TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.6, Jun
e
201
4, pp. 4747 ~ 4
7
5
5
DOI: 10.115
9
1
/telkomni
ka.
v
12i6.549
5
4747
Re
cei
v
ed
De
cem
ber 2
9
, 2013; Re
vi
sed
March 8, 201
4; Acce
pted
March 24, 20
14
Total Variation Differential Equation with Wa
velet
Transform for Image Restoration
Dongh
ong Z
h
ao
Dep
a
rtment of Appl
ied Mat
h
e
m
atics,
Schoo
l of Mathematics
and Ph
ysics,
Univers
i
t
y
of Scienc
e an
d T
e
chno
log
y
B
e
ij
ing
,
100
08
3, Chin
a
email: zd
h75
11
11@
ustb.ed
u
.cn
A
b
st
r
a
ct
T
o
overco
me
the staircasi
n
g
effects and simult
a
neo
usly
avoi
d edg
e bl
urring,
w
e
pre
s
ent a
n
ada
ptive
parti
al
differenti
a
l
equ
atio
n co
mbini
n
g
the
total variation wi
th wavelet
tran
sform for
image
restoratio
n. A nois
e
remova
l
algorit
hm
bas
ed on var
i
atio
n meth
od a
n
d
partial differ
e
nce eq
uati
ons
is
prop
osed
in
th
is pa
per. T
h
e
combi
n
in
g a
l
g
o
rith
m takes
the a
d
va
ntag
e
of both fi
lter
s
since
it is able to
preserv
e
ed
ge
s w
h
ile avo
i
di
ng the stairc
a
s
e effect
in smo
o
th reg
i
o
n
s
.
T
he T
V
method pr
ovi
des f
a
st
ada
ptive w
a
vel
e
t-base
d
solve
r
s for the T
V
mo
de
l.
Our approac
h not onl
y emp
l
oys a w
a
vel
e
t colloc
a
ti
o
n
meth
od a
p
p
lie
d to the T
V
mode
l usin
g tw
o-di
mens
io
nal
a
n
isotro
pic tens
or prod
uct of w
a
velets, but als
o
prop
oses th
e d
i
fferentia
l e
qua
tion for i
m
ag
e r
e
storatio
n.
Mo
st conventi
o
n
a
l
imag
e pr
ocess
o
rs cons
ider
littl
e
the infl
ue
nce
of hu
ma
n vis
i
on psyc
ho
logy
. T
he al
gorit
h
m
i
n
h
e
rently
n
o
t only c
o
mb
i
nes the
restor
atio
n
prop
erty of w
a
velet co
mpres
s
ion a
l
g
o
rith
ms w
i
th that
of the T
V
mode
l, but als
o
giv
e
s a relativ
e
n
e
w
T
V
function
al w
h
ic
h consi
ders th
e influ
ence
of hu
ma
n vi
sio
n
p
sychol
ogy. W
e
present a det
a
iled
descri
p
tion
of
our metho
d
w
h
ich in
dicates th
at a combin
ati
on of w
a
ve
let
base
d
restorati
on techn
i
q
ues
w
i
th the T
V
mode
l
prod
uces su
pe
rior results. Experi
m
e
n
tal res
u
lts illustr
a
te the effectiven
ess
of t
he mod
e
l i
m
a
ge restor
ati
on.
Ke
y
w
ords
: is
otropic
diffusio
n
, T
V
mod
e
l,
w
a
velet trans
form, i
m
a
ge r
e
storatio
n, W
eber
’
s
l
a
w
,
Vision,
psychophysics
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
The
noi
se
s a
l
ways exist t
h
roug
h im
age
acq
u
is
itio
n a
nd tran
smissi
on. Th
e p
r
e
s
ence of
noise serio
u
sly affects the
visual effe
cts of
the imag
e
and follo
w-u
p
image
processing.
Wave
let
transfo
rmatio
n and the to
tal variation
(TV) eq
uatio
n are the m
o
st pop
ular i
m
age d
enoi
sing
method
s in
rece
nt years. I
n
this pa
per,
we
pr
e
s
e
n
t a
n
ad
aptive m
u
ltilevel total
variation
met
hod
for imag
e d
enoi
sing
whi
c
h utili
ze
s T
V
partial
differential
eq
ua
tion mod
e
l a
nd exploit
s
t
he
prop
ertie
s
of wavelet
s
.In this pap
er, we
use t
he termi
nology wavelet resto
r
ation
and propo
se
d
to
use t
he n
onl
ocal total
variation for t
h
is appli
c
ati
on.
