TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.6, Jun
e
201
4, pp. 4438 ~ 4
4
4
3
DOI: 10.115
9
1
/telkomni
ka.
v
12i6.547
9
4438
Re
cei
v
ed
De
cem
ber 2
2
, 2013; Re
vi
sed
Febr
uary 15,
2014; Accept
ed Feb
r
ua
ry
28, 2014
Underground Image Denoising
Zhang Ye, JIA Meng*
Dep
a
rtment of Electrical E
ngi
neer
ing, Xi
n
x
ia
ng
Un
iversit
y
,
East Jin Sui Street, Xi
n
x
ia
ng
cit
y
, Hen
an pr
o
v
ince, Ch
ina
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: tianshi
_cd@
163.com
A
b
st
r
a
ct
A Mixed W
i
n
d
o
w
Shrink a
n
d
BayesS
hrink
Imag
e D
eno
i
s
ing A
l
gor
ith
m
Based
on C
u
rve l
e
t
T
r
ansform is
pr
opos
ed
in t
h
is
pap
er. Curv
e l
e
t transfor
m
is
effective i
n
pr
e
s
entin
g l
i
ne
a
n
d
surfac
e pr
op
erty
of i
m
a
ge. In
th
e pr
opos
ed
a
l
g
o
rith
m, C
u
rvel
e
t
transfo
r
m
is
e
m
p
l
oy
ed for
th
e first stag
e, th
en
accord
in
g t
h
e
theory of
im
age dem
ising
m
e
thod
bas
ed on Wav
e
let
transform
, we
com
b
ine Window Shrink
and
BayesShr
ink d
eno
isin
g al
gori
t
hm to perfor
m
noise re
ducti
o
n
. Experi
m
e
n
t results show
that the pro
pos
e
d
a
l
go
ri
thm
i
s
co
mp
e
t
i
t
i
v
e
to
Wa
ve
l
e
t tra
n
s
fo
rm
in
te
rm
s o
f
Pe
a
k
Si
gna
l
to
N
o
i
s
e
Ra
ti
o
(PSN
R
)
and
den
oisi
ng i
m
ag
e qua
lity.
Ke
y
w
ords
: cur
v
elet transfor
m
, imag
e de
no
is
e, hard thres
h
o
l
d, ada
ptive co
efficient
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
Wavelet tra
n
s
form
ha
s be
en wid
e
ly used in the tra
d
itional meth
ods to
remov
e
noise
from ima
ge .
T
houg
h the
wavelet tran
sform h
a
ve th
e be
st b
a
se
s
whe
n
it re
pre
s
ent
s ta
rg
e
t
function
s
whi
c
h h
a
s dot
singula
r
ity [1], it can
ha
rdl
y
get the be
st ba
se
s
wh
en it present
the
sing
ularity of line and hyper-plan
e
. This ma
ke
s the traditional two-dime
nsio
nal wav
e
let
transfo
rm in
dealin
g with t
he imag
e ha
ve some
li
mitations. To
overcome th
e a
bove-m
ention
e
d
sho
r
tco
m
ing
s
that Wavelet
tran
sform
h
a
s,
Dono
ho, and some
ot
her peopl
e
p
r
omote Cu
rvelet
transfo
rm th
e
o
ry. The a
n
isotropy of
Curvelet tr
an
sform theory i
s
v
e
ry c
ondu
civ
e
to present the
edge of the i
m
age.
2. Res
earc
h
Method
Cruvelet Tran
sform wa
s d
e
veloped on
the bas
is
of Ridegelet Tra
n
sform, and
Ridgelet
transform is introduced first.
