Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
Vol
.
4
,
No
. 5, Oct
o
ber
2
0
1
4
,
pp
. 74
1~
75
0
I
S
SN
: 208
8-8
7
0
8
7
41
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Medical Image Compression usin
g Lifting b
a
sed New Wavelet
Transforms
A H
a
z
a
ra
th
ai
ah
1
, B
Pr
abh
a
kar
a
Rao
2
1
Dept. of
EC
E, SV Colleg
e
of
En
gg., Tirup
a
ti, IN
DIA
2
Jawaharlal Neh
r
u Technologi
cal University
, Kak
i
nada
(JNTUK), Kakinada,
INDI
A
Em
ail:
a
.
haz
a
rth
@
gm
ail.com
1
,
d
r
bpr@rediffm
ail
.
com
2
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Mar 13, 2014
Rev
i
sed
Jun
5
,
2
014
Accepted
Jun 25, 2014
In this p
a
per
,
th
e construction of
ne
w lif
ting b
a
sed wavelets b
y
a
new method
of calcu
lat
i
ng l
i
fting coef
fic
i
en
ts
is
presented. First of all, new basis
functions ar
e uti
lized to ease new
orthogonal tr
adition
a
l wav
e
lets. Then
b
y
us
ing the de
co
m
pos
ing pol
y
-
p
h
as
e m
a
trix
the
lift
i
ng s
t
eps
ar
e ca
lcu
l
at
ed
using a simplified method. Th
e inter
e
sti
ng feature of lif
ting
scheme is that
the
construction
of wav
e
let is
deri
ved
in sp
atial domain
only
;
hence
th
e
difficu
lty
in
th
e design
of traditional
wa
v
e
le
ts is
avoided
.
Lifting scheme
was
used to gen
e
rat
e
second g
e
ner
a
tion wav
e
l
e
ts
which are
not
necessar
i
l
y
translation and dilation
of
one partic
ular
function. Short
and s
h
arp basis
functions
are
ch
osen so as to
ob
tain
the non-uniform nature o
f
usual image
clas
s
e
s
.
Im
plem
ented wav
e
le
ts
a
r
e appl
ied on a n
u
m
b
er of m
e
dica
l im
ages
. I
t
was found that the compression
ratio
(CR) and
Peak Signal to
Noise Ratio
(PSNR) are far ahead of th
at are
obtained
with the popular tradition
a
l
wavele
ts
as
wel
l
as
the s
u
cc
es
s
f
ul
5/3 and 9/7
lifting b
a
sed wavelets. Set
Partition
i
ng in
Hierarch
ica
l
Trees (SPIHT) is used to incorpora
t
e
c
o
mpre
ssion.
Keyword:
Basis Functions
Com
p
ression
Li
ft
i
ng Sch
e
m
e
Po
ly-p
h
a
se represen
tation
Copyright ©
201
4 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
A Hazarat
haiah
,
Dept
.
o
f
EC
E
,
SV
C
o
l
l
e
ge of
En
gg
.,
Karak
a
m
b
ad
i Ro
ad,
Tirup
a
ti,
A
ndh
r
a
Pr
ad
esh
,
I
NDI
A Pin
:
5
175
07
.
Em
a
il: a.hazarth@gm
a
il.co
m
1.
INTRODUCTION
From
the past three deca
des
many researchers are
wo
rk
ing
on
wav
e
lets an
d
its app
licatio
n
s
.
A v
e
ry
fi
ne re
vi
ew o
f
appl
i
cat
i
o
ns of
wavel
e
t
s
such
as bi
om
edi
cal appl
i
cat
i
ons
, wi
rel
e
ss com
m
uni
cat
i
o
ns, c
o
m
put
er
gra
p
hi
cs or t
u
rb
ul
ence i
s
gi
ven i
n
[1]
.
I
m
age com
p
ressi
on i
s
o
n
e o
f
t
h
e m
o
st
d
o
m
i
nant
and vi
si
bl
e
ap
p
lication
s
of wav
e
lets. Nowad
a
ys th
e
d
i
gital
i
m
ag
in
g
h
a
s al
m
o
st su
p
e
rsed
e th
e prin
t imag
in
g
,
wh
ich in
sists
l
a
rge
num
ber o
f
t
ech
ni
q
u
es o
p
erat
i
n
g
on
di
g
i
t
a
l
im
ages. M
eanw
h
i
l
e
, t
h
e
m
e
di
cal
im
aging
has t
a
ken i
t
s sha
p
e
from
print to
digital since the last
decade
.
Hence
,
a
n
attention for t
h
e
design
of com
p
ression techniques i
s
requ
ired
for
prov
id
i
n
g lesser m
e
m
o
ry req
u
irem
en
t with
g
ood
qu
ality i
n
d
i
fferen
t app
licatio
n
s
. A t
y
p
i
cal
i
m
ag
e u
s
u
a
lly co
n
t
ains larg
e
sp
atial red
und
an
cy in m
o
st o
f
th
e reg
i
on
s in
i
m
ag
e [2
]. In ad
d
ition
t
o
th
e sp
atial
red
u
nda
ncy
,
a
n
i
m
age cont
ai
ns s
u
bject
i
v
e
r
e
du
n
d
ancy
,
w
h
i
c
h i
s
det
e
rm
i
n
ed
by
p
r
o
p
e
r
t
i
e
s o
f
a
h
u
m
a
n vi
sual
sy
stem
(HVS)
[3]
.
A
n
HV
S p
e
rm
its so
m
e
tolerance b
a
sed
on t
h
e content
s
of the im
age and
viewpoints
. The
red
u
nda
ncy
(
b
ot
h st
at
i
s
t
i
cal
and/
or
su
b
j
ect
i
v
e), t
hus
, ca
n
be elim
inated to achieve
com
p
ression
of the i
m
age
dat
a
.