Our
main
co
ntribution
in t
h
is p
ape
r i
s
to
extend the
total the total
va
riation
ba
sed
wavelet
re
sto
r
ation
to the
n
onlo
c
al total
variation
ba
se
d
model, in ord
e
r to re
cover
textures an
d geomet
ry stru
cture
s
simult
aneo
usly.
The presen
ce of noise i
n
image is u
n
a
voidable. It may be intro
duced by the
image
formation p
r
o
c
e
ss, ima
ge reco
rdin
g, ima
ge tran
sm
i
ssi
on, etc. The
s
e ran
dom di
st
ortion
s ma
ke
it
difficult to pe
rform a
n
y re
quire
d pi
cture pro
c
e
s
sing.
The convent
ional ima
ge restoration m
odel
use
s
Partial
Differential E
quation (P
DE
). Based on
Bayesian the
o
ry and varia
t
ion probl
em, the
resto
r
atio
n m
odel i
s
d
eem
ed a
s
a
n
e
n
e
rgy fun
c
ti
on
of imag
e, a
nd by mi
nimi
zing t
he e
nergy
function th
e
model
re
stores the
target
regi
on.
It re
store
s
th
e im
age by mi
ni
mizing
the le
ngth
energy functi
on of ima
ge.
The TV m
o
del diffuses
only cross th
e iso
phote,
a
nd it re
stores the
unkno
wn re
gi
on as
strai
ght
lines. So the
inpai
nted im
age is n
o
t a smooth imag
e, and the targ
et
region
c
ontour is
left. Bertalmio, et al. [1]
in
troduced another
inpaint
i
ng PDE
which diffuses only
along the isophote. Thi
s
equation smoothly rest
ore
s
the targ
et region wh
ile pre
s
ervin
g
the
isop
hote
s
in
image. It ad
opts the P
D
Es whi
c
h
diffuse
only alo
ng the i
s
op
h
o
te dire
ction
to
inpaint. The
inpainting
re
sult of this
model i
s
not
good, a
nd
some
linea
r
stru
ctures i
n
the
inpainte
d
ima
ge a
r
e
not
prese
r
ved.
Lysake
r, [4
] al
so
propo
se
d a
fourth
ord
e
r
PDE inp
a
inti
ng
approa
ch to i
m
age inte
rp
olation mod
e
l a
c
cordi
ng to th
e axiomatic
T
o
tal variation
(TV) in
ord
e
r
to
remove n
o
ise while
retai
n
ing impo
rta
n
t feature
s
, su
ch a
s
edg
es. Thi
s
ha
s been stu
d
i
ed
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TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4747 – 4
755
4748
extensively i
n
[2, 3] The
TV re
storation mod
e
l
was first p
r
op
ose
d
by Rudin.
The mo
del i
s
to
minimize the followin
g
ene
rgy:
2
00
()
(
)
.
2
E
f
f
f
dx
f
f
dx
Ho
wever, th
e diffusi
on
resultin
g fro
m
minimi
zi
ng t
he TV
norm
is
stri
ct
ly orthogon
al to t
h
e
gradi
ent
of the im
age,
an
d
tang
ent t
o
the
edg
es.
That i
s
to
say, both fro
m
theo
retical
and
experim
ental
point of view,
it has
been
sho
w
n
th
at the TV-n
orm t
r
ansfo
rm
s sm
ooth si
gnal i
n
to
piecewi
s
e co
nstant
s,
the so-call
ed stai
rca
s
e
effec
t.
In ord
e
r to
re
solve thi
s
p
r
o
b
lem, there a
r
e
almost t
w
o
solution
s. On
e
is thi
s
m
odel
[7, 8] which
use
d
a
combi
nation
of TV
diffusion
whe
r
e
there a
r
e likely edge
s
()
f
and isotropi
c diffusion in
more hom
ogen
eou
s re
gion
s
()
f
. This
minimiz
a
tion problem is
:
22
0
()
1
mi
n
.
22
fB
V
ff
f
d
x
f
dx
f
f
dx
Our id
ea
s in
dicate th
at u
s
ing
wavelet
s
to comp
re
ss TV de
noi
se
d image
s
results in a
highe
r co
mpression ratio than the re
gul
ar wavel
e
t method
s [9]. Superi
o
r d
enoi
sed ima
g
e
s
are
obtaine
d fro
m
the adapti
v
e TV method whe
n
co
m
par
e
d
to those obtaine
d from wavel
e
t or TV
denoi
sing
alo
ne. In additio
n
, we note th
at solving th
e PDE in the
wavelet do
main [10] is l
e
ss
expen
sive th
an
solving
th
e PDE
in th
e
image
dom
ai
n on
the
full g
r
id.
