Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
Vol
.
4
,
No
. 3,
J
une
2
0
1
4
,
pp
. 41
1~
42
1
I
S
SN
: 208
8-8
7
0
8
4
11
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
An Opti
mal HSI Image Comp
ressi
on using DWT an
d CP
D. Narmadha,
K. Gayathri, K.
Thilaga
va
thi, N. Sa
rda
r
Ba
sha
Departement of
Electronics and
Co
mmunication Engineering,
C. Abdul
Ha
keem
College of
Engineering
and
Techno
log
y
, Vellore, India
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Feb 13, 2014
Rev
i
sed
May
6, 201
4
Accepted
May 23, 2014
The com
p
res
s
i
on of h
y
pers
pe
ctr
a
l im
ages
(HS
I
s
)
has
recent
l
y
bec
o
m
e
a ve
r
y
attr
act
ive is
s
u
e f
o
r rem
o
te s
e
ns
ing applic
ations
becaus
e
of th
eir
volum
etric
data. An efficien
t method for h
y
persp
ectr
a
l imag
e compression is presented
.
The proposed
algorithm, b
a
sed on Discrete Wavelet Tr
an
sform and
CANDECOM/P
A
RAFAC
(DWT-CP),
exploi
ts both the spectral and the
spatial
information in the images
. The cor
e
idea beh
i
nd ou
r proposed
techn
i
que
is to apply
CP on the
DWT co
efficien
ts of spectr
a
l b
a
nds of HSIs.
W
e
us
e DW
T to effect
ivel
y s
e
p
a
rate HS
Is
into different s
ub-im
ag
es
and CP
to effi
cien
tl
y co
m
p
act the
energ
y
of s
ub-im
ages
. W
e
evalu
a
t
e
t
h
e effe
ct of
the proposed method on real HSIs and al
so comp
are the r
e
sults with the well-
known compression methods. The obtai
ned
results show a b
e
tter
performance wh
en comparing
with the existin
g
method PCA with JPEG
2000 and 3D
SPECK.
Keyword:
Com
p
ression
CP
DW
T
Hype
rspect
ral Im
ages
Ten
s
o
r
Deco
mp
o
s
ition
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
:
D. Narm
adha
,
Depa
rt
em
ent
of El
ect
r
oni
cs
a
n
d
C
o
m
m
uni
cat
i
on E
n
gi
nee
r
i
n
g
,
C.Abdul Hak
e
em College of
En
gineer
ing
, Vellore, India.
Em
a
il: srin
an
a1
822
@g
m
a
il.c
o
m
1.
INTRODUCTION
Hype
rspect
ral im
ages
are use
d
i
n
diffe
re
nt practical
applications s
u
c
h
as
the
detection
of t
h
e ea
rth’s
surface, soil type a
n
alysis, agricultu
re
a
n
d forest m
onitoring, e
n
vironm
ental studies a
n
d s
o
on [1].
There are two
types of re
dundancy
in
th
e HSIs i.e. sp
atial an
d
sp
ectral re
dundancies
. Howe
ve
r, the
spect
ral
c
o
r
r
el
at
i
on (
b
a
n
d
re
du
n
d
ancy
) i
s
gene
ral
l
y
but
n
o
t
al
ways stro
ng
er th
an
spatial co
rrelation
[2
].
Seve
ral com
p
ression m
e
thods ha
ve
recentl
y been propos
e
d
whic
h ca
n
be classifie
d
i
n
to t
w
o m
a
in types as
l
o
ssl
ess an
d l
o
ssy
com
p
ressi
o
n
m
e
t
hods.
Lo
ssl
ess com
p
res
s
i
on ca
n
onl
y
pr
o
v
i
d
e l
i
m
i
t
e
d com
p
ressi
on
rat
i
o
s,
an
d th
e m
a
x
i
m
u
m
ach
iev
a
ble o
r
d
e
r is arou
nd
t
h
ree tim
e
s
(3
:1)
[3
]. This ratio
is no
t
a reason
ab
le
valu
e i
n
man
y
p
r
actical app
licatio
n
s
, esp
ecially in
rem
o
te sen
s
in
g
.
Trad
ition
a
l com
p
ressio
n
algo
rith
m
s
fo
r HSIs
h
a
v
e
on
ly co
nsid
ered
t
h
e sp
ectral
v
a
lu
e in
a feat
u
r
e
space whose dim
e
nsions were
spectral bands.
T
h
e
n
, di
mension
reduction was often applied
(by
means of
Pr
in
ci
p
a
l Co
mp
on
en
t A
n
alysi
s
(
P
CA)
o
r
I
ndep
e
nd
en
t Co
mp
on
en
t A
n
alysi
s
(
I
C
A)
[4
]. Ho
w
e
v
e
r
,
th
ey did
n
o
t
co
nsid
er
th
e spatial
co
rrelation
wh
en
fo
cu
si
ng
o
n
th
e sp
ectral
d
ecorelatio
n.
Th
erefo
r
e, 3D wav
e
let-b
a
sed
techn
i
qu
es
su
ch as
Set
Partitio
n
i
ng
in
Hierarch
ical Tress (SPHIT)
alg
o
rith
m
an
d
an
d Set
Partitio
n
e
d Em
b
e
d
d
e
d
b
l
o
c
k
(SPECK) fo
r h
y
p
e
rspectral im
ag
e co
m
p
ressio
n
h
a
ve b
een
p
r
op
o
s
ed
to
exp
l
o
it a j
o
i
n
t con
s
id
eratio
n
o
f
th
e sp
atial an
d sp
ectral correl
a
tio
n
s
[5
]. It has b
een
sho
w
n in
[5
]
th
at 3
D
-SPECK is b
e
tter th
an
3D-SPIHT t
o
ach
iev
e
an
efficient com
p
ression. In [6
]
,
a PC
A-
base
d m
e
t
h
o
d
i
n
co
nju
n
c
tion
w
i
th
JPEG20
00
fo
r
co
m
p
r
e
ssi
ng
H
S
I
s
w
a
s in
t
r
oduced. The results reveal
that the perform
a
nce of
t
h
e m
e
t
hod i
s
sup
e
ri
o
r
t
o
t
h
a
t
of t
h
e
spect
r
a
l
D
W
T
,
an
d t
h
e be
st
PC
A
p
e
rf
orm
a
nce oc
curs
w
h
en
a re
duce
d
num
ber
of
PC
s
are
ret
a
i
n
ed
a
n
d
enc
o
ded
.