The
basi
c m
e
asure
f
o
r t
h
e
pe
rf
orm
a
nce o
f
a
com
p
ressi
o
n
a
l
go
ri
t
h
m
i
s
com
p
ressi
on
rat
i
o
,
defi
ned
as
a ratio betwee
n ori
g
inal data
size and compress
ed da
ta size. In a
n
im
age com
p
ression schem
e
, the image
com
p
ression algorithm
shoul
d attain
a tradeoff
betwee
n com
p
ression rati
o
and im
age quality (in the form
of
PSNR
)
[4
]-[6
]. Usu
a
lly, if an
im
ag
e co
m
p
ression
algo
rit
h
m
p
r
o
d
u
ces
h
i
gh
CR v
a
lues, th
en
th
e qu
ality
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 4
,
N
o
. 5
,
O
c
tob
e
r
20
14
:
741
–
7
50
74
2
(PSNR) will be less an
d
v
i
ce v
e
rsa. Im
ag
e co
m
p
ression
is v
e
ry im
p
o
r
tan
t
for effecti
v
e tran
sm
issio
n
and
st
ora
g
e
of
i
m
ages.
Dem
a
nd
f
o
r
com
m
uni
cat
i
on
o
f
m
u
l
t
i
m
e
di
a dat
a
t
h
r
o
ug
h t
h
e
t
e
l
ecom
m
uni
cat
i
ons ne
t
w
o
r
k
an
d
retrev
i
n
g
th
e
m
u
lti
m
e
d
i
a d
a
ta th
roug
h In
tern
et is
g
r
o
w
i
n
g
ex
pon
en
tially [7
]. W
i
th
th
e u
s
e of d
i
g
ital
cam
e
ras, re
q
u
i
r
em
ent
s
for
st
o
r
age
,
m
a
ni
pul
a
t
i
on, a
n
d t
r
a
n
sf
er o
f
di
gi
t
a
l
i
m
ages,
has
g
r
o
w
n e
xpl
osi
v
el
y
.
These
im
age files can be
very large
and can
occ
u
py a lot of
m
e
mory
. A gray
sc
al
e
im
age
of di
m
e
nsi
on 2
5
6
x
2
5
6
ha
s
65,536 elem
ents to store, and
a
t
y
pi
cal
col
o
u
r
i
m
age
of di
m
e
nsi
o
n 6
4
0
x
4
8
0
has nearl
y
a m
i
ll
i
on.
Down
lo
ad
ing
o
f
th
ese files fro
m
in
tern
et can
b
e
v
e
ry tim
e
co
nsu
m
in
g
task
. Im
ag
e d
a
ta co
m
p
rise o
f
a
sig
n
i
f
i
can
t
p
o
rtio
n
of
th
e
m
u
lti
med
i
a d
a
ta an
d
th
ey o
c
cupy th
e
m
a
j
o
r
p
a
r
t
o
f
th
e co
m
m
u
n
i
catio
n
b
a
n
d
w
i
d
t
h
fo
r m
u
l
t
i
m
e
di
a com
m
uni
cat
i
on. T
h
e
r
ef
ore
d
e
vel
o
pm
ent
of
effi
ci
ent
t
e
c
h
ni
q
u
es
fo
r i
m
age c
o
m
p
ressi
o
n
has
becom
e
quite necessa
ry
[8]. The ob
jective
of im
age compre
ssi
on
is to
o
b
t
ain an
im
a
g
e rep
r
esen
tatio
n
i
n
whic
h
pixels
are less c
o
rrelated. JPEG a
n
d JPE
G
20
0
0
are
t
w
o i
m
po
rt
ant
t
e
c
hni
que
s u
s
ed
f
o
r
im
age
co
m
p
ression
.
Work
on
i
n
tern
atio
n
a
l
standard
s fo
r im
ag
e co
m
p
ression started
i
n
th
e late 1
970
s
with
th
e
CCITT (c
u
rre
n
tly
ITU-T
)
nee
d
to
stan
dar
d
iz
e bina
ry
im
age com
p
ression algorith
m
s
fo
r G
r
ou
p
3
facs
im
il
e
com
m
uni
cat
i
ons.
Si
nce t
h
e
n
,
m
a
ny
ot
her
c
o
m
m
i
tt
ees and
st
anda
rd
s ha
v
e
been
f
o
rm
ed t
o
ge
nerat
e
d
e
ju
re
stan
d
a
rds (su
c
h
as JPEG),
wh
ile sev
e
ral commercial
l
y su
ccessfu
l
in
itiati
v
e
s
h
a
v
e
effect
iv
ely b
eco
m
e
d
e
facto
st
anda
rd
s (s
uc
h as G
I
F
)
. I
m
age com
p
ressi
on st
a
nda
r
d
s
bri
n
g a
b
o
u
t
m
a
ny
bene
fi
t
s
, suc
h
as:
(
1
)
easi
e
r
excha
n
ge o
f
i
m
age fi
l
e
s bet
w
een
di
f
f
ere
n
t
devi
ces a
nd
appl
i
cat
i
o
ns;
(
2
) re
use
of e
x
i
s
t
i
ng ha
rd
war
e
an
d
soft
ware
fo
r a
wider a
rray
o
f
p
r
o
d
u
cts; (3
)
existence
o
f
benc
hm
arks a
nd
refe
renc
e d
a
t
a
set
s
for
ne
w an
d
altern
ativ
e d
e
velo
p
m
en
ts.
In the
field of
medical, with m
o
re sophisticated m
e
dical e
qui
pm
en
t
and i
m
agi
ng
devi
ce
s, t
h
e i
m
age
processi
ng
has
a tradem
ark im
portance.
W
i
th the requ
i
r
e
m
ent
of o
n
-
d
e
m
and ser
v
ices, teleconfe
r
enci
ng a
nd
vi
de
o c
o
n
f
ere
n
ci
ng i
m
age co
m
p
ressi
on i
s
e
ssent
i
a
l
fo
r
fas
t
er com
m
uni
cat
i
on a
nd
q
u
i
c
k
deci
si
o
n
on m
e
di
cal
treat
m
e
n
t
.
W
ith
th
e m
e
n
tio
n
o
f
Wav
e
let b
y
Haar in
h
i
s doc
to
ral th
esis in 1
909
, th
e era
o
f
wav
e
let h
a
s
started.
The dec
o
m
pos
i
t
i
on of a si
g
n
a
l
i
n
t
e
r
m
s of no
n-si
nu
soi
d
a
l
funct
i
o
n i
s
t
h
e co
ncept
of
a wavel
e
t
.