We
note t
hat the
den
oi
sed
image
s
obtai
ned f
r
om
u
s
i
ng the
meth
o
d
of
nonl
ocal mea
n
s a
r
e
sup
e
rio
r
, b
u
t
this al
go
rithm
is
vastly more
com
putatio
nally inten
s
i
v
e. The
p
a
per i
s
o
r
g
a
n
ize
d
a
s
fol
l
ows: Sectio
n two
introdu
ce
s th
e total vari
ation mo
del
an
d di
scusse
s t
he n
u
me
rical tech
niqu
e u
s
ed
to
solve
the
asso
ciated P
D
E. Section t
h
ree
revie
w
s
the ba
ckgro
u
nd be
hind
Da
ube
chie
s-typ
e
wavelet
s
a
nd
indicates ho
w
wavelet
co
efficients ma
y be
used to
gene
rate
spa
r
se
g
r
id
s fo
r
use
in
num
erical
PDE comp
utations. Sectio
n four pre
s
e
n
t
s results
fro
m
several n
u
m
eri
c
al expe
riments involvi
n
g
the TV model
, wavelet-ba
sed image d
e
n
o
isin
g, and th
e wavelet sol
u
tion of the TV model.
For si
mplicity, we introd
uce
the notation
1
2
2
xx
y
y
Du
u
u
u
u
2
2
2
2
,
,,
,1
,
jk
L
jk
uu
u
(1)
If the wavelet
in space
pp
BL
，
then there i
s
an
equivalen
c
e
relation mm
m
1
2
,
,,
22
,
pp
p
p
kp
kp
jk
BL
jk
uu
Her
e
o
n
ly
c
onsi
der
t
h
e
spe
c
ial
ci
rc
u
m
st
an
ce
s
1
p
，
1
that
sp
ace
11
BL
.For
conve
n
ien
c
e,
write
,
,,
jk
uu
, then:
11
1
1
2
k
BL
uu
(2)
In which,
,,
1
,
2
,
3
0
,
0
,
1
,
.
.
.
2
1
k
Si
j
k
i
k
j
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TELKOM
NIKA
ISSN:
2302-4
046
Total Variatio
n Differential
Equation with
Wavelet Tra
n
sform
for Image… (Do
n
g
hong Zh
ao
)
4749
Furthe
rmo
r
e,
to improve
the edge-p
r
eser
vin
g
ca
pability, Strong and
Cha
n
[7, 8
]
pre
s
ente
d
the
adaptive TV approa
ch to image resto
r
a
t
ion:
2
2
0
()
mi
n
(
)
.
2
uB
V
L
x
u
dx
u
u
dx
Whe
r
e
x
is a spatially
a
n
d
scale ada
ptive
function,
0
1
1
x
kG
u
as an ed
ge
stoppi
ng fun
c
tion use
d
for
controlling th
e spe
ed of the diffusion, where
k
r
ep
re
se
nts a thre
sh
ol
d
para
m
eter, a
nd
2
22
1
ex
p
22
x
Gx
denote
s
the Ga
ussia
n
filter with p
a
ram
e
ter
.
Th
e
corre
s
p
ondin
g
theory of viscosity sol
u
tions
wa
s inve
stigated the
r
e
in detail.
The
p
ape
r
i
s
orga
nized as follows:
In se
cti
on
2, we
p
r
esent the
ad
aptive total v
a
riation
model
and th
e asso
ciate
d
partial
differential eq
uatio
ns. In
se
ctio
n 3, we intro
duce the
unif
i
ed
model an
d g
eneral algo
rit
h
m frame
w
o
r
k ba
sed o
n
variation an
d
wavelet tran
sform d
enoi
si
ng
algorith
m
.
In se
ction 4,
furt
her nume
r
i
c
al
exampl
e
s
hi
ghlight the
re
markabl
e rest
oration
qu
alities
of non local total variation
regul
ari
z
ation
for natural im
age
s.
2. Models an
d Related Al
gorithms
We all kn
ow that all ima
ges a
r
e eve
n
t
ually percei
v
ed and inte
rpreted by the human
visual sy
stem
. As a result, vision psy
c
h
o
logy
and p
sychop
hysi
cs
play an impo
rtant role in t
he
su
ccessful communi
catio
n
of image
informati
on.
