TELKOM
NIKA
, Vol.13, No
.3, Septembe
r 2015, pp. 9
96~100
5
ISSN: 1693-6
930,
accredited
A
by DIKTI, De
cree No: 58/DIK
T
I/Kep/2013
DOI
:
10.12928/TELKOMNIKA.v13i3.1809
996
Re
cei
v
ed Fe
brua
ry 27, 20
15; Re
vised
June 25, 20
15;
Accept
ed Jul
y
13, 201
5
A Fractal Image Compression Method Based on Multi-
Wavelet
Yan Feng*, Hua Lu, XiLiang 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
:489
91
293
@q
q.com
A
b
st
r
a
ct
How
to effectiv
ely store
an
d t
r
ans
mit suc
h
mu
lt
i-
med
i
a fi
l
e
s as
imag
e a
nd vi
de
o h
a
s
beco
m
e a
re
se
a
r
ch
ho
tspo
t. Th
e
tra
d
i
t
i
o
n
a
l
com
p
re
ssi
on
a
l
g
o
r
i
t
hm
s ha
ve
a
rel
a
ti
vel
y
l
o
w com
p
re
ssio
n
ra
ti
o
a
n
d
bad
qua
lity of
dec
o
ded
i
m
a
ge,
at
prese
n
t, the fr
actal
i
m
ag
e c
o
mpr
e
ssio
n
method
w
i
th a
hi
g
her c
o
mpress
i
on
ratio fails to
meet the req
u
ire
m
e
n
ts of the pr
actical
a
ppl
icat
ions i
n
the q
u
a
lity of the co
mpresse
d i
m
ag
e
as
w
e
ll as the cod
i
ng a
nd dec
odi
ng time.T
his p
aper int
egr
ates
fractal thought
and multi-w
a
v
e
let transfor
m
an
d
prop
oses a fra
c
tal imag
e co
mpressi
on al
gorit
hm b
a
se
d
on m
u
lti-wavelet
tr
ansform
.
To transform
the image
mo
de
l i
n
to a
c
o
mbi
natio
n
of r
e
lev
ant e
l
e
m
e
n
ts in
t
he
fre
q
uency
do
mai
n
instea
d
of mer
e
ly bui
ldi
ng on
the
found
atio
n of the nei
gh
borh
ood gr
ay
-scal
e
correlati
on
has the ab
ility
to code larg
er image b
l
oc
ks,
eli
m
i
nates
the
possi
bil
i
ty of g
l
oba
l corr
elati
o
n in
the
i
m
a
g
e
an
d i
m
proves
the co
din
g
s
p
e
ed
of the
existi
ng
fractal i
m
ag
e c
o
mpressi
on
al
gorith
m
. T
he e
x
peri
m
e
n
tal r
e
sult show
s tha
t
the alg
o
rith
m propos
ed
in this
pap
er ca
n acc
e
ler
a
te the
co
din
g
sp
eed
of
the pr
es
ent fra
c
tal i
m
a
ge c
o
mpr
e
ssio
n
and
have
certa
i
n
self
-
ada
ptivity w
h
ile
slightly re
duci
ng the q
ual
ity of decodi
ng i
m
a
ge.
Ke
y
w
ords
: mu
lti-w
a
velet, fractal theory, i
m
a
ge co
mpr
e
ssio
n
Copy
right
©
2015 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
The ima
ge
compressio
n tech
nolo
g
y is a techniq
u
e
to use a
s
fe
w bits a
s
po
ssi
ble to
expre
ss th
e image
sign
al from the information s
ource to lowe
ra
s
much
re
sou
r
ce con
s
um
ption
su
ch
as the
freque
ncy
ba
ndwi
d
th o
c
cu
pied
by
the
i
m
age
data, t
he
storage
space a
nd th
e
transmissio
n time as po
ssi
ble for the sa
ke of
the tran
smissio
n
and
storag
e of image si
gnal.
In
fact, there exi
s
ts st
ron
g
co
rrelation b
e
tween the
imag
e pixels an
d su
ch correlati
on ha
s broug
ht
plenty of red
unda
nt information to the image,
which make
s ima
ge com
p
re
ssi
on possibl
e [1]. As
a ne
w im
age
com
p
ression
algo
rithm d
e
v
eloped i
n
th
e pa
st de
ca
d
e
, fractal
ima
ge
comp
re
ssion
method atta
ches g
r
e
a
t importan
c
e to d
i
gging the
se
l
f
-simila
rity in most imag
es
and reali
z
es t
h
e
codi
ng of an i
m
age
with co
mplicate
d
visual ch
arac
te
ri
stics on the
surface via lim
ited coeffici
en
ts
by usin
g the i
t
erated fun
c
ti
on sy
stem an
d som
e
simp
l
e
iteration
rul
e
s. By usi
ng t
hese rul
e
s, th
e
decode
r can
reali
z
e the it
erative
de
co
ding of the o
r
iginal im
age,
therefo
r
e, th
e fractal im
a
ge
comp
re
ssion
algorith
m
can
achi
eve a hi
gherco
mp
ression
ratio th
a
n
othe
r imag
e co
mpressio
n
algorith
m
s [2]
.
