Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
7,
No.
2,
April
2017,
pp.
850
ā
857
ISSN:
2088-8708
850
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
p
-Laplace
V
ariational
Image
Inpainting
Model
Using
Riesz
Fractional
Differ
ential
Filter
G
Sride
vi
1
and
S
Srini
v
as
K
umar
2
1
Department
of
ECE,
Aditya
Engineering
Colle
ge,
Kakinada,
AP
,
India
2
Department
of
ECE,
JNTUK,
Kakinada,
AP
,
India
Article
Inf
o
Article
history:
Recei
v
ed
May
25,
2016
Re
vised
Mar
14,
2017
Accepted
Mar
29,
2017
K
eyw
ord:
Fractional
calculus
Image
inpainting
P
artial
Dif
ferential
Equations
Riesz
fractional
deri
v
ati
v
e
V
ariational
models
ABSTRA
CT
In
this
paper
,
p
-Laplace
v
ariational
image
inpainting
model
with
symmetric
Riesz
fractional
dif
ferential
ļ¬lter
is
proposed.
V
ariational
inpainting
models
are
v
ery
useful
to
restore
man
y
smaller
damaged
re
gions
of
an
image.
Inte
ger
order
v
ariational
image
inpainting
models
(especially
second
and
fourth
order)
w
ork
well
to
complete
the
unkno
wn
re
gions.
Ho
we
v
er
,
in
the
process
of
inpainting
with
these
models,
an
y
of
the
unindented
visual
ef
fe
cts
such
as
staircasing,
speckle
noise,
edge
blurring,
or
loss
in
contrast
are
introduced.
Recently
,
fractional
deri
v
ati
v
e
operators
were
applied
by
researchers
to
restore
the
damaged
re
gions
of
the
image.
Experimentation
with
these
operators
for
v
ariational
image
inpainting
led
to
the
conclusion
that
second
order
symm
etric
Riesz
fractional
dif
ferential
operator
not
only
completes
the
damaged
re
gions
ef
fecti
v
ely
,
b
ut
also
reducing
unintended
ef
fects.
In
this
arti-
cle,
The
ļ¬lling
process
of
damaged
re
gions
is
based
on
the
fractional
cent
ral
curv
ature
term.
The
proposed
model
is
compared
with
inte
ger
order
v
ariational
models
and
also
Grunw
ald-
Letnik
o
v
fractional
deri
v
a
ti
v
e
based
v
ariational
inpainting
in
terms
of
peak
signal
to
noise
ratio,
structural
similarity
and
mutual
information.
Copyright
c
ī
2017
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
G
Sride
vi
Department
of
Electronics
and
Communication
Engineering
Aditya
Engineering
Colle
ge
Surampalem,
Andhra
Pradesh,
India
sride
vi
g
amini@yahoo.com
1.
INTR
ODUCTION
Image
inpainting,
is
an
art
of
implementing
untraceable
modiļ¬cations
on
images.
It
is
used
to
restore
the
damaged
re
gions
of
an
im
age
based
on
the
pix
el
information
from
the
kno
wn
re
gions.
It
is
not
only
used
to
reco
v
er
the
damaged
parts
b
ut
also
used
to
discard
the
o
v
erlaid
te
xt
and
undesired
objects.
Inpainting
is
most
useful
in
reco
v
ering
the
old
photographs
and
images
in
ļ¬ne
art
museums.
It
can
be
used
as
a
pre-processing
step
for
other
image
processing
problems
lik
e
image
se
gmentation,
pattern
recognition
and
image
re
gistration.
In
this
w
ork,
image
inpainting
model
for
te
xt
remo
v
al
and
scratch
remo
v
al
are
demonstrated.
The
image
inpainting
techniques
are
mainly
classiļ¬ed
into
three
cate
gories:
te
xtural
inpainting,
structural
inapinting
and
h
ybrid
inpainting
(combination
of
tw
o
approaches).
T
e
xtural
inpainting
is
mainly
connected
with
the
te
xture
synthesis.
Man
y
te
xture
inpainting
methods
ha
v
e
been
proposed
since
a
f
amous
te
xture
synthesis
algorithm
w
as
de
v
eloped
by
Efros
and
Leung
[1].
Man
y
other
te
xture
synthesis
algorithms
are
proposed
with
the
impro
v
ement
in
speed
and
ef
fecti
v
eness
of
the
Efros-Leung
method.
Structure
inpainting
is
the
process
of
introducing
smoothness
priors
to
dif
fuse
(propag
ate)
local
structured
information
from
source
re
gions
to
unkno
wn
re
gions
along
the
isophote
directi
on.
