Indonesian
J
our
nal
of
Electrical
Engineering
and
Computer
Science
V
ol.
23,
No.
3,
September
2021,
pp.
1440
1450
ISSN:
2502-4752,
DOI:
10.11591/ijeecs.v23.i3.pp1440-1450
r
1440
Image
mixed
gaussian
and
impulse
noise
elimination
based
on
sparse
r
epr
esentation
model
Ahmed
Abdulqader
Hussein
1
,
Sabahaldin
A.
Hussain
2
,
Ahmed
Hameed
Reja
3
1,3
Department
of
Electromechanical
Engineering,
Uni
v
ersity
of
T
echnology
,
Baghdad,
Iraq
2
Computer
Science
Department,
Colle
ge
of
Science,
Aljufra
Uni
v
ersity
,
Libya
Article
Inf
o
Article
history:
Recei
v
ed
Dec
7,
2020
Re
vised
Jul
28,
2021
Accepted
Aug
4,
2021
K
eyw
ords:
Gaussian
noise
Image
denoising
Impulse
noise
Mix
ed
noise
Sparse
representation
model
ABSTRA
CT
A
modified
mix
ed
Gaussian
plus
impulse
image
denoising
algorithm
based
on
weighted
encoding
with
image
sparsity
and
nonlocal
self-similarity
priors
re
gulariza-
tion
is
proposed
in
this
paper
.
The
encoding
weights
and
the
priors
imposed
on
the
im-
ages
are
incorporated
int
o
a
v
ariational
frame
w
ork
to
treat
more
comple
x
mix
ed
noise
distrib
ution.
Such
noise
is
characterized
by
hea
vy
tails
caused
by
impulse
noise
which
needs
to
be
eliminated
through
proper
weighting
of
encoding
residual.
The
outliers
caused
by
the
impulse
nois
e
has
a
significant
ef
fect
on
the
encoding
weights.
Hence
a
more
accurate
residual
encoding
error
initialization
pl
ays
the
important
role
in
o
v
erall
denoising
performance,
especially
at
high
im
pulse
nois
e
rates
.
In
this
paper
,
outliers
free
initialization
image,
and
an
easier
to
implement
a
parameter
-free
procedure
for
updating
encoding
weights
ha
v
e
been
proposed.
Experimental
results
demonstrate
the
capability
of
the
proposed
strate
gy
to
reco
v
er
images
highly
corrupted
by
mix
ed
Gaus-
sian
plus
impulse
noise
as
compared
with
the
state
of
art
denoising
algorithm.
The
achie
v
ed
results
moti
v
ate
us
to
implement
the
proposed
algorithm
in
practice.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Ahmed
Abdulqader
Hussein
Department
of
Electromechanical
Engineering
Uni
v
ersity
of
T
echnology
,
Baghdad,
Iraq
Email:
ahmedabdulqaderhussein@gmail.com
1.
INTR
ODUCTION
In
v
arious
image
applications,
an
image
is
ine
vitably
polluted
by
noise
of
dif
ferent
types.
Mix
ed
noise
could
occur
when
an
image
that
has
already
been
contaminated
by
Gaussian
noise
in
the
procedure
of
image
acquisition
with
f
aulty
equipment
suf
fers
impulsi
v
e
corruption
during
its
transmission
o
v
er
noisy
channels
successi
v
ely
[1].
The
main
objecti
v
e
of
an
y
denoising
algorithm
is
to
remo
v
e
noise
as
totally
as
possible
and
to
preserv
e
image
edges
as
completely
as
possible.
In
a
vie
w
of
rele
v
ant
studies,
se
v
eral
denoising
algorithms
ha
v
e
been
implemented
for
denoising
a
corrupted
images
by
either
impulse
noise
or
Gaussian
noise
[2]-[6].
While,
fe
w
rele
v
ant
w
orks
ha
v
e
been
proposed
in
order
to
remo
v
e
mix
ed
noise
ef
ficaciously
due
to
the
distinct
characteristics
of
both
kinds
of
de
gradation
processes.
Generally
,
enhanced
Gaussian
denoising
algorithms
are
considered
to
be
insuf
ficient
for
suppressing
the
impulse
noise
due
to
the
noisy
pix
els
are
translated
as
an
edges
and
should
be
preserv
ed.
Meanwhile,
the
strate
gies
for
impulse
noise
elimination
detect
the
impulse
pix
els
and
replace
them
with
estimated
v
alues
while
lea
ving
the
remaining
pix
els
unchanged.
Along
these
lines,
most
Gaussian
noise
in
the
restored
images
wil
l
be
k
ept
leading
to
grain
y
and
unsatisf
actory
visual
results.
Thus,
a
specific
algorithm
needs
to
be
designed
for
standing
ag
ainst
mix
ed
Gaussian
plus
impulse
noise
[7],
[8].
The
sparse
representation
(SR)
has
pro
v
ed
its
ef
fecti
v
eness
in
dealing
with
v
arious
image
processing
J
ournal
homepage:
http://ijeecs.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
1441
fields
[9]-[11].
The
algorithms
base
d
on
SR,
benefit
from
the
sparse
nature
of
the
images
which
can
be
well
restricted
by
a
linear
combination
of
only
fe
w
atoms
of
transformed
domain
called
dictionary
.
The
main
weak-
ness
of
SR
model
based
denoising
algorithms
is
its
incapability
in
dealing
with
mix
ed
noise
due
to
complicated
properties
of
mix
ed
noise
distrib
ution.