However, the fractal image
compressi
on algorithm
still
has many problems in both
theory and
appli
c
ation. For exampl
e
,
during th
e
compressio
n, the com
putation is too
compli
cate
d, the comp
re
ssion time is too long, t
he converg
e
n
c
e
pro
c
e
ss i
s
difficult to predi
ct
and control
a
nd there i
s
bl
ock effect in the hi
gh
com
p
re
ssi
on ratio
.
The bigge
st
proble
m
of the
basi
c
fractal i
m
age
com
p
ression
algo
rithm is th
at it
s
high
comp
re
ssion
ratio i
s
a
t
the co
st of the
huge codin
g
time. It requires glob
al se
a
r
ch o
n
all the domain bl
ocks for every R
block to sea
r
ch
for the optima
l
matchin
g
do
main blo
c
k, theref
o
r
e, the
codi
ng ph
ase
deman
ds m
u
ch
comp
utation
time [3]. It u
s
ually takes
hours to
cod
e
a co
mmon
256x256 im
age, whi
c
h
g
r
eatly affects the
practicability of fractal i
m
age
compressi
on, ther
efore,
numerous i
m
proved algorithms
are try
i
ng
to find a quick
way to accele
rate the codi
ng
sp
ee
d,and neve
r
thele
ss,
the increa
sed
co
ding
spe
ed
come
s togeth
e
r
with the d
e
c
re
ased im
a
ge-rep
r
od
ucti
on qu
ality. Tosu
rmo
unt
the
sho
r
tco
m
ing
s
of the traditional fra
c
tal i
m
age
comp
ression al
gorit
hms, this p
a
per ma
ke
so
me
resea
r
ch on t
he co
ding m
e
thod integ
r
a
t
ing fractal
a
nd wavel
e
t transfo
rm. In e
s
sen
c
e, wav
e
let
transfo
rm
is to an
alyze
th
e si
gnal
in
m
u
lti-re
sol
u
ti
on
or multi-scali
ng, which i
s
very suitable
for
the loga
rith
mic
cha
r
a
c
te
ristic
s
of hu
man-eye visual sy
stem
on the f
r
eq
u
ency p
e
rce
p
tion.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
A Fractal Im
age Com
p
ression Method B
a
se
d on Multi
-
Wavel
e
t (Ya
n
Feng
)
997
Therefore, p
eople
start to
apply wavel
e
t transfo
rm
in the image
comp
re
ssion
and they ha
ve
achi
eved
so
me results. A
s
a
develo
p
m
ent of
singl
e wavelet, m
u
lti-wavel
e
t h
a
s
su
ch
exce
llent
prop
ertie
s
a
s
symmetry,
orthog
onality, sho
r
t
supp
o
r
t and
se
co
n
d
-o
rde
r
vani
shing mo
ment
s,
therefo
r
e, it h
a
s the
advant
age
s ov
er sin
g
le wavelet i
n
the ima
ge
pro
c
e
ssi
ng a
nd it ca
n p
r
ov
ide
a more a
c
curate analysi
s
method [4, 5].
Based
on
the
re
sea
r
ch in
the relevant li
tera
ture, this
pape
r inve
sti
gates the int
egrate
d
method by introdu
cing the
multi-wavele
t transform
a
nd integratin
g the sepa
rat
e
advantage
s of
wavelet
codi
ng an
d fra
c
ta
l codi
ng. Thi
s
pape
r, firs
tly, analyze
s
th
e pri
n
ci
ple a
nd reali
z
ation
of
fractal ima
g
e
comp
re
ssi
o
n
codi
ng. Th
en, it
discu
sse
s
the multi
-
wavel
e
t decompo
sition a
nd
recon
s
tru
c
tio
n
of 2D im
age
s. Base
d
on the a
b
o
ve-me
n
tione
d re
sea
r
ch,
the experi
m
ent
simulatio
n
an
d analysi
s
verify the effectivene
ss a
nd
practicability of this algo
rithm
in this pape
r.
2. Fractal Image Comp
re
ssion Codin
g
2.1. Basic Pr
inciple of Image Comp
re
ssion
The imag
e i
n
formatio
n compressio
n codi
ng is
co
ndu
cted a
c
cordin
g to the
intrinsi
c
statistical
cha
r
acte
ri
stics of
the im
age
si
gnal
and
the visual ch
ara
c
t
e
risti
c
s
of hu
man
b
e
ing
s
. The
statistical ch
ara
c
teri
stics
have sho
w
n
that
there
exists
stro
ng correlatio
n between
the
neigh
borhoo
d
pixels, the
neigh
borhoo
d
lines
and th
e neig
hbo
rho
od fram
es. T
o
use certai
n
codi
ng meth
od to rem
o
ve su
ch
corre
l
ation ca
n re
alize the d
a
ta com
p
re
ssi
on of the imag
e
informatio
n.
This
process is to
re
du
ce
as mu
ch
no
n-correlative
redu
nda
nt inf
o
rmatio
n to t
h
e
image qu
ality and it is an informatio
n-p
r
ese
r
ving
com
p
re
ssi
on codi
ng. Anot
her consi
deration
i
s
that the image is finally watched o
r
j
udge
d by
hu
man eyes
or the obse
r
va
tion instru
me
nt.
Acco
rdi
ng to
the visual
ph
ysiology a
nd
physiol
ogi
cal
cha
r
a
c
teri
stics, ce
rtain im
a
ge di
stortion i
s
allowed in th
e re
store
d
i
m
age
whi
c
h
unde
rgo
e
s t
he co
mpression co
ding a
s
long
as
such
distortio
n
i
s
d
i
fficult to
see
for the
ge
ne
ral au
dien
ce.
