TELKOM
NIKA
, Vol. 13, No. 4, Dece
mb
er 201
5, pp. 1330
~1
336
ISSN: 1693-6
930,
accredited
A
by DIKTI, De
cree No: 58/DIK
T
I/Kep/2013
DOI
:
10.12928/TELKOMNIKA.v13i4.1901
1330
Re
cei
v
ed
De
cem
ber 2
3, 2014; Re
vi
sed
Jan
uar
y 29, 2
015; Accepte
d
March 12, 2
015
A
Sparse Representation
Image
Denoising Method
Based on Orthogonal Matching Pursuit
Xiaojun
Yu
1
,
Def
a
Hu*
2
1
School of Co
mputer Eng
i
n
e
e
rin
g
, Jiangs
u
Univers
i
t
y
of
T
e
chn
o
lo
g
y
, Ch
angz
ho
u 213
0
01, Jian
gsu, C
h
in
a
2
School of Co
mputer an
d Informatio
n
Engi
n
eeri
ng,
Hun
an
Univers
i
t
y
of Commerce, Ch
a
ngsh
a
41
020
5,
Hun
an, Chi
n
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: hdf666
@1
63.
com
A
b
st
r
a
ct
Imag
e
den
oisi
ng is
a
n
i
m
p
o
rtant res
earc
h
as
pec
t i
n
t
he fi
eld
of d
i
gital
i
m
a
ge
pr
ocessi
ng,
ands
parse re
p
r
esentati
on th
eory is al
so
one of the re
search focus
e
s in recent years. T
he spa
r
se
repres
entati
o
n
of the
i
m
a
ge
can
better extr
act the
nature
of the
i
m
ag
e,
an
d us
e a
w
a
y as c
onc
ise
as
p
o
ssib
l
e to
e
x
pre
ss th
e im
ag
e. In
im
ag
e de
no
i
s
i
n
g
b
a
s
ed
o
n
sp
arse
repr
e
s
entatio
n, the
useful
inf
o
rmat
i
on
of the i
m
age
p
o
ssess certa
i
n
structural fe
atu
r
es, w
h
ich
mat
c
h the
ato
m
str
u
cture.
How
e
v
e
r, nois
e
d
oes
not
possess s
u
ch
prop
erty, there
f
ore, sparse r
e
prese
n
tati
o
n
c
an effective
l
y
separ
at
e the
u
s
eful i
n
for
m
ati
on
from no
ise to a
c
hiev
e the pur
pose of de
nois
i
ng. Aimin
g
at ima
ge d
eno
isi
n
g prob
le
m of lo
w
signal-to-n
o
i
s
e
ratio (S
NR)
i
m
age, c
o
mb
ine
d
w
i
th Orthog
o
nal
Matchi
ng
Pursuit
an
d sp
arse r
epres
ent
ation
the
o
ry, this
pap
er puts for
w
ard an i
m
ag
e
deno
isin
g met
hod. T
he ex
pe
riment show
s that co
mpar
ed
w
i
th the traditiona
l
imag
e den
ois
i
ng bas
ed o
n
Syml
ets, ima
g
e
den
oisi
ng
b
a
sed o
n
Co
ntourl
e
ttransfor
m
, this metho
d
ca
n
del
ete no
ise in
low
SNR imag
e and ke
ep the
useful in
for
m
a
t
ion in the
orig
i
nal i
m
age
mor
e
efficiently.
Ke
y
w
ords
: Image D
e
n
o
isi
ng,
OMP (Orthogo
nal Matc
h
i
ng P
u
rsuit), Sparse
Repr
esentati
o
n
Copy
right
©
2015 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1.
Introduc
tion
Whe
n pe
ople
re
ceiving
out
side i
nform
ation, 80%
a
r
e
visual info
rm
ation. Digital i
m
age i
s
the majo
r source
of visual informati
on. Ho
weve
r, in the me
antime, while we
re
ceivi
ng
informatio
n o
f
the image, t
here
are inev
itably in
terference of both
internal fa
cto
r
s
and exte
rn
al
factors, whi
c
h make
s the
image co
ntain many
no
ise
s
and ma
ke
s the re
ce
ived informat
ion
incom
p
lete
o
r
even i
n
correct [1]. Noi
s
e co
mmonly
refers to
th
e usele
s
s inf
o
rmatio
n. In
th
e
pro
c
e
ss
of image p
r
o
c
e
s
sing, it is req
uired to e
ffe
ctively restrict
noise and im
prove the q
u
ality
and visu
al effect of the im
age, whi
c
h
can not onl
y i
m
prove
co
rre
ct judgm
ent for the ima
ge,
but
also i
s
very meanin
g
ful for the after-pro
c
essing of the
image [2].