It
uses
partial
dif
ferential
equations
(PDE)
and
v
ariational
reconstructions
methods.
Marcelo
et
al.
[2]
introduced
ļ¬rst
PDE
based
digital
image
inpainting.
These
models
produce
good
results
in
restoring
the
non-te
xtured
or
relati
v
ely
smaller
unkno
wn
re
gions.
Na
vier
-stok
es
equations
of
ļ¬uid
dynamics
were
used
by
the
same
authors,
to
inpaint
the
unkno
wn
re
gions
by
considering
the
image
intensity
as
a
stream
and
isophote
lines
as
ļ¬o
w
of
streamlines.
Ho
we
v
er
,
these
are
slo
w
iterati
v
e
processes.
In
order
to
minimize
the
computational
time
a
f
ast
marching
technique
is
described
in
[3],
which
ļ¬lls
the
unkno
wn
re
gion
in
J
ournal
Homepage:
http://iaesjournal.com/online/inde
x.php/IJECE
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
,
DOI:
10.11591/ijece.v7i2.pp850-857
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
851
single
iteration
using
weighted
means.
First
v
ariational
approach
to
image
completion
w
as
proposed
by
Masnou
and
Morel
[4].
A
f
amous
v
ariational
w
ork
w
as
introduced
by
Chan
and
Shen
[5]
in
2001.
Their
method
completes
the
missing
re
gions
by
minimizing
the
total
v
ariation
(TV)
norm.
The
y
retain
the
sharp
edges
for
non-te
xtured
parts
using
curv
ature
term
in
the
corresponding
Euler
-Lagrange
equation.
TV
-norm
con
v
erts
smooth
(ļ¬at)
re
gions
into
piece
wise
constant
le
v
els
(staircase
ef
fect).
Meanwhile,
small
detai
ls
and
te
xtured
re
gions
are
s
moothed
out.
The
TV
inpainting
model
has
been
e
xtended
and
considerably
impro
v
ed
in
subsequent
w
orks,
such
as
curv
ature
dri
v
en
dif
fusion
[5],
Eulerā
s
elastica
equation
[6],
Gauss
curv
ature
dri
v
en
dif
fusion
[
7],
fractional
curv
ature
dri
v
en
dif
fusion
[8],
fractional
TV
inpainting
in
spatial
and
w
a
v
elet
domain
[9],
and
fractional
order
anisotropic
dif
fusion
[10].
Fractional
dif
ferentiation[11],
[12]
ļ¬nds
an
important
role
in
the
area
of
signal
and
image
processing.
Frac-
tional
dif
ferentiation
can
be
vie
wed
as
the
generalization
of
inte
ger
dif
ferentiation.
The
deļ¬nition
of
fractional
dif-
ferentiation
is
not
united.
The
commonly
used
deļ¬nitions
are
proposed
by
the
authors
Gr
ĀØ
unw
ald-Letnik
o
v
(G-L),
Riemann-Liouville
(R-L),
Caputo
and
Riesz.
Man
y
researchers
ha
v
e
applied
these
deļ¬nitions
to
man
y
image
process-
ing
applications.
Y
i
et
al.
[8]
proposed
G-L
fractional
deri
v
ati
v
e
based
curv
ature
dri
v
en
dif
fusion
for
the
minimization
of
metal
artif
acts
in
computerized
X-ray
images.
Beno
Ė
ıt
et
al.
[13]
implemented
fractional
deri
v
ati
v
e
for
the
detection
of
edges.
Y
i-Fei
et
al.
[14]
constructed
the
fractional
dif
ferential
masks
to
enhance
the
te
xture
el
ements
in
the
images.
Y
i
et
al.
[15]
proposed
tw
o
ne
w
non-linear
PDE
image
inpainting
models
using
R-L
fractional
order
deri
v
ati
v
e
with
4-directional
masks.
Stanislas
and
Roberto
[16]
proposed
fractional
order
dif
fusion
for
image
reconstruction
inspired
by
the
w
ork
of
[17].
Qiang
et
al.
[18]
applied
symmetric
Riesz
fractional
deri
v
ati
v
e
for
enhancing
the
te
xtured
images.
Y
i
et
al.
[9]
appl
ied
fractional
order
deri
v
ati
v
e
deļ¬ned
by
the
second
deļ¬nition
of
Y
i-Fei
et
al.
[14]
and
ļ¬lling
process
is
achie
v
ed
by
the
fractional
curv
ature
term.
