
ction
expand
in
different
way
s
.
Usi
ng ite
r
atio
n fun
c
tion fa
mily
and
colla
ge t
heory a
s
ba
sic theo
ry, the
rotation
after the pixel
pa
rallel m
o
ving
in the affe
cted
regio
n
as
sp
ecific fun
c
tio
n
, and the d
i
vided di
ffere
nce of the fractal pixel in
associ
ated
8
con
n
e
c
ted
re
gion
s after th
e rotatio
n
pa
rallel moving
as the
level
of simila
rity, for elimi
nating
a
few simila
r pi
xel. The pixel which i
s
eliminated is
de
termine
d
as the most simil
a
r to the recent
pixel, and it the dou
ble ed
ge from the same edg
e. The pro
c
e
d
u
r
e
in detail belo
w
:
Step 1: pi
cki
ng u
p
the
12
situatio
n in
Figur
e 3,
or
makin
g
the
suitable p
a
ttern plat
e
basi
ng dem
a
nder, calculat
ing ea
ch dire
cted pixel bla
n
ket fra
c
tal di
mensi
on in th
e image.
Step 2: analyzing the affe
cted regio
n
of pixel in cho
s
e
n
pattern plat
e.
Step 3: basi
ng on de
ma
nder, d
e
termi
n
ing thresh
ol
d, the pixel whi
c
h ha
s th
e fractal
highe
r than th
is thre
shol
d in pro
c
ed
ure 1 is rem
a
rked
as the point
of edge.
Step 4: Rota
ting the pixel which is re
marked a
s
th
e edge p
o
int
and its 8 co
nne
cted
regio
n
a
s
the
8 differe
nt
ways
of rotat
i
ng in fi
gu
re
6. After that, parallel mov
i
ng them i
n
to
its
affected regio
n
.
Cal
c
ulating
th
e nin
e
poi
nts,
itself a
nd 8
c
onne
cted
poi
nts, an
d findi
ng o
u
t the diff
eren
ce
betwe
en the
s
e 9 p
o
ints a
nd its
movin
g
pla
c
e.
Det
e
rmini
ng the
threshold. If t
he differen
c
e
is
lowe
r than th
e threshold, the pixel sh
oul
d be red
e
fine
d as no
n-e
d
g
e
point.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
The Blan
ket Fra
c
tal Dim
ensio
n ba
sed
on the Direct
ed Pattern Plate (Wang Ya
n)
5131
4.3. The Res
u
lts and Ana
l
y
z
ing
Usi
ng the
cla
ssi
cal l
ena g
r
ey level imag
e as th
e sa
m
p
le,
3 in this
case. Th
e Fig
u
r
e 9
and 10 a
r
e the re
sults of
the blan
ket fractal
ed
ge se
gmentation al
gorithm fro
m
D7 an
d D10
in
the Figure 5. Among them,
the red one i
s
the
eliminat
ed point a
c
co
rding to the st
ep 4.
Figure 9.
The
Results of the Blanket F
r
a
c
tal
Edge Segme
n
tation Algorit
hm from D7
Figure 10. Th
e Re
sults of
the Blanket Fractal
Edge Segme
n
tation Algorit
hm from D10
From
co
ntra
sting the
re
sult
s of Fi
gu
re 3
and
9,
it is
cl
ear to
see th
at the p
heno
menon
of
doubl
e edg
e i
n
Figu
re 9 i
s
more
rema
rkable than
Fig
u
re 7
be
cau
s
e Figu
re 9 h
a
s
processe
d the
eliminating. T
he cove
red
p
a
rt
of affecte
d
regi
on
of t
he bl
an
ket fractal
dime
nsi
on a
nd
adja
c
ent
pixel is very small, as Fig
u
re 11 sho
w
s. In
the
Figure 7, the affected regio
n
of A is marke
d
as
red
trian
g
le,
and B’
s i
s
m
a
rked
a
s
bl
ack
circle.
Th
e
Figure
1
1
pro
v
es
that
the affected re
gio
n
in
the 4 adja
c
e
n
t conn
ecte
d
points have
no cove
r
ed
part after p
r
o
c
e
ssi
ng the
dire
cted bla
n
ket
fractal dim
e
n
s
ion. Thi
s
is a
b
le to make t
he phe
nome
n
on of doubl
e edge
wea
k
e
r
.
Figure 11. Th
e Affected Re
gion of Pixel in Figure 7 as
3
From
cont
ra
sting the Figu
re 9 and
10, i
t
is cl
ea
r to see the differe
nce
of the re
sult of
examining th
e edge in diff
erent di
re
ctio
ns. For exa
m
ple, in the rig
h
t side of the image, the Fi
gure
9 elimin
ates
a lot
of ed
g
e
s i
n
min
u
s
45°
and
rem
a
ins the
po
sitive 45°,
but i
n
the
Figu
re
10,
most po
sitive 45° is elimi
n
a
t
ed
，
and the
negative pa
rts are
remai
n
e
d
.
The Fig
u
re
5
sho
w
s all of the 12 p
a
ttern
plates a
r
e
ab
le to examine
the edge i
n
d
i
fferent
dire
ction
s
. Since thi
s
cal
c
ulation is b
a
sed on the
the
o
ry of fractal
geomet
ry, different with o
n
ly
contrastin
g th
e g
r
ey level
s
in cl
assi
cal
e
dge
dimen
s
io
n, it is abl
e to
expan
d th
e
pattern
plate
s
in
Figure 6 to more shap
es, a
nd get more edge
s with
co
mplicate
d
direction
s
.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 9, September 201
3: 512
6 – 5132
5132
5. Conclusio
n
This a
r
ticle
p
o
ints out that
dire
cting the
dire
ction of
pixel to the calcul
ation of
fractal
dimen
s
ion
is able to
get
the fra
c
tal di
mensi
on
wi
th
different a
d
j
a
ce
nt co
nst
r
uction
s of
pi
xel.
Usi
ng
su
ch di
recte
d
fra
c
tal
dimen
s
ion
to
pro
c
e
s
s the
extraction
of
edge,
u
s
in
g the difference
of
the fractal di
mensi
on of the different ed
ge point
s in the sam
e
adja
c
ent re
gion,
and u
s
ing fra
c
tal
geomet
ry’s th
e iteratio
n fu
nction fa
mily and
colla
ge
t
heory to
elimi
nate the p
o
in
ts in the
pixel
’
s
asso
ciated af
fected region.
From ab
ove, this
cal
c
ulati
on is abl
e to achi
eve these:
(1)
Red
u
ci
ng
the phen
ome
non of dou
ble
edge an
d findi
ng out the e
dge
s in different dire
ction
s
.
(2) T
he edg
e whi
c
h is fou
n
d
out is able t
o
pro
c
e
ss the
exact locatin
g
.
(3) It is able
to provid
e
more i
n
form
ation of ed
g
e
to the de
eper
usi
ng d
u
ring th
e im
age
pro
c
e
ssi
ng.
Referen
ces
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eng W
e
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ang
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