Ho
w to d
eno
ising th
e ima
ge is on
e of
the
research focuse
s i
n
re
cent yea
r
s. Signal
pro
c
e
ssi
ng m
e
thod
cha
nge
s from
orth
og
onal tran
sform to wavel
e
t transfo
rm, th
en to multi-scale
transfo
rm. In
re
cent ye
ars, alo
ng
with
the
d
e
velop
m
ent
of com
p
re
ssed se
n
s
ing
technolo
g
y,
spa
r
se re
pre
s
entatio
n the
ory ha
s be
come a ne
w
resea
r
ch dire
ction in the
field of image
denoi
sing. Sp
arse mo
del
refers to de
scribe the exi
s
t
sign
al only
wi
th very little linear set in ba
sic
diction
a
ry [3]. It is well kn
own that ordi
nary im
ag
es
can b
e sp
arse rep
r
e
s
entat
ion in ce
rtain
transfo
rm do
main, thus to transfe
r the image to th
is tran
sform d
o
m
ain. And the fortunate thi
ng is
that noise
ca
nnot be
sparse
rep
r
e
s
ent
ed in tran
sform domai
n. Base
d on
this
premi
s
e,
spa
r
se
rep
r
e
s
entatio
n theo
ry
can
effectively d
e
lete the
noi
se i
n
th
e ima
ge. Th
e im
ag
e ove
r
com
p
let
e
sign
al sp
arse
representati
on t
heo
ry is first put forwa
r
d by Mallat in1993, an
d th
e adopte
d
im
age
spa
r
se de
co
mpositio
n alg
o
rithm i
s
the
Matchin
g
Pursuit (MP
)
alg
o
rithm p
u
t by him. Of co
urse
,
until now thi
s
theory is
still not com
p
let
e
ly ma
ture, a
nd re
quires f
urthe
r stu
dy and di
scussi
on.
Ho
wever, it is another n
e
w
thought an
d d
i
rectio
n a
fter
previou
s
ima
ge den
oisin
g
theorie
s [4, 5].
This
pap
er
mainly ba
se
d on
sparse
rep
r
e
s
e
n
tation the
o
ry to
explore the
imag
e
denoi
sing p
r
oblem. It fir
s
t cond
uct
s
a brief introdu
ction of
noisy image and spa
r
se
rep
r
e
s
entatio
n theory, then explains th
e main
pro
c
e
ss a
nd idea
of OMP. Based on the ab
ove
mentione
d re
sea
r
ch theo
ri
es an
d tech
n
ologie
s
, it
puts forward the
desig
n pro
c
e
ss of the
spa
r
se
rep
r
e
s
entatio
n imag
e d
eno
ising
metho
d
based
on O
M
P.
The la
st p
a
rt of thi
s
p
a
p
e
r i
s
expe
rime
nt
desi
gn an
d re
sult analy
s
is.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No
. 4, Decem
b
e
r
2015 : 133
0 – 1336
1331
2.
Nois
y
Ima
ge
and Spar
se
Rep
res
en
tation
Th
eor
y
2.1.
Matrix Expres
sion
of Digital Image
Digital im
age
refe
rs to th
e sample
d
a
nd q
uantified
two
dime
nsi
onal fu
nctio
n
,
whi
c
h
sampl
es th
ro
ugh e
quidi
sta
nt recta
ngul
a
r
gri
d, and
condu
cts
equ
al interval q
u
antizatio
n to
the
scope. The d
i
gital image d
i
scusse
d in this pap
er is
li
mited to two dimen
s
ion
a
l grayscal
e sta
t
ic
image
gain
e
d
thro
ugh
sam
p
ling
and
qu
a
n
tization
(th
a
t is,
digitizatio
n
o
r
A/D) of
two
dimen
s
io
nal
contin
uou
s i
m
age. Usuall
y
a digital image is oft
en ex
pre
s
sed throu
gh two dime
n
s
ion
a
l matrix.
Sample im
age
(,
)
f
xy
,
sel
e
ct
M
N
data, se
que
nce th
ese
data
into
a m
a
trix a
c
cordin
g
to the correspondi
ng po
sition of the sa
mple dot,
an
d then quanti
f
y each array position to g
a
in a
digital matrix.