2.1. Ridgelet Transform
Ridg
e let tran
sform
(RIT
) o
v
erco
me
s the
wea
k
ne
ss of wavelet tran
sform re
pre
s
e
n
ting in
two or hig
h
e
r
dimen
s
ion
s
[2]. Ridgelet transfo
rm
can
be define
d
as follows [4]. Set satisfying the
conditions:
2
2
()
d
(
1
)
Ridg
elet funct
i
on
22
,,
:
ab
R
R
of the two-dime
nsi
onal
spa
c
e i
s
defin
ed as:
1/
2
,,
1
2
()
(
c
o
s
s
i
n
)
/
ab
x
ax
x
b
a
(
2
)
And, a repre
s
ent
s the Ri
dgelet scale;
b stand
s for Ridg
elet p
o
sition;
repre
s
ent
s
Ridg
elet di
re
ction. Given
the dual
inte
gral fu
nctio
n
()
f
x
, we
ca
n defi
ne continu
o
u
s
Ridgel
et
trans
form (CRT) in the
2
R
as:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Und
e
rg
ro
und
Im
age Denoi
sing (Zha
ng
Ye)
4439
2
,,
(,
,
)
()
AB
f
R
CRT
a
b
f
x
d
x
(
3
)
For a straigh
t
line with the singul
ar m
u
lti-varia
b
le functio
n
, Ridg
elet transfo
rm has a
good a
pproximation pe
rformance [6].
2.2. Radon T
r
ansform
Ridg
elet tra
n
s
form
is a
c
hi
eved by
Ra
d
on tr
an
sform
in the
do
mai
n
of o
n
e
-
dim
ensi
onal
wavelet t
r
ansform. For the func
tion
(,
)
f
xy
in
2
(,
)
x
yR
plane, th
e
Radon
tra
n
sfo
r
m is the
function at all
angle
s
on a
straig
ht line p
r
oje
c
tion, [7, 8] as:
(,)
(
,
)
(
c
o
s
s
i
n
)
f
R
fx
y
x
y
d
d
(,)
[
0
,
2
)
R
(4)
is the
u
n
it p
u
lse
fun
c
tion
, and
the
Ridg
elet tran
sform
coefficient
(,
,
)
f
Ra
b
of
(,
)
f
xy
can be
carrie
d out on its Radon tra
n
sfo
r
m c
oefficie
n
ts of wavelet transfo
rm to be
:
12
(,
,
)
(
,
)
[
(
)
/
]
ff
R
ab
R
a
b
a
d
(
5
)
Acco
rdi
ng to the Fou
r
ier Projectio
n
Theo
rem:
(c
o
s
,
s
i
n
)
(
,
)
it
f
f
Re
d
t
(
6
)
2.3. Curv
elet Transform
The edg
es
of natural ima
g
e
are alm
o
st
in cu
rv
e, so the Rid
gelet a
nalysi
s
of the images
of the e
n
tire
singl
e-scale i
s
n
o
t very
effective.
To
si
n
gular curve
with the multi
-
variabl
e fun
c
ti
on,
its performan
ce is only cl
ose to the equivalent
of wavelet tran
sform. In ord
e
r to solve the
sing
ular curv
e with
the
mu
lti-variabl
e fu
nction
of the
spa
r
se a
p
p
r
o
x
imation p
r
ob
lem, we
can
turn
to Curvelet transfo
rm. The
basi
c
step
s a
s
sh
own in Figure 1:
a)
Sub-ban
d Decom
p
o
s
ition
.
Through the
wavelet tran
sform it divided into a number
of sub-ban
d compon
ents [3
]. For the
NN
image f, the first brea
k will be:
00
1
I
i
i
f
Pf
f
(
7
)
0
Pf
is for the l
o
w-frequ
en
cy comp
one
nts,
and
1
I
i
i
f
are fo
r the hig
h
freque
ncy
comp
one
nts.
b)
Smooth Pa
rtitioning [5]. E
a
ch
sub-ban
d
hig
h
-freque
ncy
sub
-
divid
ed into
a n
u
m
ber
of piece
s
, wit
h
different su
b-comp
one
nt divi
sion of the
sub-blo
ck
size can b
e
different.