In
this
article,
fractional
deri
v
ati
v
e
is
combined
with
inte
ger
order
v
ariational
inpainting
model.
The
sec-
ond
order
symmetric
Riesz
fractional
dif
ferential
operator
is
considered
in
this
w
ork,
because
it
possesse
s
non-local
and
anti-roational
characteristics.
The
proposed
model
gi
v
es
good
visual
ef
fects
and
superior
objecti
v
e
performance
metrics
viz.,
PSNR,
SSIM,
and
MI
with
respect
to
inte
ger
order
v
ariational
image
inpainting
models.
This
article
is
or
g
anized
in
ļ¬
v
e
sections.
In
section
2,
fractional
order
v
ariational
inpainting
model
is
pre-
sented
and
fractional
central
curv
ature
term
is
also
represented.
In
section
3,
construction
of
symmetric
Riesz
fractional
dif
ferential
ļ¬lter
is
presented.
Simulation
results
are
e
xplained
in
section
4.
Conclusions
are
gi
v
en
in
section
5.
2.
PR
OPOSED
MODEL
Gi
v
en
an
image,
f
L
2
ī
,
with
ī
R
2
an
inpainting
or
missing
domain
ha
ving
boundary
ī
,
and
E
an
surrounding
domain
nearby
ī
.
The
problem
is
to
reconstruct
the
original
image
u
from
the
observ
ed
image
f
.
A
fractional
order
v
ariational
model
is
propo
s
ed
in
this
article,
which
pro
vides
not
only
an
ef
fecti
v
e
image
inpainting,
b
ut
also
visible
reduction
of
unintended
ef
fects.
The
proposed
v
ariational
model
mnimizes
an
ener
gy
cost
functional
J
,
containing
a
mask
that
specify
kno
wn
and
unkno
wn
re
gions
of
the
image.
Therefore,
the
completed
and
enhanced
image
is
determined
as
a
result
of
the
ne
xt
minimization
J
ī
u
x;
y
1
p
M
x
1
M
y
1
r
ī
u
x;
y
p
ī
ī
2
M
x
1
M
y
1
u
x;
y
f
x;
y
2
(1)
where
ī
is
an
y
real
number
and
p
1
;
2
,
the
mask
is
based
on
the
characteristic
function
of
the
inpainting
re
gion,
which
is
represented
as
ī
ī
ī;
x;
y
ī
0
;
other
w
ise
and
the
Neumann
boundary
condition
u
n
0
is
applied.
Where
n
is
an
unit
v
ector
outw
ard
perpendicular
to
ī
.
The
ļ¬rst
term
of
(1)
is
the
fractional
re
gularization
term,
which
is
used
to
inpaint
the
damaged
parts
based
on
the
non-local
characteristics
of
the
image.
The
second
term
of
(1)
is
ļ¬delity
term,
which
is
used
to
preserv
e
the
important
features
lik
e
edges
and
ī
ī
is
a
scaling
parameter
in
the
inpainting
re
gion
ī
,
which
is
used
to
tune
the
weight
of
tw
o
terms
in
the
inpainting
re
gion
only
.
According
to
the
fractional
calculus
of
v
ariations,
the
Euler
-Lagrange
equation
is
1
ī
div
ī
r
ī
u
x;
y
r
ī
u
x;
y
2
p
ī
ī
u
x;
y
f
x;
y
0
(2)
The
computation
of
numerical
al
g
or
ithm
is
based
on
the
gradient
descent
approach.
and
the
follo
wing
fractional
v
ari-
atinal
model
is
obtained.
p-Laplace
V
ariational
Ima
g
e
Inpainting
Model
Using
Riesz
F
r
actional
Dif
fer
ential
F
ilter
(G
Sride
vi)
Evaluation Warning : The document was created with Spire.PDF for Python.
852
ISSN:
2088-8708
u
x;
y
t
1
ī
cur
v
ī
u
x;
y
ī
ī
u
x;
y
f
x;
y
(3)
The
result
of
minimization
(1),
representing
the
restored
image,
will
be
determined
by
solving
(3).
The
frac-
tional
central
curv
ature
is
introduced
to
increase
the
performance
of
image
reconstruction.