Use thi
s
mat
r
ix to re
pla
c
e
function,
(,
)
f
xy
. In other
wo
rd
s,
digital imag
e
d ca
n b
e
rep
r
e
s
ente
d
by a matrix.
The ele
m
ent
s in the
matr
i
x
are
called
the pixel
s
of the imag
e, which
can b
e de
scri
bed a
s
:
00
0
1
0
1
10
1
1
1
1
Sa
m
p
l
in
g
10
1
1
1
1
Qu
ant
i
fi
ca
t
i
on
(,
)
(
,
)
(,
)
(,
)
(
,
)
(,
)
(,
)
(,
)
(,
)
(,
)
(,
)
(
,
)
N
N
MM
M
N
l
MN
M
N
fx
y
fx
y
fx
y
fx
y
f
x
y
fx
y
fx
y
fx
y
fx
y
fx
y
fi
j
f
i
j
(1)
In which,
(,
)
l
f
ij
refe
rs to the quant
ified pixel value.
If the sampl
e
amou
nt is
M
N
, quanti
z
ation
grad
e i
s
2
n
Q
, then the bit n
e
e
ded fo
r
storin
g a digit
a
l image is:
BM
N
n
(
2
)
Usually a two
dimensi
onal
numbe
r set is used
to sto
r
e
the digital image data. Th
e scale
of the two di
mensi
onal n
u
m
ber
set eq
uals to that
of the digital image. Each eleme
n
t in the
numbe
r
set
is co
rrespon
din
g to
ea
ch
pixel in t
he digit
al
ima
ge, whi
c
h sto
r
e
s
co
rresp
ondi
ng pi
xel
grayscal
e value.
2.2.
Nois
y
M
odel
In reality, during th
e proce
s
s of dig
i
tiza
tion an
d
transfe
r, the digital ima
ge often
interfered
by the im
age
formatio
n d
e
vice
and
noi
se from
out
sid
e
environm
e
n
t, and
be
co
mes
noisy im
age.
Image
den
oi
sing
refe
rs to
deletin
g a
nd
red
uci
ng th
e
noi
se i
n the
digital im
ag
e.
Before im
age
den
oisi
ng
we ne
ed to
first build
a n
o
isy image
mod
e
l, whi
c
h
is created
by a
d
d
ing
a rand
om noi
se to the origi
nal image:
(,
)
(,
)
(,
)
px
y
f
x
y
qx
y
(
3
)
(,
)
f
xy
refers to
the
image,
(,
)
qx
y
refers
to the n
oise,
the noi
sy im
age i
s
de
scri
bed
as
(,
)
px
y
.
2.3.
Sparse
Decomp
ositi
on
Algorith
m
The sp
arse d
ecom
po
s
ition
of the signal
means that whe
n decom
posi
ng on th
e over-
compl
e
te dictionary, the basi
c
functio
n
s that
rep
r
ese
nt the signal ca
n be
selecte
d flexibly
according
to
the si
gnal’
s
chara
c
te
risti
c
s, and it r
equi
res
only a littl
e of ba
si
c fu
nction
s. In ot
her
words, the
signal can be
rep
r
e
s
ente
d by the pr
odu
ct of a set of sparse
coeff
i
cient
s and t
he
training di
ctio
nary. If there
are only a small part of
element
s in
a sign
al is n
on-ze
ro, then
the
sign
al i
s
defi
ned
as sparse. In re
al
sig
nal p
r
o
c
e
ssi
n
g, in o
r
de
r to
improve the
pro
c
e
ssi
ng
effect
and spe
ed,
such sp
arse d
ata
expressio
ns
a
r
e al
ways ne
ede
d. Repla
c
ing th
e
origin
al data
by
spa
r
se ap
pro
x
imation ca
n
not only re
d
u
ce th
e si
gn
al pro
c
e
s
sing
co
st funda
m
entally, but a
l
so
greatly impro
v
e the compa
c
t efficien
cy [6, 7].
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
A Sparse
Re
pre
s
entatio
nIm
age Denoi
si
ng Method B
ase
d on Orth
ogon
al …
(Xi
aojun Yu
)
1332
An importa
nt premi
s
e
of si
gnal
spa
r
se d
ecom
po
s
ition
is to find the
spa
r
se do
mai
n of the
sign
al, whi
c
h
dire
ctly relates
to the
recon
s
tru
c
tion accu
ra
cy
of the co
mpact p
erce
ption.