A
not
her
com
p
re
ssi
on
al
g
o
ri
t
h
m
based o
n
JP
EG
20
0
0
f
o
r
H
S
Is
was
pr
o
pose
d
i
n
[
7
]
.
The al
g
o
r
i
t
h
m
can be ap
pl
i
e
d f
o
r l
o
ssy
a
nd
nea
r-l
ossl
es
s com
p
ressi
o
n
ap
pl
i
cat
i
ons i
n
o
n
e
si
ngl
e t
o
ol
. It
was al
so s
h
ow
n t
h
at
t
h
e p
r
o
pos
ed sc
hem
e
has a ne
gligibl
e
effect on
th
e resu
lts of sel
ected
appl
i
cat
i
o
ns (e
.g.
,
ha
r
d
cl
ass
i
fi
cat
i
on, s
p
ect
ral
u
n
m
i
xi
ng,
and
an
om
al
y
det
ect
i
o
n
)
.
In
[8]
a
new l
o
s
s
y
-
t
o
-
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
4, No
. 3,
J
u
ne 2
0
1
4
:
41
1 – 4
2
1
41
2
l
o
ssl
ess
c
ode
r base
d on
3
D
Tarp
-
b
ased
C
o
di
n
g
wi
t
h
cl
as
si
fi
cat
i
on fo
r Em
beddi
n
g
(T
C
E
)
c
o
u
p
l
e
d
wi
t
h
t
h
e
reve
rsi
b
l
e
i
n
t
e
ger
-
val
u
ed
Ka
r
h
u
n
e
n
-L
oè
ve
Tran
sf
orm
(K
LT) w
a
s i
n
t
r
o
duce
d
.
The
pr
op
ose
d
m
e
t
hod cl
osel
y
match
e
s th
e lossy p
e
rf
or
m
a
n
ce o
f
JPEG20
00
and
ou
t
p
erf
o
r
m
ed
JPEG
2000
at lo
ssless com
p
r
e
ssio
n
.
A
n
H
S
Is
com
p
ressi
o
n
m
e
t
hod b
a
sed
on
ban
d
-
o
f
i
n
t
e
rest
B
O
I
-
p
r
es
ervi
ng
- base
d
was p
r
o
p
o
se
d i
n
[9]
.
S
o
m
e
band
s of
HSIs are m
o
re im
portant in the specific applications,
and BOI selection
m
e
thods ca
n be chose
n
according to
ap
p
lication
requ
irem
en
ts. BOI an
d non
-B
OI b
a
n
d
s are
respectiv
ely co
m
p
ressed
with low d
i
stortion
an
d h
i
gh
di
st
ort
i
o
n.
I
n
[
10]
t
h
e i
m
pact
of l
o
ssy
com
p
ressi
o
n
on s
p
e
c
t
r
al
unm
i
x
i
n
g
and
su
per
v
i
s
e
d
cl
assi
fi
cat
i
o
n
usi
n
g
Support Vector
Machine (SVM
) was i
nvesti
g
ated.
It was
s
h
own that fo
r
certain com
p
ression techniques,
a
hig
h
er
com
p
re
ssion
ratio
(C
R) m
a
y
lead to m
o
re
acc
ura
t
e cl
assi
fi
cat
i
on r
e
sul
t
s
.
A
n
e
w
gr
o
up
an
d
re
gi
o
n
base
d
paral
l
e
l
com
p
ressi
o
n
m
e
t
h
o
d
fo
r
hy
pe
rspect
ral
i
m
agery
was
p
r
o
p
o
se
d i
n
[
1
1]
.
Som
e
co
m
p
res
s
i
on m
e
t
hods
cur
r
ent
l
y
co
nsi
d
er
HS
Is as 3
D
dat
a
. T
h
ose
com
p
ressi
on
m
e
t
hods a
r
e
called
a th
ird-ord
e
r ten
s
o
r
: two
sp
atial d
i
m
e
n
s
ion
s
an
d
on
e sp
ectral d
i
m
e
n
s
ion
.
Th
ey try to
tak
e
in
to
acco
un
t
th
e sp
atial and sp
ectral co
rrelatio
n
o
f
HSIs
si
m
u
ltan
e
o
u
s
ly
an
d no
t alternativ
ely as
is the case for t
h
e
above
t
echni
q
u
es
[
1
3
]
. Seve
ral
t
e
ns
or
dec
o
m
posi
t
i
ons
ha
ve
bee
n
i
n
t
r
o
d
u
ced
i
n
[
14]
.
O
n
e
of t
h
e
m
o
st
pop
ul
ar
t
e
ns
or
decom
positions is the Canoni
cal decom
position (CP
)
, whic
h has bee
n
use
d
for the com
p
ression of HS
Is. CP
allo
ws t
h
e selectio
n
o
f
an
y
v
a
lu
es for each
di
m
e
n
s
io
n
of the core tensor,
wh
ich
will b
e
d
e
fi
n
e
d as
i
n
t
h
e
ne
xt
sect
i
on a
n
d
hel
p
s
t
o
obt
ai
n
a
hi
g
h
e
r c
o
m
p
ressi
on
rat
i
o
.
Fi
gu
re
1.
Thi
r
d
-
di
m
e
nsi
on m
odel
i
n
g
o
f
HS
Is.
We
pr
o
pose
a
ne
w
HS
Is c
o
m
p
ressi
on al
g
o
ri
t
h
m
base
d
on
di
sc
ret
e
wa
vel
e
t
t
r
an
sf
or
m
(D
W
T
)
an
d
CP. App
l
ying
2
D
WT to
each
sp
ectral
b
a
nd
will tak
e
care o
f
first stag
e co
m
p
ression
b
y
u
s
i
n
g
the (9
/
7
)
b
i
ortho
gon
al wav
e
let.
