TELKOM
NIKA
, Vol. 13, No. 4, Dece
mb
er 201
5, pp. 1312
~1
318
ISSN: 1693-6
930,
accredited
A
by DIKTI, De
cree No: 58/DIK
T
I/Kep/2013
DOI
:
10.12928/TELKOMNIKA.v13i4.1897
1312
Re
cei
v
ed Au
gust 27, 20
15
; Revi
sed O
c
t
ober 2
4
, 201
5; Acce
pted
No
vem
ber 1
2
,
2015
Image Denoising Based on K-means Singular Value
Decomposition
Jian Ren*, Hua Lu, Xilian
g
Zeng
Hun
an Un
ivers
i
t
y
of Internati
o
nal Eco
nomics,
Chan
gsh
a
410
205, Hu
na
n, Chin
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
:
9461
565
@q
q
.
com
A
b
st
r
a
ct
T
he i
m
age
is
u
s
ually
p
o
ll
uted
by n
o
ises
i
n
its ac
q
u
isiti
o
n
a
n
d
trans
missio
n
and
n
o
ises
ar
e
of gr
eat
importa
nce in t
he i
m
a
ge q
ual
i
t
y, t
herefore, ima
ge d
e
-no
i
si
ng has b
e
co
me
a sign
ificant
techni
qu
e in i
m
a
g
e
ana
lysis
and
p
r
ocessi
ng. In t
he i
m
ag
e d
e
-n
oisin
g
base
d
o
n
spars
e
re
pre
s
entatio
n,
on
e
of the h
o
t sp
ots i
n
recent y
ears, t
he
usefu
l
i
m
ag
e i
n
fo
r
m
atio
n
h
a
s certa
i
n
stru
ctural fe
atures,
w
h
ich c
o
i
n
cid
e
w
i
th th
e at
o
m
ic
structure w
h
il
e
no
ises
do
n
’
t
h
a
ve s
u
ch
featu
r
es, ther
efor
e,
sparse
repr
ese
n
tation
ca
n s
e
parate
the
us
e
f
ul
infor
m
ati
on fro
m
th
e n
o
ises
e
ffectively so
as
to achi
eve t
h
e
purp
o
se
of de
-noisi
ng. In v
i
e
w
of the ab
ove
-
me
ntio
ned
the
o
retica
l b
a
sis,
this p
a
p
e
r pr
o
poses
a
n
i
m
ag
e d
e
-n
oisi
ng
al
gorith
m
of s
p
a
r
se re
prese
n
tat
i
o
n
base
d
on K-
means Si
ngu
lar
Valu
e Deco
mp
ositio
n (K-SVD
). T
h
is method
can int
egr
ate the constructio
n
and
o
p
ti
mi
z
a
ti
on
of ov
er-co
m
p
l
ete
dicti
o
n
a
ry, train
the
atom d
i
ctio
nar
y w
i
th the
i
m
a
ge s
a
mpl
e
s to
b
e
deco
m
pose
d
a
nd effectively b
u
ild th
e ato
m
d
i
ction
a
ry
that reflects vario
u
s imag
e features
to enhanc
e th
e
de-n
o
isi
ng p
e
r
f
orma
nce of t
he al
gorit
h
m
i
n
this
pa
per.
T
h
roug
h si
mul
a
tion a
n
a
l
ysis,
this meth
od
can
cond
uct no
ise
filtration
on th
e
imag
e
w
i
th
dif
f
erent
no
ise de
nsities and it
s de-n
o
isi
ng
effect is also
bette
r
than oth
e
r methods.