Ha
ar has
em
pl
oy
ed a
S
qua
re t
y
pe
wa
vef
o
rm
as t
h
e
basi
s
f
unct
i
o
n, a
n
d
Dau
b
ec
hi
es
has
use
d
a di
f
f
ere
n
t
s
p
i
k
e l
i
k
e
wav
e
fo
rm
as th
e
b
a
sis
fun
c
tio
n, all targ
eted to
ex
tract
a
n
d
rep
r
ese
n
t
i
n
fo
r
m
at
i
on i
n
n
o
n
-
st
at
i
onary
a
n
d
si
gnal
s
with
sh
arp
ed
ges and
d
i
scon
tin
u
ities,
wh
ich
was
n
o
t
cap
t
ured
b
y
t
h
e Fo
urier tran
sfo
r
m
.
A larg
e nu
m
b
er of
trad
itio
n
a
l
wavelets are th
en
propo
se
d and
us
ed
on im
age com
p
ression.
Di
ffe
re
nt
l
i
f
t
i
ng schem
e
s are pr
o
pose
d
i
n
[
9
]
-[1
3]
. I
n
[1
4]
,
fo
ur ne
w o
r
t
h
og
o
n
al
wavel
e
t
s
are devi
se
d
and use
d
for i
m
age com
p
res
s
ion
with SPIHT.
In this
pa
p
e
r t
h
e lifting
v
e
rsi
o
n of th
ese wav
e
lets is deriv
e
d
.
A
sim
p
l
i
f
i
e
d cal
cul
a
t
i
on
o
f
l
i
f
t
i
n
g
st
eps i
s
p
r
o
pos
ed a
n
d a
ppl
i
e
d t
o
deri
ve t
h
e st
ep
s o
f
t
h
e
fo
u
r
ne
w
ort
h
og
o
n
al
wavel
e
t
s
. T
h
e
rest
of t
h
e pa
p
e
r i
s
or
ga
ni
sed
as fol
l
o
ws.
I
n
t
h
e ne
xt
sect
i
on t
h
e l
i
f
t
i
ng s
c
hem
e
over
v
i
e
w wa
s
p
r
esen
ted. In
t
h
e th
ird
section
,
th
e
po
ly-phase repres
en
tatio
n
o
f
lifting
sch
e
m
e
is p
r
esen
ted
.
In
th
e fo
urth
sect
i
on n
e
w l
i
f
t
i
ng f
o
rm
ul
at
i
on i
s
gi
ve
n. T
h
e
fi
ft
h sect
i
o
n
presen
ts th
e
simu
latio
n
resu
lts
an
d
t
h
e last sectio
n
concl
ude
s t
h
i
s
pape
r.
Lifting Schem
e
Design
ing
wavelets with
liftin
g
sch
e
m
e
in
clu
d
e
s of
three step
s: Th
e
first, sp
lit p
h
a
se that sp
lit d
a
ta
in
to
odd
and
ev
en
sets, second
pred
ict step, in
wh
ich
odd
set is esti
mated
fro
m
ev
en
set an
d
t
h
e th
ird
up
d
a
te
p
h
a
se th
at
will up
d
a
te ev
en
set u
s
ing
wav
e
let co
effici
en
t t
o
calcu
late scalin
g
fun
c
tion
.
Pred
ict ph
ase
mak
e
su
re th
e
po
lyno
m
i
a
l
can
cellatio
n
i
n
h
i
gh
p
a
ss. Upd
a
te stag
e en
su
res
p
r
eserv
a
tio
n of m
o
men
t
s in low
p
a
ss.
Lifting schem
e
of wa
velet transform
is being use
d
for digi
tal speech compressi
on and
digital image
com
p
ressi
o
n
f
o
r t
h
e
f
o
l
l
o
wi
ng
ad
vant
a
g
es
o
v
e
r co
nv
en
tio
n
a
l wav
e
let
tr
an
sf
or
m
tech
n
i
qu
e.
It
perm
i
t
s
a fast
er im
pl
em
entat
i
on o
f
t
h
e w
a
vel
e
t
t
r
ansf
o
r
m
.
It
requi
r
e
s hal
f
n
u
m
b
er of
com
put
at
i
ons
as
co
m
p
are to
trad
itio
n
a
l co
nvo
lu
tio
n b
a
sed
d
i
screte wav
e
le
t tran
sfo
r
m
.
Th
is is v
e
ry attracti
v
e for real time
lo
w po
wer
applicatio
n
s
.
The lifting
sc
hem
e
allows a
fully
in-place calculation
of the
wa
velet
trans
f
orm
.
In other
words,
no
au
x
iliary m
e
mo
ry is
req
u
i
red
an
d its wav
e
let tran
sf
o
r
m
p
r
ov
id
e a su
b
s
titu
t
e
fo
r t
h
e
o
r
i
g
inal sig
n
a
l.
Liftin
g
sch
e
m
e
allo
ws us to
i
m
p
l
e
m
en
t rev
e
rsib
le in
teg
e
r
wav
e
let tran
sfo
r
m
s
. In
con
v
e
n
tio
n
a
l sch
e
m
e
it
in
vo
lv
es fl
o
a
tin
g
p
o
i
n
t
o
p
e
ratio
n
s
, wh
ich
in
tro
d
u
ces ro
und
ing
erro
rs due to
flo
a
tin
g
po
in
t arith
m
e
tic
.
Wh
ile in
case
o
f
lifting
sch
e
me p
e
rfect reco
n
s
t
r
u
c
tion
is
p
o
s
sib
l
e fo
r l
o
ss-less co
m
p
ressio
n. It is easier to
st
ore a
n
d
pr
oce
ss i
n
t
e
ge
r
n
u
m
b
ers
com
p
are
d
t
o
fl
oat
i
n
g
p
o
i
n
t
n
u
m
b
ers.
Easier to understand and im
ple
m
ent.
It can
be
use
d
f
o
r
irre
g
u
lar sa
m
p
ling.
Th
e i
d
ea
o
f
wav
e
let tran
sfo
r
matio
n
is to
ob
tain
co
rr
elation structure
present
in real life signals t
o
b
u
ild
sp
arse ap
pro
x
i
m
a
tio
n
.
Th
e correlation
structu
r
e is
lo
cal in
b
o
t
h
frequ
e
n
c
y an
d
sp
atial/ti
me d
o
main
.