This fact implies that
any ideal image
pro
c
e
s
sor
sh
ould ta
ke into
accou
n
t the con
s
e
que
nce
s
of visio
n
psycholo
g
y and
psychop
hysi
cs.
The current p
aper m
a
kes
an attempt in
this dire
ction
.
We develop
an image re
storatio
n mo
del
that intend
s t
o
in
corpo
r
ate
one
of the
most
well kn
own
an
d
infl
uential psy
c
h
o
logi
cal re
sul
t
s—
Web
e
r’
s la
w for
sou
nd
and lig
ht pe
rce
p
tion.We
study the
co
mputational
strategy fo
r
the
asso
ciated n
online
a
r PDE
.
Most co
nvent
ional imag
e p
r
ocesso
rs [9, 10]
con
s
id
er l
i
ttle the influence of hum
an
visio
n
psycholo
g
y and many con
v
entional re
st
oration mo
de
ls don’t take
into account
that our visu
al
sen
s
itivity to the local fluct
uation d
epe
n
d
s o
n
t
he
am
bient inten
s
it
y level. That is, mod
e
ls
su
ch
as (2) a
s
sum
e
that a local
variation sh
oul
d be treat
ed equ
ally indepe
ndent of
the backg
ro
und
intensity level
.
So in this paper the minim
i
zation p
r
obl
e
m
:
2
0
()
2
fD
u
u
x
d
x
dy
u
u
dx
dy
u
(3)
To simplify the com
p
lication of the problem
, we p
r
opo
se the
much
simpl
e
r model:e
manu
script with other pap
ers, that
it is innovative, it
are u
s
ed
in the ch
apter "
R
e
s
ea
rch Me
thod"
to describ
e the step of rese
arch and u
s
e
d
in the
chapt
er "Re
s
ults a
nd Discu
s
sio
n
" to suppo
rt the
analysi
s
of t
he results [5
]. If the man
u
script
wa
s written re
ally
have
hi
gh origin
ality,
which
prop
osed a n
e
w:
2
0
2
Du
u
x
d
x
dy
u
u
dxdy
u
(
4
)
The foll
owin
g
will
put fo
rward
the
differential e
quatio
n from
the
a
bove m
odel.
Whe
r
e
x
is a spatially
and scal
e a
daptive functi
on,
0
1
1
x
kG
u
as an e
dge sto
ppin
g
function
use
d
for
cont
rol
ling the
spe
ed of the
di
ffusion, whe
r
e
k
r
ep
re
se
nts a thre
sh
old
para
m
eter, a
nd
2
22
1
ex
p
22
x
Gx
den
otes th
e Gau
ssi
an filter with pa
ra
meter
.
The co
rrespo
nding the
o
ry of visco
sity solu
tion
s wa
s i
n
vestigate
d
there in d
e
tail.
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TELKOM
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Vol. 12, No. 6, June 20
14: 4747 – 4
755
4750
2.1. The Eular-Lagla
nge E
quation
w
i
th
the Model
Most convent
ional resto
r
ati
on mod
e
ls(6,
7
) do
not
take acco
unt tha
t
our visu
al sensitivity
to the local fluctuatio
n de
pend
s o
n
the
ambient i
n
te
nsity level. T
herefo
r
e, te
chnically we
should
stay away fro
m
the black hole and a
ssume that
.
0
u
T
he black hol
e
similarly imp
o
se
s some
natural
restri
ctions
on th
e n
o
ise
mod
e
l:
u
u
0
. Since
0
u
al
so
re
pre
s
ent
s the
i
n
tensity valu
e,
we
mu
st hav
e
0
0
u
,
whi
c
h i
m
plies that
u
. The
con
d
ition
u
is e
quivale
nt t
o
u
.
Therefore
.
2
0
u
u
u
u
u
The combi
nation of all th
e
element
s di
scu
s
sed a
bov
e lead
s
to the followin
g
natural a
d
m
i
ssi
ble spa
c
e
for the new
Web
e
r TV re
storatio
n (4
):
0
1
0(
)
,
,
2
Du
Su
u
B
V
x
d
x
d
y
u
u
u
This is th
e sp
ace that we shall wo
rk
with
from now o
n
.
Theo
rem 1.
