Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
5, N
o
. 5
,
O
c
tob
e
r
201
5, p
p
. 1
018
~102
6
I
S
SN
: 208
8-8
7
0
8
1
018
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Wavelet and FFT Based Image
Denoising Using Non-Linear
Filters
S. Go
pinat
h
an*
,
R. Ko
kila*
,
P. Tha
n
ga
vel*
*
Department o
f
C
o
mputer Scien
c
e, University
of
Madras, Ch
epau
k, Chenn
a
i - 600
005, India
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Feb 13, 2015
Rev
i
sed
Jun
28,
201
5
Accepte
d
J
u
l 14, 2015
We propose a station
a
r
y
and dis
c
rete
wavelet based imag
e denoising scheme
and an FFT based image denoising scheme
to remove Gaussian noise. In th
e
first
appro
ach
, high
subbands are adde
d
with
each o
t
her
an
d then
soft
thresholding is
perform
ed. The
sum
of lo
w subbands is filter
e
d
with either
piec
ewise l
i
ne
ar
(PW
L
) or Lagr
a
nge or splin
e
int
e
rpola
t
ed PW
L f
ilter
.
In
the
second appro
a
ch, FFT is
emplo
y
ed
on
the n
o
is
y
image
and
then
low
frequency
and high
frequen
c
y
coefficien
ts
are
s
e
parat
e
d with
a s
p
ecif
i
ed
cutoff fr
equen
c
y. Th
en th
e
inver
s
e of
low fr
equency
components
is filter
ed
with one of
the
PWL filters and
the
inv
e
rse of h
i
gh frequen
c
y
co
m
ponents is
filte
red with
soft threshold
i
ng.
The exp
e
rim
e
nt
al resul
t
s are
co
m
p
ared wit
h
Liu
and
Liu’s tensor-based diffu
sion model (TD
M
) approach.
Keyword:
Cu
b
i
c sp
lin
e i
n
terpo
l
atio
n
Fast Four
ier
tran
sfor
m
Lagra
n
ge i
n
t
e
r
pol
at
i
o
n
Piecewise line
a
r filter
Soft
t
h
re
sh
ol
di
ng
Copyright ©
201
5 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
P. T
h
a
nga
vel
,
Head
o
f
t
h
e
de
part
m
e
nt
, De
p
a
rt
m
e
nt
of C
o
m
put
er Sci
e
nc
e, U
n
i
v
e
r
si
t
y
o
f
M
a
d
r
as,
C
h
e
p
au
k,
C
h
e
n
nai
-
6
000
05
, Ind
i
a.
Em
a
il: th
an
g
a
velp
@yahoo
.com
1.
INTRODUCTION
Im
ages acqui
r
e
d by
i
m
age sens
ors m
a
y
be cor
r
u
p
t
e
d
by
noi
se
due t
o
n
o
i
s
y
sens
ors
,
f
a
ul
t
y
C
C
D
ele
m
en
ts an
d
du
st on
th
e len
s
[1-2
].
No
ise cou
l
d
d
e
grad
e the q
u
a
lity o
f
th
e i
m
ag
e an
d
also
m
i
g
h
t
resu
lt in
lo
ss
of i
m
port
a
nt
dat
a
. F
u
rt
her
m
ore, n
o
i
s
e c
a
n
be i
n
t
r
o
d
u
ced
by
t
r
a
n
s
m
i
ssi
on er
ro
rs
an
d al
s
o
by
di
f
f
ere
n
t
com
p
ressi
o
n
m
e
t
hods
. The
r
efo
r
e,
den
o
i
s
i
n
g i
s
oft
e
n vi
t
a
l
and fi
rst
st
ep
t
o
be pe
rf
o
r
m
e
d be
fo
re i
m
ag
es are
being analyze
d
. It has
rem
a
ined a
fundamental problem
for resea
r
che
r
s
because
noise
rem
oval introduce
s
artifacts and ca
uses
blurri
ng of the
im
age.
M
a
ny
im
age d
e
noi
si
ng m
e
t
hods t
h
at
ha
ve
been
p
r
o
p
o
sed
and st
udi
e
d
.
They
can
be c
l
assi
fi
ed i
n
t
o
two
g
r
ou
ps [3
]
:
th
e sp
atial do
m
a
in
filterin
g
and
th
e
tran
sfo
r
m
d
o
m
ain
filterin
g
.
In
sp
at
ial d
o
m
ain
filtering
,
sp
atial filters are app
lied
to
th
e i
m
ag
e d
a
ta in
o
r
d
e
r t
o
rem
o
v
e
th
e n
o
i
se. Th
ese
filters rem
o
v
e
n
o
i
se to
a
reasona
b
le ext
e
nt but at the cost
of
bl
u
rri
ng i
m
ages wh
i
c
h i
n
t
u
r
n
m
a
kes t
h
e ed
ges
i
n
pi
ct
ure i
nvi
si
bl
e.
Vari
o
u
s
no
n-lin
ear filters
su
ch
as weigh
t
ed
med
i
an
[4
], ran
k
cond
itio
n
e
d rank
section
[5
] and
relax
e
d
med
i
an
[6]
we
re i
n
t
r
o
duce
d
t
o
o
v
er
com
e
t
h
i
s
dra
w
bac
k
. R
u
ss
o
[7]
p
r
op
ose
d
a
n
i
m
age enha
n
c
em
ent
sy
st
em
usi
n
g
piecewise line
a
r filter
for
directional smoot
hing.
Th
a
n
gavel
and
Gopinatha
n
[8]
propose
d
a
n
image
enha
ncem
ent schem
e
using
wavelet transform and sm
ooth
approxim
a
tion of a
piecew
ise
linear (PW
L
) filter.
In
frequ
en
cy
d
o
m
ain
[9
] t
h
e rem
o
v
a
l of
n
o
i
se is
ach
i
ev
ed b
y
d
e
signin
g
a
frequ
ency d
o
m
ain
filter and
adapting a c
u
toff
fre
quency
whe
n
t
h
e noise
com
p
onents
are decorrelate
d
from
the us
eful si
gnal domain.