This ki
nd
of compressio
n
coding
is a
no
n-
pre
s
e
r
ving
co
ding
be
cau
s
e
it ca
uses certain imag
e inf
o
rmatio
n lo
ss. The
re
sea
r
ch dem
on
strat
e
s
that the mo
re re
gula
r
th
e grayscale
distrib
u
ti
on o
f
the origi
nal
image i
s
, t
he st
ron
ger
the
stru
ctural of the imag
e
co
n
t
ents is, the
more
co
rrel
a
tive the pixels
are a
nd the
more
com
p
re
ssed
the data are.
The ba
sic p
r
i
n
cipl
e of image com
p
ressi
on co
ding i
s
indicated in Fi
gure 1 [6].
Figure 1. Prin
ciple of imag
e informatio
n comp
re
ssion
2.2. The Co
mmon Algor
ithm and Re
alizati
on o
f
F
r
actal Image Compre
ssio
n
Coding
Curre
n
tly, the core of fra
c
t
a
l image com
p
re
ssi
on codi
ng is a sub-bl
ock iterated f
unctio
n
system ba
se
d
on
th
e cont
ractive affine
tran
sfor
m
a
tion. The
ba
si
s of the
codi
ng p
r
o
c
e
s
s is the
colla
ge theo
rem.To co
ndu
ct fractal codi
ng on an ima
ge is to find a prop
er com
p
re
ssi
on affin
e
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No. 3, September 20
15 : 996 – 100
5
998
transfo
rmatio
n to m
a
ke its fixed poi
nts t
o
be
an
ap
proximation a
s
good
to the
o
r
iginal
imag
e
a
s
possibl
e. The
n
re
cord do
wn the co
rresp
ondin
g
param
eters
and u
s
e
them as the f
r
actal
co
de
s of
the image fo
r the storage
and tran
smi
ssi
on. The d
e
co
ding p
r
o
c
ess is, firstly, to determin
e
a
grou
p of co
mpre
ssion af
fine tran
sformations vi
a t
he sto
r
ag
e o
r
tran
smi
ssi
o
n
paramete
r
s to
con
s
titute an
iterated function sy
stem a
nd then se
ek the attractor
of this system. According
to
the attra
c
tor t
heorem,
su
ch
attra
c
tor is th
e ap
prox
im
ation of th
e
origi
nal ima
ge. T
h
e ba
si
c
codi
n
g
prin
ciple
will be introd
uced
in the form of figure and te
xt [7].
1) Image mo
del
A
ssu
me t
hat
(,
)
NN
Rd
is the gray
scale imag
e
spa
c
e of
NN
and the grayscale
valuera
nge
i
s
{0
,
1
,
2
,
,
1
}
l
(
l
is no
rma
lly 256, n
a
m
ely the q
uantization
of
8
bi
t
). In its
appli
c
ation
s
,
N
is u
s
ually th
e po
we
r
of 2,
(i.e. 2
5
6
×
25
6, 512
×5
12
e
t
c). T
herefore
,
an im
age
I
can b
e
expre
s
sed a
s
a ma
trix
()
ij
N
N
I
,
ij
I
mean
s th
e gray
scale value of the im
age at
(,
)
ij
.
d
is a
compl
e
te me
tric to b
e
u
s
ed in the
distortion jud
g
m
ent and it i
s
usu
a
lly take
n as
ro
ot mean
squ
a
re (RMS
):
2
1/
2
2
,1
1
(,
)
(
,
)
(
)
,
,
N
NN
ij
ij
ij
dx
y
R
M
S
x
y
x
y
x
y
R
N
(1)
2) Image
seg
m
entation
Adopt fixed block segme
n
tation meth
od and
seg
m
ent image
I
into a se
rie
s
of
BB
pixel su
b-blo
c
k (2
D a
r
ray) rul
e
rs
(1
,
2
,
,
)
ir
Ri
N
, of fixed si
ze. Th
ey won’t
ove
r
lap
and th
e
y
cover the e
n
tire imag
e (Fig
ure 2
)
. In other wo
rd
s,
1
,(
)
,
,
1
,
2
,
r
N
ii
i
r
i
I
RR
R
i
j
i
j
N
Such
sub-blo
c
k is
called
Range
blo
c
k
(
R
block fo
r
sho
r
t) and
its si
ze
s
inclu
de: 4
×
4,
8×8
,
16×16 et
c. In the su
b-bl
ock form
ed by
R
block, code t
hem on
e by
one a
c
cordin
g to the orde
r of
c
o
lumns
,
namely to lis
t the
R
blocks by the followin
g
order:
11
12
1
2
1
2
2
2
1
2
,,
,
,
,
,
,
,
,
,
,
,
(
/
)
nn
n
n
n
n
RR
R
R
R
R
R
R
R
n
N
B
B
e
side
s,
ima
g
e
I
is divided into a serie
s
of sub-blo
c
ks
1
{}
d
N
ii
D
with large
r
si
ze and the
s
e
sub
-
blo
c
ks can ove
r
lap
a
nd they
won’t
need
to
cov
e
r the
entire i
m
age. T
hey
are
call
ed d
o
m
ain
block (
D
block f
o
r
sh
ort). In
t
he a
ppli
c
atio
ns, the
size o
f
D
block corre
s
pondi
ng to
th
e
R
block
wit
h
a si
ze of
BB
is usually
22
BB
(i.e.8×8, 16
×1
6 a
nd 32
×32 et
c). They can b
e
gene
rated
by
moving a
22
BB
win
dow f
r
om l
e
ft to rig
h
t and
from
up to
do
wn
with a
ho
rizontal
step
-l
ength of
h
and a verti
c
a
l
step-l
ength
of
v
. Obviously
, two neigh
b
o
rho
od blo
cks have
h
(or
v
) pix
e
ls
overlap
ped in
the horizo
n
ta
l (or vertical)
dire
cti
on.In the application,
the horizo
n
ta
l step-le
ngth is
the same
a
s
t
he ve
rtical
st
ep-le
ngth,
na
mely
hv
,and th
e
side
len
g
th of
R
blo
c
k i
s
B
(A
s
indicated in Figure 3,
3
D
blocks a
r
e dra
w
n he
re). T
herefo
r
e, the
numbe
r of
D
blocks is
2
2
(1
)
d
NB
N
.