T
o
o
v
ercome
SR
model
limitations,
a
dedicated
algorithm
need
to
be
carefully
designed
to
deal
with
hea
vy
tails
of
mix
ed
noise
caused
by
impulse
noise.
Inte
grating
the
SR
into
total
v
ariation
frame
w
ork
with
proper
encoding
residual
weighting,
open
the
door
for
appearing
algori
thms
capable
of
eli
minating
mix
ed
noise
ef
ficiently
.
A
weighted
encoding
with
sparse
nonlocal
re
gularization
model
named
(WESNR)
w
as
successfully
applied
in
[9].
The
main
goal
of
this
paper
is
to
enhance
the
performance
of
WESNR
algorithm
firstly
by
initial
izing
the
algorithm
with
a
more
accurate
image
to
ensure
proper
residual
encoding
error
initialization
and
secondly
by
suggesting
simpler
weight
updating
procedure
that
ensures
proper
weighting
of
encoding
residual
to
guarantee
cancelling
the
hea
vy
tails
of
mix
ed
noise
distrib
ution
induced
by
impulse
noise.
The
achie
v
ed
results
sho
w
that
the
proposed
algorithm
e
xhibits
impro
v
ed
performance
in
dealing
with
images
highly
corrupted
by
mix
ed
Gaussian
plus
impulse
noise
as
well
as
f
acilitating
the
practical
implementation
conditions.
The
rest
of
the
paper
is
or
g
anized
as
follo
ws.
Related
w
ork
is
introduced
in
Section
2.
Section
3
describes
the
mix
ed
Gaussian
plus
impul
se
noise
model
used
in
this
paper
.
Section
4
briefly
re
vie
ws
the
core
of
sparse
denoising
model.
The
proposed
image
denoising
al
gorithm
is
e
xplained
In
Section
5.
In
section
6,
the
results
of
the
proposed
denoising
algorithm
is
compa
red
with
WESNR
[9].
Finally
,
the
concluding
remarks
are
gi
v
en
in
section
7.
2.
RELA
TED
W
ORK
A
general
adapti
v
e
frame
w
ork
for
detection
and
elimination
dif
ferent
types
of
noise,
in
v
olving
Gaus-
sian
noise,
impulse
noise
and
more
importantly
their
mixtures
has
been
proposed
in
[12].
The
method
is
implemented
based
on
maximum
lik
elihood
estimation
(MLE)
approach
as
well
as
sparse
representati
on
s
o
v
er
a
trained
dictionary
.
A
pat
ch-based
approach
for
remo
ving
mix
ed
noise
based
on
sparse
representation
has
been
presented
in
[13].
An
optimization
problem
is
formulated
based
on
`
1
-norm
and
`
0
-quasiy-norm
penalties.
The
sparse
representation
in
a
dictionary
and
sparsity
of
residual
is
enforced
by
`
0
-
`
1
penalties
respecti
v
ely
.
The
dictionary
is
learned
using
independent
component
analysis
(ICA).
The
image
is
denoised
iterati
v
ely
using
a
combination
of
soft
and
hard
thresholding.
Results
sho
w
that
the
proposed
method
g
a
v
e
good
results
in
terms
of
SSIM
quantitati
v
e
measure
for
images
rich
with
fine
details.
Ho
we
v
er
,
the
performance
of
the
proposed
algorithm
is
not
v
erified
under
hea
vily
mix
ed
noise
de
gradation.
A
weighted
encoding
with
sparse
nonlocal
re
gularization
(WESNR)
that
can
deal
with
images
corrupted
by
mix
ed
noise
has
been
implemented
in
[9].
W
eighted
encoding
is
utilized
in
order
to
handle
the
impul
se
and
Gaussian
noise
jointly
.
The
prior
image
sparsity
as
well
as
prior
nonlocal
self-similarity
ha
v
e
been
consolidated
by
means
of
a
re
gularization
term
and
inserted
into
the
v
ariational
encoding
mechanism.
The
major
dra
wbacks
of
this
algorithm
is
its
incapability
of
dealing
with
highly
corrupted
images.
A
nonparametric
Bayesian
model
to
solv
e
sparse
outlier
has
been
proposed
in
[14],
[15].
This
strate
gy
pro
v
es
its
ef
fecti
v
eness
in
treating
mix
ed
Gaussian
plus
impulse
(salt
and
pepper)
noise.
The
noisy
image
is
considered
to
be
composed
of
three
terms
namely
clean
image,
Gaussian
noise
image,
and
sparse
impulse
noise
image.
The
model
emplo
ys
spik
e-slob
sparse
prior
to
recognize
sparse
coef
ficients
of
the
real
data
term
and
outlier
noise.
An
algorithm
for
jointly
denoising
h
yperspectral
images
corrupted
by
mix
ed
Gaussian
plus
impulse
noise
has
ben
proposed
in
[16]-[18].
The
proposed
algorithm
a
v
ails
the
inherent
spatial
and
spectral
correlation
of
such
images.
The
denoising
problem
is
formulated
based
on
synthesis
prior
(SP)
and
solv
ed
using
Split-Bre
gman
algorithm.
A
ne
w
statistical
re
gularization
term
called
joint
adapti
v
e
statistical
prior
(J
ASP)
has
been
prese
n
t
ed
in
[19].
The
proposed
strate
gy
is
incorporated
into
v
ariational
scheme.