The
discrete
representation
of
the
fractional
central
curv
ature
term
cur
v
ī
u
x;
y
is
represented
as
cur
v
ī
u
x;
y
div
ī
r
ī
u
x;
y
r
ī
u
x;
y
2
p
r
ī
x
r
ī
x
u
x;
y
r
ī
x
u
x;
y
2
0
:
5
r
ī
y
c
u
x;
y
2
ī
2
p
2
r
ī
y
r
ī
y
u
x;
y
r
ī
y
u
x;
y
2
0
:
5
r
ī
xc
u
x;
y
2
ī
2
p
2
(4)
where,
ī
is
a
small
constant
to
stay
a
w
ay
di
vide
by
zero.
The
proposed
symmetric
Riesz
fractional
dif
ferential
ļ¬lter
coef
ļ¬cients
are
used
to
dif
fuse
the
pix
el
information
in
the
inpainting
re
gion
based
on
fractional
central
curv
ature
term.
The
construction
of
the
Riesz
fractional
dif
ferential
coef
ļ¬cients
will
be
e
xplained
in
the
ne
xt
section.
3.
CONSTR
UCTION
OF
RIESZ
FRA
CTION
AL
DIFFERENTIAL
FIL
TER
The
second
order
symmetric
fractional
order
deri
v
ati
v
e
of
u
x
for
the
inļ¬nite
interv
al
x
based
on
the
Riesz
deļ¬nition
is
represented
as
a
combination
of
the
right
and
left
sided
R-L
fractional
deri
v
ati
v
es
ī
x
ī
u
x
c
v
ī
x
ī
ī
x
ī
u
x
(5)
where
c
v
2
cos
ī
ī
2
1
with
ī
1
;
m
1
ī
m
2
for
m
N
ī
x
ī
u
x
1
ī
m
ī
x
m
x
u
m
ī
ī
x
ī
m
1
dī
(6)
ī
x
ī
u
x
1
ī
m
ī
m
x
m
x
u
ī
x
ī
ī
m
1
dī
(7)
The
symmetric
Riesz
fractional
order
deri
v
ati
v
e
is
represented
based
on
second
order
fractional
centered
dif
ference
method
[19]
with
step
h,
ī
u
x
x
ī
1
h
ī
k
1
k
ī
ī
1
ī
ī
2
k
1
ī
ī
2
k
1
u
x
k
h
(8)
By
noting
Eulerā
s
reļ¬ection
formula
for
Gamma
function,
ī
ī
2
ī
1
ī
2
ī
sin
ī
ī
2
and
ī
ī
ī
1
ī
ī
sin
ī
ī
ī
2
sin
ī
ī
2
cos
ī
ī
2
gi
v
es
ī
ī
ī
1
ī
ī
ī
2
ī
1
ī
2
1
2
cos
ī
ī
2
(9)
By
substituting
equ.
(9)
in
equ.(8),
one
can
has
ī
u
x
x
ī
1
2
cos
ī
ī
2
h
ī
k
0
w
ī
k
u
k
xh
0
k
w
ī
k
u
k
xh
(10)
where
w
ī
0
ī
1
ī
2
ī
ī
1
ī
2
ī
ī
(11)
IJECE
V
ol.
7,
No.
2,
April
2017:
850
ā
857
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
853
(a)
(b)
(c)
(d)
(e)
(f)
Figure
1.
Fractional
dif
ferential
ļ¬lters
a
W
ī
x
b
W
ī
x
c
W
ī
y
d
W
ī
y
e
W
ī
xc
f
W
ī
y
c
w
ī
k
1
k
1
ī
ī
2
ī
1
ī
2
ī
ī
2
k
1
ī
ī
2
k
1
ī
ī
;
k
1
;
2
;
:::
(12)
In
the
perspecti
v
e
of
images,
the
smallest
distance
between
the
tw
o
pix
els
in
x
-direction
and
y
-direction
is
one.
F
or
a
2-D
image,
u
x;
y
at
a
pix
el
x
1
;
y
1
,
in
the
positi
v
e
x
-direction
N
1
pix
els
are
considered.
Therefore,
u
k
x
1
;
y
1
u
x
1
k
h;
y
1
,
where
h
x
1
N
;
0
k
N
;
and
N
is
the
numbe
r
of
di
visions.
Similar
procedure
is
considered
in
other
directions,
lik
e
positi
v
e
y
-direction,
ne
g
ati
v
e
x
-direction,
ne
g
ati
v
e
y
-direction.
Consider
h
1
and
the
anterior
forw
ard
N
1
equi
v
alent
fractional
order
dif
ference
of
the
fractional
partial
dif
ferentiation
in
the
positi
v
e
x
-direction
is
ī
u
x
x
ī
1
2
cos
ī
ī
2
h
ī
N
k
0
w
ī
k
u
k
xh
;
0
ī
2
;
ī
1
(13)
F
or
the
central
dif
ference
in
x
-direction
of
the
image
u
x;
y
at
a
pix
el
x
1
;
y
1
,
N
1
pix
els
are
considered
in
the
positi
v
e
x
-direction
and
N
pix
els
are
considered
in
the
ne
g
ati
v
e
x
-direction.