is usually
B
(or
2
B
), at this time, the num
ber of
blocks
is
2
(1
)
r
N
and
it is half-overl
appe
d in the vert
ical (hori
z
ontal) di
re
ctio
n [8].
3) Sea
r
ch fo
r the optimal
matchin
g
bl
o
c
k
()
mi
D
of
R
block
and d
e
termi
n
e the ma
ppin
g
para
m
eter.
i
R
, every
R
block is a
pproximate with the
si
ze ratio re
setting
and b
r
ightn
e
ss
transfo
rmatio
nof
()
mi
D
, a cert
ain
D
block
(Figure 4).
The map
p
in
g
i
w
usu
a
lly choo
se
s
comp
re
ssion
affine transfo
rmation and it
s co
mmon fo
rm is:
D
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
A Fractal Im
age Com
p
ression Method B
a
se
d on Multi
-
Wavel
e
t (Ya
n
Feng
)
999
()
()
()
(
)
((
)
)
((
)
)
im
i
i
im
i
i
k
i
m
i
i
wD
D
s
t
D
o
(2)
Figure 2. Partition scheme
(R blo
c
k)
Figure 3. Pro
duci
ng D bl
ocks
The comp
re
ssion affine tra
n
sformation
i
w
inclu
d
e
s
sp
atial cont
ra
ctive transfo
rmati
on
i
and g
r
ay
scal
e modifi
catio
n
tran
sformat
i
on
i
. It can be
see
n
fro
m
Fi
gure
4 that th
e map
p
ing
i
transl
a
tes fro
m
the
sub
-
im
age
()
mi
D
to the
sub-im
age
i
R
.
Th
en it
ma
ke
s i
t
s si
ze t
o
t
a
lly
ov
erla
p
the size of
i
R
through pixel m
ean value o
r
decim
ation contra
ction. The mappi
ng
i
modifies th
e
grayscal
e inf
o
rmatio
n of
()
mi
D
to get a
better app
roximatio
n
of the g
r
ay
scale of
i
R
by in
trodu
cing
the grayscal
e
adjustme
n
t and the offset para
m
eters
,
ii
s
o
as Figu
re 5.
Figure 4 .Ef feet of transfo
rmation
i
w
Figure 5. Sca
ling and offse
t
of image
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No. 3, September 20
15 : 996 – 100
5
1000
(1
,
2
,
,
)
j
d
D
jN
, every
D
blo
c
k ad
opts
4-nei
ghb
orh
o
od pixel
m
ean val
ue
or
decim
ation to get the pixel block
ˆ
j
D
of
BB
. Use
S
to symbolize this computation
(spa
ce
contractio
n o
perato
r), n
a
m
e
ly
ˆ
()
j
j
SD
D
. For exam
ple, acco
rdin
g to
the pixel
expre
ssi
on, the 4-
neigh
borhoo
d
pixel mean value is:
,
2
,2
2
1
,2
2
,
2
1
2
1
,2
1
ˆ
()
/
4
k
l
kl
k
l
kl
k
l
d
d
ddd
(3)
In this
formula,
,,
ˆ
,
kl
k
l
dd
are the grayscale valu
es of
j
D
and
ˆ
j
D
at the pixel poin
t
(,
)
kl
.
All such con
t
raction
su
b-blocks form
a “v
irtual
co
debo
ok” an
d
mark this
codeb
oo
k as
,
namely
ˆ
{(
)
:
1
,
2
,
,
}
j
jd
DS
D
j
N
[9].
3. Multi-Wav
e
let De
comp
osition and
Reco
nstruc
tion of 2D Ima
g
e
The de
comp
o
s
ition and
reconstructio
nal
gorithm
s
of discrete multi-wavelet tran
sform are
the develop
ment of sin
g
le wavelet
and the
difference is
that the decomp
o
sitio
n
and
recon
s
tru
c
tio
n
filters are
vector f
ilters; therefore, vect
or
sign
al is req
u
ire
d
to be input into the
filter. And a probl
em in th
e algo
rithm realization
is t
he vecto
r
ization of the inp
u
t scalar
dat
a;
corre
s
p
ondin
g
ly, the vector data i
s
re
qu
ired to be
re
stored to
scalar data in th
e
recon
s
tru
c
tio
n
.