Moreo
v
er
,
the
proposed
v
ariational
model
is
solv
ed
using
Split-Bre
gman
iterati
v
e
algorithm
to
reco
v
er
images
corrupted
by
mix
ed
Gaussian
plus
impulse
noise.
A
weighted
joint
sparse
represe
ntation
model
called
WJSR
to
reco
v
er
images
corrupted
by
mix
ed
noise
has
been
presented
in
[10].
The
model
group
similar
image
patches
and
solv
ed
for
global
optimal
solution
using
weighte
d
simultaneous
orthogonal
matching
pursuit
(W
-SOMP).
The
weights
are
included
to
stands
ag
ainst
impulse
noise
which
abolishes
image
patch
similarities.
The
WJSR
is
inte
grated,
with
the
global
and
sparse
error
priors,
into
v
ariational
frame
w
ork
to
ef
ficiently
deal
with
mix
ed
noise
corruption.
The
major
dra
wback
of
this
algorithm
is
its
computation
comple
xity
due
to
the
usage
of
greedy
algorithm
W
-SOMP
.
Ima
g
e
mixed
gaussian
and
impulse
noise
elimination
based
on
spar
se
...
(Ahmed
Abdulqader
Hussein)
Evaluation Warning : The document was created with Spire.PDF for Python.
1442
r
ISSN:
2502-4752
An
iterati
v
e
non-con
v
e
x
lo
w
rank
matrix
approximation
(NonLRMA)
model
for
denoising
h
yper
-
spectral
image
(HIS)
corrupted
by
mix
ed
noise
has
been
proposed
in
[20]-[22].
NonLRMA
decomposes
the
de
graded
HSI
image
into
a
lo
w
rank
component
and
a
sparse
term.
Results
pro
v
e
the
capability
of
the
proposed
algorithm
to
preserv
e
lar
ge-scale
image
structures
and
small-scale
details
o
v
er
a
wide
range
of
image
bands
de
gradations.
An
adapti
v
e
Median
based
Non-local
Lo
w
Rank
Approximation
(AMNLRA)
approach
for
de-
noising
images
polluted
by
mix
ed
g
aussian
plus
impulse
noise
has
been
proposed
in
[23].
The
proposed
method
is
based
on
eliminating
the
ef
fect
of
impulsi
v
e
noise
on
the
mix
ed
noise
probability
distrib
ution
to
reco
v
er
the
Gaussian
probabili
ty
distrib
ution.
T
o
locate
the
position
of
pix
els
contaminated
by
impulse
noise,
an
adapti
v
e
median
filter
AMF
[24]
w
as
used.
According
to
data
redundanc
y
,
these
identified
pix
els
are
then
processed
using
non-local
mean
filtering
(NLM)
to
con
v
ert
the
residual
noise
into
a
Gaussian
noise
distrib
ution.
Finally
,
the
pre-processed
image
has
been
processed
using
reduced
rank
optimization
to
obtain
the
denoised
image.
3.
PR
OBLEM
FORMULA
TION
In
a
vie
w
of
image
denoising
process,
the
elimination
of
mix
ed
noise
is
more
complicated
than
the
standardized
si
n
gl
e
noise.
As
a
rule,
this
noise
is
sampled
from
v
arious
distrib
utions.
In
this
paper
,
the
proposed
strate
gy
gi
v
es
more
attention
for
image
denoising
of
mix
ed
noise
which
specified
by
a
combination
of
Gaussian
and
impulse
noise.
Ob
viously
,
it
is
a
considerable
problem
that
can
be
modeled
as
(1):
y
=
I
M
P
(
x
+
n
)
(1)
where,
x
,
y
and
n
denotes
the
original
image
pix
el,
corrupted
image
pix
el
and
additi
v
e
white
Gaussian
noise
respecti
v
ely
.
The
symbol
I
M
P
,
denotes
the
corruption
process
by
salt
and
pepper
impulse
noise.
4.
SP
ARSE
DENOISING
MODEL
Let
us
denote
an
image
by
x
2
R
N
.
The
e
xtracted
image
patch,
can
be
represented
as
(2):
x
j
=
R
j
x
(2)
where
x
j
the
stretched
v
ector
of
an
image
is
a
patch
of
size
p
n
p
n
and
R
j
is
the
matrix
e
xtraction
op-
erator
at
location
j
.
The
goal
of
sparse
representati
on
theory
is
to
find
an
o
v
er
complete
dictionary
D
=
[
d
1
;
d
2
;
:::::::
:d
n
]
2
R
n
m
to
sparsely
code
x
j
where
d
j
2
R
n
is
the
j
th
atom
of
D
.
Based
on
the
deri
v
ed
dictionary
D
,
the
image
patch
can
be
re
written
according
to
(3)
[9]:
x
j
=
D
j
(3)
where
j
is
a
sparse
coding
v
ector
.
The
least
square
solution
of
x
can
be
e
xpressed
as
(4):
x
=
D
(4)
where
is
the
set
of
all
coding
v
ectors
j
.
In
case
of
A
WGN
corrupted
image
y
,
can
be
e
xpressed
as
(5)
[9]:
^
=
ar
g
min
k
y
D
k
2
2
+
R
(
)
(5)
where
R
(
)
and
are
the
re
gularization
term
and
re
gularization
parameter
that
ensure
maximum
a
posteriori
probability
(MAP)
solution
of
for
A
WGN
model.