De
comp
osi
ng
the si
gnal
on
the over-co
m
plete di
ct
io
nary
can
gain
the sparse
e
x
pressio
n of t
he
origin
al sig
n
a
l
. Signal sparse expressio
n
unde
r t
he over-com
plet
e diction
a
ry is more effect
ive,
and m
atrix
combine
d fro
m
so
me type
s of
ran
dom
matrix
a
nd diction
a
rie
s
with certai
nty
ha
s
small re
stri
cted
i
s
ometry con
s
tant
s.
T
he spa
r
se
d
ecom
po
s
ition
pro
c
e
s
s of t
he
signal
ca
n be
descri
bed
a
s
: for the
D
dimensi
onal
Hi
lbert
spa
c
e
D
HR
,
offer a set
,1
,
2
,
l
Gg
l
L
,
LD
, s
e
t
G
refers to
the dictionary[8].
The redunda
ncy of the dictiona
ry
()
LD
,
make
v
e
ct
or
l
g
no long
er li
near i
nde
pen
dent. For
an
y signal
vH
with
a length of
D
, approxim
ate by
automatically sel
e
ctin
g the
be
st
m
atoms from
G
, ma
ke
the a
pp
r
oxima
t
e erro
r
2
vv
, (
refers to a su
fficiently smal
l pos
itive num
ber), gain
its approximatio
n
v
:
1
0
m
ll
l
vc
g
v
.
Sparse ap
pro
x
imation refe
rs to wh
en th
e appr
oximation error i
s
d
e
termin
ed, selectin
g
the
m
with
small
e
st valu
e fro
m
the
m
set
that meet
s t
he a
bove
e
quation. Sig
nal
spa
r
se
decompo
sitio
n
is a typical
NP
(Non-De
termini
s
ti
c P
o
lynomial
)
wi
th extremely
high
cal
c
ulati
o
n
compl
e
xity, beca
u
se the i
n
volved L0
n
o
rm i
s
n
on-convex. Und
e
r the over-co
mplete
con
d
ition,
decompo
sitio
n
algo
rithm
of polynomia
l time is
co
mpletely un
realisti
c. The
r
efore,
subo
ptimal
approximatio
n algorith
m is requi
red [9, 10].
3.
Main Process and
Con
cept
o
f
The
OMP
(Or
t
ho
gonal
Matching Pursuit)
The MP (Mat
chin
g Pursuit
)
algo
rithm is a spa
r
se
sig
nal rep
r
e
s
e
ntation algo
rith
m base
d
on a re
dun
da
nt dictiona
ry. MP is a gre
e
dy signal
a
p
p
r
oximation al
gorithm, sele
cting at lea
s
t one
atom at each
iteration to be
st match the i
nner [11].
OMP po
sse
s
se
s the
prop
erty of optim
al iterat
io
n, which
ha
s a
smaller ite
r
atio
n amo
unt.
It only select
one atom to update the set duri
ng ea
ch
iteratio
n. Therefore, its iteration time is
c
l
os
ely related to spars
i
ty
K
and sa
mple amount
M
.
The
ba
sic
co
nc
e
p
t
of
OMP
i
s
t
o
sel
e
ct
s
e
ns
ing matrix c
o
lumns
through
greedy
iteration. The s
e
lec
t
ed
columns
of eac
h
iteration and
the current
re
dund
an
cy are
co
rrelated
to
the l
arg
es
t
e
x
tent. Subtra
cting th
e
relat
ed p
a
rt f
r
om
the
measuri
ng ve
ctor an
d iterat
e repe
atedly unt
il the iteration time rea
c
hes the
spa
r
sity
K
.
Input: over-complete di
ctionary
12
,
[,
,
,
]
L
Gg
g
g
, original signal
y
, initialized sparsi
ty
M
, redund
an
cy
0
ry
, suppo
rt inde
x set
0
A
, initial iteration
ml
.
Process: itera
t
ion, in the
m
circul
ation, ope
rate ste
ps (1)-(5
).
(1) Fin
d
ing
o
u
t the
co
rrespondi
ng fo
ote
r
of the
maxi
mum of
the
p
r
odu
cts of
re
sidu
al
r
and sen
s
ing
matrix colum
n
i
g
, calculate the su
ppo
rt index:
1
1,
,
ar
g
m
ax
,
mm
i
iN
rd
.