In
de
pen
d
e
n
t
co
m
ponent
a
n
al
y
s
i
s
m
e
t
hod
wa
s
im
pl
em
ent
e
d i
n
[1
0]
f
o
r
de
n
o
i
si
ng
of
n
o
n
G
a
ussi
an
dat
a
.
An
i
m
age res
o
l
u
t
i
o
n e
n
han
c
em
ent
t
echni
que
usi
n
g
di
s
c
ret
e
wa
vel
e
t
t
r
an
sf
orm
an
d st
at
i
o
nary
w
a
v
e
let tr
an
sfo
r
m
w
a
s pr
opo
sed
b
y
D
e
m
i
r
e
l et al. [
1
1
]
.
Ch
ouh
an
et al. [
1
2
]
pr
opo
sed w
a
v
e
let
b
a
sed b
l
i
nd
wat
e
rm
arki
ng
i
n
or
der
t
o
o
v
e
r
com
e
t
h
e p
r
o
b
l
e
m
of
fal
s
e m
a
t
c
hi
ng
of
fi
n
g
e
r
p
r
i
n
t
.
Jai
s
wal
et
al
. [
1
3]
de
n
o
i
s
ed
the im
age by using filtering
m
e
thod a
n
d t
h
en a
pplie
d
wa
velet based technique
using t
h
reshold. Cha
n
a
n
d Ma
[14
]
pro
p
o
s
ed
a bo
x co
n
s
t
r
ain
e
d m
u
ltip
licat
iv
e iterativ
e al
g
o
rith
m
fo
r box
co
nstrain
e
d
i
m
ag
e reso
lu
tion
.
A
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 5
,
N
o
. 5
,
O
c
tob
e
r
20
15
:
101
8
–
10
26
1
019
l
o
cal
and co
nt
e
x
t
u
al
co
nt
r
o
l
l
e
d (LC
C
)
f
o
urt
h
orde
r Part
i
a
l
d
i
ffere
nt
i
a
l
equa
t
i
on m
e
t
hod
fo
r t
h
e noi
se rem
oval
was p
r
ese
n
t
e
d
by
Zen
g
et
al
.
[1
5]
. La
ng et
al
. [1
6]
presented
a n
e
w nonlin
ear no
ise red
u
c
tion
m
e
th
od
th
at
u
s
es t
h
e
d
i
screte wav
e
let tran
sform
.
Clyd
e et al. [17
]
d
i
scu
s
sed
abou
t b
a
yesian
meth
od
s
for m
u
ltip
le
shri
nka
ge est
i
m
a
t
i
on i
n
wa
vel
e
t
s
. Da
b
o
v
et
al
. [1
8]
p
r
op
ose
d
a
n
i
m
age
den
o
i
s
i
n
g
st
rat
e
gy
ba
se
d o
n
a
n
enha
nce
d
s
p
ar
se rep
r
ese
n
t
a
t
i
on i
n
t
r
a
n
sf
o
r
m
dom
ai
n. Y
a
ng et
al
.
[
1
9
]
pro
p
o
se
d an
im
age enha
n
c
em
ent
app
r
oach
usi
n
g wa
vel
e
t
t
r
an
sfo
r
m
and Haa
r
t
r
ans
f
orm
for
m
e
di
cal
im
age den
o
i
s
i
n
g.
D
i
ffere
nt
t
h
res
h
ol
di
n
g
proce
d
ures we
re
analysed by Lázaro
et al.
[20]
using t
h
e
discrete wa
ve
let transform
and dec
o
m
position level
depe
ndent thre
shol
d. Donoho [21] propo
se
d a soft thres
hol
ding m
e
thod
for
de
noising an im
age. Kha
r
e et al.
[22]
proposed a
m
e
thod for
denoisi
ng m
e
dical im
ages us
ing s
o
ft threshol
ding in
wa
velet dom
ain on m
u
ltiple
l
e
vel
s
. L
u
i
s
i
e
r
et
al
. [
23]
i
n
t
r
o
duce
d
a
ne
w a
p
p
r
oach
base
d
on
i
n
t
e
rscal
e
o
r
t
h
on
orm
a
l
wavel
e
t
t
h
res
h
ol
d
i
ng.
Lu et al.
[24] inve
stigated a
n
adaptive
sc
he
me
based o
n
t
h
e no
nl
ocal
m
eans (NL
-
m
e
a
n
s) algorithm
fo
r m
e
di
cal
im
age de
noi
si
ng
.
Shan
g an
d H
u
an
g [
2
5]
pr
op
ose
d
a
m
e
t
hod
for
den
o
i
s
i
n
g
usi
n
g ext
e
nde
d
no
n-
negat
i
v
e
spa
r
s
e
codi
ng
ne
ura
l
net
w
o
r
k sh
ri
nka
ge al
g
o
r
i
t
h
m
.
Adl
e
r et
al
. [2
6]
pr
o
p
o
s
ed
a new a
p
pr
oa
ch t
h
at
opt
i
m
i
zes t
h
e sha
p
e o
f
t
h
e s
h
ri
nka
ge
fu
nct
i
ons a
n
d m
a
xi
m
i
zes den
o
i
s
i
n
g
per
f
o
r
m
a
nce by
em
phasi
zi
ng
t
h
e
cont
ri
b
u
t
i
o
n
o
f
spa
r
se
ove
rc
o
m
pl
et
e repre
s
e
n
t
a
t
i
ons
. Li
u
a
n
d
Li
u
[
1
]
ha
v
e
pr
o
p
o
s
ed
i
m
age
den
o
i
s
i
n
g
base
d
on
di
ff
usi
o
n t
e
nso
r
s.
Ji
an
wei
[2
8]
p
r
o
p
o
se
d
an al
g
o
r
i
t
h
m
based
o
n
pi
xel
pr
ocessi
ng
f
o
r
t
h
e n
o
i
s
e
rem
oval
o
f
col
o
r im
ages. Jai
n
and Ty
a
g
i
[2
9]
pr
op
ose
d
l
o
cal
l
y
adapt
i
v
e pat
c
h base
d (
L
AP
D) m
e
t
hod w
h
i
c
h re
duc
es t
h
e
noi
se
w
h
i
l
e
pr
eservi
ng
t
h
e
r
e
l
eavant
features of t
h
e im
age.