Ho
we
v
er
,
for
mix
ed
noise
remo
v
al,
MAP
solution
is
unreachable
due
to
hea
vy
tail
caused
by
impulse
noise.
T
o
deal
with
mix
ed
noise
remo
v
al,
a
ne
w
model
named
weighted
encoding
with
sparse
nonlocal
re
gularization
(WESNR)
has
been
proposed.
According
to
WESNR,
The
(5)
is
re
written
as
[9]:
^
=
ar
g
min
W
1
2
(
y
D
)
2
2
+
R
(
)
(6)
where
W
is
a
diagonal
weights
matrix
its
elements
W
j
j
is
deri
v
ed
from
residual
error
e
j
=
(
y
D
)
j
as
[9]:
W
j
j
=
exp
(
c
e
2
j
)
(7)
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
23,
No.
3,
September
2021
:
1440
–
1450
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
1443
where
c
is
a
po
s
iti
v
e
constant
to
control
the
decreasing
rate
of
W
j
j
.
The
weights
W
are
adapti
v
ely
updated
in
each
iteration.
The
pix
el
corrupted
by
impulse
noise
is
assigned
small
weight
close
to
zero,
while
the
uncorrupted
pix
el
is
left
unchanged
through
assigning
weight
close
to
1.
This
process
will
eliminate
the
ef
fect
of
hea
vy
tails
caused
by
impulse
noise
and
hence
ensure
MAP
solution
of
(6)
to
reco
v
e
r
images
corrupted
by
mix
ed
Gaussian
plus
impulse
noise.
The
updating
of
W
depends
on
the
coding
residual
error
e
.
In
[9],
the
initialized
image
x
(0)
for
salt
and
pepper
impulse
noise
is
obtained
by
applying
AMF
[24]
to
y
.
Thus,
the
residual
error
is
initialized
according
to
(8).
e
(0)
=
y
x
(0)
(8)
Inspired
by
the
w
ork
in
[25],
local
sparsity
and
nonlocal
self-similarity
are
the
tw
o
priors
adopted
in
WESNR
to
model
re
gularization
term
R
(
)
yields
to
(9):
^
=
ar
g
min
W
1
2
(
y
D
)
2
2
+
k
k
1
(9)
as
proposed
in
[9],
the
(9)
w
as
solv
ed
via
iterati
v
ely
re
weighted
least
square
minimization
scheme
which
has
been
proposed
in
[26]
using
the
(10):
v
j
j
(
k
+
1)
=
p
(
j
(
k
)
j
)
2
+
"
(
k
)
2
(10)
where
j
(
k
)
and
"
(
k
)
is
the
j
th
element
of
coding
v
ector
and
numerical
stability
parameter
in
the
k
th
iteration
respecti
v
ely
.
Hence,
"
(
k
)
as
well
as
are
updated
according
to
(11)
and
(12)
respecti
v
ely
[9].
"
(
k
+
1)
=
min
(
"
(
k
)
;
median
(
j
(
k
)
j
))
(11)
(
k
+
1)
=
(
D
T
W
D
+
V
(
k
+
1))
1
(
D
T
W
y
D
T
W
D
)
+
(12)
5.
THE
PR
OPOSED
ALGORITHM
In
WESNR
[9],
the
weights
W
ha
v
e
the
important
role
in
attenuating
the
ef
fect
of
hea
vy
tails
triggered
by
impulse
noise
and
hence
retrie
v
e
back
the
distrib
ution
of
mix
ed
Gaussian
plus
impulse
noise
to
resemble
Gaussian
noise
distrib
ution.
This
will
pa
v
e
the
w
ay
for
sparse
modeling
technique
to
implement
denoising
algorithms
that
can
deal
with
images
corrupted
by
mix
ed
noise
without
e
xplicit
impulse
detection
phase.
Re-
ferring
to
(6),
the
accurac
y
of
estimated
sparse
v
ector
^
and
t
he
diagonal
weight
matrix
W
are
highly
af
fected
by
residual
encoding
error
which
in
turn
depends
upon
the
ini
tialized
image
x
(0)
.
Thus,
better
performance
is
e
xpected
when
more
accurate
image
is
used
for
initialization.
Generally
,
the
salt
and
pepper
impulse
noise
tak
e
either
minimum
or
maximum
v
alues
in
the
dynamic
range
of
the
image
(e.g.
[0,255]).
Thus,
we
need
to
process
the
pix
els
that
tak
e
these
e
xtreme
v
alues.
The
question
raised
no
w
is
ho
w
to
replace
the
corrupted
pix
el
in
such
a
w
ay
that
ensures
resemblance
with
its
noise
free
neighbor
pix
els
and
hence
dimensions
the
ef
fect
of
outliers.
Intuiti
v
ely
,
critical
replacements
occurs
when
the
test
windo
w
u;z
centered
at
pix
el
p
u;z
filled
with
impulses.
T
o
solv
e
this
problem,
we
suggest
to
adopt
the
Rob
ust
Outlier
Exclusion
(R
OE)
proposed
in
[27].
In
this
w
ork
the
test
windo
w
size
is
e
xpanded
to
the
ne
xt
higher
one
(typically
4
4)
and
apply
the
trimmed
mea
n
which
is
characterized
by
its
rob
ustness
ag
ainst
out
liers.
The
conception
for
limiting
the
maximum
test
windo
w
size
to
4
4
is
to
boost
the
estimated
accurac
y
without
influencing
the
significant
correlation
characteristics
of
neighbor
pix
els.