Nex
t
, CP is ap
p
lied to
th
e four
wav
e
let su
b-im
a
g
es of th
e
HSIs in
o
r
d
e
r to
ach
i
eve
m
o
re CR. Fin
a
lly, ad
ap
tiv
e arith
m
e
tic co
d
i
ng
(AAC) is u
s
e
d
fo
r c
odi
ng
t
h
e el
em
ent
s
o
f
t
h
e c
o
re
t
e
ns
ors
.
We com
p
are t
h
e pr
op
ose
d
m
e
t
hod wi
t
h
t
h
e best
kn
o
w
n t
ech
ni
q
u
es
, such as t
h
e
3D-
SPEC
K
alg
o
r
ith
m
[
5
],
an
d
b
o
t
h
PCA an
d
JPEG
2000
[
6
]. Our
exp
e
r
i
m
e
n
t
al r
e
su
lts o
v
e
r
th
e tw
o
m
o
st u
s
ed H
S
I
s
(A
VIR
I
S dat
a
s
e
t
s
:
C
upri
t
e
an
d M
o
ffet
t
Fi
el
d) dem
onst
r
ate and confi
r
m
the effectiv
enes
s of our algorithm
in
p
r
ov
id
ing
a
mu
ch
sm
aller M
S
E for th
e d
e
sired
set o
f
CRs
, especially when the CR is
selected higher than
16
0, i
n
t
h
i
s
ca
se t
h
e pe
rf
orm
a
nce o
f
t
h
e
pr
o
pos
ed al
go
ri
t
h
m
i
s
si
gni
fi
can
t
l
y
bet
t
e
r t
h
an
t
h
e ot
he
r t
ech
n
i
ques
.
The
pr
o
p
o
s
ed
m
e
t
hod al
s
o
i
n
creases t
h
e
pi
x
e
l
-
base
d s
u
per
v
i
s
ed
cl
assi
fi
ca
t
i
on acc
uracy
.
2.
3
D
REPRESE
N
TATION
OF
HY
PERSPEC
T
RA
L IMA
G
ES
HSIs are
the i
m
ages ge
nerat
e
d
by the im
aging s
p
ect
rom
e
ter
by collecting im
age data si
m
u
ltaneously
in
hun
dr
ed
s
o
f
sp
ectr
a
l
b
a
nds or
f
r
e
q
u
en
ci
es (
e
.g
.
5
–10
nm
spectral wi
dth) that
reac
h a
nearly c
onti
g
uous
spect
ral
reco
r
d
. T
h
e l
a
r
g
e a
m
ount
o
f
ban
d
s
i
n
crea
ses t
h
e
com
p
l
e
xi
t
y
and
t
i
m
e
of
p
r
o
cessi
ng
.
HS
Is
can
be
rep
r
ese
n
t
e
d
as
a
t
h
i
r
d di
m
e
nsional
dat
a
∈
as s
h
ow
n i
n
Fi
gu
re
1.
Un
lik
e n
a
t
u
ral
i
m
ag
es, HSIs h
a
ve two
types o
f
correlatio
n
sim
u
ltan
e
o
u
s
ly, wh
ich
are th
e spatial
correlation wit
h
in im
ages
and the
s
p
ectral
correlation bet
w
een s
p
ect
ral
b
a
nd
s. Th
e spectral correlatio
n is
g
e
n
e
rally strong
er t
h
an sp
atial co
rr
elation. T
h
e a
v
era
g
e c
o
rrelation c
o
ef
fi
c
i
ent
bet
w
ee
n t
w
o
spect
ral
ba
nds
o
f
3D-data cube c
a
n
be m
easure
d
as
follows:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
An
O
p
t
i
m
al
H
S
I I
m
age
C
o
m
p
r
e
ssi
on
usi
n
g
D
W
T a
n
d
C
P
(
D
. N
a
r
m
ad
h
a
)
41
3
For
= 1 t
o
= 1
:,:
,
,
:,:
,
1
:,:,
,
:,:
,
1
End
∑
(1
)
In m
o
st HSIs
datasets, suc
h
as
AVIRIS
data
cube
,
th
e v
a
l
u
e of is
h
i
gh
er than
0
.
9
0
wh
ich
mean
s th
at
t
h
e s
p
ect
ral
c
o
r
r
el
at
i
on am
on
g
ba
nd
s i
s
very
hi
g
h
.
Exp
l
o
itin
g
bo
th
of sp
ectral
an
d
sp
atial correlatio
ns is the k
e
y fo
r th
e
su
ccess
o
f
a co
m
p
ression
al
go
ri
t
h
m
.
In t
h
i
s
pa
per
,
a h
y
b
ri
d sc
hem
e
base
d o
n
D
W
T an
d C
P
f
o
r
com
p
ressi
o
n
o
f
HS
Is i
s
i
n
t
r
o
duc
e
d
.
D
W
T a
n
d
C
P
a
r
e
bri
e
fl
y
i
n
t
r
o
duce
d
i
n
t
h
e
ne
xt
sect
i
o
ns.
2.2. Discrete
Wavelet
Tr
ansfor
m
(DWT
)
The DWT
ha
s success
f
ully been
use
d
in
m
a
ny im
age
processing a
pplications inc
l
uding
nois
e
red
u
ct
i
o
n, e
d
g
e
det
ect
i
o
n
,
an
d com
p
ressi
on
[1
5]
. I
n
deed
,
t
h
e D
W
T i
s
a
n
effi
ci
ent
deco
m
posi
t
i
on of
s
i
gnal
s
in
to
lo
wer reso
lu
tion
and
d
e
tails. Fro
m
th
e th
e d
e
termin
istic i
m
ag
e p
r
o
c
essin
g
p
o
i
n
t
o
f
v
i
ew, DWT
may b
e
viewe
d
as succ
essive low-pass and
high-pass filtering
of t
h
e disc
rete ti
me-dom
ain signal. At each level, the
h
i
gh
p
a
ss
filter p
r
od
u
c
es d
e
t
a
iled
in
form
ati
o
n
(ho
r
izon
ta
l (H), v
e
rtical (V) an
d
d
i
ago
n
al (D) inform
a
tio
n
)
,
wh
ile th
e l
o
w p
a
ss
filter asso
ciated
with
scalin
g
fun
c
tion
pro
d
u
ces t
h
e app
r
o
x
i
m
a
te
(A) inform
ati
o
n.