In
sp
ite of t
h
ese m
e
th
o
d
s, th
e
p
r
ob
lem
attracts
many researchers beca
use of
its practical importance a
nd e
v
en a sm
all improvem
ent in results would y
i
eld a
hi
g
h
or
der
o
f
si
gni
fi
ca
nce.
S
o
we
have
co
nsi
d
e
r
ed
t
h
e
p
r
o
b
l
e
m
i
n
gen
e
ral
n
o
t
s
p
eci
f
i
c t
o
any
part
i
c
ul
ar
ap
p
lication
.
In
t
h
i
s
pa
per
we
pr
o
pose
t
w
o a
p
p
r
oache
s
,
t
o
rem
ove
n
o
i
s
e fr
om
an i
m
age.
The
fi
r
s
t
a
p
p
r
oach
use
s
stationary
wa
v
e
let transfo
r
m
(S
WT) an
d dis
c
rete wave
let tran
sfo
r
m
(DWT) wh
ere
as the second approach
uses
fast
F
o
ur
i
e
r t
r
an
sf
orm
(FFT
).
The
p
r
op
ose
d
sc
hem
e
s use
P
W
L
or
i
t
s
sm
oot
h
ap
pr
o
x
i
m
at
i
on
fo
r
sm
oot
hi
ng sl
o
w
l
y
vary
i
n
g co
m
ponent
s a
nd
soft
t
h
res
hol
di
ng
fo
r s
h
ar
pe
n
i
ng e
dge
rel
a
t
e
d pi
xel
s
.
W
e
d
i
scuss
ab
ou
t p
i
ecewise lin
ear filter an
d
its sm
o
o
t
h
ap
pro
x
i
m
a
tio
n
s
in
Sectio
n
2
.
Th
e so
ft thresh
o
l
d
i
ng
techn
i
q
u
e
is
prese
n
t
e
d
i
n
Se
ct
i
on
3.
W
e
de
scri
be
i
m
age denoi
si
ng
ap
p
r
o
ach
usi
n
g
S
W
T an
d
D
W
T i
n
Sect
i
o
n
4
.
F
F
T
base
d
im
age denoi
si
ng sc
hem
e
i
s
prese
n
t
e
d i
n
S
ect
i
on 5. E
xpe
ri
m
e
nt
al resul
t
s
are di
scusse
d
i
n
Sect
i
on 6.
Fi
nal
l
y
the conclusions are
drawn i
n
Section
7.
2.
PIECEWISE
LINEAR FIL
TER AND
IT
S S
M
OOT
H
APP
R
O
X
I
M
ATIO
NS
In
th
is section
,
we rev
i
ew PWL filter propo
sed
b
y
Ru
sso [7
] for sm
o
o
t
h
i
ng
no
n
ed
g
e
p
i
x
e
ls. Th
e
pr
ocessi
ng
dea
l
s
wi
t
h
fo
u
r
di
f
f
ere
n
t
s
ubset
o
f
pi
xel
s
W
1
,
W
2
,
W
3
and
W
4
.
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
(a)
(b
)
(c)
(d
)
Figure
1. (a
) Pi
ecewise linea
r
(PWL) filter; (b) Lagra
nge
interpolated PWL filter;
(c
) Spl
i
ne interpolated
PW
L filter; (d
) Soft thresh
o
l
d
fun
c
tion
for
T
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
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8-8
7
0
8
Wa
velet and
FFT
b
a
s
ed
i
m
age d
e
n
o
i
si
n
g
u
s
i
n
g non
-lin
ea
r f
ilters (P. Tha
n
g
a
vel)
1
020
The output
,
o
f
th
is filter is
g
i
ven
b
y
th
e
relatio
n
,
,
1
4
,
,
,
,
,
,
,
(1
)
whe
r
e
,
,
,
,
,
,
i
s
t
h
e
f
unct
i
o
n
o
f
t
h
e
t
h
re
e
param
e
ter se
ve
n se
gm
ent
P
W
L
f
u
nc
t
i
on s
h
ow
n
i
n
Fi
gu
re
1(a
)
a
n
d t
h
e
set
W
p
is
selected
su
ch
t
h
at th
e
fo
llo
wi
n
g
qu
an
tity is min
i
mized
.
min
1,2
,
3,4
|
,
,
,
,
,
,
,
|
(2
)
Based
on
th
e filterin
g
sch
e
me d
e
fin
e
d
b
y
(1
) and
(2
) on
ly
o
n
e
p
i
x
e
l su
b
s
et is se
lected
. Th
e filtering
ope
rat
i
o
n i
s
p
e
rf
orm
e
d by
r
e
st
ori
n
g t
o
t
h
e
m
i
nim
u
m
op
erat
or
. To
fu
rt
her r
e
d
u
ce t
h
e
noi
se, a
not
he
r pi
xel
sub
s
et
i
s
sel
ect
ed,
by
rest
o
r
i
n
g t
o
t
h
e m
a
xi
m
u
m
operat
o
r
w
h
i
c
h i
s
de
fi
ne
d
by
(
3
)
an
d
(
4
).
,
,
1
4
,
,
,
,
,
,
,
(3
)
max
1,2
,
3,4
|
,
,
,
,
,
,
,
|
(4
)
Ru
sso
[7
] d
e
sig
n
e
d
th
e PWL
filter b
a
sed
on
th
e rang
e
o
f
lumin
a
n
ce u
s
ing
th
e p
a
ram
e
ters
a
;
b
and
c
.
Th
e leng
th
s
of all seg
m
en
ts
o
f
th
e PWL
fun
c
tion
are co
n
t
ro
lled
b
y
param
e
ter
a
. Fo
r sm
all
lu
m
i
n
a
n
c
e
diffe
re
nces
|
|
wh
ich
are in
terp
reted
as no
ise, th
e filter sho
u
l
d
p
e
rform
stron
g
sm
o
o
t
hin
g
.