This p
r
o
b
lem
is u
s
ually
so
lved throu
gh
the pre
-
filter and
the co
rresp
ondi
ng po
st-filter a
nd t
h
e
desi
gn of the pre
-
filter is rel
a
t
ed to the multi-wavel
e
t used [10].
Assu
me that the co
rrespon
ding 2
D
matri
x
to an image is:
0,
0
0
,
1
1,
1
1
,
1
N
NN
N
aa
A
aa
(4)
Then the
step
s to perfo
rm
multi-wavelet transfo
rm on
Image
A
are a
s
follows:
(He
r
e,
N
is the integer p
o
wer of 2. Take
2
r
, namely 2-level
multi-wavel
e
t transfo
rm)
(1) Li
ne pre-fi
ltering
Firstly, form a
line vector si
gnal with eve
r
y line of
A
ac
cording to the following way.
,2
,2
1
()
ik
iR
ik
a
An
a
0
,
1,
1,
0
,
1
,
.
.
1
/
2
iN
k
N
(5)
Then, pe
rform line pre
-
filtering o
n
iR
A
.
,
,
2
()
(
)
in
iR
k
i
R
N
k
in
b
Bn
p
A
n
k
b
[
]
,
0
,1
,
1
,
0
,1
,
.
.
1
/
2
ij
Bb
i
N
k
N
(6)
In this fo
rmul
a, and
k
p
is the
matrix of 2
×
2
,
indicatin
g
th
e pre-filter
co
rre
sp
ondi
ng t
o
the
wavelet u
s
ed.
(2) Colum
n
pre-filterin
g
Form a
col
u
mn vecto
r
si
gnal with
every col
u
mn
of
B
according
to the following
approa
ch.
2,
21
,
()
ni
iC
ni
b
Bn
b
0
,
1,
1,
0
,
1
,
.
.
1
/
2
iN
k
N
(
7
)
Then, pe
rform colum
n
pre
-
filtering o
n
iC
B
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
A Fractal Im
age Com
p
ression Method B
a
se
d on Multi
-
Wavel
e
t (Ya
n
Feng
)
1001
,
,
2
()
(
)
ni
iC
k
i
C
N
k
ni
c
Cn
p
B
n
k
c
[
]
,
0
,1
,
1
,
0
,1
,
.
.
1
/
2
ij
Cc
i
N
k
N
(8)
(3) Multi
-
wavelet decompo
sition
Step 1: Multi-wavelet de
co
mpositio
n in the line directi
on.
Firstly, form the vector
sig
nal with every
line of
C
acco
rding to the followin
g
mean
s.
,
,
2
()
in
iR
N
in
c
Cn
c
0
,
1,
1,
0
,
1
1
/
2
iN
n
N
(9)
Then, pe
rform multi-wavel
e
t transfo
rm
on every line
of
()
iR
Cn
.
,
,2
,
4
,
L
im
L
L
im
n
m
i
R
N
n
im
d
Dn
G
C
n
d
1
0
,
1,
1,
0
,
1,
4
N
iN
m
(10
)
,
,2
,
4
,
H
im
H
H
im
n
m
i
R
N
n
im
d
Dn
H
C
n
d
1
0
,
1,
1,
0
,
1,
4
N
iN
m
(11
)
In this
formula,
k
G
is the m
a
tri
x
of 2×2, indi
cating th
e correspon
ding l
o
w-frequ
en
cy filter
to multi-wavelet while
k
H
, a matrix of 2×2, is the corresp
ondi
ng hi
gh-frequ
en
cy filter to the
wavelet [11].
Make
,,
,,
,
LL
H
H
L
H
ij
ij
D
DDD
D
D
D
.
Step 2: Multi-wavelet de
co
mpositio
n in the col
u
mn di
rection.
Similarly, form the vecto
r
sign
al on eve
r
y colu
mn of
D
in the same
way as the lin
e
and
then perfo
rm
colum
n
wavel
e
t transfo
rm.
,,
,,
22
,,
1
,
2
,
,
1
,
2
,
,
2
LH
ni
ni
LH
LH
iC
iC
NN
ni
ni
DD
N
D
nD
n
n
i
N
DD
(12
)
Perform m
u
lti-wavel
e
t tran
sform o
n
L
iC
D
and
H
iC
D
respec
tively [
12].
,
,2
,
4
,
,2
,
4
,
,2
,
4
11
,0
,
1
,
,
0
,
1
,
24
11
,0
,
1
,
,
0
,
1
24
1
,0
,
1
,
,
2
LL
mj
LL
L
LL
im
n
m
i
R
N
n
mj
LH
mj
LH
L
LH
im
n
m
i
R
N
n
mj
HL
mj
HL
L
HL
im
n
m
i
R
N
n
mj
E
NN
En
G
D
n
i
m
E
E
NN
En
H
D
n
i
m
E
E
N
En
H
D
n
i
m
E
,
,2
,
4
1
0,
1
4
11
,0
,
1
,
,
0
,
1
24
HH
mj
HH
L
HH
im
n
m
i
R
N
n
mj
N
E
NN
En
H
D
n
i
m
E
(13
)
Step 3: finally, get the multi-wavel
e
t tran
sform of
A
.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No. 3, September 20
15 : 996 – 100
5
1002
,,
,
,
,,
,
LL
LH
H
L
H
H
ij
ij
ij
i
j
L
L
E
L
HE
H
L
E
H
HE
，
namely:
11
1
2
1
1
1
2
21
2
2
2
1
2
2
11
1
2
1
1
1
2
21
2
2
2
1
2
2
LL
LL
LH
LH
LL
LL
LH
LH
LL
LH
E
HL
HL
HH
HH
HL
HH
HL
HL
HH
HH
(14
)
It is not difficult for u
s
to
find that due
to the exist
ence of seve
ral
scali
ng
(wavelet)
function
s, on
e sub
-
ba
nd a
fter the singl
e wavelet tra
n
sform is furt
her de
com
p
o
s
ed into
2
r
sub-
blocks in th
e
multi-wavele
t transfo
rm.