Therefore,
u
k
x
1
;
y
1
u
x
1
k
h
;
y
1
u
x
1
k
h;
y
1
.
Si
milar
procedure
is
considered
for
central
y
-direction.
So,
the
anterior
2
N
1
equi
v
alent
fractional
order
centeral
dif
ference
of
the
fractional
partial
dif
ferentiation
in
the
central
x
-direction
is
ī
u
x
x
ī
1
2
cos
ī
ī
2
h
ī
N
k
N
w
ī
k
u
k
xh
;
0
ī
2
;
ī
1
(14)
The
fractional
dif
ferential
ļ¬lters
along
symmetric
directions,
the
positi
v
e
x
-axis
W
ī
x
,
ne
g
ati
v
e
x
-axis
W
ī
x
,
positi
v
e
y
-axis
W
ī
y
,
ne
g
ati
v
e
y
-axis
W
ī
y
,
central
x
-axis
W
ī
xc
,
central
y
-axis
W
ī
y
c
are
constructed
p-Laplace
V
ariational
Ima
g
e
Inpainting
Model
Using
Riesz
F
r
actional
Dif
fer
ential
F
ilter
(G
Sride
vi)
Evaluation Warning : The document was created with Spire.PDF for Python.
854
ISSN:
2088-8708
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
Figure
2.
Comparison
of
v
ariational
inpainting
models
for
te
xt
remo
v
al
(a)
Ground
truth
image,
(f)
Image
with
o
v
erlaid
te
xt
(PSNR
=
17.75
dB),
(b)
Inpainted
image
using
TV
model
[20]
(PSNR
=
30.45
dB),
(c)
Inpainted
image
using
fourth
order
PDE
model
[21]
(PSNR=31.6
dB),
(d)
Inpainted
image
using
Y
i
et
al.
model
[9]
(PSNR
=
31.6
dB),
(e)
Inpainted
image
using
proposed
model
(PSNR
=
32.8
dB),
(g)
Enlar
gement
of
parrotā
s
f
ace
of
(b),
(h)
Enlar
gement
of
parrotā
s
f
ace
of
(c),
(i)
Enlar
gement
of
parrotā
s
f
ace
of
(d),
(j)
Enlar
gement
of
parrotā
s
f
ace
of
(e)
and
sho
wn
in
Figure
1.
These
fractional
dif
ferential
ļ¬lters
possess
non-local
and
anti-rotational
properties.
In
Figure
1,
C
ī
u
0
is
the
ļ¬lte
r
coef
ļ¬cient
corresponding
with
the
interested
pix
el.
The
size
of
the
ļ¬lter
is
2
N
1
,
where
N
is
an
y
positi
v
e
inte
ger
and,
one
implements
2
N
1
2
N
1
fractional
dif
ferential
ļ¬lter
.
Airspace
ļ¬ltering
technique
is
performed
on
the
symmet
ric
directions
with
2
N
1
2
N
1
fractional
dif
ferential
ļ¬lter
.
The
usage
of
the
airspace
ļ¬lter
is
to
mo
v
e
the
windo
w
pix
el
by
pix
el
and
these
are
computed
using
r
ī
l
u
x;
y
N
i
N
N
j
N
W
ī
l
i;
j
u
x
i;
y
j
(15)
where
l
x
;
x
;
y
;
y
;
xc;
y
c
The
fractional
dif
ferential
ļ¬lter
coef
ļ¬cients
are
C
ī
u
0
1
2
cos
ī
ī
2
ī
1
ī
1
ī
2
ī
1
ī
2
ī
2
ī
(16)
C
ī
u
k
1
2
cos
ī
ī
2
1
k
ī
ī
1
ī
ī
2
ī
1
ī
2
ī
ī
2
k
1
ī
ī
2
k
1
ī
2
ī
;
k
1
;
2
;
:::
(17)
4.
RESUL
TS
AND
DISCUSSION
The
proposed
technique
described
here
has
been
tested
on
lar
ge
collecti
ons
of
images
af
fected
by
missing
re
gions.
The
USC-SIPI
database
is
used
in
our
e
xperiments.