Ke
y
w
ords
: Image D
e
-no
i
si
ng
, K-mea
n
s Sin
gul
ar Valu
e De
compos
ition, S
parse R
epres
e
n
tation
Copy
right
©
2015 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
The imag
e in
reality usu
a
l
l
y contain
s
n
o
ise
s
, whi
c
h
not only gre
a
t
ly affect the image
quality, but also bri
ng ma
ny difficulties to t
he subse
quent imag
e pro
c
e
ssi
ng; therefo
r
e, ima
ge
de-n
o
isi
ng h
a
s be
co
me very impo
rtant
in image
proce
s
sing. No
ise
s
ca
n be
any factors that
prevent p
eopl
e from u
nde
rstandi
ng o
r
a
nalyzin
g t
he i
m
age
sou
r
ce
s they re
ceiv
e with thei
r visual
orga
n o
r
th
e
system
sen
s
ors [1]. Th
e
common
noi
se
s a
r
e
un
pre
d
i
c
table
ra
ndo
m si
gnal
s,
which
can o
n
ly be known via pro
bability statist
i
cs. Nois
es a
r
e very signifi
cant in image
pro
c
e
ssi
ng a
nd
they affect every link of the
input, colle
ction and
p
r
o
c
essing of ima
ge processin
g
as
well a
s
the
entire process of the resul
t
out
put. Ima
ge de-noi
sing
can red
u
ce or eliminate t
he noises mi
xed
in the ima
ge
to ce
rtain
extent and
p
r
eserve th
e
deta
iled ima
ge inf
o
rmatio
n so
as to
sto
r
e t
h
e
image to hig
her qu
ality. It still remain
s
an impo
rtant topic in ima
g
e
pre
-
p
r
ocessing h
o
w to
use
certai
n tech
n
o
logy to remo
ve image noi
se
s and p
r
e
s
erve the imag
e details [2].
Variou
s de
-noisi
ng met
hod
s ari
s
e i
n
acco
rda
n
ce with the
rule
s an
d statistical
cha
r
a
c
teri
stics of
the
sp
ect
r
al
di
stri
butio
n of th
e n
o
ise
s
a
s
well
a
s
t
he im
age
feat
ure
s
.
Comp
uter
image p
r
o
c
e
ssi
ng mai
n
ly adopt
s two
main ki
nd
s o
f
methods: o
ne is to p
r
o
c
ess in the
sp
atial
domain,
nam
ely to process the
imag
e i
n
vario
u
s ma
nners i
n
the
i
m
age
sp
ace
and th
e othe
r is to
orthog
onally t
r
an
sform
the
image i
n
the
spatial
dom
ai
n into the
fre
quen
cy do
ma
in, to co
ndu
ct
variou
s p
r
o
c
e
ssi
ng i
n
the
freque
ncy
dom
ain a
nd i
n
ve
rsely tran
sform to the
spati
a
l do
main
an
d to
form a pro
c
e
s
sed imag
e [3]. Corre
sp
o
ndingly, man
y
application
methods a
p
pear, in
cludi
ng
mean filter,
median
filter, low-pa
ss filter, wi
ene
r filter a
nd mi
nim
u
m di
stortion.
The
s
e m
e
th
od
s
have be
en
widely u
s
e
d
in promoting
the develo
p
m
ent of digit
a
l sig
nal p
r
o
c
e
ssi
ng g
r
ea
tly,
however, th
e tra
d
itional
imag
e d
e
-noisi
ng i
s
t
o
p
r
oje
c
t th
e ima
ge
sig
nals to a
certain
transfo
rmatio
n dom
ain
wh
ere th
e n
o
ise
s
a
r
e
se
pa
rat
ed fro
m
the
signal
s but
su
ch
se
paration
is
not do
ne th
o
r
oug
hly, therefore,
dama
g
e
s
ca
n b
e
caused to
the
origi
nal i
m
a
ge info
rmatio
n in
su
ch im
age
de-noi
sing,
neverth
ele
s
s, the im
ag
e de
-noi
sin
g
method
ba
sed
on
sp
a
r
se
rep
r
e
s
entatio
n can
sep
a
ra
te the signal
s from the n
o
ise
s
comple
tely since th
e noises a
r
e
not
spa
r
e comp
o
nents of the signal
s [4, 5].