Suppo
se
2
1
2
)
(
y
y
x
x
u
u
u
u
u
D
, Then the va
ri
ation fun
c
tion
al will
become:
2
0
2
xx
y
y
uu
u
u
u
x
dx
dy
u
u
dx
dy
u
Then the formal Euler-La
gran
ge differential of
u
is
:
0
1
0
x
u
xu
u
uD
u
2.2. Total Variation and
Wav
e
let Inform Image De
noising
In [12], the a
u
thor p
r
opo
sed one TV regul
a
r
ized wavelet resto
r
a
t
ion model
s d
epen
ding
on
wheth
e
r
or n
o
t noi
se
is
con
s
id
ere
d
.
The id
ea i
s
to combin
e
a re
gula
r
i
z
ati
on term in
the
image
dom
a
i
n. For a
two-dime
nsio
nal ima
g
e
u
, let u
s
d
enote th
e
standard
wav
e
let
rep
r
e
s
entatio
n as:
,,
,
,
jk
jk
jZ
k
Z
uu
x
x
,
Whe
r
e
1,
1,
,
x
MN
and
,
j
k
d
enote
s
wavel
e
t coeffici
ents of u at level j
and lo
cation
k.
And for si
mplicity
,
j
k
denotes a given
orthog
onal o
r
biorthog
onal
wavelet ba
si
s
function. If we use an o
r
t
hogo
nal ba
si
s, the coefficient
,,
,
j
kj
k
. In discr
ete case, let
I
be the un
corrupted
kno
w
n ind
e
x set
,
,
()
j
k
f
or
,
j
kI
den
otes me
asure
d
coeffici
ents, the followin
g
two mod
e
ls, resp
ective
ly for the noisele
ss ca
se a
nd th
e noisy on
e,a
r
e
con
s
id
ere
d
in
the paper [1
2]:
ar
g
m
i
n
,
a
r
g
mi
n
,
x
x
TV
u
x
u
x
,,
.,
,
jk
j
k
st
j
k
I
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TELKOM
NIKA
ISSN:
2302-4
046
Total Variatio
n Differential
Equation with
Wavelet Tra
n
sform
for Image… (Do
n
g
hong Zh
ao
)
4751
Lemma: Let
be a
Hilbe
r
t space a
set of orthog
onal
wavelet ba
se
s,
,
f
u
are
the ba
sis F
u
nction
s
f
and
u
i
n
this g
r
oup
und
er t
he wavelet coefficient
s, the small
e
st
s
o
lution of func
tional is
:
21
2
2
Fu
x
u
f
u
,
0,
,
ff
uW
f
f
ff
In whic
h,
W
is
for the
wavel
e
t soft thre
sh
old op
erato
r
.
Basov
spa
c
e is the
spa
c
e of
boun
ded va
ri
ation ne
ar th
e minimum
[11] and
sp
ac
e image
s i
s
not allo
wed i
n
the bo
rd
er.
So
with a smoo
th of orde
r, Basov spa
c
e
12
CL
. Space to
descri
be the
image of the
regul
arity, G
e
t a
ne
w variation fun
c
tio
nal, then
by
minimizi
ng th
e vari
ation fu
nction
resto
r
ed
image.
Using
Basov no
rm
of
wavelet
coeffici
ent
s can d
e
scribe
the n
a
ture of
the
equivale
nt
norm
s
will solve
the problem
i
s
transform
ed
to t
he
wavelet
domain mini
mum by iterative
cal
c
ulatio
n. High freq
uen
cy
comp
one
nts
of wavelet
tra
n
sform ha
s a
wealth
of det
ails of the
ed
ge
informatio
n, so it can
re
co
nstru
c
t
high
q
ualit
y image
s, and th
e int
r
odu
ction
of wavelet alg
o
rit
h
m
make
s the te
xt of the new runnin
g
time is sh
o
r
t, and fast speed.
This mod
e
l o
v
erco
me
s slo
w
Cham
bolle i
m
age resto
r
a
t
ion, the sho
r
tcoming
s
of a
long time, a
nd have
signi
ficantly impro
v
ed
the quality of
the imag
e. Not only that, the pa
pe
r ta
ki
ng into
acco
u
n
t the p
s
ych
o
logy of the visual
effect of imag
e resto
r
atio
n, prop
osed a n
e
w variatio
n functio
nal, thu
s
noi
se mod
e
l
:
2
11
2
0
mi
n
2
L
uX
BL
u
uu
x
u
Among them,
Unde
r the eq
uivalen
c
e rel
a
tion (1
) and
(2), the
r
e:
22
22
00
L
uu
u
u
11
1
1
2
k
BL
u
u
uu
The firs
t two
wavel
e
t co
e
fficients fun
c
t
i
on, re
sulting
in the follo
wing
se
que
n
c
e of
convex fun
c
tional eq
uivale
n.