The
proposed
technique
pro
vides
an
ef
fecti
v
e
restoration
of
the
de
graded
image,
completing
successfully
the
missing
zones.
It
also
preserv
es
the
image
details,
lik
e
edges,
and
reduces
the
unintended
ef
fects,
such
as
image
blurring,
staircasing
and
speckle
ef
fects.
The
optimal
image
reconstruc-
tion
results
are
achie
v
ed
by
the
proper
selection
of
fractional
order
.
This
v
alue
is
detected
by
trial
and
e
rror
,
through
emprical
observ
ation.
In
this
w
ork,
when
ī
1
:
4
the
proposed
model
produces
optimal
reconstruction
result.
The
performance
of
this
fractional
order
v
artional
model
has
been
quantiļ¬ed
by
using
well-kno
wn
measures,
such
as
peak
Signal
to
Noise
Ratio
(PSNR),
Structural
Sim
ilarity
(SSIM)
[22],
and
Mutual
Information
(MI)[23].
This
approach
outperforms
numereous
state
of
the
art
inpainting
methods.
This
fractional
order
v
ariational
image
IJECE
V
ol.
7,
No.
2,
April
2017:
850
ā
857
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IJECE
ISSN:
2088-8708
855
inpainting
technique
is
able
to
restore
multiple
missing
re
gions.
F
or
this
reason,
it
can
be
successfully
used
for
some
important
tasks,
such
as
remo
ving
the
superimposed
te
xt,
remo
ving
the
scrat
ches,
or
remo
ving
the
w
atermarks
from
the
digital
images.
A
te
xt
remo
ving
e
xample
using
proposed
technique
described
in
Figure
2,
wheresome
method
comparison
results
are
displayed.
The
images
of
that
ļ¬gure
depict
the
inpainting
results
achie
v
ed
by
v
arious
inpainting
techniques
on
the
parrots
color
image
collected
from
LIVE
image
database
and
cropped
to
[256
X
256]
.
The
te
xt
is
superimposed
on
the
image
and
the
inpainting
techiniques
are
are
applied.
These
inpainting
techniques
are
carried
out
in
YCbCr
color
space.
The
te
xt
is
almost
remo
v
ed
by
all
the
models.
Ho
we
v
er
,
one
can
observ
e
that,
the
te
xture
part
near
the
parrotā
s
e
ye
is
not
res
tored
well
by
state
of
the
art
methods,
such
as
TV
inpainting
b),
fourth
order
PDE
model
c),
and
Y
i
et
al.
model
d)
[9].
These
models
do
not
preserv
e
edges
and
produce
loss
in
contrast.
The
zoomed
v
ersion
of
these
techniques
are
sho
wn
in
Figure
2(g)-(j).
The
inpainting
re
gions
after
applying
the
proposed
model
are
ļ¬lled
ef
fecti
v
ely
than
the
other
thr
ee
models.
Inpainting
models
for
te
xt
remo
v
al
with
the
same
damaged
mask
are
applied
on
dif
ferent
images
and
re
gisterd
in
T
able
1.
As
one
could
observ
e
in
that
table,
the
performance
measures
of
proposed
inpainting
technique
achie
v
e
the
highest
v
alues.
One
more
observ
ation
is
that,
the
proposed
model
w
orks
well
e
v
en
if
the
image
ha
ving
partially
te
xtured
re
gions,
b
ut
t
he
other
three
models
are
not.
The
logic
is
that,
the
total
v
ariation
model
is
second
order
PDE,
Y
i
et
al.
model
(
ī
=1.8)
is
closed
to
fourth
order
PDE,
where
as
the
proposed
model
(
ī
=1.4)
is
closed
to
third
order
PDE.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
Figure
3.
Inpainting
of
artiļ¬cial
lines
on
pepperā
s
image
(a)
Ground
truth
image,
(f)
Damaged
image
(PSNR=16.60dB)
(b)
Inpainted
image
using
TV
model
[20]
(PSNR
=
33.97
dB,
SSIM
=
0.9001,
MI
=
3.6850),
(c)
Inpainted
image
using
fourth
order
PDE
model
[21]
(PSNR
=
33.89
dB,
SSIM
=
0.9312,
MI
=
3.8402),
(d)
Inpainted
image
using
Y
i
et
al.
model
[9]
(PSNR
=
35.9
dB,
SSIM
=
0.9497,
MI
=
4.3149)
(e)
Inpai
n
t
ed
image
using
proposed
model
(PSNR
=
36.16
dB,
SSIM
=
0.9712,
MI
=
5.4789),
(g)
Residual
image
of
(b),
(h)
Residual
image
of
(c),
(i)
Residual
image
of
(d),
(j)
Residual
image
of
(e)
T
able
1.