Trad
ition
a
l
wav
e
let tran
sfo
r
m
u
s
e wav
e
let fil
t
ers to
bu
ild
ti
me frequ
en
cy l
o
calizatio
n
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Med
ica
l
Imag
e Co
mp
ression
u
s
ing
Lifting
ba
sed
New Wa
velet Tran
sfo
r
ms
(A H
a
za
ra
thaia
h
)
74
3
2.
POLYP
H
ASE
REPRESE
N
TATION
Let u
s
con
s
id
er th
e sequ
en
ce of sam
p
les o
f
si
g
n
a
l
x(k)
. Z t
r
a
n
sf
orm
of
t
h
i
s
seq
u
ence
can
b
e
gi
ve
n a
s
,
k
k
z
k
x
z
X
)
(
)
(
(1
)
Let u
s
co
n
s
i
d
er fi
n
ite i
m
p
u
l
se resp
on
se
(FIR) filter
h
h
a
v
i
ng
filter co
efficien
ts
h=
{hk1,.
. .
. .
,
hk
2}.
Z
tran
sform
o
f
this filter is Lau
r
en
t po
lyno
m
i
al
with
d
e
g
r
ee |
k2-k
1
|
gi
ve
n by
,
2
1
)
(
k
k
k
k
k
k
z
h
z
H
(2
)
Filterin
g
o
f
sign
al
x(k)
b
y
filte
r
h
ca
n
be easi
l
y
descri
bed
i
n
z
t
r
an
sf
orm
by
t
h
e (
3
)
Y(z
)
=
H(
z)
X(
z)
(3
)
Su
b-sam
p
l
i
ng
of t
h
e si
g
n
al
x(k)
is co
rrespo
n
d
i
ng
to
k
e
ep
ing
on
ly th
e ev
en
sam
p
les i.e
. xe=x(2k)
.
Z tra
n
sform
of
suc
h
s
u
b-s
a
m
p
l
e
d si
gnal
c
a
n
be
gi
ve
n as
k
k
e
z
k
x
z
X
)
2
(
)
(
(4
)
X
(
z)
=
x(
0)
z
0
+
x(
1)
z
1
+
x(2)z
2
+ x(
3)
z
3
+ .
.
. .
. .
X(
-z)
=
x(
0)z
0
- x(
1)z
1
+ x(2
)
z
2
-
x(
3)z
3
+
.
. .
. .
.
k
k
e
z
k
x
z
X
z
X
z
X
2
2
)
2
(
)]
(
)
(
[
2
1
)
(
(5
)
Si
m
ilarly
k
k
o
z
k
x
z
X
z
X
z
z
X
2
2
)
2
(
)]
(
)
(
[
2
)
(
(6
)
From
(5) a
n
d (6), it is clear t
h
at the
signal X(z)
ca
n be decom
posed
int
o
X
e
(z
2
) a
n
d
X
o
(z
2
) as
gi
ve
n i
n
(7
).
X(z
)
=
X
e
(z
2
) +
z
-1
X
o
(z
2
) (
7
)
Now let u
s
co
nsid
er th
at sig
n
a
l
X(z
)
d
eco
m
p
o
s
ed
in
to
two
p
a
rts u
s
i
n
g
h
i
gh
p
a
ss filter
g
and l
o
w pass fi
l
t
e
r
h
,
then it ca
n
be
represe
n
ted as:
)
(
)
(
)
(
)
(
)
(
z
X
z
G
z
H
z
hp
z
lp
(8
)
Sub
-
sam
p
lin
g
step
co
rr
espond
s t
o
2
)
(
)
(
)
(
)
(
2
)
(
)
(
)
(
)
(
2
2
z
X
z
H
z
X
z
H
z
lp
z
lp
z
lp
z
LP
e
(9
)
2
)
(
)
(
)
(
)
(
2
)
(
)
(
)
(
)
(
2
2
z
X
z
G
z
X
z
G
z
hp
z
hp
z
lp
z
HP
o
(1
0)
Th
e abo
v
e
equatio
n
s
can
b
e
written
i
n
m
a
trix
fo
rm
as
)
(
)
(
)
(
)
(
)
(
)
(
2
1
)
(
)
(
)
(
)
(
2
2
2
2
z
X
z
X
z
G
z
G
z
H
z
H
z
lp
z
lp
z
HP
z
LP
o
e
(1
1)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 4
,
N
o
. 5
,
O
c
tob
e
r
20
14
:
741
–
7
50
74
4
In
th
is case we first calcu
late
all th
e co
efficien
ts and
th
en
t
h
ro
w away h
a
l
f
o
f
th
e
work
do
n
e
. It will b
e
m
o
re
effectiv
e
if we p
e
rform
sa
m
p
li
n
g
b
e
fo
re
filterin
g
,
m
ean
th
at
we co
m
p
u
t
e only ev
en
p
a
rt of
lp
and
hp
.
)
(
)
(
)
(
)
(
)]
(
)
(
[
)
(
1
z
X
z
H
z
z
X
z
H
z
X
z
H
z
lp
o
o
e
e
e
e
(1
2)
Si
m
ilarly,
)
(
)
(
)
(
)
(
)]
(
)
(
[
)
(
1
z
X
z
G
z
z
X
z
G
z
X
z
G
z
hp
o
o
e
e
e
e
(1
3)
Let u
s
d
e
no
te o
u
t
p
u
t
o
f
sub
-
sa
m
p
ler and
low p
a
ss filter as
λ
(z) and
ou
tput o
f
su
b-sam
p
ler and
h
i
gh
p
a
ss filter
as
γ
(z
). The
n
a
b
ove t
w
o equat
i
ons
can be
re
presente
d as
,
)
(
)
(
)
(
)
(
)
(
0
1
z
X
z
z
X
z
P
z
z
e
(1
4)
Wh
ere, P(z) is
a po
ly-ph
a
se m
a
trix
is
g
i
v
e
n
by
)
(
)
(
)
(
)
(
)
(
z
G
z
G
z
H
z
H
z
P
o
e
o
e
(1
5)
In
o
r
de
r t
o
at
t
a
i
n
pe
rfect
rec
onst
r
uct
i
o
n,
fi
l
t
er
h
and
g
mu
st b
e
co
m
p
lemen
t
ary filters th
at will resu
l
t
u
n
ity
d
e
term
in
an
t of
p
o
l
y-ph
ase m
a
t
r
ix
. Po
lyp
h
a
se
matrix
co
rrespo
nd
ing
t
o
lazy
wa
v
e
let tran
sform
will b
e
1
0
0
1
)
(
z
P
(1
6)
Th
is
p
o
l
y-ph
ase m
a
trix
will sp
lit in
pu
t sam
p
les in
to
od
d and
ev
en set.