We
apply a two
di
mensional DWT to each band of HSIs
. If
ea
ch im
age band has rows
a
n
d colum
n
s,
then after
ap
p
l
ying
th
e
2DW
T
,
w
e
ob
tain
fo
ur
su
b-
b
a
nd
im
ag
es (
A
,
H, V
,
D)
, each
hav
i
ng
2
⁄
ro
ws a
n
d
2
⁄
colum
n
s.
The
A s
u
b-
ba
n
d
i
m
ages ha
ve
t
h
e hi
ghe
st
ene
r
gy
am
on
g al
l
t
h
e c
o
ef
fi
ci
ent
s
of
t
h
e
ot
he
r s
u
b-
ba
nd
i
m
ages.
Th
e 2DW
T
of
f
u
n
c
tion
,
with
o
f
size
and
can
be s
h
ow
n as
,
∑∑
,
,
,
(2
)
,
,
∑∑
,
,
,
,
(3
)
,
,
(4
)
,
,
,
∑
2
√
2
2
∑
2
√
2
2
(5
)
,
,
2
2
,
2
2
∑
2
√
2
2
∑
2
√
2
2
(6
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
4, No
. 3,
J
u
ne 2
0
1
4
:
41
1 – 4
2
1
41
4
Fig
u
re 2
.
New d
eco
m
p
o
s
ition
sch
e
m
e
fo
r
HSIs.
Tab
l
e 1
.
9
/
7
Bio
r
t
h
ogo
n
a
l Wav
e
let
Filter
N
φ
[n]
ψ
[n]
0 0.
7885
0.
8527
±1 0.
4181
0.
3774
±2 -
0
.
04069
-
0
.
111
±3 -
0
.
06454
-
0
.
02385
±4
0.
0378
3
φ
is called
scalin
g
fun
c
tion
.
Th
e
φ
,
coef
fi
ci
ent
s
de
fi
ne a
n
a
p
pr
o
x
i
m
at
i
on o
f
,
. T
h
e
,
,
coefficients add horiz
o
ntal, verti
cal and di
agonal details. Norm
ally
2
is selected and
so that
0,1
,
2,
…
.
,
1
[16
]
.
Th
e
φ
and
are called wav
e
let filters.
Fo
r t
h
e
cho
i
ce of wav
e
let
filterin
g
, t
h
e
(
9
/
7
)
b
i
o
r
t
h
ogon
al w
a
v
e
let w
h
ich
is u
s
ed
in
JPEG2
000
, is selected
in
o
r
d
e
r
to
im
p
r
o
v
e
the co
m
p
r
e
ssio
n
r
a
tio
[17
]
. Filter co
efficien
ts
o
f
(9
/
7
) b
i
o
r
thog
on
al wav
e
let are sh
own
in Tab
l
e
1
[18
]
,
[19
]
.
The
i
nve
rse 2
D
-
D
WT
i
s
gi
v
e
n by
,
,
∑∑
,
,
,
)
+
∑∑
∑∑
,
,
,
,
,
,
(7
)
In l
o
ssy
com
p
ressi
on
, we ca
n
i
g
n
o
re t
h
e
det
a
i
l
e
d sub
-
ban
d
s
(f
or e
x
am
pl
e H, V a
nd
D)
or
prese
r
ve
o
n
l
y im
p
o
r
tan
t
d
e
tailed
in
formatio
n
.
In
order to
ach
i
ev
e a h
i
gh
er co
m
p
ressio
n
ratio
,
we can
still d
e
co
m
p
o
s
e
th
e im
ag
es in
m
o
re th
an
on
e
lev
e
l.
B.
Nonn
eg
ativ
e Ten
s
o
r
Deco
m
p
o
s
itio
n
In t
h
i
s
sect
i
o
n
a b
r
i
e
f
re
vi
ew
of t
h
e t
e
nso
r
m
ode
l is presen
ted. Im
p
o
rtan
t no
tatio
ns are sh
own
i
n
Tabl
e 2.
The t
h
i
r
d-
or
de
r T
D
t
e
ns
or
i
s
de
scri
be
d
as
a
decom
posi
t
i
o
n
o
f
a
gi
ve
n t
h
i
r
d
or
der
t
e
nso
r
∈
in
to an
un
know
n co
r
e
ten
s
or
∈
m
u
ltip
lie
d
b
y
a set
of th
ree
u
nkn
own co
m
p
on
en
t
ma
tr
ic
e
s
w
h
er
e
,
,……
…
∈
(n =
1
,
2,
3)
,
r
e
prese
n
t
c
o
m
m
o
n
fact
ors
[
20]
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
An
O
p
t
i
m
al
H
S
I I
m
age
C
o
m
p
r
e
ssi
on
usi
n
g
D
W
T a
n
d
C
P
(
D
. N
a
r
m
ad
h
a
)
41
5
Tabl
e 2. N
o
t
a
t
i
ons
o
f
Tens
o
r
Decom
posi
t
i
o
n
Notation Descr
i
ption
R
n
n-
dim
e
nsional r
eal vector
space
Y
T
h
ir
d or
der
T
e
nsor
Y
(n
)
n-m
ode
m
a
tr
icization of tens
or
Y
A
(n
)
n-m
ode
m
a
tr
ix in tucker
m
odel
°
Outer
pr
oduct
X
n
n-m
ode pr
oduct of
a tensor
by
m
a
tr
ix
Fi
gu
re
3.
Fi
be
r
s
o
f
t
h
e
3
-
or
der
t
e
ns
or
Fi
gu
re
4.