A
g
r
adu
a
l tran
sitio
n is
req
u
i
red
wh
en th
e lu
m
i
n
a
n
c
e
d
i
fferences are m
e
d
i
u
m
|
|
an
d
m
e
di
um
-
larg
e
|
|
. Su
ch
tran
sitio
n can
be ach
iev
e
d b
y
u
s
ing
t
h
e trap
ezo
i
d-sh
ap
ed
fun
c
tio
n so
th
at
the filter
ca
n perform
wea
k
sm
oothing.
Sm
oot
hi
ng is
not
necessa
ry for l
a
rge
lum
i
nanc
e
|
|
. T
h
e
ran
g
e f
o
r
the p
a
ram
e
ters
b
and
c
are
1
b
2
a
n
d 2
c
5 resp
ect
i
v
el
y
.
The
param
e
t
e
r
a de
pen
d
s
on
t
h
e
noi
se
va
ri
ance
.
He
re we have
use
d
b
=
1.
5 a
n
d
c
= 5.
2.
1. L
a
gra
n
ge
Interp
ol
a
t
i
o
n
Let
x
0
,
x
1
;
x
2
, …,
xn
be
n
+1 d
i
stin
ct po
in
ts o
n
t
h
e real axis an
d
let f
(
x
) be a real
valued function
defi
ned
o
n
so
m
e
i
n
t
e
rval
I
= [
a
,
b
]
co
nt
ai
ni
ng t
h
ese
poi
nt
s
.
W
e
ca
n c
onst
r
uct
a
pol
y
n
o
m
i
al
p
(
x
) of de
gr
ee
n
wh
ich
in
terpo
l
ates
f
(
x
) at t
h
e
n
+1
di
st
i
n
ct
p
o
i
nt
s
x
0
,
x
1
,
x
2
, …,
x
n
[27
]
as fo
llo
ws:
⋯
(
5
)
(6
)
Usi
n
g (
5
) t
h
e P
W
L f
u
nct
i
o
n i
s
sm
oot
hed f
o
r t
h
e pa
ram
e
t
e
rs
b
= 1
.
5 a
n
d
c
=
5 an
d i
s
s
h
ow
n i
n
Fi
gu
r
e
1(
b)
. T
h
e sm
oo
t
h
ed
ve
rsi
o
n
of
P
W
L f
u
nct
i
o
n
i
s
use
d
t
o
sm
oot
h
n
o
n
ed
ge
p
i
xel
s
o
f
t
h
e
i
n
p
u
t
i
m
age.
2.
2.
Cu
bi
c Spl
i
n
e Inter
p
ol
ati
o
n
We can al
so
obt
ai
n t
h
e sm
oot
h ap
pr
o
x
i
m
at
i
on o
f
pi
ece
wi
se l
i
n
ear fu
nct
i
on
usi
n
g cubi
c spl
i
ne
in
terpo
l
atio
n.
Fo
r th
e abov
e selected
p
a
ram
e
ters
the i
n
terpolated curve
is
sh
own
in Figur
e 1(
c)
.
3.
SOFT TH
RE
SHOL
DIN
G
A s
o
ft
t
h
res
h
ol
d i
s
a
p
r
e
p
r
o
ce
ssi
ng
t
o
ol
t
h
at
red
u
ces
t
h
e
ba
ckground in an im
age, s
o
that
the
pixel
s
with
in
ten
s
ity v
a
lu
es
b
e
low t
h
e thresho
l
d
valu
e are
redu
c
e
d.
Du
ri
n
g
vi
s
u
al
i
zat
i
on, t
h
e
s
e t
h
res
h
ol
de
d
pi
xel
s
becom
e
m
o
re
trans
p
are
n
t.
Soft t
h
res
h
old sets zero
th
e ele
m
en
ts who
s
e ab
so
lu
te
v
a
lu
es are l
o
wer t
h
an
thres
hol
d, t
h
en
sh
rin
k
s t
h
e
oth
e
r c
o
-e
fficients
towa
r
d
s ze
ro
.
The s
o
ft
t
h
res
hol
d
was
pr
o
pos
ed
by
Do
n
o
h
o
[2
1]
. Lat
e
r Ya
n
g
et
al
. [
19]
use
d
t
h
e fol
l
owi
n
g
th
resh
o
l
d
v
a
lu
e to
calcu
late t
h
e soft thresho
l
d
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 5
,
N
o
. 5
,
O
c
tob
e
r
20
15
:
101
8
–
10
26
1
021
2log
.
(7
)
1
.
,
̅
(8
)
Fi
gu
re
2.
B
l
oc
k
di
ag
ram
of i
m
age de
noi
si
n
g
usi
n
g
S
W
T
a
n
d
D
W
T
sche
m
e
.
whe
r
e
W and
H are t
h
e wi
dth and
height of
S
th
s
u
bba
n
d
a
nd
̅
is th
e m
e
a
n
of
S
th
su
b
b
an
d. We have us
ed
t
h
e
abo
v
e f
o
rm
ul
a t
o
com
put
e sof
t
t
h
resh
ol
d i
n
o
u
r ex
pe
ri
m
e
nt
s. Soft
t
h
resh
ol
d i
s
used t
o
re
m
ove t
h
e noi
se usi
n
g
soft
s
h
ri
n
k
age
rul
e
w
h
i
c
h
i
s
g
i
ven by
,
i
f
T
0,
i
f
|
|
T
i
f
T
(9
)
For
e
x
am
pl
e, t
h
e
gra
p
h f
o
r t
h
resh
ol
d
f
u
nct
i
o
n
wi
t
h
T
is illu
strated
i
n
Fi
g
u
re
1
(
d
)
.
4.
IMA
G
E DEN
O
ISI
N
G USI
N
G SWT AN
D DWT
We co
nsi
d
er i
m
ages ad
ded
wi
t
h
Ga
ussi
a
n
noi
se
fo
r di
f
f
ere
n
t
val
u
e
s
of st
a
nda
r
d
de
vi
at
i
on
(
).