3.
CAL
CUL
ATI
O
N
OF
LIFTING
SC
H
E
M
E
FOR
NEW
WAVELETS
In
th
is section
th
e liftin
g
steps for th
e n
e
w
o
r
t
h
ogo
n
a
l wav
e
lets p
r
op
o
s
ed
in
[1
4
]
will b
e
calcu
lated
by
a si
m
p
l
i
f
i
e
d m
e
t
hod
of c
a
l
c
ul
at
i
n
g
l
i
f
t
i
n
g
schem
e
.The
b
a
si
s f
unct
i
o
ns
of
t
h
e
o
r
t
h
og
o
n
al
wa
vel
e
t
s
a
r
e gi
ve
n
in
th
e figu
re 1
.
Th
e wav
e
let
filters
fo
r
th
ese
wav
e
lets are calcu
lated
an
d fo
r
si
m
p
licit
y let
h = {
h
-4
, h
-3
, h
-2
, h
-1
, h
0
, h
1
, h
2
, h
3
, h
4
} a
n
d
g = {
g
-4
, g
-3
, g
-2
, g
-1
, g
0
, g
1
, g
2
, g
3
, g
4
}
Fo
r th
e propo
sed
wav
e
lets
h
-4
=0, g
-4
=0.
)
(
'
)
(
'
)
(
'
)
(
'
)
(
'
z
G
z
G
z
H
z
H
z
P
o
e
o
e
The
p
o
l
y
-p
has
e
m
a
t
r
i
x
be
f
o
re
t
h
e ci
rc
ul
ar
co
nv
ol
ut
i
o
n
3
3
1
1
1
3
3
4
4
2
2
0
2
2
3
3
1
1
1
3
3
4
4
2
2
0
2
2
)
(
'
z
g
z
g
z
g
z
g
z
g
z
g
g
z
g
z
h
z
h
z
h
z
h
z
h
z
h
h
z
h
z
P
We ob
tain
th
e
Po
ly-p
h
a
se m
a
trix
P(z) by
ap
p
l
ying
cir
c
u
l
ar
co
nvo
lu
tion
to
the elem
ents of P’(z
).T
h
ere
f
ore
1
3
1
1
2
3
2
4
1
2
0
2
1
3
1
1
2
3
2
4
1
2
0
2
)
(
z
g
g
z
g
z
g
z
g
z
g
g
z
g
z
h
h
z
h
z
h
z
h
z
h
h
z
h
z
P
)
(
)
(
)
(
)
(
z
G
z
G
z
H
z
H
o
e
o
e
(1
7)
No
w
P(z
)
ca
n
be
decom
pose
d
i
n
t
o
t
w
o m
a
t
r
i
ces as
1
0
)
(
1
)
(
)
(
)
(
)
(
)
(
z
S
z
G
z
G
z
H
z
H
z
P
New
o
e
New
o
e
(1
8)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Med
ica
l
Imag
e Co
mp
ression
u
s
ing
Lifting
ba
sed
New Wa
velet Tran
sfo
r
ms
(A H
a
za
ra
thaia
h
)
74
5
From
eq
uat
i
o
n
s
(
1
7
)
a
n
d (
1
8)
,
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
0
0
z
G
z
s
z
G
z
G
z
H
z
s
z
H
z
H
e
New
o
e
New
o
(1
9)
One ca
n obtain S(z) and H
0
Ne
w
(z) by
di
vi
di
n
g
H
0
(z
) by
H
e
(z). S(z) will b
e
th
e q
u
o
tien
t
an
d
H
0
New
(z)
will b
e
th
e
remain
d
e
r.
Simil
a
rly H
0
New
(z)
will b
e
th
e
remain
d
e
r in th
e d
i
v
i
sion
G
0
(z
)
by
G
e
(z)
.
S(z
)
,
H
0
New
(z)
and G
0
New
(z
) c
a
n cal
cul
a
t
e
d a
n
d
gi
ven
by
2
9
1
8
7
2
6
1
5
4
3
3
2
2
1
1
)
(
)
(
)
(
z
C
z
C
C
z
G
z
C
z
C
C
z
H
z
C
z
C
z
C
z
S
New
o
New
o
(2
0)
Whe
r
e the C
1
, C
2
, C
3
, …., C
9
are con
s
tan
t
s in
term
s o
f
wavelet filter co
efficien
ts.Th
e
equ
a
tio
n
(18
)
becom
e
s,
Fi
gu
re 1.
Ne
w Wavel
e
t
s
:
phi
and
p
s
i
f
unct
i
o
ns of
Ne
w
W
a1
,
Ne
w
W
a2
, Ne
w
W
a
3
a
n
d Ne
w
W
a
4
1
0
1
)
(
)
(
)
(
3
3
2
2
1
1
2
9
1
8
7
2
6
1
5
4
z
C
z
C
z
C
z
C
z
C
C
z
G
z
C
z
C
C
z
H
z
P
e
e
(2
1)
Now t
h
e
first
matrix
in
th
e ab
ov
e equ
a
tion
can
furth
e
r
b
e
d
eco
m
p
o
s
ed
i
n
to
two
m
a
trice
s
as,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 4
,
N
o
. 5
,
O
c
tob
e
r
20
14
:
741
–
7
50
74
6
1
)
(
0
1
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
z
T
z
G
z
G
z
H
z
H
z
G
z
G
z
H
z
H
New
o
New
e
New
o
e
New
New
o
e
New
o
e
(2
2)
From
eq
uat
i
o
n
(2
2)
,
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
z
G
z
T
z
G
z
G
z
H
z
T
z
H
z
H
New
o
New
e
e
New
o
New
e
e
(2
3)
As
we ha
ve ca
lculated S(z),
H
0
New
(z) a
n
d
G
0
New
(z) we ca
n
calculate T(z),
H
e
New
(z) a
n
d
G
e
New
(z) by
per
f
o
r
m
i
ng di
v
i
si
ons.