For
a
2
D
im
age,
N-level
wavel
e
t de
compo
s
ition
will
gene
rate
2
31
rL
sub
-
imag
es. Figu
re 6 is the de
comp
osed co
efficient map
whe
n
2,
2
Lr
.
L-level
multi-wavelet t
r
an
sform d
e
comp
ose
s
th
e im
a
ge into
2
31
rL
sub
-
b
l
oc
ks.
The
recon
s
tru
c
tio
n
pro
c
e
ss i
s
the inverse p
r
ocess of
the
above step
s. That is to say, performth
e
inverse m
u
lti-wavelet t
r
ansform in the column
di
recti
on. Th
en th
e
line
direction
.
And finally t
he
post-filte
r
ing i
n
the line dire
ction, after th
at, the image recon
s
tru
c
tio
n
is co
mplete
d [13, 14].
4. Fractal Image Comp
re
ssion Algor
ithm in Multi-Wav
e
let Domain
The ba
si
c ide
a
of the fract
a
l image
com
p
re
ssi
on alg
o
rithm in the
multi-wavelet
domain
is: de
comp
ose the ori
g
inal
image into t
he sub-i
m
ag
es at diffe
ren
t
spatial freq
uen
cie
s
thro
u
g
h
multi-wavelet
transfo
rm, p
e
rform f
r
act
a
l codin
g
on t
he high
-level
wavelet coe
fficients only
by
usin
g the co
rrelation bet
we
en the wavel
e
t coefficie
n
ts
at different scale
s
an
d esti
mate the fract
a
l
cod
e
of the l
o
w-l
e
vel wavelet coeffici
e
n
ts from
that
of the uppe
r-tiere
d
wavel
e
t coefficie
n
t
s
. In
this way, it greatly re
du
ces the codin
g
ti
me and improve
s
the
comp
re
ssi
o
n
ratio witho
u
t
signifi
cantly redu
cing th
e q
uality of the d
e
co
ded i
m
ag
e. Figure 7 i
s
the ba
si
c pro
c
ed
ure
of mul
t
i-
wavelet fra
c
tal image codi
ng algo
rithm.
Figure 6. Multi-wavel
e
t tran
sform
whe
n
2,
2
Lr
Figure 7. Ske
t
ch map of m
u
lti-wavel
e
t coding al
gorith
m
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
A Fractal Im
age Com
p
ression Method B
a
se
d on Multi
-
Wavel
e
t (Ya
n
Feng
)
1003
Its
s
p
ec
ific
s
t
eps
are as
follows
:
(i)
Perform 3-le
vel
wavelet decompo
sitio
n
on
the ori
g
inal
ima
ge and
g
e
t
10 wavelet sub
-
image
s.
(ii)
Take the 4
*
4
non-ove
r
la
p
p
ing sub-blo
c
ks
whi
c
h are
divided from
the low-freq
uen
cy part
LL3 after the
lifting wavelet transfo
rm.
Perfor
m a
n
o
t
her lifting wavelet decom
positio
n on
LL3. Ta
ke t
he coefficie
n
t
blocks wit
h
a
si
ze
of
4*4 from t
he same
po
sition in th
e
decompo
se
d
4 part
s
(a, b,
c, d) an
d re
store
a
ll the
possibl
e 8*8
D blo
c
ks in t
he ori
g
inal
image throug
h wavelet re
con
s
tru
c
tion
algorith
m
. Th
e D’, the sa
mpled
D blo
c
ks
can b
e
found in
a, therefo
r
e, mat
c
h all the 4*
4
sub
-
blo
c
ks i
n
the tran
sformed lo
w-freq
uen
cy part a
as D’
with the R bl
ocks divided from the LL3.
The range t
o
search the optimal D’ i
s
narro
wed d
o
w
n to the half
of the origina
l
range,
thu
s
greatly sh
orte
ning the codi
ng time.
(iii)
Con
s
id
erin
g the positive and neg
ative wave
let coe
fficients, whi
c
h is not go
od for th
e
simila
rity matchin
g
bet
wee
n
the pa
rent
block
a
nd the
sub
-
blo
c
k, we extract th
e
symbol
(+
or -) of the wavelet coeffi
cients for
se
pa
rate codin
g
a
nd we
only p
e
rform f
r
a
c
tal
codin
g
on
the absolute
value of the wavelet coefficient
s.
(iv)
Perform f
r
a
c
tal co
ding
on
the wavel
e
t coeffici
ent
s
of different scal
es
a
nd
fini
sh the
co
ding
of the entire image.