Comparison
of
inpainting
models
for
te
xt
remo
v
al
on
dif
ferent
images
Image
I/P
PSNR
TV
[20]
F
ourth
order
PDE
[21]
Y
i
et
al.
[9]
Pr
oposed
model
PSNR
SSIM
MI
PSNR
SSIM
MI
PSNR
SSIM
MI
PSNR
SSIM
MI
Cameraman
16.16
30.38
0.9374
3.74
30.58
0.9379
3.75
31.05
0.9396
3.79
32.94
0.9473
5.23
Elaine
18.69
38.16
0.9420
4.89
38.56
0.9478
4.92
38.72
0.9587
4.94
39.60
0.9694
5.95
Lena
19.60
34.21
0.9288
4.51
34.32
0.9327
4.57
34.52
0.9369
4.64
35.02
0.9532
5.75
Mandrill
19.53
31.18
0.8275
3.03
31.20
0.8349
3.04
31.23
0.8455
3.06
33.53
0.9516
4.80
The
proposed
model
is
also
applied
to
remo
v
e
the
unw
anted
scratches
from
the
image.
The
simulation
results
on
pepperā
s
image
are
sho
wn
in
Figure
3.
This
inpainting
technique
outperforms
the
TV
inpainting,
fourth
order
PDE
model,
Y
i
et
al
.
model.
The
e
xperiment
sho
ws
the
loss
of
contrast
after
applying
the
inpainting
techniques.
In
order
p-Laplace
V
ariational
Ima
g
e
Inpainting
Model
Using
Riesz
F
r
actional
Dif
fer
ential
F
ilter
(G
Sride
vi)
Evaluation Warning : The document was created with Spire.PDF for Python.
856
ISSN:
2088-8708
T
able
2.
Comparison
of
inpainting
models
for
scratch
remo
v
al
on
dif
ferent
images
Image
I/P
PSNR
TV
[20]
F
ourth
order
PDE
[21]
Y
i
et
al.
[9]
Pr
oposed
model
PSNR
SSIM
MI
PSNR
SSIM
MI
PSNR
SSIM
MI
PSNR
SSIM
MI
Cameraman
17.47
26.68
0.8020
3.05
26.55
0.8532
3.22
27.64
0.9290
3.82
29.20
0.9234
5.28
Man
18.24
32.21
0.9050
3.48
31.76
0.9145
3.42
32.10
0.9277
3.64
33.52
0.9493
5.11
Lena
16.64
31.84
0.8762
3.94
32.42
0.9124
3.60
32.68
0.9381
3.96
33.26
0.9590
5.15
House
16.97
34.74
0.8522
3.08
34.94
0.8865
3.22
34.87
0.9043
3.42
35.91
0.9211
4.93
to
understand
the
loss
of
contrast,
the
residual
images
f
u
100
are
sho
wn
in
Figure
3(g)-(j).
Figure
3(g),
sho
ws
the
result
of
total
v
ariation
inpainting.
It
produces
loss
in
contrast
and
edges
are
also
blurred.
F
ourth
order
PDE
model
ļ¬lls
the
damaged
re
gions
ef
fecti
v
ely
than
TV
model.
Ho
we
v
er
,
edges
are
smoothed.
Y
i
et
al.
model
preserv
es
the
contrast
to
some
e
xtent
only
because
the
fractional
curv
ature
term
is
applied,
which
is
based
on
forw
ard
and
backw
ard
fractional
dif
ferences.
The
proposed
model
uses
fractional
central
dif
ferences.
Hence,
there
is
no
loss
in
contrast
and
edges
are
also
not
blurred
by
the
proposed
model.
When
the
fractional
order
is
1.4,
the
proposed
model
gi
v
es
higher
results
than
other
models
in
terms
of
PSNR,
SSIM
,
and
MI
also
in
visual
quality
.
The
inpainting
techniques
on
dif
ferent
images
with
the
same
mask
are
applied
and
the
simulation
results
are
re
gistered
in
T
able
2.
One
could
observ
e
that,
the
performance
measures
of
proposed
inpainting
technique
achie
v
e
the
highest
v
alues.
5.
CONCLUSIONS
In
this
article,
symmetric
Riesz
fractional
dif
ferential
ļ¬lter
is
applied
to
p
-Laplace
v
ariational
image
in-
painting.
Fractional
order
v
ariational
inpainting
models
restored
superior
to
inte
ger
order
v
ariational
models.