(2) Intro
du
c
in
g the signal
suppo
rt set,
1
()
mm
m
, find the re
con
s
tru
c
ted atom
set
1
[,
]
t
tt
from the se
nsing matrix.
(3) G
ainin
g
2
ˆˆ
arg
m
i
n
tz
t
x
yx
through the le
ast squares.
(4)
Upd
ating the re
sidu
al:
1
()
mm
m
m
TT
m
ry
G
G
G
G
y
.
(5)
1
mm
if
mM
, then
end the ite
r
at
ion, if not, ret
u
rn to
step
(1
),until the iteration
y=termin
ation
conditio
n
:
mM
is
met[12].
Output: sup
p
o
rt index set
1
mm
, sparse
coeffi
cient
1
()
mm
m
m
TT
bG
G
G
G
y
.
4.
Sparse
Repres
entatio
n
Image De
noising
Alg
orithm
Ba
se
d
on Ortho
gonal
Match
i
ng
Pursuit
(1) Imag
e blo
ck
cla
ssifi
cati
on, divide the image
into three
stru
ctures of smo
o
th
, texture
and ed
ge. Di
fferent stru
ct
ure
s
po
sse
s
s different vi
sual inform
atio
n. The image
classification
is
rep
r
e
s
ente
d
by the followi
ng equ
ation
s
:
s
te
GG
G
G
(
4
)
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ISSN: 16
93-6
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TELKOM
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Vol. 13, No
. 4, Decem
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e
r
2015 : 133
0 – 1336
1333
In equation (4),
G
refe
rs to
the entire im
age,
s
G
,
t
G
and
e
G
res
p
ec
tively refer to the
smooth, texture and ed
ge
part of the image. Image bl
ock cla
ssifi
ca
tion is the basis an
d pre
m
i
s
e
of building th
e sub
-
content
dictiona
ry in the
smooth, t
e
xture and e
d
ge part
s
of the image [13].
(2) Ba
sed
on
the bl
ock cl
assificatio
n
t
o
the
task im
age, g
a
in
different
sub-blo
c
k sets,
and then
co
ndu
ct spa
r
se
deco
m
po
siti
on to thes
e
sub
-
blo
c
ks i
n
each
co
rrespondi
ng trai
n
i
ng
diction
a
ry wit
h
OMP algorit
hm [14].
2
0
20
arg
m
i
n
{
}
.
,
,
1
,
2
,
,
ii
i
yG
x
s
t
i
x
T
i
N
(5)
Gain ea
ch type of spa
r
se coeffici
ent matrix
Q
through
equatio
n (5
).
(3) After g
a
ini
ng the
sparse
matrix of th
e
trainin
g
di
cti
onary
and
im
age to
be
pro
c
e
s
sed,
denoi
se t
he t
ask ima
ge.
Here,
con
s
id
er
i
y
as the
colu
mn vecto
r
tra
n
sformed
fro
m
the
sign
al
of
the noi
sy im
age to b
e p
r
oce
s
sed,
ˆ
i
y
as
the useful inf
o
rmatio
n of the imag
e,
is the
ˆ
i
y
freque
ncy b
a
ndwi
d
th,
ˆ
is th
e noi
se in
sid
e the freq
ue
ncy ba
nd
,
ˆ
is the noi
se o
utsid
e
the freq
uen
cy band
.
ˆ
ca
n
be
co
nsi
dered a
s
i
s
ort
hogo
nal
with
all the
ato
m
s in
the
diction
a
ry, then the sp
arse
decom
po
sition equ
ation o
f
step
k
is
[15]:
22
2
2
2
22
2
ˆ
ˆ
ˆ
ˆ
()
(
)
kk
k
ii
i
Ry
R
y
R
y
(6)
The gain
ed sparse r
epresentation matri
x
'
Q
ofthe den
oi
sed ima
ge.
(4) Fi
nally, gain the den
o
i
sed ima
ge o
f
each pa
rt of the image
'
s
P
,
'
t
P
and
'
e
P
th
r
o
ugh
equatio
n(7
)
,
recon
s
tru
c
t the three ima
g
e
s
to gain the
denoi
se
d targ
et image
'
P
.
''
PG
Q
(7)
5.