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 : 131
2 – 1318
1313
This
pap
er first illu
strate
s t
he tra
d
itional
im
age
an
d n
o
ise
s
, introdu
ce
s the
rel
e
vant de
-
noisi
ng meth
ods a
nd te
ch
nique
s an
d e
x
plore
s
the
relevant sp
arse rep
r
e
s
entat
ion algo
rithm
s
and te
ch
niqu
es
ba
sed
on
the theo
ry of
sp
arse
re
pre
s
entatio
n a
n
d
Di
scre
te Co
sine
Tran
sform
(DCT) di
ction
a
ry. Then, b
a
se
d on suffi
cient theo
ry and techniqu
es, it raise
s
new ima
ge d
e
-
noisi
ng
algo
ri
thm ba
se
d o
n
K-SVD an
d
multiple
di
ct
ionari
e
s.
Final
ly, it verifies t
he effe
ctiven
ess
of this algorit
hm throu
gh e
x
perime
n
tal simulation.
2. Image and Noises
Image, the vision of two o
r
three-dimen
s
ional
scen
e in
human eye
s
, is the main medium
for peo
ple to convey info
rmation.
With i
t
s advantag
e
s
su
ch a
s
la
r
ge amo
unt of informatio
n, rapid
transmissio
n and long op
e
r
ating di
stan
ce, image has becom
e an importa
nt sou
r
ce a
nd mea
n
s
for pe
ople
o
b
tain an
d u
s
e inform
ation
.
Since ima
g
e
is inte
rfere
d
by vario
u
s noises i
n
th
e
gene
ration
a
nd tran
smissi
on, the
imag
e qu
ality will
be d
a
ma
ged,
whi
c
h
i
s
h
a
rmful to the
fo
llow-
up high
er-lev
el image
pro
c
essing. Ma
ny factors will
affect the imag
e quality in th
e ba
sic
step
s of
image a
c
qui
si
tion, codin
g
, tran
smi
ssi
on
and resto
r
atio
n. For examp
l
e, the usel
ess inform
ation
in
real
imag
e a
r
e noi
se
s
and
su
ch
elem
ent
s a
s
the
equi
pment, the
en
vironme
n
t an
d the
a
c
qui
sition
method
s ca
n also b
r
ing in
many noises,
includin
g
the
electrom
agn
etic interfe
r
en
ce, the gra
nul
ar
noises,
th
e sensor noi
se
s
in colle
cting
image
si
gna
ls, the i
n
ter-chann
el n
o
ise
s
a
nd
even t
h
e
filter noi
se
s.
So, in
ord
e
r to
imp
r
ov
e t
he i
m
ag
e
quality a
n
d
the
sub
s
e
q
uent hi
ghe
r-l
eve
l
pro
c
e
ssi
ng, i
t
has
bee
n
an imp
o
rtant
link to
de
-noise the i
m
age a
nd
pe
ople h
a
ve b
een
sea
r
ching fo
r a feasibl
e
de-noisi
ng meth
od [6, 7].
Noi
s
e
s
can
be see
n
as the elemen
ts whi
c
h ha
mper pe
ople’
s se
nse o
r
g
an from
unde
rsta
ndin
g
the
inform
ation they
receive.
The
interfe
r
en
ce
the im
age
suffers f
r
om
its
gene
ration a
nd tran
smi
ssi
on ha
s playe
d
gre
a
t influe
nce o
n
si
gnal
pro
c
e
ssi
ng, tran
smi
ssi
on
and
stora
ge.
Ima
ge denoi
sin
g
is an impo
rta
n
t image pro
c
e
ssi
ng task,
both as a proce
s
s itself, and
as a compo
n
ent in other
pro
c
e
s
ses. V
e
ry many wa
ys to denoi
se an image
or a set of d
a
ta
exists. The m
a
in pro
p
e
r
ties of a good image de
noisi
n
g
model a
r
e that it will rem
o
ve noise whi
l
e
pre
s
e
r
ving e
dge
s. Assu
ming that the deg
ene
ra
tive image
(,
)
f
xy
can b
e
ob
tained by
dege
nerating
the input image
(,
)
g
xy
, set a d
egen
erate fu
nction in the
origin
al imag
e and add
s
an additive noise item
(,
)
mx
y
.