Di
scret
e
wa
ve
l
e
t
t
r
ansfo
r
m
is appl
i
e
d t
o
d
ecom
pose t
h
e
noi
sy
i
m
age int
o
di
ffe
rent
s
u
b
b
a
nd i
m
ages. LH
(Low-Hi
g
h),
HL (High-Low), HH
(Hi
g
h-High) a
r
e t
h
e t
h
ree hi
gh s
u
bba
n
ds
, each
of
which c
ontains
the hi
gh
fre
que
ncy
c
o
m
p
o
n
e
n
t
s
o
f
t
h
e
i
n
p
u
t
i
m
age. LL (L
ow
-L
ow
)
sub
b
a
n
d co
nt
ai
ns t
h
e l
o
w
f
r
e
que
ncy
c
o
m
ponent
s
o
f
t
h
e inpu
t imag
e. Bicub
i
c in
terpo
l
ation
with
in
terpo
l
atio
n
factor
o
f
2 is app
lied
to
all frequ
en
cy su
bb
and
i
m
ag
es o
f
DWT. Sim
ilarly
SW
T is app
lied
to
th
e no
i
s
y
im
age, w
h
i
c
h dec
o
m
pos
es
the im
age into four
d
i
fferen
t
sub
b
an
d
s
n
a
m
e
ly
SLL, SLH, SHL
an
d
SHH.
In
this d
eco
m
p
o
s
itio
n
all th
e sub
b
an
ds will b
e
o
f
sam
e
si
ze. N
o
w
we a
d
d
t
h
e c
o
rres
p
on
di
n
g
su
b
b
an
ds
of
S
W
T a
n
d
D
W
T
wi
t
h
eac
h
ot
he
r.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Wa
velet and
FFT
b
a
s
ed
i
m
age d
e
n
o
i
si
n
g
u
s
i
n
g non
-lin
ea
r f
ilters (P. Tha
n
g
a
vel)
1
022
For
t
h
e e
s
t
i
m
a
t
e
d l
o
w f
r
e
que
ncy
su
b
b
an
d t
h
e P
W
L
o
r
La
gra
n
ge o
r
c
u
bi
c spl
i
n
e i
n
t
e
r
p
ol
at
ed
P
W
L
filter is ap
p
lied
to
i
m
p
r
ov
e th
e sm
o
o
t
h
n
e
ss an
d
rem
o
v
e
n
o
i
se. So
ft th
resho
l
d
i
ng
is u
s
ed
to
enh
a
n
ce th
e
shar
p
n
ess
of t
h
e rem
a
i
n
i
ng t
h
ree
hi
g
h
f
r
eq
uency
s
u
b
b
a
n
d im
age. Invers
e SW
T is appl
ied for the
res
u
lting
fre
que
ncy
s
u
bb
and
s
t
o
ret
r
i
e
v
e
t
h
e
den
o
i
s
e
d
im
age. The
bl
o
c
k
di
ag
ram
of t
h
e sc
hem
e
i
s
sho
w
n i
n
Fi
g
u
r
e
2.
5.
FFT BASE
D IMAGE
DENOISING
First th
e
fast
Fo
urier t
r
an
sfo
r
m
is ap
p
lied to th
e
n
o
i
se add
e
d
im
ag
e to
tran
sfo
r
m
th
e im
a
g
e
fro
m
th
e
sp
atial do
m
a
i
n
to freq
u
e
n
c
y d
o
m
ain
.
A lo
wp
ass f
ilter an
d a
h
i
ghp
ass
filter are d
e
si
g
n
e
d
.
Then
th
e
tran
sform
e
d
i
m
ag
e is filtered
with
lowp
ass fi
lter an
d
h
i
ghp
ass filter. Now th
e inv
e
rse
fast
Fou
r
ier t
r
an
sform
i
s
e
m
p
l
o
y
ed
to th
e lowp
ass
filtered
im
ag
e an
d h
i
g
h
p
a
ss
fi
ltered
im
ag
e. So
ft t
h
resho
l
d
i
n
g
is app
lied
to
th
e
in
v
e
rse
o
f
h
i
gh
p
a
ss im
ag
e in
o
r
d
e
r to en
han
ce t
h
e sh
arpn
ess
of t
h
e imag
e. Th
e PWL filter
o
r
Lag
r
an
g
e
or
sp
lin
e i
n
terpo
l
ated
PWL
filter is ap
p
lied to
th
e inv
e
rs
e of
lo
wp
ass im
ag
e, to
enh
a
n
c
e the sm
o
o
t
h
n
e
ss
o
f
t
h
e
im
age. Then t
h
e res
u
lting t
w
o im
ages
are co
m
b
in
ed
to
retriev
e
th
e d
e
no
ised im
age. The bloc
k
diagra
m
of
FFT
base
d i
m
age
den
o
i
s
i
n
g s
c
hem
e
i
s
sho
w
n i
n
Fi
g
u
r
e
3.
Fi
gu
re
3.
B
l
oc
k
di
ag
ram
of F
F
T
based
i
m
ag
e de
noi
si
ng
sc
hem
e
5.
1. L
o
w
p
a
ss
Fi
l
t
er
Th
e tran
sformed
im
ag
e in
th
e frequ
ency d
o
m
ain
n
ear th
e cen
ter
will h
a
v
e
l
o
w frequ
e
n
c
y
com
pone
nts. T
h
e lowpass
filter [2] ca
n
be
c
onst
r
ucte
d
using the
following form
ula:
,
1,
i
f
,
0,
i
f
,
(1
0)
whe
r
e D(
,
) i
s
the
distance
betwee
n the
ce
nter
of
the fre
quency rectangl
e
and
a point (
,
) i
n
t
h
e f
r
e
q
uency
dom
ai
n:
,
/
2
/
2
(
1
1
)
here
W
and
H
are t
h
e wi
dt
h a
nd
hei
g
ht
of t
h
e im
age resp
ectively. W
e
ha
ve set the cutoff freque
n
cy,
f
c
=
20
0,
as w
e
h
a
v
e
u
s
ed
im
ag
es o
f
size 51
2 x 512
.
5.
2. Hi
gh
pa
ss Fi
l
t
er
In the tra
n
sform
e
d im
age, the region nea
r
the ed
ge will have high frequency com
pone
nts than the
cen
tral
reg
i
o
n
. Figu
re
4
sh
ows t
h
e
p
i
cto
r
i
a
l repres
en
tatio
n
o
f
t
h
ese fi
lters. Th
e
h
i
gh
p
a
ss
filter can
b
e
co
nstru
c
ted
b
y
u
s
ing
t
h
eir co
rrespo
n
d
i
ng
l
o
wp
ass
filter:
,
1
,
(
1
2
)
6.