Experiment De
sign
an
d
Resul
t
Ana
l
y
s
is
In ord
er to p
r
ove the
effici
ency
and
adv
antage
of the
algo
rithm of t
his
pape
r i
n l
ow S
NR
image
den
oi
sing, t
w
o ot
her
den
oisi
n
g algo
rithm
s
are
compa
r
ed in te
rm
s of noi
sy im
age
denoi
sing. Th
ere alg
orithm
s
inclu
de wavelet hard thre
shol
d image
denoi
sing al
g
orithm ba
sed
on
Symlets and i
m
age de
noi
si
ng algo
rithm
based on
Co
ntourlet tra
nsf
orm.
5.1.
Introduc
tion
to Sy
mlets
an
d
Con
t
ourlet
Tra
nsform
Symlets is th
e discrete wa
velet transfo
rm based on
multi-re
sol
u
tion and m
u
lti-sampli
ng
filter theo
ry, whi
c
h i
s
wid
e
ly appli
ed i
n
en
gine
erin
g. It is a
co
mpactly
sup
p
orted
orth
og
onal
wavelet funct
i
on improve
d
from db wa
velet. db
wavelet does n
o
t possess the pro
perty of
s
y
mmetry, but Symlets
h
as
better
s
y
mmetr
y. Symletsis often rep
r
e
s
ented
by sym
2,
3
,
.
.
.
NN
.
The wavel
et and scali
ng functio
n of sym4 are
sh
o
w
n in figure 1
-
3, It has pro
pertie
s
of ne
ar
symmetri
c
, orthogon
al and
biortho
gon
al.
Conto
u
rlet transfe
r h
a
s t
he p
r
op
ertie
s
of multi
-re
solutio
n
, local po
sitionin
g
, multi-
dire
ction, n
e
a
r
-critical
sam
p
ling a
nd
ani
sotro
p
y.
Its b
a
si
c fun
c
tion
s are
di
stribute
d
on
multi-scale
and multi-dire
ction. It can e
ffectiv
ely capture the e
dge
conto
ur of the
image with
small amou
nt of
coeffici
ents.
Edge contou
r is the mai
n
cha
r
a
c
te
ri
stic of a nature image.
Th
e b
asi
c con
c
ept
of
Conto
urlet transfe
r i
s
to f
i
rst in
sp
ect t
he ed
ge
sin
gularity throu
gh multi-scal
e de
com
p
o
s
ition
simila
r to wav
elet, then gat
her the
sing
ul
arities
clo
s
e i
n locatio
n int
o the co
ntour
according to t
he
dire
ction info
rmation.
Conto
u
rlet tra
n
sform first u
s
e LP tran
sfo
r
m to
cond
uct
multi-scale a
nalysi
s
to the image.
Each g
r
a
de
of decompo
si
tion gain
s
a l
ow fr
e
que
ncy
image a
nd
a high frequ
ency ima
ge.
LP
decompo
sitio
n first ge
ne
ra
tes a lo
w-pa
ss ap
prox
im
ation of the o
r
ig
inal imag
e an
d a differe
nce
image
betwe
en the
ori
g
in
al imag
e a
n
d
the lo
w-pa
ss
pre
dicte
d im
age. F
urthe
r
decompo
sitio
n of
the low-pa
ss image ca
n gain the next
level lo
w-p
a
ss ima
ge an
d differen
c
e i
m
age, like th
is,
gainin
g the multi-re
sol
ution
decom
po
sition of t
he image throu
gh filtration step b
y
step. Applyin
g
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
A Sparse
Re
pre
s
entatio
nIm
age Denoi
si
ng Method B
ase
d on Orth
ogon
al …
(Xi
aojun Yu
)
1334
the two
dim
e
nsio
nal
Direct
ional Filte
r
B
a
nk
(DFB
) i
n t
he hi
gh f
r
equ
ency
co
mpon
ent of e
ach le
vel
gaine
d throu
gh LP d
e
co
mpositio
n
ca
n gain
2
n
di
rection
a
l subb
and
s at a
n
y
scale
s
. Input
the
high-pa
sssu
b
band
gain
ed
by ea
ch L
P
sub
ban
d
decompo
sitio
n
of the im
age to th
e
DFB,
grad
ually
co
nne
ct the
sin
gularitie
s i
nto
a lin
ear st
ructure to
ca
pture th
e im
age
co
ntour.
This
pape
r sel
ect
s the 4level Contourl
et decompo
si
tion i
m
age de
noi
si
ngalg
orithm a
s
the co
mpa
r
i
s
on
algorith
m.