(v)
In the de
co
di
ng recon
s
tru
c
tion, estimate
t
he fra
c
tal
code of
scale
1 ba
sed
on t
hat of scal
e
2. And the estimation form
ula is:
12
12
/
av
r
a
v
r
ss
rr
(15
)
Her
e
,
1
s
and
2
s
are the
scala
r
coeffici
ents o
f
scale
s
1
an
d 2 i
n
the
wa
velet co
effici
ent
matrix res
pec
tively,
1
av
r
r
and
2
av
r
r
are the
grayscale
offset
coefficient
s of
scale
s
1
a
nd 2
respe
c
tively and
5
.
Rec
ons
truc
t the wavelet c
oeffic
i
ents
of differe
nt scal
e
s with
the fractal
cod
e
s of the
scale
s
. Th
e
n
ad
d the
sy
mbol of
wav
e
let coefficie
n
t. Integrate
the de
co
ded
low-
freque
ncy
pa
rt with th
e h
i
gh-frequ
en
cy part. Pe
rform inverse
wavelet tran
sf
orm, the i
m
a
ge
decodin
g
re
constructio
n
is complete
d.
5. Experiment Simulation and Analy
s
is
Acco
rdi
ng to
the al
gorit
hm ba
se
d
on
wavelet
fractal
imag
e comp
re
ssi
on
codi
ng
,
experim
ent o
n
the imag
es of Lena a
n
d
pper
and
co
mpare with t
he fra
c
tal im
age
comp
re
ssion
codi
ng. Th
e
experim
ent t
e
st pl
atform i
s
CPU:
Intel
(
R) Co
re
(TM
)
2CPU, 1.35G
HZ,
RA
M
：
2.
0G,
the operating
system is:
Wind
ows 7 a
nd the experi
m
ental env
ironment is: M
a
tlab20
12a.
The
test perfo
rma
n
ce pa
ram
e
ters in
clu
de compressio
n
ratio, peak si
g
nal to noise ratio (PSNR) and
codi
ng time (s).
It can be see
n
from Figure
8 and Table
1 that
compa
r
ed with FIC,
multiwavelet
-FIC has
greatly imp
r
o
v
ed in the compressio
n time by
red
u
cing 2
2
.23%
on the ave
r
age, that th
e
maximum co
mpre
ssion
ra
tio redu
ce i
s
1.347 and t
hat the sig
n
a
l to noise
ratio decre
ases
slightly. Thro
ugh algo
rithm
analysi
s
, multiwavelet-F
I
C
redu
ce
s the codi
ng co
mpl
e
xity in contrast
to FIC a
nd
it sho
r
ten
s
t
he mat
c
hin
g
sea
r
ch time
and
accel
e
rates th
e cod
i
ng spee
d vi
a
experim
ent.
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ISSN: 16
93-6
930
TELKOM
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Vol. 13, No. 3, September 20
15 : 996 – 100
5
1004
(a) De
codi
ng image
of
mult
iwavelet-FIC
(b) De
codi
ng image
of
FIC
(c) De
co
ding
image of mult
iwavelet-FIC
(d) De
codi
ng image
of
FIC
Figure 8. Co
mpari
s
o
n
bet
wee
n
multiwa
v
elet-FIC an
d
FIC
Table 1. Te
st Re
sults
Com
pari
s
on
Bet
w
een Multiwavelet-FIC a
nd
FIC
FIC
Multiw
avelet-FI
C
Time(S)
Compression
rati
o
PSNR(dB)
Time(S)
Compression
rati
o
PSNR(dB)
Pears
27.358
13.556
29.53
19.861
12.209
28.87
Peppers
41.476
11.268
31.86
33.668
10.241
30.92
6. Conclusio
n
The imag
e chara
c
te
risti
c
s are clo
s
ely
related to th
e comp
re
ssi
on effect. In orde
r to
better mat
c
h
the image
ch
ara
c
teri
stics
with comp
ression al
go
rith
m, this pap
er has p
r
o
p
o
s
e
d
a
new multi-wa
velet fractal image comp
ression al
g
o
rit
h
m. Throu
g
h
the experim
ent perfo
rma
n
ce
simulatio
n
a
nd compa
r
ison expe
rime
nt, it can be
see
n
that compa
r
ed
with FIC, this
ne
w
algorith
m
ha
s excellent
effects a
nd i
t
effectively improve
s
the
comp
re
ssio
n perfo
rma
n
ce,
sho
r
ten
s
the time of the matching
sea
r
ch
and increa
se
s the co
ding
spe
ed.
Referen
ces
[1]
Ming-S
hen
g W
u
. Genetic Algorit
hm Bas
ed on D
i
scret
e W
a
velet T
r
ansformatio
n
for F
r
actal Image
Compress
ion.
Journ
a
l of Visu
al Co
mmu
n
ic
at
ion a
nd Imag
e Repr
esentati
o
n
. 2014; 25(
8): 1835-
184
1.
[2]
Rob
e
rto de Qu
adros Gomes, Vlad
imir Guerr
e
iro,
Ro
drig
o d
a
Rosa Ri
gh
i, Luiz Gonz
aga
da Silv
eira Jr,
Jin
y
o
ung Ya
ng
.
Analy
z
i
n
g Per
f
orma
nce of th
e Parall
el-
base
d
F
r
actal Imag
e Co
mpress
io
n
Proble
m
o
n
Multicore Syst
ems
. AASRI Pr
oced
ia. 20
13; 5(2): 140-
14
6.