As
stated
in
[27],
25
%
of
the
outliers
e
xcl
usion
accomplishes
a
po
werful
estimatation
for
reco
v
ering
the
corrupted
pix
els.
Algorithm
1
sho
ws
the
initialization
algorithm
steps
according
to
R
OE
[27].
Ima
g
e
mixed
gaussian
and
impulse
noise
elimination
based
on
spar
se
...
(Ahmed
Abdulqader
Hussein)
Evaluation Warning : The document was created with Spire.PDF for Python.
1444
r
ISSN:
2502-4752
Algorithm
1
Initialization
algorithm
Input:
y
,
an
image
corrupted
by
impulse
noise
u;z
Slide
windo
w
centered
at
p
u;z
of
Size
M
M
Where
M
is
an
Odd
number
typically
3
Output:
x
(0)
,
the
initialized
image
f
or
each
non
border
pix
el
p
(
i;
j
)
2
y
do
if
0
<
p
u;z
<
255
then
The
pix
el
is
uncorrupted.
Hence,
left
unchanged
else
The
pix
el
is
corrupted,
do
the
follo
wing:
Exclude
all
noisy
pix
els
in
u;z
Compute
median
v
alue
p
median
Set
p
u;z
=
p
median
if
u;z
is
filled
of
impulse
noise
then
Replace
u;z
by
u;z
of
size
M+1
M+1
Exclude
all
noisy
pix
els
in
u;z
Compute
T
rim
Mean
in
u;z
with
25
%
e
xclusion
p
tr
im
Set
p
u;z
=
p
tr
im
end
if
if
u;z
is
filled
of
impulse
noise
then
Compute
the
Mean
in
u;z
p
mean
Set
p
u;z
=
p
mean
end
if
end
if
end
f
or
Output
the
initialized
image
x
(0)
The
initialization
image
x
(0)
is
no
w
used
to
calculate
the
residual
encoding
error
which
plays
i
mpor
-
tant
role
in
updating
the
weight
matrix
W
.
Referring
to
(7),
the
assigned
weights
W
is
firstly
sensiti
v
e
to
the
v
alue
of
the
constant
c
which
controlli
ng
the
decreasing
rate
of
the
weights
and
secondly
use
trigonometric
e
xponential
function.
T
o
modify
the
algorithm
to
suit
the
implementation
requirements,
it
is
usually
best
to
a
v
oid
the
use
of
mathematical
functions,
in
this
case,
e
xponential
functions.
Hence,
our
goal
is
to
replace
the
e
xponential
function
while
retaining
the
properties
of
its
e
x
ecution.
Referring
to
(7),
impulse
noise
will
cause
a
significant
increase
in
residual
error
,
which
will
lead
to
the
assignation
of
weight
close
to
zero
(
W
!
0
),
thereby
eliminating
the
impact
of
hea
vy
tails
caused
by
impulse
noise.
Otherwise,
if
t
he
pix
el
has
no
impulse
noise,
the
weight
will
be
close
to
one
(
W
!
1
).
No
w
to
achie
v
e
our
goal,
assuming
a
is
a
positi
v
e
v
alue
greater
than
zero
and
less
or
equal
to
1,
then
(1
a
)
2
!
0
when
a
!
1
and
(1
a
)
2
!
1
when
a
!
0
.
The
function
(1
a
)
2
has
the
same
properties
as
the
e
xponential
function.
Consequently
,
the
diagonal
weight
matrix
W
can
be
updated
according
to
(13).
W
=
(1
a
)
2
0
6
a
6
1
(13)
The
constraints
imposed
on
the
v
ariable
a
in
(13)
can
be
r
elated
to
the
absolute
normalized
residual
coding
error
(
j
e
nor
m
j
)
.
Accordingly
,
the
weight
matrix
W
is
updated
through
the
follo
wing
steps:
1.
Calculate
absolute
residual
error:
e
(
k
)
=
j
(
y
D
(
k
))
j
(14)
2.
Normalize
the
error
by
element
wise
di
vision:
e
nor
m
(
k
)
=
e
(
k
)
max
(
e
(
k
))
(15)
3.
Update
weight
diagonal
matrix:
W
j
j
(
k
)
=
(1
e
nor
m
(
k
)
)
2
(16)
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
23,
No.
3,
September
2021
:
1440
–
1450
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
1445
Re
g
arding
to
the
case
of
impulsi
v
e
noise
which
is
specified
by
(16),
the
normalized
error
(
j
e
nor
m
j
)
approaches
1
and
thus
a
v
alue
close
to
zero
will
be
assigned
to
the
encoding
weights
W
j
j
which
eliminate
the
ef
fect
of
the
impulsi
v
e
noise.
Hence,
ensuring
MAP
solution
of
(6)
in
order
to
reco
v
er
images
corrupted
by
mix
ed
Gaussian
plus
impulse
noise.
Moreo
v
er
,
compared
with
(7)
and
(16)
has
the
adv
antage
of
being
easier
to
implement
in
practice
by
a
v
oiding
the
use
of
an
e
xponential
function.
Therefore,
there
is
no
need
to
calibrate
the
parameter
c
.
By
follo
wing
the
steps
outlined
abo
v
e,
Algorithm
2
demonstrates
the
modification
of
the
proposed
WESNR
based
mix
ed
Gaussian
plus
impulse
image
denoising
algorithm.