Sl
i
ces o
f
a
3
-
o
r
de
r t
e
ns
or
Fig
u
re 5
.
CP deco
m
p
o
s
itio
nof
a 3
way
array
The CP
dec
o
m
position
factorizes a tens
or int
o
a
sum
of
c
o
m
ponent
rank-one
tens
or
s.
For e
x
am
ple, given a
th
ird-ord
e
r tenso
r
X
R
I×J×K
, we
wish t
o
wri
t
e it as
X
1
rr
r
R
r
ab
c
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
4, No
. 3,
J
u
ne 2
0
1
4
:
41
1 – 4
2
1
41
6
Fi
gu
re
6.
Thi
r
d
o
r
de
r Te
ns
or
Decom
posi
t
i
o
n
(8
)
He
re t
e
ns
or i
s
a
n
est
i
m
a
t
i
on of
t
e
ns
or
, an
d i
t
de
pe
nds
o
n
t
h
e val
u
es,
w
h
i
c
h a
r
e
t
h
e di
m
e
nsi
ons
of
t
h
e
core
t
e
ns
or
,
and
t
e
ns
or
d
e
not
es
t
h
e e
s
t
i
m
at
i
on er
ror. Mo
st algorith
m
s
fo
r th
e
Nonn
eg
ativ
e
Ten
s
o
r
Decom
posi
t
i
o
n
(NT
D
) m
odel
are base
d
on
Al
t
e
rnat
i
v
e L
east
Sq
uare
(
A
LS
) m
i
nim
i
zat
i
on o
f
t
h
e s
qua
re
d
Eucl
i
d
ea
n
di
st
ance
use
d
as a
g
l
obal
c
o
st
f
u
nc
t
i
on s
u
bject
t
o
no
n
n
egat
i
v
e
c
o
nst
r
ai
nt
s
,
t
h
at
i
s
:
ǁ
ǁ
ǁ
(9
)
Here th
e
obj
ectiv
e is to fi
n
d
the op
tim
a
l
co
mp
on
en
t m
a
trice
s
∈
and the
core
tensor
∈
.
Alm
o
st all
th
e ex
istin
g
al
g
o
rith
m
s
fo
r Ten
s
or d
e
co
m
p
o
s
ition
s
[12
]
req
u
i
re certain
pro
cessin
g
b
a
sed
on t
h
e f
u
l
l
t
e
nsor
d
u
ri
n
g
t
h
e
est
i
m
a
ti
on. T
h
e real
-
w
orl
d
dat
a
oft
e
n co
n
t
ai
n
m
i
ll
i
ons o
f
el
em
ent
s
. Fu
l
l
dat
a
pr
ocessi
ng
(s
u
c
h as t
h
e i
n
v
e
r
s
e com
put
at
i
on) i
s
t
h
ere
f
ore im
practical. We
use
the
Hierarchical Nonnegative
Tens
or Dec
o
m
position al
gorith
m
in propose
d
m
e
thod.
3.
PROP
OSE
D
CO
MP
RESSI
ON ALG
O
RI
THM
The c
o
m
p
ressi
on i
s
per
f
o
rm
ed usi
n
g f
o
u
r
st
eps.
In
t
h
e firs
t step, 2DWT
is applied t
o
e
ach s
p
ectral
ban
d
o
f
t
h
e
H
S
Is i
n
or
de
r t
o
o
b
t
a
i
n
4
su
b-i
m
ages (a
p
p
r
o
x
i
m
a
t
e
, di
ago
n
a
l
, ve
rt
i
cal
an
d
ho
ri
zo
nt
al
)
fo
r
eac
h
spectral ba
nd (see Figure 2).
In the sec
o
nd
step, the CP
al
gorithm
is applied to
the four tens
ors.
For
each
tensor, t
h
e size
of the c
o
re
tensor
, i.e.
(J
1
, J
2
, J
3
),
was s
e
l
ect
ed m
a
nual
l
y
. The a
p
pr
oxi
m
a
t
e
t
e
nso
r
h
a
s
t
h
e
l
o
west
f
r
e
que
n
c
y
com
ponent
s
cont
ai
ni
ng m
o
st
o
f
t
h
e
wav
e
l
e
t
coeffi
ci
ent
ener
gy
, s
o
t
h
e
val
u
es
of
(J
1A
, J
2A
,
J
3A
) were
set
h
i
ghe
r t
h
a
n
t
h
os
e of
ot
he
r t
e
ns
ors
.
T
h
e val
u
e
s
of
(J
1D
, J
2D
, J
3D
) fo
r t
h
e
di
a
g
o
n
al
t
e
ns
or
,
whi
c
h
contains the
di
agonal i
n
form
ation,
were also
set
h
i
gh
er than
the (J
1H
, J
2H
, J
3H
) a
n
d (
J
1V
, J
2V
, J
3V
) v
a
lu
es
of
ho
ri
zo
nt
al
an
d
vert
i
cal
t
e
ns
or
s
.
3.
1. C
o
mpu
t
i
n
g CP Dec
o
mp
osi
t
i
o
n.
As m
e
nt
i
oned
previ
o
u
s
l
y
, t
h
ere i
s
no fi
ni
t
e
al
gori
t
h
m
for det
e
rm
i
n
i
ng t
h
e ran
k
o
f
a t
e
nso
r
[1
4
3
,
1
0
1
]
; con
s
eq
u
e
n
tly, th
e
first i
ssu
e t
h
at arises in
co
m
put
i
ng
a C
P
d
ecom
p
o
s
i
t
i
on i
s
ho
w t
o
c
h
o
o
se
t
h
e
n
u
m
b
er
of ra
nk-one
c
o
m
pone
nts.
M
o
st
proce
d
ures
fit m
u
ltiple CP dec
o
m
posi
tions with differe
n
t num
bers
of
com
pone
nts until one is “good.” Ideally, if the data are
noi
s
e-free and we
have a
procedure for calculating CP
wi
t
h
a gi
ven
n
u
m
b
er of c
o
m
p
o
n
e
n
t
s
, t
h
e
n
we can
d
o
t
h
at
com
put
at
i
on f
o
r
R
= 1
,
2
,
3
,
. . .
c
o
m
pone
nt
s an
d
stop at the
first
value
of
R
th
at
g
i
v
e
s a
f
it of
10
0%.