The
symmetric
Riesz
ļ¬lter
possesses
non-local
property
,
anti-rotational
property
,
and
inpainting
re
gion
is
ļ¬lled
based
on
the
fractional
central
curv
ature
term.
It
uses
forw
ard,
backw
ard,
and
fractional
central
dif
ferences.
Therefore,
this
model
pro
vides
the
ef
fecti
v
e
image
inpainting
and
o
v
ercomes
the
unintended
visual
ef
fects.
The
simulation
results
display
that
the
performance
of
the
proposed
model
is
e
xceeding
inte
ger
order
v
ariational
models
and
Y
i
et
al.
model
[9].
REFERENCES
[1]
A.
A.
Efros
and
T
.
K.
Leung,
āT
e
xture
synthesis
by
non-parametric
sampling,
ā
in
The
Pr
oceedings
of
the
Se
venth
IEEE
International
Confer
ence
on
Computer
V
ision
,
v
ol.
2.
IEEE,
1999,
pp.
1033ā1038.
[2]
B.
Marcelo,
S.
Guillermo,
C.
V
incent,
and
B.
Coloma,
āImage
inpainting,
ā
in
Pr
oceedings
of
the
27th
annual
confer
ence
on
Computer
gr
aphics
and
inter
active
tec
hniques
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A
CM
Press/Addison-W
esle
y
Publishing
Co.,
2000,
pp.
417ā424.
[3]
T
.
Ale
xandru,
ā
An
image
inpainting
technique
based
on
the
f
ast
marching
method,
ā
J
ournal
of
gr
aphics
tools
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v
ol.
9,
no.
1,
pp.
23ā34,
2004.
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S.
Masnou
and
J.-M.
Morel,
āLe
v
el
lines
based
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ā
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ence
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Ima
g
e
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ocessing
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1998
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IEEE,
1998,
pp.
259ā263.
[5]
C.
F
.
T
on
y
and
S.
Jianhong,
āNonte
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BIOGRAPHIES
OF
A
UTHORS
G
Sride
vi
recei
v
ed
B.T
ech
de
gree
in
Electronics
and
Communic
ation
Engineering
from
Nag
arjuna
Uni
v
ersity
,
Andhr
a
Pradesh,
India
and
Masters
de
gree
from
Ja
w
aharlal
Nehru
T
echnological
Uni-
v
ersity
,
Kakinada,
Andhra
Pradesh,
India
in
2000
and
2009
respecti
v
ely
.
She
is
currently
pursuing
her
Ph.D
in
Ja
w
aharlal
Nehru
T
echnological
Uni
v
ersity
,
Kakinada.
Her
areas
of
interest
are
Digital
image
processing
and
Digital
signal
processing.
She
has
more
than
14
years
of
teaching
e
xpe-
rience.
She
is
presently
w
orking
as
an
Associate
professor
in
the
department
of
Electronics
and
Communication
Engineering
in
Aditya
Engineering
Colle
ge,
Surampalem,
Andhra
Pradesh.
She
is
the
member
of
Institution
of
Electronics
and
T
elecommunication
Engineers.
S
Srini
v
as
K
umar
is
w
orking
as
a
Professor
in
the
department
of
Electronics
and
Communication
Engineering
and
Director
(Research
and
De
v
elopment),
JNTU
Colle
ge
of
Engineering,
Kakinada,
India.
He
recei
v
ed
his
M.T
ech.
from
Ja
w
aharlal
Nehru
T
echnological
Uni
v
ersity
,
Hyderabad,
India.
He
recei
v
ed
his
Ph.D.
from
E
&
ECE
Department,
IIT
Kharagpur
.
He
has
twenty
eight
years
of
e
xperience
in
teaching
and
research.
He
has
publi
shed
more
than
50
research
papers
in
National
and
International
journals.
Fi
v
e
Research
scholars
ha
v
e
com
pleted
their
Ph.
D
and
presently
11
research
scholars
are
w
orking
under
his
supervisi
on
in
the
areas
of
Image
processing
and
P
attern
recognition.
His
research
interests
are
Digital
image
processing,
Computer
vision,
and
application
of
Artiļ¬cial
neural
netw
orks
and
Fuzzy
logic
to
engineering
problems.
p-Laplace
V
ariational
Ima
g
e
Inpainting
Model
Using
Riesz
F
r
actional
Dif
fer
ential
F
ilter
(G
Sride
vi)
Evaluation Warning : The document was created with Spire.PDF for Python.