2
1
2
1
0
,2
2
k
g
u
Qu
w
u
u
x
u
(5)
About to
u
take the
small
e
st
0
uZ
, where the
wavelet
coeffi
cient
s
Z
s
o
t
h
a
t
l
o
w
-
freque
ncy pa
rt of the zero in the sp
ace
2
l
.
Equivalent to con
s
ider the follo
wing two
minimization probl
em
s:
2
2
1
2
1
0
mi
n
2
2
k
ul
u
uu
x
u
Fixed
u
, find the function
al (5) on the mini
mum sol
u
tion
u
, Ac
c
o
rding to Lemma,
(1
)
0
2
k
uQ
u
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4747 – 4
755
4752
Namely
:
2
2
0
mi
n
uZ
uu
，
sol
u
tion of the minimum:
0
L
uT
u
Whe
r
e the wavelet coefficients of the functio
n
L
T
，
that the low-f
r
equ
ency pa
rt of the thre
shol
d
value is zero,
to sum
up fu
nction
al (5)
minimization
of the solutio
n
ca
n b
e
obt
ained
by itera
t
ion,
the algorith
m
is Algorithm:
1. Initialization:
0
0
u
;
2. Iteration:
1
0
2
k
nn
uW
u
;
3. Stop condit
i
on:
n
n
uu
,
1
1
n
n
uu
;
A
ssu
mpt
i
on
is that a pre-given
small
positive
nu
mber, If you meet the co
n
d
itions
1
ma
x
nn
uu
,
to stop the iterative.
3. Test and
Resul
t
s
The follo
win
g
new alg
o
rith
m will b
e
the
text with a
sin
g
le Webe
r tot
a
l variatio
n d
enoi
sing
model com
p
arison.
In ord
e
r
to red
u
ce blurri
ng th
e e
dge
s of the
i
m
age
and
bl
ock effe
cts, t
h
e
pape
r u
s
e
s
transl
a
tion inv
a
rian
ce
wave
let transfo
rm,
only a layer
of wavelet tra
n
sform. Use
db4
wavelet,
the experim
ental results sho
w
n
in
Figu
re
1
and Figu
re
2. Figure
1 is a total variation
model
usi
ng
the metho
d
o
f
(256
× 25
6) image
de
noi
sing
re
sult
s.
It can
be
se
en, the o
r
igi
n
al
model
are mo
re a
m
big
uou
s edg
e. Figu
re
2 i
s
a
city (2
56
× 2
56) im
age
usi
ng the
origi
nal m
e
th
od
of Cultu
r
e a
n
d
the d
enoi
si
ng results. O
b
jectivel
y, you ca
n u
s
e th
e pea
k
sign
a
l
to noi
se ratio
(PSNR) an
d mean
squa
re
erro
r (MSE
) to evaluate the effects of g
ood o
r
bad i
m
age d
enoi
si
ng.
Clo
s
e to
the
pea
k si
gnal
to n
o
ise
ratio is a
mo
re
effective
evaluation
of t
he h
u
ma
n vi
sual
conte
n
t. Mean squ
a
re e
r
ror of the re
stored ima
ge
with the stan
dard u
s
e
d
to measu
r
e th
e
clo
s
en
ess of t
he imag
e. In
addition, ta
ki
ng into a
c
cou
n
t the practi
cal feasi
b
ility of the time (Ti
m
e)
is also used
as an eval
uation ind
e
x. Usin
g t
hese
three indi
ca
tors of the two meth
ods of
quantitative analysi
s
of image re
stor
ation re
sults, th
e experime
n
tal data sho
w
n in Table 1, we
can
see that the pro
p
o
s
ed
method is n
o
t only fas
t, with time short a
nd get better
quality resto
r
ed
image.
In this
se
ctio
n, we
p
r
e
s
en
t a seri
es of
nume
r
ical ex
perim
ents wh
ich
sh
ow that
image
denoi
sing
pe
rforme
d with
the adaptive
TV met
hod with wavel
e
t denoi
sing .