Algorithm
2
Proposed
mix
ed
noise
denoising
algorithm
Input:
Dictionary
D
,
noisy
image
y
,
and
number
of
iterations
L
Initialization
:
Apply
R
OE
[27]
to
y
in
order
to
obtain
the
initialized
image
x
(0)
Calculate
residual
error
e
(
k
=
0)
using
(14)
Initialize
to
zero
Output
:
Denoised
image
x
f
or
k=1,2,.......
L
do
1-
Compute
(
k
)
by
(12)
2-
Compute
x
(
k
)
=
D
(
k
)
3-
Update
the
nonlocal
coding
v
ector
=
D
T
x
(
k
)
4-
Compute
the
residual
error
using
(14)
5-
Normalize
the
residual
error
using
(15)
6-
Update
weight
matrix
W
using
(16)
end
f
or
Output
the
denoised
image
x
=
D
(
L
)
6.
EXPERIMENT
AL
RESUL
TS
The
proposed
algorithm
is
e
v
aluated
on
a
set
of
standard
images
ha
ving
distinctly
dif
ferent
features
of
size
512×512.
These
images
are
artificially
polluted
firstly
by
Gaussian
noise
with
standard
de
viation
of
10,
20,
and
30
and
then
30
%
,
50
%
,
and
70
%
of
image
pix
els
are
replaced
by
sal
t
and
pepper
impulsi
v
e
noise.
Objecti
v
ely
,
the
obtained
res
ults
are
e
v
aluated
using
peak
signal
to
noise
ratio
(PSNR)
and
the
structural
similarity
inde
x
(SSIM)
[28].
Moreo
v
er
,
subjecti
v
e
e
v
aluation
is
included
to
pro
v
e
the
quality
of
the
reco
v
ered
images.
The
proposed
algorithm
is
compared
with
WESNR
[9].
The
parameters
of
WESNR
are
set
to
its
def
ault
v
alues
as
proposed
in
[9].
The
parameters
are
c
=
0.0008,
(0)
=
0.0001
to
reduce
the
ef
fect
of
impulse
noise
on
block-matching
then
=
1
for
>
10
or
=
0.5
for
6
10
,
"
(0)
=
0
:
1
and
updated
in
accordance
with
(11).
The
image
patches
are
encoded
o
v
er
a
set
of
of
fline
learned
PCA
based
dictionaries.
The
proposed
algorithm
has
the
same
parameters
as
WESNR
e
xcept
of
the
parameter
c
which
has
been
discarded.
The
number
of
iterations
L
is
k
ept
fix
ed
at
8
for
all
e
xperiments.
6.1.
Objecti
v
e
e
v
aluation
The
denoising
relati
v
e
performance
in
terms
of
PSNR
and
SSIM
for
an
a
v
erage
of
fi
v
e
runs
using
a
set
of
images
are
recorded
in
T
able
2
(the
v
alue
in
parenthesis
is
the
SSIM).
According
to
the
performance
criteria,
the
superior
outcomes
are
referred
with
a
bold
font.
Re
g
arding
to
the
results
e
xamination
of
T
able
1,
the
proposed
algorithm
outperforms
WESNR
algorithm
most
of
the
time
in
terms
of
indi
vidual,
a
v
erage
PSNR
as
well
as
SSIM
v
alues
o
v
er
the
whole
scope
of
noise
le
v
els
and
images
under
test.
In
addition,
by
carefully
checking
the
obtained
results
in
T
able
2,
it
is
noticed
that
as
the
le
v
el
of
mix
ed
noise
increases
,
the
proposed
algorithm
e
xhibits
higher
performance
compared
to
WESNR.
As
an
illustration,
Figure
1a,
Figure
1b,
Figure
2a,
and
Figure
2b
demonstrate
the
relati
v
e
denoising
of
both
algorithms
in
terms
of
PSNR
and
SSIM
for
=30,
sp
=30
%
,
and
=30,
sp
=70
%
respecti
v
ely
.
The
achie
v
ed
results
pro
v
e
the
enhanced
performance
and
the
ef
fecti
v
eness
of
the
modification
inserted
on
the
con
v
entional
WESNR
algorithm
namely
starting
with
a
more
accurate
initialized
image
x
(0)
.
The
more
accurate
initializat
ion
of
the
proposed
algorithm
has
been
reflected
in
increasing
the
accurac
y
of
estimating
the
initialized
weight
matrix
W
.
Furthermore,
the
proposed
function
for
updating
the
diagonal
weight
matrix
W
in
(16)
pro
v
es
its
ability
to
attenuate
the
ef
fect
of
hea
vy
tails
caused
by
impulse
noise
considerably
.
Therefore,
a
MAP
solution
for
reco
v
ering
images
corrupted
by
mix
ed
Gaussian
plus
impulse
noise
has
been
realized.
Ima
g
e
mixed
gaussian
and
impulse
noise
elimination
based
on
spar
se
...
(Ahmed
Abdulqader
Hussein)
Evaluation Warning : The document was created with Spire.PDF for Python.
1446
r
ISSN:
2502-4752
T
able
1.