As for the degen
eratio
n pro
c
e
ss
with
linearity and locatio
n
invarian
ce, th
e dege
neratio
n model in th
e spatial d
o
m
a
in ca
n be in
dicate
d as (1):
(,
)
(
,
)
(,
)
(
,
)
f
xy
h
x
y
g
x
y
m
x
y
(
1
)
In Form
ula (1
),
(,
)
hx
y
is the
sp
atial de
scription
of t
he deg
en
erate fu
nction
. Throu
gh th
e
spatial
co
nvo
l
ution of the
dege
ner
ate functio
n
an
d the ori
g
inal i
m
age, the
spat
ial deg
ene
rati
ve
image
can b
e
obtaine
d a
nd the deg
en
eration
pro
c
e
ss will
be co
mpleted within
addi
ctive
noise
after the d
e
g
eneration [8,
9]. Image
restor
ation i
s
to re
store
th
e ori
g
inal im
age
(,
)
g
xy
by
analyzi
ng th
e
deg
ene
ratio
n
mod
e
l a
n
d
formul
ating
the inverse
pro
c
e
ss.
See
the im
age
de-
noisi
ng mod
e
l
in Figure 1.
Set
degener
a
t
e
func
tion
A
dd noise
Original
signa
l
De
-
n
o
i
s
i
n
g
s
i
gnal
Multi scale
decomposition
I
n
v
e
rs
e
transf
ormat
i
on
Multiscale
denoising
Signal
re
con
s
t
r
uct
i
on
Figure 1. Image de-noi
sing
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Im
age Denoi
sing Ba
sed o
n
K-m
eans Si
ngula
r
Value
De
com
positio
n
(Jian Ren
)
1314
3. Sparse Re
presen
ta
tion
Theor
y
In sign
al
sa
mpling,
spa
r
se re
pre
s
e
n
tation theo
ry co
mpletes the
spa
r
se
codi
n
g
du
ring
the samplin
g. Gen
e
rally
, spa
r
se re
pre
s
entat
io
n
uses over-compl
ete re
dund
ant
fun
c
tion
diction
a
ry a
s
the ba
se f
unctio
n
in
ste
ad of
the traditional
sta
ndard
o
r
thog
onal b
a
si
s.
The
diction
a
ry sel
e
ction shall
coin
cid
e
with
the
structu
r
e of the signals to be a
pproxim
ated as
possibl
e an
d
the ele
m
ent
s in th
e di
cti
onary
are
ca
lled atom
s.
Signal recon
s
tru
c
tion i
s
t
h
e
inverse process of sp
arse co
di
ng. When the
spa
r
se rep
r
e
s
ent
ation theo
ry recon
s
tru
c
ts
the
sign
al, it conside
r
s the o
r
i
g
inal sig
nal
s as un
kn
own
one
s and get
s the re
con
s
t
r
ucte
d sig
nal
s
throug
h the
p
r
odu
ct of
sp
a
r
se
coefficie
n
t
s an
d
the co
rre
sp
ondi
ng diction
a
ry
. Th
e sparse
si
gn
als
in the redu
nd
ant dictio
na
ry can
be
re
sto
r
ed from the
few
ran
dom
o
b
se
rvation va
lues via
certa
i
n
algorith
m
s, n
a
mely that after ado
pting t
he re
dun
dant
dictiona
ry, the spa
r
si
ty of the sign
als
can
be enh
an
ced
and the si
g
nals
can
be
resto
r
e
d
fro
m
fewer
ob
servation at h
i
gher
pro
babi
lity.
Signal spa
r
se
de
com
positi
on can
redu
ce
the co
st of
sign
al p
r
o
c
e
s
sing
while
pre
s
ervin
g
the
m
a
in
c
h
arac
teris
t
ics
[10, 11].