T
h
e T
(
z
)
,
H
e
New
(z)
an
d
G
e
New
(z) val
u
e
s
are gi
ve
n by
,
1
7
6
1
5
4
1
3
2
1
)
(
)
(
)
(
z
A
A
z
G
z
A
A
z
H
z
A
A
z
A
z
T
New
e
New
e
(2
4)
No
w t
h
e e
quat
i
on
(
2
2)
bec
o
m
e
s
1
0
1
)
(
)
(
)
(
)
(
)
(
)
(
1
3
2
1
1
7
6
1
5
4
z
A
A
z
A
z
G
z
A
A
z
H
z
A
A
z
G
z
G
z
H
z
H
New
o
New
o
New
o
e
New
o
e
(2
5)
B
y
usi
ng t
h
e e
quat
i
o
ns
(2
1)
,
(2
2) a
n
d (2
5
)
t
h
e p
o
l
y
-p
hase
m
a
t
r
i
x
P(z) ca
n be re
p
r
ese
n
t
e
d i
n
a f
o
rm
,
so
t
h
at th
e lifti
n
g
step
s bo
th prim
a
l
an
d
du
al
liftin
g
steps can
b
e
calcu
lated.
1
0
1
1
0
1
)
(
)
(
)
(
3
3
2
2
1
1
1
3
2
1
1
7
6
1
5
4
z
C
z
C
z
C
z
A
A
z
A
z
G
z
A
A
z
H
z
A
A
z
P
New
o
New
o
(2
6)
Hen
c
e th
e lifti
n
g
sch
e
m
e
is d
e
v
i
sed as
fo
llows:
F
o
rw
ar
d W
a
ve
l
e
t
Trans
f
o
r
m:
Sp
lit
:
λ
k
x(
2
k
)
γ
k
x(
2
k
+1
)
Du
al
Lifting
(Pred
i
ct)
:
γ
k
γ
k
+[A
1
γ
k-
1
+A
2
γ
k
+
A
3
γ
k+
1
]
Prim
al Lifting (Update)
:
λ
k
λ
k
+ [C
1
λ
k+1
+ C
2
λ
k+2
+ C
3
λ
k+3
]
Inverse W
a
velet Transform:
Inverse Prim
a
l
Lifting
(Updat
e)
:
λ
k
λ
k
- [
C
1
λ
k+1
+ C
2
λ
k+2
+
C
3
λ
k+3
]
In
ver
s
e
Dual L
i
fting
(P
redict)
:
γ
k
γ
k
+ [A
1
γ
k-1
+
A
2
γ
k
+
A
3
γ
k+1
]
M
e
r
g
e
:
(
2
k
)
λ
k
x(
2k
+1
)
γ
k
4.
SIMULATION
RESULTS
In th
is secti
o
n
th
e sim
u
latio
n
resu
lts
wh
ich
co
n
t
ains th
e liftin
g
step
s as
well as th
e
p
e
rfo
r
m
a
n
ce of
new l
i
f
t
i
n
g a
n
d t
r
a
d
i
t
i
onal
wavel
e
t
s
on
m
e
di
cal
im
age com
p
ressi
o
n
i
s
prese
n
t
e
d
.
Fo
ur
new
o
r
t
h
o
g
onal
wav
e
lets
p
r
o
p
o
s
ed
in
[14
]
are con
s
id
ered. Th
e liftin
g
step
s and
wavelet filters fo
r th
e first
wavelet is
p
r
esen
ted h
e
re. Th
e wav
e
let fi
lters fo
r t
h
e
first wav
e
let is g
i
v
e
n b
y
h
= {0
-
0
.1167
0
.
1
274
0
.
1
838
-0
.25
5
6
-0
.27
6
0
0
.
43
41
0
.
44
43
-0
.63
9
6
}
g
= {0
-
0
.6396
-0
.44
4
3
0
.
4
341
0
.
2
760
-0
.25
5
6
-0
.18
3
8
0
.
12
74
0
.
1
167
}
The c
o
nst
a
nt
s
C
1
, C
2
, .
. .
, C
9
fo
r t
h
e
first
wav
e
let are
g
i
v
e
n
as fo
llo
ws.
C
1
= -0
.9
155
,
C
2
= 8
.
62
76
, C
3
= 2
5
.
9
32
7,
C
4
= 2
.
74
20
, C
5
= -
5
.738
5, C
6
=
16
.5
853
,
C
7
= -5
.3
383
,
C
8
= 3
.
76
07
, C
9
= -3
.0
254
.
The c
o
nst
a
nt
s
A
1
, A
2
, .
.
. , A
7
are gi
ven as
follows.
A
1
=
0
.
04
65
, A
2
= 0.004
0,
A
3
=
-0
.1
143
,
A
4
= -
1
.362
7, A
5
=
1
.
89
61
, A
6
= 0.558
8,
A
7
=
-
0
.345
9.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Med
ica
l
Imag
e Co
mp
ression
u
s
ing
Lifting
ba
sed
New Wa
velet Tran
sfo
r
ms
(A H
a
za
ra
thaia
h
)
74
7
Th
e
fund
am
en
tal co
m
p
lex
ity
in
testing
an
i
m
ag
e co
m
p
ressio
n
system
is
ho
w to
d
ecide wh
ich
test
i
m
ag
es to
u
s
e for th
e an
alysis. Th
e im
ag
e co
n
t
en
t b
e
i
n
g
v
i
ewed
influ
e
n
ces t
h
e percep
tion
o
f
qu
ality
irres
p
ective of
technical pa
ra
meters of the s
y
ste
m
[15]
-[
1
8
]
. No
rm
ally
, a
series o
f
pictures, which a
r
e a
v
era
g
e
i
n
t
e
r
m
s of ho
w di
f
f
i
c
ul
t
t
h
ey
are for sy
st
em
bei
ng eval
u
a
t
e
d, has bee
n
cho
o
sen
.
To
obt
ai
n a bal
a
n
ce of
critical and m
oderately critical
m
a
terial we
used a
wi
de
vari
ety of m
e
dical im
ages.