()
S
s
QS
Q
Q
fw
f
(
8
)
Q
w
represented i
n
binary box
as:
11
2
2
[
/
2
,
(
1
)/
2
]
[
/
2
,
(
1
)/
2
]
s
ss
s
Qk
k
k
k
(
9
)
And it i
s
the
set of
sm
ooth
functio
n
. Thi
s
step allo
ws
ea
ch su
b-ba
nd wa
s smo
o
t
hed
by
wind
ow fun
c
ti
on blo
ck.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
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046
TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4438 – 4
443
4440
Ridg
elet de
compo
s
ition [
9
]. Each
sub
-
ban
d smoot
h partition
of
the su
b-blo
ck i
s
fo
r
Ridg
elet tran
sform.
Figure 1. Curvelet Tran
sform Flow Chat
3. Curv
elet Transform
Used
in Image Noise-remov
i
n
g
The metho
d
of Windo
w Shrin
k
-im
age
-removing noi
se is very imp
o
rtant in the theori
e
s of
wavelet. T
h
ro
ugh th
e a
dap
tive pro
c
e
s
si
ng to
the
par
ameter of
wa
velet, it ca
n a
c
hieve
the
go
al
of removing n
o
ise. Thi
s
arti
cle ap
plie
s the theor
y in Curvelet pa
ram
e
ter processi
ng for removi
ng
noise.
Set
,
ij
d
is the
p
a
ram
e
ter whi
c
h i
s
from
curvelet-tran
s
forme
d
n
o
ise-image,
ch
oo
se a
,
ij
d
cente
r
ed
wi
ndo
w of n
×
n
as the
pro
c
essing
su
bje
c
t. Figure 2 i
ndicate the
wind
owShi
n
k
w
h
en
n is 3
.
Ea
c
h
o
f
,
ij
d
following
the
pro
c
essing
bel
ow
(If
,
ij
d
is in th
e e
dge
of the
pa
ramete
r-
matrix, then ignore it):
Figure 2. Illustration of the Neig
hbo
rh
o
o
d
Wind
ow a
n
d
Curvel
et Co
efficient
B
l
oc
k
i
m
age
nn
2D
FFT
Or
ig
in
a
l
im
a
g
e
T
o
c
a
r
r
y ou
t
s
u
b-
ba
n
d
f
i
l
t
e
r
F
o
r
t
h
e
s
m
oot
h
c
o
nd
uc
t
o
f
e
a
c
h
s
u
b-
ba
n
d
bl
o
c
k
Ea
c
h
s
u
b
-
z
o
n
e
o
f
ea
c
h
b
l
o
c
k
t
o
c
a
r
r
y ou
t
an
a
l
ys
i
s
o
f
t
h
e R
i
d
g
el
et
Ra
d
o
n
f
a
ct
o
r
R
i
d
g
el
et
f
a
c
t
o
r
O
n
e
-
d
i
me
ns
i
o
na
l
in
v
e
r
s
e
FF
T
On
e
-
d
i
m
e
n
s
ion
a
l
WT
2
nn
2
nn
Po
i
n
t
o
f
vi
e
w
Fr
e
q
u
e
n
c
y
w
i
nd
o
w
s
h
r
i
nk
T
h
e
cu
rv
e
l
e
t
c
o
e
ffi
c
i
e
n
ts
to
b
e
th
r
e
s
h
o
l
d
e
d
33
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Und
e
rg
ro
und
Im
age Denoi
sing (Zha
ng
Ye)
4441
Firstly, we ca
n get the sum
of all the parameter’
s
squ
a
re in the n
×
n-wi
ndo
w [12
]
.
(1
)
/
2
(
1
)
/
2
22
,,
(1
)
/
2
(
1
)
/
2
in
j
n
ij
p
q
pi
n
q
j
n
Sd
(
1
0
)
,0
0,
0
XX
X
X
(
1
1
)
Set Symbolic function:
22
2l
o
g
n
.