Signal
spa
r
se de
com
p
o
s
i
t
ion refe
rs to
the a
c
q
u
isiti
on p
r
o
c
e
s
s o
f
the optimal
sp
arse
rep
r
e
s
entatio
n
or spa
r
se approxim
atio
n of the si
gn
als in th
e ov
er-com
plete diction
a
ry,
na
mely
that the si
gn
al can b
e
rep
r
esented
in t
he form of
th
e produ
ct of
a group
of sp
arse
coeffici
e
n
ts
and th
e trai
ni
ng di
ction
a
ry. The
mo
re
zero
s o
r
app
roximate zero
s in
the ve
ct
or valu
es of
the
spa
r
se coeffi
cient
s,
the sp
arser
th
e sig
nal
repr
esent
ation is. T
he
sign
al with
N
non-ze
ro
s in
the
vector value
s
of the sparse coeffici
ents is ca
lled
N-spa
r
se sig
nal
. According t
o
the releva
nt
conte
n
ts
of sparse th
eory,
all si
gnal
s
can be
sparse
ly represente
d
,
namely co
mpre
ssed.
T
h
e
spa
r
se d
e
co
mpositio
n al
gorithm
an
d
the de
sig
n
o
f
the spa
r
se
dictio
nary
a
r
e th
e two
main
asp
e
ct
s of
sparse
re
pre
s
entation [1
2]. Fro
m
the
a
bove a
nalysi
s
, the
gen
eral procedu
re
s of
image de
-n
oising b
a
sed o
n
spa
r
se de
compo
s
ition can be summa
rize
d as follo
ws:
(1)
Co
nstruct the over-com
plete di
cti
onary of at
oms. Th
e p
e
rform
a
n
c
e
of the
con
s
tru
c
tion
of atom dictiona
ry ca
n dire
ct
ly affect the sp
arsity of the image
sp
arse
decompo
sitio
n
and it determines the im
age de
-noi
sin
g
.
(2) Imag
e sp
arse de
comp
osition. Co
nd
uct sp
arse d
e
com
p
o
s
ition
on the image in the
over-com
plet
e dictio
nary o
f
atoms by u
s
ing
de
co
mp
osition
algo
rithms
su
ch a
s
OMP algo
rithm.
In this
pro
c
e
ss,
attention
shall
be
paid
to t
he
end
con
d
ition
s
of
de
comp
ositi
on, nam
ely
what
circum
stan
ce
can
be
se
e
n
as the fa
ct that t
he effective info
rm
ation of the i
m
age
ha
s b
een
extracted
co
mpletely. Whi
l
e gu
ara
n
teei
ng a
s
m
u
ch
as
effective i
n
formatio
n to
be
extracte
d
,
the
spa
r
sity of th
e imag
e rep
r
ese
n
tation
sh
all al
so b
e
ta
ken
into
con
s
ideratio
n. As
the core
sta
g
e in
image d
e
-n
oi
sing, the
com
putation in thi
s
sta
ge
is
usually very hu
ge and th
e sparse
coeffici
ent
matrix of the image can be
obtaine
d and
the e
ffective atom set can
be extracte
d [13].
(3)
Re
con
s
truction of d
e
-
noi
sed im
a
ge. Re
con
s
truct the ima
ge with the
spa
r
se
coeffici
ent matrix and the effective
atom set and get
the de-noi
se
d image.
Assu
me that
the sig
nal to
be de
com
p
o
s
ed is
y
and th
e over-compl
ete diction
a
ry
is
D
and then the
spa
r
se de
co
mpositio
n of the im
age
sig
nal ca
n be de
scribe
d as foll
ows:
2
20
1
arg
m
in
2
y
y
xD
y
y
(
2
)
In this
formula,
2
——
2
L
norm,
0
——
0
L
norm,
D
——over-co
m
plete dictio
nary,
12
[,
,
,
,
]
L
Dd
d
d
,
y
——sparse repre
s
e
n
tation
coefficient a
nd
0
y
is the nu
mber of non
-zero co
efficie
n
ts
in
y
,
——tra
de-off para
m
eter b
e
t
ween the
co
ntrol re
sid
ue
and sparsity.