N
ow we pre
s
e
n
t
some n
u
me
ri
cal re
sult
s. We u
s
e the stand
ard Pe
ak Signal to
Noise
(
PSNR) to qua
ntify the
performance of
wavele
t coefficient filling:
2
1
10
(,
)
1
0
l
o
g
uu
PS
N
R
u
u
(a)
(b)
(c
)
Figure 1. Wo
man Image
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Total Variatio
n Differential
Equation with
Wavelet Tra
n
sform
for Image… (Do
n
g
hong Zh
ao
)
4753
A is co
rru
pte
d
with additiv
e
white noi
se
at
a rate of 17.617
4 PSNR; B is pro
c
e
s
sed by
TV algorithm for 50 times with
=0.
0
28
，
t
=0.
2
，
h
=
1,
and th
e resul
t
is at
a rate
of 22.06
30
PSNR; C i
s
pro
c
e
s
sed by
our ne
w al
g
o
rithm for 2
0
times with
wavelet'db6
',
t
=0
.
2
，
h=2,and
the result is a
t
a rate of 22.7335 PSNR;
Table 1. Imag
e Wom
an
Image Woman
TV method
Our n
e
w
algorith
m
Iterations 50
20
Wavelet
----
db4
PSNR
22.1530
22.8450
Time
4.072016s
0.948516s
(a)
(b)
(c
)
Figure 2. City Image
The imag
e was first
corru
p
ted with ad
ditive white n
o
ise at a rat
e
of 29.6 PSNR. We
then execute
d
200time st
eps
of the TV algorithm with
0.2
,
3
,
t
and
1
h
. A
is
the
origin
al imag
e; B is resto
r
ed with TV m
odel at
a rate
of 17.2929
PSNR;
C i
s
p
r
ocesse
d by our
new al
gorith
m
for 20 time
s,
0.2
,
2
th
,and the result is at a rate
of 22.8999 P
S
NR;
TV model an
d the new al
g
o
rithm compa
r
ed the results of image re
storatio
n:
Table 2 Image City
Image
TV method
Our n
e
w
algorithm
Iterations 50
20
Wa
ve
le
t -
-
--
D
b
4
PSNR
22.3318
22.7888
Time
7.445699s
3.322172s
Furthermore,
we present the ot
her num
erical
experim
ents to
illustrate the efficiency and
feasibility of
our
novel m
e
thod. We
ha
ve test
ed th
e
pro
p
o
s
ed
m
e
thod. Thi
s
t
e
xt will comp
are
improve
d
TV model with t
r
adition
al mo
del on ima
g
e
resto
r
ation
e
ffect. The bel
ow is th
e bri
e
f
pro
c
ed
ure of repai
rin
g
algo
rithm:
Detail
s: For e
a
ch pixel in th
e regio
n
to be
repai
red
(i)
Input noise image to be p
r
ocesse
d, co
m
pute the vertical/hori
z
o
n
tal smooth
n
e
s
s
variation u
s
in
g Lapla
c
e;
(ii)
Comp
uting th
e isop
hote
s
d
i
rectio
n;
(iii) Prop
agat
ing smo
o
thne
ss va
riation
s
along to the isop
hote
s
;
(iv)
Upd
a
ting the
value of the pixel through t
he com
b
inati
on of( i)–
(
iii);
Iterate s
t
ep: (i)–(iv)
s
i
gm
a=
15;
K
=
4.
55
21;
50
/
50;
C
a
t
t
e s
m
oot
h
d
算法
i
f
f
u
s
i
on
,
S
N
R
=
17.
90
64
四算
法
：
s
i
gm
a=
15;
K
=
4
.
571;
100/
100;
S
N
R
=
17
.
955
8
四
算法中
波：
s
i
g
m
a=
1
5
;
K
=
4
.
5
7
1
;
1
00
/
1
00
;
S
N
R
=
17
.
5
9
5
4
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4747 – 4
755
4754
Duri
ng th
e re
storatio
n, diffusio
n
p
r
o
c
ed
ure
wa
s inte
rl
eaved
with o
ne pe
r te
nth repairi
ng
loop to e
n
su
re noi
se i
n
se
nsitivity and
pre
s
e
r
ve the
sha
r
pn
ess
of edge
s. We perfo
rme
d
the
resto
r
atio
n al
gorithm
until t
he pixel
valu
es i
n
th
e
re
gi
ons
to
b
e
restored did not cha
nge. We
use
Gau
s
s-Jacobi
iterate metho
d
s, this compl
e
tes the la
st con
c
lu
sio
n
.