First
o
f
all th
e co
m
p
ression
p
e
rform
a
n
ce of ex
is
tin
g trad
itio
n
a
l
wav
e
lets, new trad
itio
nal wav
e
lets
and
po
p
u
l
a
r 5/
3 an
d 9/
7 l
i
f
t
i
n
g base
d wa
vel
e
t
s
i
s
present
e
d f
o
r ‘m
ri
1.j
p
g
’
. The C
R
an
d PSNR
wi
t
h
t
h
e
abo
v
e
men
tio
n
e
d
tech
n
i
q
u
e
s are p
r
esen
ted
in
th
e
figu
re. Th
e
CR with
th
e ex
istin
g
trad
ition
a
l wav
e
lets as well as
p
r
op
o
s
ed
trad
itio
n
a
l wav
e
lets is ab
ou
t 3b
pp
.
Bu
t with
t
h
e 5/
3 an
d 9/
7 l
i
f
t
i
n
g base
d
wavel
e
t
s
t
h
e C
R
i
s
abo
u
t
1
0bp
p. Th
e PSNR with
th
e ex
istin
g
trad
itio
n
a
l wav
e
lets is
j
u
st aro
und
23
d
B
ex
cep
t
fo
r Co
iflet wav
e
let, for
whi
c
h t
h
e PS
NR
i
s
40
dB
. P
S
NR
wi
t
h
new
t
r
adi
t
i
onal
wa
vel
e
t
s
i
s
arou
n
d
3
0dB
a
nd
wi
t
h
5/
3 a
nd
9/
7
l
i
f
t
i
n
g
b
a
sed
wav
e
lets it is o
v
e
r
3
7dB. In
t
o
tal th
e
co
m
p
ression
perfo
r
m
a
n
ce o
f
n
e
w t
r
ad
ition
a
l wav
e
lets is a better i
n
term
s o
f
PSNR to
th
at
o
f
existin
g
trad
itio
nal wav
e
lets
and
th
e 5
/
3
an
d
9
/
7
lifting
b
a
sed
wav
e
lets com
p
le
tely
out
per
f
o
r
m
bot
h t
h
e
t
r
a
d
i
t
i
ona
l
schem
e
s i
n
t
e
rm
s of
bot
h C
R
an
d P
S
NR
.
The
pr
o
p
o
s
ed
l
i
f
t
i
ng
ver
s
i
o
n
o
f
t
h
e
ne
w t
r
a
d
i
t
i
onal
wavel
e
t
s
has
pr
o
duce
d
e
v
en
bet
t
e
r
com
p
ressi
o
n
resu
lts. In
th
e fig
u
res
2
an
d 3,
th
e GUI
u
s
ed in
MATLA
B
a
n
d the
sam
p
le medical im
ages are
shown. C
T
and
MRI
ima
g
e
s
o
f
Br
a
i
n
,
H
e
ar
t an
d H
e
ad
ar
e
con
s
id
er
e
d
a
s
th
e s
a
mp
le
i
m
a
g
e
s
.
Figu
re
2.
Sam
p
le Im
ages fo
r P
r
oces
sin
g
Fi
gu
re
3.
G
U
I
of
Pr
o
pose
d
Al
go
ri
t
h
m
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 4
,
N
o
. 5
,
O
c
tob
e
r
20
14
:
741
–
7
50
74
8
Th
e PSNR
with
th
e lifting
v
e
rsion
of
n
e
w
wav
e
lets
is rang
ing
fro
m
4
0
t
o
44d
B. Th
e
CR with
th
e
l
i
f
t
i
ng
versi
o
n
o
f
new
wa
vel
e
t
s
i
s
ab
out
1
2
b
p
p
.
The
co
m
p
ari
s
on
of
p
e
rf
orm
a
nce o
f
va
ri
o
u
s t
r
a
n
s
f
orm
s
i
s
pl
ot
t
e
d i
n
fi
g
u
r
e
s 4
.
I
n
t
h
ese
fi
gu
res t
h
e P
S
N
R
an
d C
R
s
obt
ai
ned
wi
t
h
‘m
ri
1.j
p
g’
i
m
age are
pl
ot
t
e
d.
Fi
gu
re 4.
C
o
m
p
ari
s
on
o
f
C
R
val
u
es
o
b
t
a
i
n
e
d
wi
t
h
vari
ous
t
r
ans
f
o
r
m
s
for
‘m
ri
1.jp
g
’
Fi
gu
re
5.
C
o
m
p
ari
s
on
o
f
PSN
R
val
u
es
f
o
r
‘
m
ri
1.jp
g’
o
b
t
a
i
n
ed
wi
t
h
va
ri
o
u
s t
r
a
n
s
f
o
r
m
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Med
ica
l
Imag
e Co
mp
ression
u
s
ing
Lifting
ba
sed
New Wa
velet Tran
sfo
r
ms
(A H
a
za
ra
thaia
h
)
74
9
5.
CO
NCL
USI
O
N &
FUT
U
R
E
SCOPE
In th
is
p
a
p
e
r a
n
e
w liftin
g m
e
ch
an
ism
was
propo
sed.
Fi
rst, a
n
e
w
set o
f
basi
s functions
are selected
an
d
u
tilized
to
p
e
rcei
v
e
n
e
w t
r
ad
ition
a
l wavelets. Th
e wa
velet filters h
and
g
are th
en
calcu
l
ated
. Fro
m
th
ese
filters, th
e po
ly
-ph
a
se m
a
trix
p
(
z),
b
y
wh
ich
after d
e
co
m
p
osin
g
it in
to
three su
b-m
a
trice
s
, liftin
g
steps
can
b
e
calcu
lated
is fo
rm
ed
. First t
h
is m
a
trix
is p
a
rtitio
n
e
d
i
n
to
m
u
ltip
licatio
n
o
f
t
w
o
sub
-
matrices wh
ich
g
i
v
e
s
p
r
im
al
liftin
g
.
Th
en
th
e first su
b
-
m
a
trix
is ag
ain
d
i
v
i
d
e
d
in
to
m
u
ltip
l
i
catio
n
o
f
two
m
o
re su
b-m
a
trices,
w
h
er
eb
y d
e
r
i
vin
g
t
h
e
d
u
a
l li
f
tin
g. Th
e
w
a
velets af
ter
ap
p
l
yin
g
th
e pr
oposed
lif
ting
sche
m
e
ar
e app
lied
o
n
medical im
age
com
p
ression. The CR, PSNR, De
codi
n
g
t
i
m
e
, Encodi
ng t
i
m
e and t
r
ansf
o
r
m
i
ng t
i
m
e
s are
calcu
lated
.