E
X
PERI
MEN
T
AL RES
U
L
T
S
The si
m
p
l
e
st
and
m
o
st
wi
del
y
use
d
per
f
o
r
m
a
n
ce m
easures are
pea
k
signal to noise
rat
i
o (
P S
N R
)
and m
ean squared
error (
M S
E
). Let
f
(
x
,
y
) an
d
,
are t
h
e refere
nce a
n
d test im
ages res
p
ectively. Le
t
e
(
x
,
y
)
be t
h
e e
r
ror signal
bet
w
een
f
(
x
,
y
) and
,
. It
i
s
c
o
m
put
ed
pi
x
e
l
-
by
-pi
x
el
by
a
d
di
n
g
up
t
h
e
s
q
uar
e
d
ifferen
ces
of al
l th
e p
i
x
e
ls an
d d
i
v
i
d
i
ng
b
y
the to
tal p
i
x
e
l cou
n
t
. If M
x
N i
s
th
e size
of the i
m
ag
e, th
en
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 5
,
N
o
. 5
,
O
c
tob
e
r
20
15
:
101
8
–
10
26
1
023
1
,
1
,
,
(1
3)
(a)
(b
)
Fig
u
re
4
.
(a) Lo
wp
ass filter; (b
) Hi
g
h
p
a
ss filter
(a)
(b
)
(c)
(d
)
(e)
(
f)
(g
)
(h
)
Fi
gu
re
5.
Im
age de
noi
si
ng
sc
hem
e
wi
t
h
= 4
0
fo
r B
a
rba
r
a i
m
age: (a)
Original
Im
age; (b) Noisy Im
age;
Enh
a
n
c
ed
im
a
g
es u
s
ing
FFT b
a
sed
sch
e
m
e
with
(c) Lagr
an
g
e
in
terpo
l
ated
PWL filter,
(d
) sp
lin
e in
terpo
l
ated
PW
L filter,
(e) PWL filter;
En
h
a
n
c
ed
im
ag
es u
s
i
n
g SWT-DWT
b
a
sed sc
h
e
m
e
with
(f)
Lag
r
ang
e
in
terp
o
l
ated
PW
L filter,
(g
) sp
lin
e in
terpo
l
at
ed
PWL
filter,
(h) PWL filter
The
P S N R
qu
ality assessmen
t m
e
tric is d
e
fin
e
d
as
fo
llo
ws:
10log
(1
4)
here
,
MA
X
I
i
s
t
h
e m
a
xim
u
m
pos
si
bl
e
pi
xel
val
u
e
o
f
t
h
e i
m
age. T
h
e
hi
g
h
er
t
h
e
P S
N
R
,
th
e c
l
o
s
er
th
e
te
s
t
i
m
ag
e is to
th
e
o
r
i
g
in
al.
To e
v
aluate
the effe
ctivenes
s
of the
propos
ed im
age
de
no
i
s
i
ng
a
p
pr
oac
h
es, we use
d
1
2
gray
scal
e
im
ages. All the test images are of size 512
x 51
2
.
Ga
ussi
a
n
n
o
i
s
e wi
t
h
st
anda
r
d
de
vi
at
i
on
(
) is ad
d
e
d
to
th
e
i
n
p
u
t
im
age. Fi
rst
,
S
W
T a
nd
D
W
T bas
e
d i
m
age denoi
sing
m
e
th
o
d
is ap
p
lied
to
th
e no
isy imag
e.
Sim
u
l
t
a
neo
u
sl
y
FFT
based
i
m
age de
noi
si
n
g
sc
hem
e
i
s
a
l
so
app
lied
t
o
th
e no
is
y image. T
o
m
easure t
h
e
perform
a
nce of each schem
e
, we use
P S N R
m
e
t
r
i
c
bet
w
een t
h
e de
noi
se
d im
ag
e and original im
age. Figure
5 di
spl
a
y
s
t
h
e resul
t
s
o
f
FFT
and S
W
T-
D
W
T base
d i
m
age de
noi
si
n
g
s
c
hem
e
s for B
a
rba
r
a im
age. The fi
rst
ro
w co
nsi
s
t
s
of
ori
g
i
n
al
im
age, n
o
i
s
y
im
age an
d res
u
l
t
s
of FF
T bas
e
d im
age den
o
i
s
i
n
g schem
e
usi
n
g
Lag
r
ang
e
in
terp
o
l
ation
,
sp
line in
terp
o
l
ation
,
PW
L filter
resp
ectiv
ely. The secon
d
row
co
n
t
ains th
e resu
lts of
SW
T-DWT
b
a
sed
im
ag
e d
e
no
ising
sch
e
m
e
u
s
ing
Lag
r
ange
in
terpo
l
atio
n, sp
lin
e in
terpo
l
atio
n
and
PWL filter
respectively.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Wa
velet and
FFT
b
a
s
ed
i
m
age d
e
n
o
i
si
n
g
u
s
i
n
g non
-lin
ea
r f
ilters (P. Tha
n
g
a
vel)
1
024
The followi
ng grayscale im
ag
es w
ith
Gau
ssi
an
no
ise o
f
sp
ecified
f
r
om
{20, 25, 30,
35,
40, 45} are
use
d
i
n
o
u
r e
x
peri
m
e
nt
s:
Ai
r
p
l
a
ne
(
I
1
), Le
n
a
(
I
2
), Go
l
d
h
ill
(
I
3
), M
ona
rch
(
I
4
), Elaine
(
I
5
), Cam
e
ram
a
n (
I
6
),
Barb
ara (
I
7
), P
e
ppe
r (
I
8
)
,
Coup
le (
I
9
),
Br
idg
e
(
I
10
), Boat (
I
11
) an
d 5.
2.
09
(
I
12
). In t
h
e Ta
bl
e
1, we ha
ve sh
ow
n P
SNR
val
u
es
f
o
r t
h
e
de
noi
se
d
im
ages usi
n
g
F
F
T an
d
S
W
T-
D
W
T b
a
sed
sc
hem
e
s. Each c
o
l
u
m
n
co
nsi
s
t
s
of
t
w
o
cel
l
s
, one
fo
r F
F
T an
d ot
her
f
o
r S
W
T
-
D
W
T
.