PSNR
(SSIM)
denoising
results
Images
sp
30
%
sp
50
%
sp
70
%
WESNR
Pr
oposed
WESNR
Pr
oposed
WESNR
Pr
oposed
Lena
10
33.31
(0.879)
33.84
(0.881)
31.79
(0.863)
32.50
(0.867)
29.31
(0.830)
29.85
(0.835)
20
30.68
(0.810)
31.03
0.818)
29.61
(0.796)
30.19
(0.807)
27.59
(0.769)
28.17
(0.787)
30
27.23
(0.634)
27.97
(0.657)
26.00
(0.603)
27.94
(0.633)
24.01
(0.577)
26.69
(0.620)
Boat
10
30.56
(0.831)
31.39
(0.845)
28.73
(0.799)
29.56
(0.808)
26.12
(0.736)
26.87
(0.739)
20
28.27
(0.747)
28.68
(0.751)
26.89
(0.719)
27.54
(0.721)
24.86
(0.657)
25.49
(0.667)
30
25.83
(0.619)
26.39
(0.642)
24.34
(0.582)
25.93
(0.635)
22.29
(0.532)
24.45
(0.612)
Hill
10
30.88
(0.815)
31.45
(0.830)
29.77
(0.785)
30.11
(0.791)
27.98
(0.724)
28.13
0.728)
20
28.74
(0.725)
29.01
(0.731)
27.86
(0.699)
28.11
(0.704)
26.39
(0.652)
26.57
(0.654)
30
26.27
(0.599)
26.81
(0.622)
25.22
(0.558)
26.54
(0.618)
23.60
(0.516)
25.62
(0.604)
F16
10
32.47
(0.893)
29.46
(0.889)
30.61
(0.878)
28.84
(0.878)
27.72
(0.846)
27.20
(0.851)
20
30.14
(0.824)
28.18
(0.816)
28.72
(0.813)
27.68
(0.819)
26.32
(0.787)
26.21
(0.806)
30
26.79
(0.627)
26.32
(0.653)
25.65
(0.607)
26.20
(0.635)
23.28
(0.564)
25.22
(0.602)
Monarch
10
35.36
(0.935)
35.78
(0.937)
33.09
(0.930)
34.29
(0.933)
28.72
(0.900)
29.83
(0.913)
20
31.66
(0.860)
32.04
(0.880)
30.03
(0.861)
31.16
(0.870)
26.68
(0.830)
28.10
(0.849)
30
27.51
(0.674)
28.31
(0.804)
25.92
(0.644)
28.24
(0.787)
23.00
(0.600)
26.15
(0.754)
Barbara
10
29.18
(0.874)
31.44
(0.899)
26.99
(0.831)
28.82
(0.860)
23.97
(0.728)
25.21
(0.759)
20
27.13
(0.793)
28.65
(0.822)
25.44
(0.744)
26.65
(0.773)
22.75
(0.651)
23.88
(0.677)
30
24.57
(0.644)
25.96
(0.692)
22.64
(0.587)
25.02
(0.681)
20.49
(0.505)
23.20
(0.624)
House
10
36.97
(0.912)
36.87
(0.902)
36.02
(0.904)
36.21
(0.900)
33.51
(0.886)
33.62
(0.890)
20
33.67
(0.843)
33.76
(0.862)
32.78
(0.837)
33.48
(0.859)
30.84
(0.825)
31.45
(0.848)
30
28.90
(0.630)
29.70
(0.777)
27.83
(0.610)
29.04
(0.713)
25.83
(0.598)
28.31
(0.654)
Couple
10
30.34
(0.846)
31.17
(0.860)
28.77
(0.813)
29.50
(0.824)
26.18
(0.738)
26.77
(0.744)
20
28.04
(0.757)
28.52
(0.765)
26.84
(0.725)
27.32
(0.728)
24.77
(0.641)
25.20
(0.654)
30
25.56
(0.626)
26.12
(0.643)
24.37
(0.592)
25.63
(0.637)
22.51
(0.536)
24.19
(0.596)
Man
10
30.71
(0.841)
31.18
(0.849)
29.36
(0.814)
29.80
(0.818)
27.18
(0.758)
27.61
(0.761)
20
28.51
(0.753)
28.74
(0.755)
27.65
(0.734)
27.91
(0.737)
25.69
(0.679)
26.14
(0.690)
30
26.09
(0.613)
26.60
(0.653)
24.86
(0.577)
26.25
(0.644)
22.97
(0.529)
25.23
(0.636)
Zelda
10
34.73
(0.884)
34.99
(0.886)
33.83
(0.872)
34.06
(0.875)
32.10
(0.847)
32.23
(0.855)
20
31.87
(0.815)
32.14
(0.818)
31.05
(0.802)
31.53
(0.815)
29.52
(0.779)
30.09
(0.807)
30
28.10
(0.645)
29.76
(0.747)
27.14
(0.630)
28.48
(0.717)
25.60
(0.613)
27.71
(0.705)
A
v
erage
10
32.45
(0.871)
32.76
(0.878)
30.90
(0.849)
31.37
(0.855)
28.28
(0.800)
28.73
(0.808)
20
29.87
(0.793)
30.08
(0.802)
28.69
(0.773)
29.16
(0.783)
26.54
(0.727)
27.13
(0.744)
30
26.69
(0.631)
27.39
(0.690)
25.40
(0.600)
26.93
(0.670)
23.36
(0.557)
25.68
(0.641)
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
23,
No.
3,
September
2021
:
1440
–
1450
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
1447
In
order
to
complete
the
picture
as
a
whole,
Figures
3
and
Figures
4
sho
w
the
relati
v
e
a
v
erage
P
SNR
and
SSIM
performance
for
all
images
under
the
influence
of
dif
ferent
mixing
noise
le
v
els.