(12)
is the varia
n
c
e of Ga
ussi
an white
noi
se
in the im
age, then
sh
rinki
n
g
-
proce
ssi
ng
para
m
eter i
s
:
22
,,
1/
ij
ij
S
(
1
3
)
After removin
g
noise, the param
eter can
be cal
c
ulate
d
as:
'
,,
,
ij
ij
ij
dd
(
1
4
)
The metho
d
of Windo
wShi
nk de
pen
ds
o
n
the wi
n
d
o
w
’s scale. If the
windo
w i
s
too small,
it fails to re
move noi
se.
While if th
e
wind
ow i
s
to
o big, the
n
it
ca
uses
re
co
nstru
c
ted
im
age
distorted. Usually the wi
ndow will be set
at 3×3
，
5
×
5,
or 7×7 scale.
3.1. Summarize of Ba
y
esShrink-remo
v
i
ng Noise Metho
d
We set
2
D
as the varian
ce of an image
con
t
aining noi
se,
2
is the varian
ce of noi
se,
and
2
X
is the Original imag
e’s variance. We
also that noi
se varia
n
ce:
,
0
.
67
45
ij
M
ed
ia
n
d
(
1
5
)
,
ij
M
ed
i
a
n
d
is media
n
in
the param
eters of
the low-f
r
eq
uen
cy
sub-ban
d after the
transfo
rmatio
n . We can g
e
t the variance of a
noise
containe
d M×N image th
ro
ugh the form
ula:
22
,
,
1
D
ij
ij
d
MN
(
1
6
)
The varia
n
ce of original im
age:
22
XD
(
1
7
)
Setting Thre
shold is
2
X
, then begin the p
r
o
c
e
ssi
ng of re
moving noi
se
[9].
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ISSN: 23
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046
TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4438 – 4
443
4442
3.2. Images Denoising b
y
the Combination of
Wi
ndo
w
S
hrink
and Bay
esShrink
Thoug
h the
Wind
ow Sh
ri
nk the
o
ry h
a
ve sh
rin
k
fa
cto
r
s to
dispo
s
e
the co
efficien
t to clean
noise u
s
ing
a ada
ptive way, the theory have a
di
sa
dvantage th
at the noi
se m
u
st be the
Ga
u
s
s
Noi
s
e a
nd
we
sho
u
ld
kn
ow the Vari
an
ce
first
[11]. Thi
s
ma
ke
s the
method
have
some
limits.T
he
BayesShri
n
k theory ca
n concl
ude
the Varian
ce
by
t
he tran
sform
ed fa
ctors, b
u
t it u
s
e
s
a
v
e
ry
simple
ha
rd t
h
re
shol
d valu
e method to
clea
n noi
se,
whi
c
h me
an
s it use
s
the
same threshol
d to
deal with all factors, and it can’t deal with the
noise very well [10]. In order to overcom
e
the
disa
dvantag
e
s
of the two
method
s ab
o
v
e, we
acce
p
t
both the adv
antage
s of Windo
wShrin
k
and
BayesShri
n
k
to filter the noise.
Firs
tly, we estimate Varianc
e
2
X
of the orig
inal pictu
r
e u
s
ing Baye
sS
hrin
k theo
ry, then
we calculate
usin
g
2
X
instea
d of the noise Varian
ce
2
，
su
ch a
s
:
22
2l
o
g
X
n
(
1
8
)
A
t
last
we ca
n get
sh
rink f
a
ct
or
s
,
ij
and figure o
u
t the noise co
effici
ent by taking
advantag
e of
,
ij
.