In Formula (2
), the first item is a squa
re
erro
r, meani
ng the error b
e
twee
n the produ
ct of
the spa
r
se vector afte
r the spa
r
se de
comp
os
itio
n and the over-co
m
plet
e di
ctiona
ry and
the
origin
al sign
a
l
x
. Acco
rdi
ng t
o
the d
e
finition of
0
L
no
rm,
the second it
em in th
e formula i
s
the
numbe
r of no
n-zero eleme
n
ts in the sp
arse re
pre
s
e
n
tation co
efficient
y
. The smaller
0
y
is,
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TELKOM
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Vol. 13, No
. 4, Decem
b
e
r
2015 : 131
2 – 1318
1315
the fewer
no
n-zeros and
the mo
re
sp
arse th
e p
r
e
s
e
n
tation of
y
is.
Seek the
ab
ove-me
ntione
d
optimizatio
n probl
em an
d get the spa
r
se rep
r
e
s
entati
on of the origi
nal sig
nal [14
,
15].
4. The Image De-n
oising Process
Bas
e
d on K-SV
D
K-SVD alg
o
ri
thm train
s
th
e over-compl
ete dictio
nary
whi
c
h i
s
suit
able to
rep
r
e
s
ent the
sema
ntic st
ructure of the image fro
m
the
natural image lib
rary. The m
a
in idea for th
e
optimizatio
n and upd
ate o
f
K-SVD dictionary is:
ba
sed on the over-com
plete
diction
a
ry, up
date
and
adju
s
t th
e atom
s i
n
th
e di
ctiona
ry
continuo
usly
so
a
s
to
mat
c
h
the
sign
al
se
t to be
train
e
d
to
the utmo
st
extent. Additionally, K-SV
D al
go
ri
thm
can
be
u
s
e
d
togeth
e
r
with m
any o
t
her
decompo
sitio
n
algo
rithm
s
and different
approximatio
n
algo
rithm
s
can
be u
s
e
d
flexibly in train
i
ng
the diction
a
ry
in K-SVD alg
o
rithm. The
main
procedu
res of K-SV
D algorith
m
are
as follo
ws:
Input: extract
pixel blo
c
ks
and form trai
ning
sampl
e
set in the
noi
sy image f
r
o
m
the top
point to the e
nd point of th
e image m
a
tri
x
accordi
ng t
o
a certain
st
ep length. T
h
e trainin
g
sa
mple
set
is
12
,,
,
N
D
dd
d
, the original imag
e si
gnal is
x
and the sp
arsity is
K
.
(1) Initiali
zati
on
In the diction
a
ry
(0
)
,
nK
DR
n
K
, mak
e
1
J
.
Initialize the dictionary
0
DD
and
the initial iteration is
1
k
.
(2) Spa
r
se re
pre
s
entatio
n of signal
s
The algo
rithm
pe
rform
s
sp
a
r
se
d
e
co
mpo
s
ition
on th
e
noisy im
age
i
n
the i
n
itial di
ctiona
ry
D
and an over-compl
ete dict
ionary matrix
nK
D
R
can be o
b
tained. Every column represents
the atoms
of an ori
g
inal
sign
al. Given
a sig
nal
y
, it can
be rep
r
esented
as
the sp
arsity
combi
nation
of these ato
m
s. The ima
ge sig
nal
y
can be re
pre
s
e
n
ted as
yD
x
or
yD
x
,
sat
i
sf
y
i
ng
p
yD
x
. The so
-calle
d over-com
plet
ene
ss in th
e diction
a
ry ma
trix means th
at
the num
ber
o
f
atoms i
s
mu
ch bi
gge
r tha
n
the len
g
th o
f
the image
si
gnal
y
(obvio
u
s
ly the len
g
th
is
n
), na
mely
nk
. The
di
ctiona
ry up
date i
s
con
d
u
c
ted
by col
u
mn.
Wh
en u
pdatin
g
a certai
n
colum
n
of di
ctionary elem
e
n
t
i
d
, assume th
at the coeffi
cient matrix
X
a
nd the di
ction
a
ry
D
are
kno
w
n a
nd fixed.