(a) T
he scalin
g function of
sym4
(b) T
he wavel
e
t function of sym4
Figure 1. The
scali
ng fun
c
tion and
wavel
e
t function of sym4
(a)
Decomposition low-pass filter
(
b
)
D
e
c
o
mpos
itio
n
h
i
gh
-
pas
s
filte
r
Figure 2. The
decom
po
sition low-pa
ss
and hig
h-p
ass filters of sy
m4
(
a
)
R
e
c
o
ns
tr
uc
tio
n
low
-
pa
ss
filte
r
(
b
)
R
e
c
o
ns
tr
uc
tio
n
h
i
gh
-
pas
s
filte
r
Figure 3. The
reco
nst
r
u
c
tio
n
low-pa
ss a
nd high
-pa
s
s filters of sym
4
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ISSN: 16
93-6
9
30
TELKOM
NIKA
Vol. 13, No
. 4, Decem
b
e
r
2015 : 133
0 – 1336
1335
5.2.
Experiment Result a
nd
Analy
s
is
In ord
e
r to
test the vali
di
ty of the alg
o
ri
thm, the
d
enoi
sing
co
m
pari
s
on
of th
e noi
sy
image i
s
co
n
ducte
d. Figure 4 sh
ows th
e denoi
se
d
image
s by the
above menti
oned two different
denoi
sing m
e
thods.
(a) Noi
s
y
image
(b)
Den
o
ised
image by Symlets wavel
e
t
(c) De
noi
sed
image by Co
ntourlet
(d)
Den
oised
image by ou
r prop
osed met
hod
Figure 4. Den
o
ise
d
pep
pers image
s by different den
o
i
sing meth
od
s
It can be
se
en from fig
ure 4 that the
sign
al of the re
con
s
truct
ed imag
e ba
sed
on
Symlets tran
sform and
Con
t
ourlet tran
sfo
r
m will be
attenuate
d
and
distorte
d. The
recon
s
tru
c
tio
n
accuracy
is p
oor and
the
d
enoi
sing
re
su
lt is u
nsa
tisfying, which i
s
n
o
t only
cau
s
e
d
by m
u
lti-scal
e
decompo
sitio
n
and the re
con
s
tru
c
tion t
ool themse
lves, but also
becau
se these two algorit
hms
posse
ss the
deficie
ncy of
huge
red
und
ancy, an
d will
con
s
id
er the
edge
of the i
m
age a
nd ot
her
high fre
quen
cy informatio
n as the n
o
ise and filter
them, therefo
r
e, cann
ot ma
intain the ba
si
c
coeffici
ent di
stributio
n
rule
. On the
cont
rary, th
e
de
n
o
isin
g alg
o
rit
h
m put fo
rwa
r
d by thi
s
pa
per
can d
enoi
se
effectively, and gain bette
r visual effe
ct. It is beca
u
se the two majo
r
tasks of sp
arse
rep
r
e
s
entatio
n
are dictio
n
a
ry
buildi
ng and spa
r
se decompo
sitio
n
.
Whe
n
co
ndu
c
ting spa
r
se
rep
r
e
s
entatio
n of the sig
nal with a
n
a
l
ytic dict
ion
a
ry, the expre
ssi
on form o
f
the sign
al is
automatic an
d is m
o
re
a
daptive to di
fferent im
ag
e
data. Moreo
v
er, it sep
a
rated the u
s
e
f
ul
informatio
n from noi
se,
whi
c
h is o
nl
y
slight
ly affected by noi
se inten
s
ity and ba
nd
wid
t
h,
therefo
r
e, is
still effective
in den
oisi
n
g
high
noi
se
and lo
w S
NR i
m
age
s,
and
can
extract
recon
s
tru
c
tio
n
info
rmation
from the
n
o
isy image
mo
re
thoroug
hly. Its im
age
re
co
nstru
c
tion
eff
e
ct
and a
c
cura
cy
are quite satisfying.
6.