The bel
ow two experim
ent
is ba
sed
on tradi
tional TV
model a
nd T
V
model p
r
op
ose
d
by
this pap
er fo
r image
re
storation. In th
e follo
win
g
, the si
ze of L
ena ima
ge is 256×256; g
r
ey
degree: 256;
depth de
gre
e
:
8. The contrast
on the effect of image restoration:
(a)
(b)
(c
)
Figure 3. Len
a Image
Table 3. Time
and Squa
re
on Tra
d
itional
TV Model
T(time/second)
MSE
MSE1
98.13767
246.60331
2.4761e+004
Table 4. Time
and Squa
re
on Improve
d
TV Model
T(time/second)
MSE
MSE1
23.326316
6.5603e+003
2.2880e+004
The othe
r ex
ample i
s
the
size
of toys
image is
64
0×6
40; gr
ey degree: 25
6; depth
degree: 8. Th
e contrast on
the
effect of image resto
r
a
t
ion:
(a)
(b)
(c
)
Figure 4. Plane Image
Table 5 Time
and sq
uare o
n
traditional T
V
model
T(time/second)
MSE
MSE1
422.21138
4.0568e+003
3.1086e+004
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Total Variatio
n Differential
Equation with
Wavelet Tra
n
sform
for Image… (Do
n
g
hong Zh
ao
)
4755
Table 6. Time
and Squa
re
on Improve
d
TV Model
T(time/second)
MSE
MSE1
278.535649
8.1456e+003
3.1056e+004
From th
e ab
o
v
e two te
st ef
fect of two
re
stor
atio
n al
go
rithm, we
can
kno
w
th
e restoration
time on imp
r
oved TV mo
del is le
ss than the tradi
tional mod
e
l. The re
sto
r
at
ion efficien
cy
is
highe
r, con
s
u
m
ing time is
sho
r
ter. In a
word, t
he re
storation effe
ct of the impro
v
ed algorith
m
is
much b
e
tter than the traditi
onal alg
o
rith
m.
4. Conclusio
n
This pa
per p
r
esents a
ad
aptive differe
ntial eq
uation
co
mbinin
g t
he total
varia
t
ion with
Wavelet t
r
an
sform
for im
age
den
oisi
n
g
is p
r
e
s
ent
ed. Th
e p
r
e
s
ente
d
a
dapt
ive TV meth
od
provide
s
fa
st adaptive wavelet-ba
se
d
solvers
for
the TV mod
e
l. This ap
proach empl
oys a
wavelet
coll
o
c
ation
meth
o
d
ap
plied
to t
he TV
mod
e
l
usi
n
g
two
-
di
mensi
onal
an
isotro
pic ten
s
or
prod
uct
of wavelets. Thi
s
algo
rithm in
here
n
tl
y not
only combin
e
s
the
de
noisi
ng p
r
op
erty
of
wavelet
com
p
re
ssi
on
algo
rithms with
that of the
T
V
model,
but
also give
s a
relative
ne
w TV
function
al, a
nd p
r
od
uces re
sults su
perio
r to
e
a
ch
metho
d
whe
n
impl
emented
alo
ne.
Furthe
rmo
r
e t
h
is p
ape
r
con
s
ide
r
s the
hu
man p
s
yc
hol
ogy sy
stem.
Of co
urse, thi
s
p
o
int ad
ds the
difficult exten
t
of the
propo
sed
p
r
oble
m
, be
cau
s
e
thi
s
pap
er ad
d th
e influe
nce of
hum
an vi
sio
n
psycholo
g
y for the re
gula
r
ity item. It exploits the
edg
e pre
s
e
r
vatio
n
prop
erty of the TV model
to
redu
ce the
o
scill
ation
s
tha
t
may be gen
erated a
r
o
u
n
d
the edge
s i
n
wavelet co
mpre
ssion.
W
e
pre
s
ent
a det
ailed d
e
script
i
on of o
u
r m
e
thod a
nd
re
sults whi
c
h
i
ndicate that
a co
mbin
atio
n of
wavelet ba
se
d denoi
sing t
e
ch
niqu
es
with the TV mod
e
l prod
uces
superi
o
r results.
Ackn
o
w
l
e
dg
ements
This
wo
rk was
su
ppo
rte
d
by the F
undam
ental
Re
sea
r
ch Fu
nds for th
e
Cent
ral
Universitie
s
o
f
University of
Science and
Tech
nolo
g
y of Beijing unde
r Gra
n
t No.06
1080
41.
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