Ass
u
m
i
ng t
h
e num
ber of c
o
m
pone
nt
s i
s
fi
xed, t
h
e
r
e
are
m
a
ny
al
gori
t
h
m
s
t
o
com
put
e a
C
P
d
eco
m
p
o
s
ition
.
Here
we fo
cu
s
o
n
wh
at is to
d
a
y th
e “wo
r
kh
orse” algorith
m
fo
r CP:
th
e altern
ating least
squ
a
res (AL
S
) m
e
t
hod pr
o
pos
ed
i
n
t
h
e ori
g
i
n
al
pa
pers by
C
a
rr
ol
l
and
C
h
ang [3
8]
an
d
H
a
rshm
an.
F
o
r
e
a
se
o
f
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
An
O
p
t
i
m
al
H
S
I I
m
age
C
o
m
p
r
e
ssi
on
usi
n
g
D
W
T a
n
d
C
P
(
D
. N
a
r
m
ad
h
a
)
41
7
prese
n
t
a
t
i
o
n
,
we o
n
l
y
deri
ve
t
h
e
m
e
t
hod i
n
t
h
e t
h
i
r
d-
or
de
r case, but
t
h
e
ful
l
al
go
ri
t
h
m
i
s
present
e
d fo
r an
N
-
way
t
e
ns
or
i
n
Fi
gu
re
3
CR in
th
is step
can
b
e
con
s
idered
as th
e rati
o
b
e
tween
th
e
to
tal n
u
m
b
e
r of b
its in
th
e o
r
i
g
in
al in
pu
t d
a
t
a
,
an
d th
e
nu
m
b
er
o
f
b
its m
u
st be tran
sm
itted
an
d is sh
own
as
2
2
2
2
2
2
(1
0)
In th
e t
h
ird step
, t
h
e
4
co
re ten
s
ors
and
12
matrices
sh
ou
l
d
b
e
tran
sm
itte
d
.
Most ele
m
ents of t
h
e core te
nsors
(es
p
ecially diagonal,
ve
rtical a
nd
ho
ri
zont
al
c
o
re t
e
n
s
ors
)
a
r
e
nearl
y
ze
r
o
.
T
h
e
bi
t
p
l
a
ne c
o
di
n
g
p
r
oce
d
ur
e i
s
use
d
to tran
sm
it th
e elemen
ts of th
ese core ten
s
o
r
s
.
T
h
e
bi
t
p
l
a
ne c
odi
n
g
i
n
cl
u
d
es t
w
o
passes:
t
h
e si
gni
fi
cant
pass
and t
h
e refi
ne
m
e
nt
. Fi
rst
,
w
e
defi
ne a si
g
n
i
f
i
cant
m
a
p o
f
a
gi
ve
n t
h
res
h
ol
d
T
and
t
h
e
el
em
ent
.Su
ppo
se
|
represen
ts
t
h
e
ab
so
lu
te v
a
lu
e o
f
th
e
core te
nsor ele
m
ent at the location (
,
,
) an
d
represe
n
t
s
t
h
e s
i
gni
fi
ca
nt
state for the thres
h
old T
(wh
e
re T is an
in
teg
e
r power
o
f
2).
1
|
|
2
0
.
(1
1)
For
1
, the
is conside
r
ed as the
significa
nt el
ement. The si
gnificant elem
ent
m
u
s
t
be enc
o
ded a
n
d rem
ove
d fr
o
m
t
h
e core t
e
n
s
or
, an
d t
h
e i
n
si
gni
fi
ca
nt
el
em
ent
s
are pre
s
erve
d f
o
r t
h
e
nex
t
bi
t
p
l
a
ne. A
f
t
e
r t
h
at
, t
h
e si
gni
fi
cant
t
h
res
h
ol
d i
s
di
vi
de
d i
n
hal
f
, a
nd t
h
e
p
r
oces
s i
s
repea
t
ed fo
r t
h
e ne
x
t
pass.
Th
is
p
r
o
cess is rep
e
ated
u
n
til
th
e en
erg
y
of t
h
e en
cod
e
d
elemen
ts equ
a
l to
o
r
h
i
gh
er th
an
9
9
.5% of t
h
at
o
f
t
h
e
ori
g
inal core t
e
ns
or.
The sel
ected elem
ents
and t
h
eir
po
sitions m
u
st be trans
f
erre
d. T
h
ere are m
a
ny possible
ap
pro
ach
es fo
r cod
i
ng
a sign
ifican
t m
a
p
.
M
o
st
wav
e
let-b
a
sed
cod
e
rs u
s
e ad
ap
tiv
e arithmetic co
d
i
ng
(AAC
)
fo
r lossless e
n
tro
p
y
co
din
g
.
We also
use
AAC
fo
r co
di
n
g
t
h
e sign
ifican
t ele
m
en
t. Arith
m
e
tic co
d
i
ng
is a
vari
a
b
l
e
-l
en
gt
h
l
o
ssl
ess c
odi
n
g
.
Ari
t
h
m
e
t
i
c
codi
ng
d
o
es
n
o
t
re
qui
re t
h
e
t
r
ansm
i
ssi
on o
f
co
de
bo
o
k
an
d s
o
achi
e
ves a hi
g
h
er com
p
ressi
on t
h
a
n
H
u
f
f
m
a
n co
di
n
g
. A
r
i
t
h
m
e
t
i
c
codi
ng
essent
i
a
l
l
y
am
ount
s t
o
com
put
i
n
g
th
e cu
m
u
lativ
e d
i
stribu
tio
n
fun
c
tio
n
(CDF)
o
f
th
e
p
r
ob
ab
ility o
f
a seq
u
e
nce o
f
sym
b
o
l
s an
d
t
h
en
represen
ting
th
e resu
lting
nu
m
e
rical v
a
lu
e in
a
b
i
n
a
ry cod
e
.