From
t
h
i
s
t
a
bl
e, we ca
n o
b
se
rve t
h
at
FFT m
e
t
h
o
d
pe
rf
o
r
m
s
wel
l
with
resp
ect
t
o
P S N R
m
easures.
We ha
ve al
s
o
com
p
ared S
W
T-D
W
T a
n
d F
F
T base
d i
m
age de
n
o
i
s
i
n
g s
c
hem
e
resul
t
s
of Le
na a
n
d
B
o
at
im
ages wi
t
h
Li
u a
nd
Li
u [
1
]
schem
e
of t
e
n
s
o
r
-
b
a
s
ed di
ff
usi
o
n
m
odel
(TDM
)
fo
r di
f
f
ere
n
t
val
u
e
s
.
Tabl
es
2 an
d
3,
sh
ow t
h
e
P S N
R
val
u
es
co
r
r
esp
o
nds
t
o
T
D
M
,
S
W
T
-
D
W
T a
nd
FFT
b
a
sed sc
hem
e
s. From
t
h
e
resu
lts it can
be o
b
serv
ed th
at fo
r
d
i
fferen
t
val
u
es,
FFT
base
d schem
e
per
f
o
r
m
s
bet
t
e
r t
h
a
n
t
h
e
ot
he
r t
w
o
schem
e
s.
We
al
s
o
have
use
d
di
ffe
rent
cut
o
ff fre
q
u
en
ci
es
suc
h
as
1
80;
1
90;
2
00;
21
0
a
n
d 22
0 f
o
r
FFT
ba
sed
sch
e
m
e
with
Lag
r
ang
e
in
terpo
l
ated
PWL fil
t
er with
Ga
ussi
an noise adde
d im
ages su
c
h
as
A
i
rp
la
n
e
,
Bar
b
ar
a
,
El
ai
ne,
Gol
dhi
l
l
and
Lena
wi
t
h
t
h
e
st
an
dar
d
de
vi
at
i
o
n
o
f
3
0
a
n
d
5
0
.
T
h
e
P S
N
R
v
a
l
u
es
fo
r
di
ffe
re
nt
c
u
t
o
ff
fre
que
nci
e
s ar
e sho
w
n i
n
Ta
bl
e 4 an
d Ta
b
l
e 5 fo
r
= 30,
= 50 res
p
ectively, and their corres
pondi
ng
pi
ct
ori
a
l
rep
r
es
ent
a
t
i
ons are s
h
o
w
n i
n
Fi
g
u
r
e
6. Fr
om
t
h
e pictorial repres
entations
, it ca
n be obse
rve
d
that at
cut
o
ff
fre
q
u
enc
y
20
0,
t
h
e
pe
rf
orm
a
nce o
f
t
h
e
Tabl
e
1.
P S
N
R
val
u
e
f
o
r
de
noi
se
d i
m
ages usi
n
g F
F
T a
n
d
S
W
T
-
D
W
T
m
e
t
h
o
d
.
Each
cel
l
has
P S
N
R
va
l
u
e o
f
FFT a
n
d S
W
T
-
D
W
T
den
o
i
s
i
n
g sc
hem
e
usi
n
g La
gra
n
ge i
n
t
e
rp
ol
at
i
o
n
.
Her
e
represe
n
ts t
h
e sta
nda
rd
devi
at
i
o
n
of
t
h
e Ga
ussi
an
n
o
i
s
e adde
d to Grayscale im
age.
20
25
30
35
40
45
Noisy
26.
88
24.
94
23.
36
22.
02
20.
86
19.
84
I
1
33.
08
25.
09
31.
86
25.
18
30.
99
24.
92
30.
36
25.
70
29.
33
26.
80
26.
88
26.
78
I
2
31.
08
27.
41
30.
54
26.
79
31.
15
26.
27
29.
74
27.
44
27.
05
27.
67
27.
94
26.
79
I
3
34.
04
27.
93
30.
39
27.
88
29.
80
28.
28
29.
95
29.
57
27.
27
27.
91
28.
18
28.
89
I
4
27.
38
25.
02
27.
94
26.
01
29.
21
25.
18
29.
74
26.
77
27.
60
26.
07
28.
34
26.
68
I
5
29.
70
27.
08
29.
41
26.
35
27.
83
25.
05
27.
51
25.
49
27.
51
26.
24
27.
67
27.
27
I
6
34.
43
32.
14
33.
10
32.
47
31.
15
31.
91
29.
87
30.
98
28.
86
30.
75
28.
08
30.
05
I
7
30.
76
24.
83
30.
36
25.
08
29.
02
25.
17
29.
18
25.
00
28.
64
25.
13
28.
07
25.
41
I
8
27.
37
24.
63
25.
34
24.
00
24.
36
23.
68
25.
15
23.
83
24.
27
23.
84
24.
08
22.
96
I
9
31.
13
27.
66
30.
00
27.
55
29.
15
27.
37
28.
68
27.
09
27.
84
26.
92
27.
38
26.
55
I
10
30.
28
26.
85
29.
42
26.
57
28.
57
26.
43
27.
85
26.
21
27.
10
26.
10
26.
46
25.
88
I
11
34.
43
28.
52
33.
02
28.
67
31.
45
27.
43
29.
44
26.
83
29.
82
26.
76
28.
81
26.
82
I
12
30.
76
26.
64
29.
29
26.
86
28.
83
26.
49
28.
69
25.
42
26.
42
26.
43
25.
86
26.
32
Tabl
e
2. C
o
m
p
ari
s
o
n
of
P
SN
R
val
u
es
bet
w
e
e
n T
D
M
,
S
W
T
-
D
W
T a
n
d F
F
T
base
d m
e
t
h
o
d
s
fo
r B
o
at
i
m
age
TDM
SWT
-
D
W
T
FFT
PW
L Spline
L
a
gr
ange
PW
L
Spline
L
a
gr
ange
20
27.
84
27.
750
7
27.
701
3
27.
578
2
31.
386
3
33.