Ex
cep
t
CR an
d
PSNR th
e rem
a
in
in
g
d
e
si
g
n
m
e
t
r
ics are
m
o
re or less th
e sa
m
e
with
th
at o
f
ex
i
s
tin
g
trad
itio
n
a
l, n
e
w trad
itio
n
a
l an
d th
e
p
opu
lar
5
/
3
and
9
/
7
lift
i
n
g
b
a
sed
wavelets.
Th
e
CR with
ex
istin
g
an
d
new
tr
ad
itio
n
a
l
w
a
velets is aro
und 3b
pp
,
w
ith
5
/
3
an
d
9
/
7
lif
ti
n
g
b
a
sed w
a
v
e
lets it is 1
0
to
1
2bp
p, and
w
ith
th
e
n
e
w lifting
wav
e
lets it is aro
u
n
d
1
2bp
p. Th
e
PSNR
with
ex
i
s
tin
g
trad
itio
n
a
l wav
e
lets is less th
an
2
5
d
B
ex
cep
t
with
co
i
f
let wav
e
let for
wh
i
c
h
th
e
PSNR is aroun
d
40d
B.
W
ith
t
h
e n
e
w trad
itio
n
a
l
wav
e
lets th
e
PSNR is
aro
u
nd
35
dB
,
wi
t
h
5/
3 an
d
9
/
7 l
i
f
t
i
ng
base
d wa
vel
e
t
s
t
h
e
PSNR
i
s
ra
n
g
i
ng f
r
o
m
35 t
o
40
dB
.
W
i
t
h
t
h
e ne
w
l
i
f
t
i
ng
wavel
e
t
s
t
h
e P
S
NR
i
s
bet
w
ee
n
40
an
d
45
dB
.
He
nce
o
n
e ca
n say
t
h
at
t
h
e ne
w l
i
f
t
i
ng
bas
e
d
wa
ve
l
e
t
s
are
co
m
p
arativ
ely b
e
st am
o
n
g
th
e m
e
n
tio
n
e
d
wav
e
lets.It is e
xpected
th
at
with m
o
d
i
fied
SPIHT
b
e
tter
resu
l
t
s will
be obt
ai
ne
d.
AC
KO
NOW
LEDGEME
N
T
S
Fi
rst
an
d f
o
re
m
o
st
, t
h
e fi
rst
aut
h
or
wo
ul
d l
i
ke t
o
t
h
a
nk
D
r
. P
VN R
e
ddy
fo
r hi
s m
o
st
sup
p
o
rt
an
d
enco
u
r
agem
ent
.
He
ki
ndl
y
rea
d
m
y
paper
an
d
of
fere
d i
nval
u
abl
e
det
a
i
l
e
d advi
ces
o
n
gra
m
m
a
r,
or
gani
z
a
t
i
o
n
,
and the t
h
em
e of the
pa
per.
The
fi
rst
a
u
t
h
or
w
o
ul
d l
i
k
e
t
o
t
h
an
k
D
r
.
N S
u
dha
ka
r R
e
ddy
-Pri
nci
p
al
,
an
d M
a
nage
m
e
nt
of
S
r
i
Venk
ateswara
Co
lleg
e
o
f
Engin
eering
,
Tiru
pati fo
r th
eir en
co
urag
em
en
t in
d
o
i
n
g
th
is
wo
rk
.
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S
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BIOGRAP
HI
ES OF
AUTH
ORS
A
H
a
z
arathaia
h
received Engi
neering Gradua
t
i
on from
Institut
i
on of Engineers
,
Kolkatt
a
. He
rece
ived M
a
s
t
er
Degree M
.
Te
ch
(Ins
t
rum
e
ntation
and Con
t
rol) fr
om
J
N
T Univers
i
t
y
,
Kakinad
a
,
Andhrapradesh in the
y
e
ar 2005. Currently
h
e
is
pursuing Ph. D in JNTU, K Kakinada. He h
a
s
m
o
re than 15
ye
ars
tea
c
hing Exp
e
rien
ce
. P
r
es
entl
y
he is working
as a professor in Dept of ECE
in SV College o
f
Engineering,
Ti
rupati
. His
res
e
arch int
e
res
t
is
i
n
Im
age proces
s
i
ng .He is
a l
i
fe
me
mbe
r
of IST
E
,
IE,
a
nd me
mber of IE
E
E
Dr B Prabhak
a
raRao
obtain
e
d B.Te
ch &
M.Tech from
S.V. Universit
y
, Tirup
a
ti with
S
p
ecia
liz
ations
i
n
Elect
ronics
&
Comm
unicatio
ns
Engineering
,
Electronic Instrumentation &
Communications Sy
stems in th
e
y
ears 1979
an
d
1981 respectively
. He receiv
e
d the Do
ctoral
degree from
Indian Institut
e
o
f
Science, Ba
n
g
alore in1995
.He has m
o
re than 31 y
ears of
Experi
ence
in
T
each
ing &
Rese
a
r
ch. He
he
ld
dif
f
e
re
nt
positions
i
n
his c
a
ree
r
such
as Head
of th
e
Dept &Vice Pri
n
cipa
l, Dire
ctor
(Institute of Sc
ienc
e & Te
chno
log
y
)
,
Dire
ctor
of Evalua
tion
,
Director of Foreign Universities
Relations, Di
rector –Adm
ission
s during in the
y
e
ars 2001 to
2013. Presently he is working as Rector
an
d
Director-Admissions from
July
2013 in JNT
Universit
y
, Kak
i
nada. He
is S
e
ni
or IEEE (US
A
) m
e
m
b
er, F
e
llow of IE, IE
TE
, Li
fe Mem
b
er of
ISTE, (
E
MC)
Association, an
d Indian
Scien
ce Congr
ess Association
(Kolakatta). He was
honored with
th
e “State Best
Teacher Award
” for
the
y
e
ar
2010 b
y
the Govt. of An
dhra Pradesh
.
Evaluation Warning : The document was created with Spire.PDF for Python.