Th
e co
lu
m
n
s
o
f
m
a
trices
are called
i
n
t
h
e
pr
o
pose
d
a
l
go
ri
t
h
m
,
whi
c
h a
r
e
no
rm
al
ized
vectors. T
h
e a
b
sol
u
te val
u
e
of
ele
m
ents are in the ra
nge [0, 1], a
nd
t
h
e
y
are very close to each
other.
There
f
ore,
i
n
or
der
t
o
tra
n
s
f
er t
h
e
12 m
a
trices
, a un
ifo
r
m
q
u
a
n
tizatio
n is
u
s
ed
. In the fift
h step
, the
t
r
ansm
i
t
t
e
d dat
a
are deco
de
d.
Fi
nal
l
y
i
n
t
h
e si
xt
h st
ep, t
h
e i
nverse
of
2
D
WT i
s
appl
i
e
d t
o
rec
onst
r
uct
t
h
e
im
ages.
3.
2. Pro
p
ose
d
Al
g
o
ri
thm
(
D
WT
-C
P)
3.
2.
1. Pro
p
ose
d
Al
gori
t
hm f
o
r Co
der
Inpu
t: Ori
g
in
al
HSIs
(t
he siz
e
)
1-Apply DWT
to each spectral band to
obt
ain 4 sub-im
ages
(approxim
a
te, diagonal,
ve
rtical and hori
zontal
ten
s
ors).
2-
Ap
pl
y
C
P
al
go
ri
t
h
m
t
o
t
h
e
fo
ur
t
e
ns
or
s i
n
di
vi
d
u
al
l
y
. Eac
h
t
e
n
s
o
r
has t
h
e si
ze o
f
(
2
⁄
2
⁄
)
Inpu
t
: Data ten
s
or
Y: (
I
1
×
I
2
× ·
·
· ×
I
N
),
r
a
n
k
R
,
Un
fol
d
in
g rule
l
= [
l
1
,
l
2
,
.
. .
,
l
M
] where
l
m
= [
lm
(1),
.
. .
,
lm
(
Km
)]
Outp
ut
:
λ
∈
R
N
,
N
m
a
trices
A
(
n
)
∈
R
In
×
R
Beg
i
n
St
ag
e
1:
,
,
…
.,
,
min
,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
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:
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08
IJEC
E V
o
l
.
4, No
. 3,
J
u
ne 2
0
1
4
:
41
1 – 4
2
1
41
8
St
ag
e
2:
;
,
…
.,
,
1
,
2
,
…..
←
1
,
2
,
…..
1,2,
…
,
,
,
…
.,
,
,…,
,
1
←
End CP
4-Quantize
∈
an
d e
n
cod
e
co
r
e
te
n
s
o
r
∈
Usi
n
g
A
A
C
3.
2.
2. Pro
p
ose
d
Al
gori
t
hm f
o
r Deco
der
5-
Dec
ode
t
h
e e
l
em
ent
s
=
̅
6-Calculate t
h
e
inve
rse
2DWT to
reconstruc
t im
ages.
Outp
ut
: Recon
s
tru
c
ted
im
ag
es
.
4.
CO
MP
UTAT
ION
A
L CO
M
P
LE
X
I
TY
In th
e fo
llo
wi
ng
,
we an
alyze t
h
e co
m
p
lex
ity o
f
th
e algo
rithm
.
W
e
con
s
id
ered an -tap
filter
b
a
nk and
d
e
no
ted
as the nu
m
b
er
of
wav
e
let
d
eco
m
p
o
s
itio
n lev
e
ls
i
n
t
h
e sp
atial
ban
d
.
Th
e co
m
p
lex
ity o
f
app
l
ying
2
D
WT to th
e ten
s
o
r
with t
h
e size
of
is
O(8
N
(1-
2
)
/
6)
.
In
t
h
e
p
r
o
p
o
se
d al
go
ri
t
h
m
we are
usi
ng t
h
e (
9
/
7
) w
a
vel
e
t
and al
s
o
o
n
e
l
e
vel
of
dec
o
m
posi
t
i
on, s
o
t
h
e com
p
l
e
xi
t
y
of
ou
r al
g
o
ri
t
h
m
i
s
O(
9
/7.
Aft
e
r
ap
pl
y
i
ng
2
D
WT, eac
h t
e
ns
or
(a
pp
ro
xi
m
a
t
e
, di
ag
onal
,
ve
rt
i
cal
an
d
ho
ri
zo
nt
al
t
e
ns
ors
)
has t
h
e
size of
2
⁄
2
⁄
.
Th
e max
co
m
p
lex
ity o
f
th
e TD is of ord
e
r O(
), where
is the ave
r
age
num
ber o
f
pi
x
e
l
s
of t
h
e a
p
p
r
oxi
m
a
t
e
t
e
nsor
, an
d
is the avera
g
e of the
dim
e
ns
ions
of t
h
e approxim
ate core
ten
s
or
.
Table I
I
I
.
S
N
R
(
dB
) V
A
L
U
ES
FOR,
D
W
T
-
C
P
, PC
A+JPE
G
2000,
3D-
SPE
CK
M
e
thod
Bpppb
0.
05
0.
1
0.
2
0.
5
1.
0
Cupr
ite
DWT
-
CP 49.
9
52.
9
53
54.
9
58.
8
DWT
-
TD 49.
5
52.
1
52.
6
54.
2
58.
1
PCA+JPE
G
2000
43.
1
45.
3
48.
2
50.
5
54.
2
3D SPE
CK
34.
7
37.
1
40.
8
46.
6
50.
1
M
o
ffe
tt F
i
e
l
d
DWT
-
CP 42.
6
43.
9
48.
1
52.
5
55.
9
DWT
-
TD 40.
1
43.
7
47.
8
51.
3
55.
3
PCA+JPE
G
2000
34.
7
39.
3
43.
6
47.
2
51
3D SPE
CK
24.
3
28.
2
32.
3
39.
6
45.