(3)
Upd
a
te the diction
a
ry
D
by column
Assu
ming th
at the coeffici
ent
X
and the diction
a
ry
D
are fixed, if the
k
d
in the
th
k
colum
n
of the
dictiona
ry is
to be upd
ate
d
, K-SVD alg
o
rithm is th
e
th
k
colum
n
of the pro
d
u
c
t of
the s
p
arse matrix
X
and
k
d
, then the obje
c
ti
ve function is
as follo
ws:
2
2
1
2
2
.
K
j
jT
F
j
F
jk
jT
k
T
jk
F
k
kk
T
F
YD
X
Y
d
x
Yd
x
d
x
Ed
x
(
3
)
(4) SV
D me
thod de
com
poses the
matrix
k
R
E
and
gets
kT
R
EU
V
. Update the
dictionary, select
k
d
as
the firs
t
c
o
lumn of
U
, update th
e sp
arse ve
ctor
k
R
X
and
sele
ct th
e
produc
t of the firs
t c
o
lumn of
V
and
(1
,
1
)
.
(5) After o
b
taining the di
ctiona
ry
D
, rep
eat Step (2)-(4) a
nd up
da
te the diction
a
ry by
colum
n
until all colum
n
s h
a
ve been u
p
d
a
ted and t
he
end conditio
n
s
of iteration
are a
c
hieve
d
.
(6) O
u
tput the over-com
pl
ete diction
a
ry
D
.
The de-noi
se
d
ima
ge sp
arse re
pre
s
e
n
tation coe
ffici
ent matrix
ca
n be
obtai
ne
d from
th
e
above ste
p
s.
In the image de-noi
sin
g
based on
sparse
rep
r
ese
n
tation, the useful image
informatio
n
h
a
s ce
rtain structural
featu
r
es,
whi
c
h
coi
n
cid
e
with
th
e atomi
c
stru
cture
while
n
o
ise
s
don’t have
su
ch featu
r
e
s
, therefo
r
e,
spa
r
se
re
pr
ese
n
tation can
sep
a
rate th
e u
s
e
f
ul informati
o
n
from the noi
ses effectively so a
s
to achi
eve the purp
o
s
e of de
-noi
si
ng.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Im
age Denoi
sing Ba
sed o
n
K-m
eans Si
ngula
r
Value
De
com
positio
n
(Jian Ren
)
1316
5. Experimental De
sign and Re
sult Analy
s
is
In orde
r to verify the effectivene
ss a
n
d
sup
e
rio
r
ity of this algorit
hm of this pa
per in the
low-sig
nal-to
-
noise-ratio im
age, this pap
er co
mpa
r
e
s
the de-n
o
isi
n
g effect by the algo
rithm of
this pap
er
an
d the effect
s by anothe
r 2
de-n
o
isi
ng a
l
gorithm
s: the
image d
e
-n
o
i
sing al
go
rith
m
based o
n
Symlets wavele
t hard th
re
sh
old an
d
met
hod b
a
sed o
n
DCT ove
r
-compl
ete ato
m
diction
a
ry. T
he symlet
s
are n
e
a
r
ly symmetr
i
c
al wavelet
s
pro
posed
by Daube
chi
e
s a
s
modifications to the db family [16]. The properties
of the two
wavelet families are
similar.
Here
are the wavel
e
t function
s a
s
sh
own in the followin
g
Figure 2.
Figure 2. Symmetrical wa
velets
With the
Ca
meram
a
n
im
age
as exam
ple, this
pa
p
e
r co
ndu
cts experim
ental simulatio
n
and resea
r
ch
analysi
s
. Th
e image
Cam
e
ram
an h
a
s
more
smo
o
th
regio
n
s
and
abun
dant det
ail
textures a
nd i
t
also ha
s st
rong
represen
tativeness in
de-n
o
isi
ng p
r
oce
s
sing. Th
e followin
g
is
the
analysi
s
of th
e simul
a
tion result. Figu
re
3 is t
he d
e
-n
oise
d imag
es by the foreg
o
ing de
-n
oisi
ng
method
s.