4. Simulation
We
ca
n u
s
e
the meth
od
d
e
scrib
ed i
n
th
e pa
per to di
spo
s
e
imag
e
Lena
an
d Pe
pper to
see
the validi
t
y of the met
hod, an
d
we
can
also
co
mpared the
result
of it to
the re
sult
of
the
wavelet m
e
th
od. We a
dopt
a 5/3
Double
Qua
d
ratu
re
Wavelet Filter
and
5
×
5 win
dows of
Wi
nd
ow
Shrin
k
filter. The re
sult a
s
the followin
g
table:
Table 1. The
Re
sult of PSNR of Di
sp
osi
ng Vario
u
s
Noise
s
in Different Ways
Transform
metho
d
s
=20
=30
Wavelet
transform
(PSNR)
Window
Sh
rink 27.0654
26.5289
Ba
y
e
sShrink
28.3811
27.6625
Proposed metho
d
29.6792
29.0285
Curvelet
Transform
(PSNR)
Window
Sh
rink 27.8318
26.9634
Ba
y
e
sShrink 28.9546
28.1325
Proposed metho
d
30.0965
29.5694
From Ta
ble 1, we can
se
e that the result
s of the method de
scribed in the
article a
r
e
better than
result
s of Wi
ndo
wShri
n
k
and Baye
sS
hrin
k. Thou
g
h
Wind
owS
h
rink
Theo
ry
have
adopt
ada
ptive way to
rem
o
ve noi
se, it
only con
s
ide
r
ed the
trait
of noi
se
rath
er
than that
of t
h
e
image to calculate and di
spose the co
e
fficient by
noise Vari
an
ce; it can’t remo
ve the noise
of
each ima
ge
of their o
w
n t
r
ait. BayesSh
r
ink ma
ke
use of the Va
ri
ance of the i
m
age, b
u
t it use
s
the same h
a
rd threshold t
o
the transfo
rm coeffici
e
n
ts to remove
noise and it can’t deal with
th
e
transfo
rm co
efficient ba
se
d on their different fre
quen
cy band coefficient, whi
c
h
can lea
d
to key
informatio
n coefficient lo
st
in sm
all a
r
e
a
and
ma
ke i
m
age
disto
r
tion. The
meth
od de
scrib
e
d
in
the pap
er u
s
e the Va
rian
ce
of ori
g
inal
image
to
cal
c
ulate,
but th
e Varia
n
ce o
f
origin
al ima
ge
come
s from t
he estimatio
n
of Variance
of noi
se cont
ained ima
ge and the Vari
ance of noise
image. It include
s both the
information o
f
the image and noise, so i
t
can rea
c
h a
better result by
usin
g ad
aptive thre
sh
old t
o
rem
o
ve noi
se. We
can
a
l
so
see th
at
whi
c
heve
r
m
e
thod
we
cho
s
e,
the PSNR of
Cu
rvelet tra
n
sform
a
r
e
h
i
gher than
Wavelet tran
sf
orm. Thi
s
i
s
partly be
ca
u
s
e
Curvel
et tran
sform n
o
t onl
y inherit the multiple
re
sol
v
ing rate and
dot odd ch
aracter
of Wav
e
let
transfo
rm, bu
t also represent the line a
nd plan
e, so
i
t
can do
clo
s
er to the ima
ge. The Fig
u
re
3
have sh
own
different met
hod
s in
=20. We can see that the re
sult of noise rem
o
ving are a
s
the same a
s
what we sai
d
above.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Und
e
rg
ro
und
Im
age Denoi
sing (Zha
ng
Ye)
4443
Figure 3. Disposed Image
s
5. Conclusio
n
In this pape
r, a new met
hod of co
mb
ination of the Wind
owSh
rink
and Bay
e
sSh
r
in
k
based o
n
Curvelet transfo
rm is u
s
ed to
remove noi
se
from imag
e. By experime
n
t, we can
se
e it
has b
e
tter P
S
NR a
nd we
can al
so g
e
t
a more di
stinct imag
e. So the imag
e we get by
our
method is b
e
tter and that o
f
the traditional Wavelet m
e
thod
s.
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wavelet
curvelet
W
i
nd
ow
shr
i
nk
Bayes sh
r
i
n
k
sPro
m
o
ted
m
e
t
h
od
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