[3
]
Yi
Zh
an
g
,
Xi
n
g
y
u
a
n
Wa
ng
. Fra
c
ta
l
C
o
mp
re
ssi
o
n C
o
d
i
n
g
Ba
se
d on
Wa
vele
t T
r
a
n
s
fo
rm
w
i
th
Di
a
m
ond
Search.
No
nli
n
ear Ana
l
ysis: R
eal W
o
rl
d Appl
i
c
ations
. 20
12;
13(1): 10
6-1
1
2
.
[4]
Kamel Be
llo
ul
ata, Amina B
e
lal
i
a, Shi
p
i
n
g
Z
hu. Object-base
d
Stereo
Video C
o
mp
ressio
n
usin
g
F
r
actals and
Shap
e-Ad
aptiv
e DCT
.
AEU-Internati
o
n
a
l Jo
urna
l of
Electr
onics a
nd C
o
mmu
n
icati
ons
.
201
4; 68(7): 68
7-69
7.
[5]
Ren
a
to Br
anca
t
i, Ernesto
Roc
c
a, Serg
io
Sav
i
no.
A
Gear
R
a
ttle Metric
Ba
sed
on
T
he W
a
vel
e
t Multi
-
Resol
u
tio
n
A
n
al
ysis: E
x
peri
m
ental
Investi
gatio
n.
Mec
h
a
n
ical
Syste
m
s
and
Si
gna
l Pr
ocessi
ng
. 201
5;
50(1): 16
1-1
7
3
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
A Fractal Im
age Com
p
ression Method B
a
se
d on Multi
-
Wavel
e
t (Ya
n
Feng
)
1005
[6]
Arikatla H
a
zar
a
thai
ah, B Pr
abh
akara
Ra
o. M
edica
l Imag
e Compr
e
ssi
o
n
usi
ng L
i
fting
base
d
Ne
w
Wavelet T
r
ansforms.
Internati
ona
l Jo
urna
l of
Electrica
l
a
n
d
Co
mp
uter En
gin
eeri
n
g
. 20
14
; 4
(
5
)
: 74
1-
750.
[7]
Qi-gao F
e
ng,
Hao-
yu
Z
h
o
u
. Rese
arch
of Image C
o
mpr
e
ssio
n
Bas
e
d
on Qu
antum
BP Net
w
ork
.
T
E
LKOMNIKA Indon
esi
an Jou
r
nal of Electric
al Eng
i
ne
eri
n
g
.
2014; 1
2
(1): 1
97-2
05.
[8]
Yang
R
an, Z
u
o Ya
nj
un, L
e
i
W
anju
n
. Res
e
ar
chi
ng
of Ima
ge
Compr
e
ssi
on B
a
se
d o
n
Quantum B
P
Net
w
ork.
T
E
LK
OMNIKA Indon
esia
n Journ
a
l o
f
Electrical Eng
i
ne
erin
g
. 201
3; 11(11): 68
89-
689
6.
[9]
F
a
temeh D
a
ra
ee, Sa
eed
Mo
zaffari. W
a
termarking
in
Bin
a
r
y
Doc
u
me
nt Images
usin
g
F
r
actal Co
des
.
Pattern Recognition Letters
. 201
4; 35(1): 12
0-12
9.
[10]
S F
u
jimot
o, A
Rinos
hik
a
. W
a
velet M
u
lti-res
o
luti
on A
n
a
l
ysi
s
on
T
u
rbule
n
t W
a
kes
of As
ymmetric Bluff
Body
.
Internati
ona
l Journ
a
l of
Mechan
ical Sc
ienc
es
. 201
4; 92(3): 121-
13
2.
[11]
B Andò, S B
a
glio,
A Pistor
io.
A L
o
w
Cos
t
Multi-sens
or
Strategy
for E
a
rly Warning in Structural
Monitori
ng
Exp
l
oitin
g
A W
a
v
e
l
e
t Multires
ol
uti
on P
a
rad
i
g
m
. Proced
ia
E
ngi
neer
ing. 20
14;
87(5): 12
82-
128
5.
[12]
W
u
Xia
ng, Qi
an J
i
an-S
h
e
n
g
.
Comb
ine
Mu
ltipre
dictor
of
Gas Co
nce
n
tration
Pre
d
ictio
n
Bas
e
d
o
n
Wavelet T
r
ansforms.
T
E
LKOMNIKA Indone
sian Jo
urna
l
of Electrical En
gi
neer
ing
. 20
14;
12(1
0
): 736
1-
736
8.
[13]
Z
huan
g W
en W
u
, Liangr
on
g Z
hu. Car Informa
tio
n
Bu
s Image Rest
oratio
n Usin
g
Multi
w
av
elet
T
r
ansform Algorithm.
T
E
LKOMNIKA Indon
e
s
ian Jo
urn
a
l of
Electrical E
n
g
i
neer
ing
. 2
0
1
3
; 11(1
0
): 615
8-
616
5.
[14]
Ming
w
e
i
She
n
g
, Yon
g
ji
e P
a
n
g
, Le
i W
an,
et
al
. U
nder
w
a
ter
Images
En
han
cement
usin
g
Multi W
a
ve
le
t
T
r
ansform and Media
n
F
ilter.
T
E
LKOMNIKA Indones
ia
n Journ
a
l
of Ele
c
trical Engi
ne
erin
g
. 2
0
14;
12(3): 23
06-
23
13.
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