After being
p
r
ocesse
d by t
he alg
o
rithm
of th
is pa
pe
r, the edg
es
a
nd texture
s
o
f
Image
Came
ram
an
are
cle
a
r. In t
he alg
o
rithm
of this
p
ape
r
and a
c
co
rdin
g to the
und
e
r
stating
of
sp
arse
decompo
sitio
n
in the noisy signals, wit
h
the incr
e
a
se of noisy co
mpone
nts (th
e
signal to n
o
ise
ratio re
du
ce
s grad
ually), the useful si
g
nal co
mpo
n
e
n
ts incre
a
si
n
g
ly redu
ce,
namely that the
comp
one
nts
with structu
r
a
l
prop
erty red
u
ce, the
r
efore, in spa
r
se
decompo
sitio
n
, the matchi
ng
atoms to
sign
als be
co
me fewe
r an
d fewer, the
si
gnal
representati
on is mo
re a
nd more sparse
and the
com
putation al
so
falls d
r
am
atically. In
this
ca
se, to u
s
e
K-SVD al
gori
t
hm to optimi
z
e
diction
a
ry structure ha
s demon
strated
a huge
po
tential in proce
s
sing lo
w SNR imag
es,
sug
g
e
s
ting t
he supe
riorit
y of the algorithm in thi
s
pa
per. T
h
is alg
o
rithm
can
pre
s
e
r
ve
and
enha
nce the image ed
ge
s and textures and improve
the subje
c
tive effect and
obje
c
tive qua
lity
of the image while removin
g
ringin
g
and
blurring effe
ct.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
9
30
TELKOM
NIKA
Vol. 13, No
. 4, Decem
b
e
r
2015 : 131
2 – 1318
1317
(a) Noi
s
y
image
(b)
Den
o
ised
image by Symlets wavel
e
t
(c) De
noi
sed
image by DCT method
(d)
Den
o
ised
image by this method
Figure 3. Den
o
ise
d
Cam
e
raman ima
g
e
s
by different denoi
s
ing met
h
od
s
6. Conclusio
n
This pa
per h
a
s
co
me
up
with p
r
o
b
lem
s
in
ima
ge
d
e
-noi
sin
g
, u
s
ed the
ide
a
of sp
arse
decompo
sitio
n
, tried
by int
egrat
io
n
DCT
over-complet
e di
ctiona
ry
o
f
atoms
and
K-SVD al
gorit
hm
and co
mpa
r
e
d
the advantage
s and di
sadvantag
es i
n
variou
s aspec
t
s
of the de-n
oisi
ng a
nd
pre
s
e
r
vation
of image
det
ails th
rou
gh
theoreti
c
al a
nalysi
s
a
nd experim
ental
simul
a
tion. The
result ha
s sh
own that the
algorith
m
of this pa
pe
r ha
s bee
n imag
e de-noi
sing
effect and
strong
robu
stne
ss.
Referen
ces
[1]
YT
Shih, CS Chie
n CY. Chu
ang. An
Adaptiv
e Para
meterize
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ck-base
d
Sin
gul
ar Valu
e
Decom
positi
o
n
for Imag
e
D
e
-Nois
i
n
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a
n
d
Com
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ressi
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lie
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a
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ematics and
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Nasrin M Mak
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e
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w
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d Sec
u
re Di
gital Ima
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a
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ng Schem
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he Integer W
a
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ngu
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e Ee
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rmarkin
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r
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hang Ye, Ji
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TELKOM
NIKA
ISSN:
1693-6
930
Im
age Denoi
sing Ba
sed o
n
K-m
eans Si
ngula
r
Value
De
com
positio
n
(Jian Ren
)
1318
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ali
n
g
a
m
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na T
hambi
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ad Mu
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peck
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ilter.
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l
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o
ngg
ui Z
h
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g
jin
g L
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E
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om
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[8]
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a
r
d
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ahes
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a
n
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d Improv
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e
xt
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