Conclusio
n
Takin
g
ima
g
e
den
oisi
ng
as th
e sta
r
tin
g
point, thi
s
pape
r
h
a
s a
nalyze
d
the t
r
adition
al
image de
noi
sing algo
rithm
and image
denoi
sing al
g
orithm ba
se
d
on spa
r
se repre
s
e
ntation
. It
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
A Sparse
Re
pre
s
entatio
nIm
age Denoi
si
ng Method B
ase
d on Orth
ogon
al …
(Xi
aojun Yu
)
1336
have focused
on o
r
thog
on
al matching
pursuit an
d
sparse
rep
r
e
s
entation the
o
r
y, with certa
in
improvem
ent, it put fo
rward an
efficie
n
t lo
w SNR im
age
den
oisin
g
meth
od. T
he exp
e
rim
e
nt
simulatio
n
h
a
s
p
r
oved
tha
t
this m
e
tho
d
can
effecti
v
ely denoi
se
and
maintai
n
mo
re
detai
led
image texture
information.
Ackn
o
w
l
edg
ments
This
work
wa
s su
ppo
rted
by School Yo
uth Re
sea
r
ch
Found
ation
of Jiang
su
University
of Te
chn
o
log
y
(No. KYY13029
) a
n
d
National
Natu
ral Sci
ence F
ound
ation
of
Chin
a (Grant
No:
6120
2464
).
Referen
ces
[1]
Yu
yin
g
Shi, Y
o
ngg
ui Z
hu, Ji
n
g
jin
g L
i
u. Semi
i
m
p
licit Imag
e
Den
o
isi
ng Al
go
rithm for Differ
ent Bou
n
d
a
r
y
Con
d
itio
ns.
T
E
LKOMNIKA Indon
esia
n Jour
nal
of Electric
al
Engin
eeri
n
g
. 2
013; 11(
4): 205
8-20
63.
[2]
Cha
ngd
on
g W
u
, Z
h
igan
g Li
u, Hua Jia
ng. T
he Conto
u
rl
et T
r
ansform
w
i
th
Multipl
e
C
y
cl
es
Spinn
i
n
g
for
C
a
te
na
ry
Image
D
e
no
i
s
in
g
.
T
E
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nesi
an Jo
u
r
nal
of Electric
al Eng
i
n
eeri
n
g
.
2014;
12(5):
388
7-38
93.
[3]
T
ana
y
a
Guha
, Ehsan
Nez
had
ar
ya, R
a
b
ab
K. W
a
rd.
Sparse
Re
p
r
esentati
on-
ba
sed Ima
g
e
Qualit
y
Assess
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l Processi
ng: Ima
ge Co
mmunic
a
tion
. 201
4; 29(
10): 113
8-1
148
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[4]
Rob
e
rto Ros
a
s-Romero,
He
mant D. T
agar
e. S
egme
n
tati
on of E
ndoc
ar
dium
in U
l
tras
oun
d Imag
es
Based
o
n
S
par
se R
epres
enta
t
ion
over
Le
arn
ed
Red
u
n
d
a
n
t
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ari
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ngi
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in
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p
licatio
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MR Moh
a
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di, E F
a
tem
i
za
deh, M
H
Ma
h
oor. PCA-
base
d
Dicti
on
ar
y B
u
ild
in
g for
Acc
u
rate F
a
c
i
a
l
Expr
essi
on Re
cogn
ition
via
Sparse
Re
pre
s
entatio
n.
Jour
nal
of Vis
ual
Co
mmun
icati
o
n a
nd I
m
a
g
e
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esentati
o
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[6]
Karthike
ya
n N
a
tesan R
a
ma
murth
y
, Ja
yar
a
man J
T
h
iagar
aja
n
, Andreas
Span
ias. Reco
verin
g
Non-
neg
ative a
nd C
o
mbi
ned S
pars
e
Repr
esent
ati
ons.
Dig
ital Sig
nal Proc
essin
g
. 2014; 26(
3): 21-35.
[7]
Ilias T
heodor
a
k
opo
ulos, D
i
mi
tris Kastani
otis
,
George Eco
nomo
u
, Spiros
F
o
topou
los. HEp-2
C
e
lls
Classific
a
tio
n
v
i
a S
parse
R
epr
esentati
o
n
of T
e
xt
ur
al
F
eatur
e
s
F
u
sed
int
o
D
i
ssimilar
i
t
y
Sp
a
c
e.
Pattern
Reco
gniti
on
. 2
014; 47(
7): 236
7-23
78.
[8]
W
L
W
oo, SS
Dla
y.
3D S
hap
e R
e
storat
ion
usi
ng S
p
arse R
epres
e
n
tation
an
d S
epar
ation
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Signal Pr
ocessi
ng
. 20
14
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[9]
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esh
Dek
a
, Prab
in
Ku
mar Bor
a
. Re
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