048
8
34.
046
8
30
26.
16
27.
517
3
27.
525
5
27.
376
0
28.
743
9
30.
365
3
31.
424
4
50
24.
04
27.
314
2
27.
550
8
27.
504
2
26.
758
2
27.
345
4
28.
051
9
Tabl
e 3.
C
o
m
p
ari
s
o
n
of
P
S N
R
val
u
es
bet
w
een T
D
M
,
S
W
T-D
W
T
an
d
F
F
T
based
m
e
t
h
ods
f
o
r
Le
na i
m
age
TDM
SWT
-
D
W
T
FFT
PW
L Spline
L
a
gr
ange
PW
L
Spline
L
a
gr
ange
20
29.
14
26.
259
1
27.
429
8
27.
413
3
26.
753
6
30.
354
9
31.
445
2
30
27.
29
26.
670
7
26.
413
3
26.
428
8
28.
468
4
29.
350
5
29.
995
8
50
25.
15
27.
828
3
28.
462
0
28.
342
6
26.
443
2
27.
414
2
27.
881
4
Tabl
e 4.
P S N
R
v
a
l
u
es
for
d
i
fferen
t
cu
toff frequ
e
n
c
ies wit
h
FFT and
Lagrang
e
i
n
terpo
l
ated
PWL
filter
b
a
sed
sch
e
m
e
with
=
30
.
Cutoff
Airplane Barbara
Elaine
GoldHill
Lena
180
23.
947
9
23.
592
6
23.
908
4
23.
975
8
23.
984
3
190
26.
554
7
29.
299
1
27.
944
9
29.
890
0
28.
676
5
200
30.
986
6
29.
020
9
27.
833
4
29.
803
2
31.
152
7
210
26.
581
6
29.
845
7
27.
229
3
30.
496
6
28.
409
9
220
25.
940
6
29.
462
9
28.
098
6
29.
887
6
27.
715
1
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 5
,
N
o
. 5
,
O
c
tob
e
r
20
15
:
101
8
–
10
26
1
025
Tab
l
e
5
.
P SNR v
a
lu
es fo
r d
i
fferen
t
cu
toff frequ
e
n
c
ie
s wit
h
FFT and
Lagrang
e
i
n
terpo
l
ated
PWL
filter
b
a
sed
sch
e
m
e
with
=
50
.
Cutoff
Airplane Barbara
Elaine
GoldHill
Lena
180
26.
476
9
27.
580
3
26.
320
9
28.
679
2
26.
341
0
190
26.
501
1
27.
646
3
26.
693
8
28.
616
6
28.
233
0
200
26.
907
1
27.
367
4
26.
670
7
27.
751
6
27.
538
8
210
26.
451
1
27.
102
2
26.
405
2
27.
958
3
28.
059
9
220
26.
373
4
27.
439
2
25.
656
7
27.
867
3
27.
195
2
(a)
(b
)
Fi
gu
re
6.
C
u
t
o
f
f
f
r
e
que
ncy
Vs
P
S N R
v
a
l
u
es fo
r
FFT b
a
sed
sch
e
m
e
with
Lag
r
ang
e
in
terp
o
l
ated PWL
filter
(a) f
o
r
=
3
0
;
(b
)
fo
r
=
50.
FFT b
a
sed
sche
m
e
is
b
e
tter.
So
, we h
a
v
e
used
2
0
0
as
cu
to
ff frequ
e
n
c
y fo
r t
h
e Gau
ssian
filter in
all th
e FFT
base
d sc
hem
e
s.
7.
CO
NCL
USI
O
N
We
ha
ve
pres
ent
e
d
t
w
o i
m
age
de
noi
si
n
g
t
echni
que
s
fo
r
noi
se
rem
oval
.
The
ex
pe
ri
m
e
nt
al
res
u
l
t
s
sho
w
F
F
T bas
e
d i
m
age deno
i
s
i
ng sch
e
m
e
g
i
ves hi
g
h
e
r
P S N R
val
u
es t
h
an S
W
T
-
D
W
T and T
D
M
sc
hem
e
.
We ha
ve al
so
con
d
u
c
t
e
d ex
peri
m
e
nt
s on
di
ffe
re
nt
cut
o
f
f
fre
q
u
enci
es
f
o
r t
h
e F
F
T ba
sed i
m
age den
o
i
s
i
n
g
sch
e
m
e
with
Lag
r
ang
e
in
terp
o
l
ation
b
a
sed app
r
o
x
i
m
a
ti
o
n
of PWL
filter. Th
is ap
pr
oach
g
i
v
e
s b
e
tter
resu
lt
th
an
t
h
e PWL filter an
d sp
li
n
e
app
r
ox
im
a
tio
n
of PWL
filter.
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BIOGRAP
HI
ES OF
AUTH
ORS
S.
Gopinathan
rec
e
iv
ed his
M
.
S
c
and P
h
.D.
from
Bharath
i
a
r
Univers
i
t
y
an
d Univers
i
t
y
of
Madras respe
c
ti
vel
y
.
Area
of sc
i
e
ntifi
c
inte
rest:
d
i
gita
l
im
age pro
c
essing.
R. Kokila
received her
M.Sc
and M.Phil from
Un
iversity
o
f
Madras during
2005 and 2006
respect
ivel
y.
Sh
e is pursing
Ph.
D
at Univ
ersit
y
of Madras
. Ar
ea of
sci
e
ntif
ic
inter
e
st: d
i
git
a
l
im
age proc
es
s
i
n
g
and
art
i
ficial n
e
ural networks.
P.
Thangavel
received h
i
s M.Tech and
Ph.D. from
Indian
Institut
e
of
T
echnolog
y
and
Bharath
i
das
a
n
Univers
i
t
y
r
e
s
p
e
c
tiv
el
y.
He
is
c
u
rrentl
y
work
in
g as
P
r
ofes
s
o
r a
nd Head of
th
e
Department of
Computer Scien
ce,
Universit
y
o
f
Madras. Ar
ea
of scien
tifi
c
in
te
rest:
algori
t
hm
s
and ar
tifi
c
i
a
l s
y
s
t
em
s.
Evaluation Warning : The document was created with Spire.PDF for Python.