1
(
)
(1
2)
(J
1A
+J
2A
+J
3A
)
(1
3)
Th
erefo
r
e, th
e
to
tal co
m
p
lex
ity o
f
th
e
p
r
op
osed
algor
ith
m
(
D
W
T
-
C
P)
is
of
o
r
d
e
r 4 x O(
)+O(
9
/7.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
An
O
p
t
i
m
al
H
S
I I
m
age
C
o
m
p
r
e
ssi
on
usi
n
g
D
W
T a
n
d
C
P
(
D
. N
a
r
m
ad
h
a
)
41
9
5.
E
X
PERI
MEN
T
AL RES
U
L
T
S
5.
1. C
o
mpress
i
on Resul
t
s
To
m
easu
r
e t
h
e p
e
rcep
t
u
al qu
ality o
f
im
ag
es, th
e sign
al-t
o
-
no
ise ratio (SNR) can b
e
well u
s
ed
. It
actu
a
lly esti
mates th
e qu
ality
of t
h
e
reco
nstru
c
ted im
ag
es
in co
m
p
ariso
n
with
t
h
e orig
in
al
on
es
.
The
SNR
i
n
dB
i
s
d
e
fi
ne
d as
1
0
(1
4)
Fi
gu
re
7.
C
l
assi
fi
cat
i
on acc
ur
acy
of
D
W
T
-
C
P
usi
n
g
S
V
M
(
R
B
F
Ker
n
el
)
.
Whe
r
e
2
=
∑∑
∑
.
Tw
o p
o
p
u
l
a
r
AV
IR
IS
dat
a
s
e
t
s
(C
u
p
ri
t
e
an
d M
o
ffet
t
)
a
r
e
used i
n
o
u
r
e
xpe
ri
m
e
nt
s. 2 These
16
-
b
i
t
radi
a
n
ce dat
a
se
t
s
ha
ve been
cr
op
pe
d
s
p
at
i
a
l
l
y
t
o
a si
ze
o
f
51
2 x 51
2
a
n
d
c
o
m
posed of 2
2
4
spect
ral
ba
nd
s
.
First, we ap
pl
y
the pro
p
o
se
d algo
rithm
(D
WT-CP
)
to these im
ages.
Next
,
we compare the SNRs
achi
e
ve
d by
t
h
e pr
op
ose
d
al
g
o
ri
t
h
m
wi
t
h
t
hose o
b
t
a
i
n
ed
fro
m
th
e well-k
nown
te
c
hni
ques, i.e.
3D-SPEC
K
algorithm
[5], com
b
ined PC
A+JPE
G
2
0
0
0
[6]
o
n
a set
of
HSIs
. Tabl
e
I
II s
h
o
w
s t
h
e S
N
R
ve
rsus
di
f
f
ere
n
t
bp
p
pb
fo
r t
h
e
C
u
p
r
i
t
e
and M
o
f
f
et
t
HSIs
. I
n
ou
r ex
peri
m
e
nt
s, D
W
T
-
C
P
has si
g
n
i
f
i
cant
l
y
im
proved S
N
R
at
h
i
gh
CRs
(
s
m
a
ll b
p
p
p
b
s
)
esp
e
cially w
h
en
t
h
e CR is h
i
g
h
e
r
t
h
an 160
o
r
bppp
b is low
e
r th
an
0
.
1
.
Figu
re 8.
S
AD (Rad
) vers
us b
p
p
p
b
fo
r, D
W
T-CP
a
n
d
JP
E
G
2
0
0
0
.
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.2
0
.4
0.6
0
.8
1
1
.2
1.4
1
.6
1.8
SAD(rad)
Rate(bpppb)
DWT
‐
CP
PCA+JPEG2000
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
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:
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08
IJEC
E V
o
l
.
4, No
. 3,
J
u
ne 2
0
1
4
:
41
1 – 4
2
1
42
0
6.
CO
NCL
USI
O
N
In t
h
i
s
pa
per
,
we i
n
t
r
o
duce
d
a new m
e
t
h
o
d
f
o
r
HS
Is co
m
p
ressi
on
usi
ng
D
W
T an
d
C
P
. Thi
s
i
s
carried
out by reducing the si
ze of
3D t
e
nso
r
s com
put
ed
fr
om
four
wa
vel
e
t
sub
-
i
m
ages
of t
h
e s
p
ect
ral
ba
n
d
s
of
HS
Is.
T
h
e
p
e
rf
orm
a
nce o
f
t
h
e p
r
op
ose
d
al
go
ri
t
h
m
on
A
V
I
R
I
S
dat
a
set
s
y
i
el
ds t
h
e
fol
l
o
w
i
n
g
resul
t
s
:
1)
DWT-CP ac
hi
eves
higher SNR in
co
m
p
ariso
n
with
two
state-o
f
-art
algo
rith
m
s
(3D-SPECK algorithm
and com
b
ine
d
PCA+JPE
G
2000) es
peci
ally at b
ppp
b low
e
r th
an 0.1.
2)
The
proposed
algorithm
achieves a
better
pi
xel
-
based
S
V
M
cl
assi
fi
cat
i
on acc
uracy
.
REFERE
NC
ES
[1]
H.F. Grahn and
P. Geladi, Techn
i
ques a
nd Applications of H
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:
BIOGRAP
HI
ES OF
AUTH
ORS
D.
Narmadha
rece
ived B.E (E
CE) in Rangana
than Engin
eerin
g College
, Coi
m
b
atore. S
h
e is
currently
pursuing Master of
Engineer
ing in A
pplied
Electron
ics, C.Abdul Hak
eem Colleg
e
of
Engineering
and
Technolog
y
M
e
lvisharam.
S
h
e has
pres
ented m
a
n
y
p
a
pe
rs
in National
and Intern
ation
a
l Conferen
ces
.
S
h
e attend
ed
workshop in Implementati
onal
Aspects of Micr
ocontrollers. He
r
Area of
research includ
es Imag
e
Processing, Networking, Embed
d
ed S
y
stem
. S
h
e
ia
an
a
c
tiv
e m
e
m
b
er of IE
TE
.
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