In
vie
w
of
the
accomplished
results
of
Figures
3
and
4.
the
proposed
algorithm
performance
has
been
dif
ferentiated
at
=
30
compared
to
the
WESNR
algorithm.
Along
these
lines,
more
accurate
processing
of
impulse
noise
through
R
OE
[27]
has
the
greatest
impact
on
the
quality
of
denoised
images,
especially
when
sp
e
xceeds
50
%
.
(a)
(b)
Figure
1.
Relati
v
e
denoising
algorithms
performance
for
=30,
sp
=30
%
;
(a)
PSNR
(dB)
and
(b)
SSIM
(a)
(b)
Figure
2.
Relati
v
e
denoising
algorithms
performance
for
=30,
sp
=70
%
;
(a)
PSNR
(dB)
and
(b)
SSIM
(a)
(b)
Figure
3.
Relati
v
e
a
v
erage
denoising
performance
for
dif
ferent
mix
ed
noise
le
v
els;
(a)
PSNR
(dB)
and
(b)
SSIM
Finally
,
in
order
to
quantitati
v
ely
summarize
the
results
in
the
case
of
high
mix
ed
noise
le
v
els
of
=30,
sp
=70
%
,
the
proposed
algorithm
achie
v
es
an
a
v
erage
PSNR
g
ain
of
approximately
2.32
dB
o
v
er
the
entire
set
of
images
under
se
v
ere
noise
conditions.
Ima
g
e
mixed
gaussian
and
impulse
noise
elimination
based
on
spar
se
...
(Ahmed
Abdulqader
Hussein)
Evaluation Warning : The document was created with Spire.PDF for Python.
1448
r
ISSN:
2502-4752
(a)
(b)
Figure
4.
Relati
v
e
a
v
erage
denoising
algorithms
performance
for
=30;
(a)
PSNR
(dB)
and
(b)
SSIM
6.2.
Subjecti
v
e
e
v
aluation
In
accordance
to
the
performance
analysis
of
the
proposed
approach,
it
is
significant
to
inte
grate
the
visual
comprehension
in
the
relati
v
e
performance
comparison.
Figure
5
illustrates
a
sample
of
images
under
test
synthetically
corrupted
by
dif
ferent
le
v
els
of
mix
ed
Gaussian
plus
impulse
noise.
The
first
column
of
Figure
5
sho
ws
the
original
image
of
Monarch,
Barbara,
and
house
respecti
v
ely
.
The
subsequent
columns
demonstrate
the
noisy
and
the
denoi
sed
images
respecti
v
ely
.
Clearly
,
for
=10
and
sp
=30
%
,
both
algorithms
achie
v
e
nearly
equi
v
alent
performance
(see
Figure
5(a1-a4).
The
performance
of
the
WESNR
is
de
graded
as
the
mix
ed
noise
le
v
el
increased.
The
de
gradations
become
more
ob
vious
when
e
xceeds
20
and
sp
e
xceeds
50
%
.
At
a
such
high
mix
ed
noise
en
vironment,
the
proposed
algorithm
produces
visually
more
pleasant
results
which
can
be
easily
noticed
through
Figure
5(b1-b4)
and
Figure
5(c1-c4).
The
proposed
algorithm
preserv
es
the
te
xture
of
the
image
in
Figure
5(b4)
and
smooths
the
sk
y
in
Figure
5(c4)
more
accurately
as
compared
with
WESNR.
These
results
demonstrate
the
ef
fecti
v
eness
of
the
proposed
procedure
for
updating
the
weights
and
to
the
replacement
of
AMF
[24]
adopted
in
WESNR
by
R
OE
[27]
which
in
combination
reduces
the
ef
fect
of
hea
vy
tails
of
the
impulse
noise,
especially
at
high
impulse
noise
density
rates.
Figure
5.
Denoising
results
of
mix
ed
g
aussian
plus
impulse
noise
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
23,
No.
3,
September
2021
:
1440
–
1450
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
1449
7.
CONCLUSION
This
paper
proposes
a
denoising
algorithm
in
order
to
reco
v
er
images
highly
corrupted
by
mix
ed
Gaus-
sian
plus
impulse
noise.
The
promising
results
sho
w
that
the
proposed
algorithm
preserv
es
image
fine
details
more
accurately
as
compared
with
WESNR
especially
for
highly
corrupted
images.
The
main
contrib
utions
of
the
proposed
algorithm
are
the
emplo
yment
of
a
more
accurate
initialization
image
through
replacing
AMF
by
R
OE
and
the
use
of
a
simple
parameter
-free
procedure
for
updating
encoding
weights
W
by
a
v
oiding
the
use
of
the
e
xponential
function
and
the
need
for
calibrating
the
decaying
rate
parameter
(
c
)
in
accordance
with
the
noise
state
le
v
el.
As
a
quantitati
v
e
e
xample
of
the
impro
v
ement
in
results,
for
=30
and
sp
=70
%
,
the
proposed
algorithm
accomplishes
an
a
v
erage
PSNR
g
ain
of
approximately
2.32dB
o
v
er
the
entire
images
under
test.The
promising
results
of
the
proposed
algorithm
open
the
w
ay
for
further
impro
v
ement,
enabling
it
to
eliminate
more
comple
x
Gaussian
plus
random
v
alued
impulse
noise.
This
issue
is
left
as
a
possible
future
w
ork.
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(Ahmed
Abdulqader
Hussein)
Evaluation Warning : The document was created with Spire.PDF for Python.