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Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
dsWav
elet
s and
Mul
tires
olut
ion
Proc
essi
ng•
Prev
iew
–G
ood
old
Four
ier t
rans
form
•A
tran
sfor
m w
here
the
basi
s fun
ctio
ns a
re si
nuso
ids,
henc
e lo
caliz
ed in
freq
uenc
y bu
t not
loca
lized
in ti
me;
in
tran
sfor
m d
omai
n th
e te
mpo
ral i
nfor
mat
ion
is lo
st–
New
bas
is fu
nctio
ns c
alle
d w
avel
ets
•Va
ryin
gfr
eque
ncy
and
limite
d du
ratio
n; a
n at
tem
pt to
re
veal
bot
h th
e fr
eque
ncy
and
tem
pora
l inf
orm
atio
n in
th
e tra
nsfo
rm d
omai
n–
Mul
tires
olut
ion
anal
ysis
•Si
gnal
(im
age)
repr
esen
tatio
n at
mul
tiple
reso
lutio
ns
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Bac
kgro
und
•Im
ages
–
Con
nect
ed re
gion
s of s
imila
r tex
ture
com
bine
d to
form
di
ffer
ent o
bjec
ts•
Smal
l siz
ed o
r low
in c
ontra
st o
bjec
ts: e
xam
ined
at h
igh
reso
lutio
n •
Larg
e si
zed
or h
igh
in c
ontra
st: e
xam
ined
at a
coa
rse
view
•Se
vera
l res
olut
ions
nee
ded
to d
istin
guis
hed
betw
een
diff
eren
t ob
ject
s
–M
athe
mat
ical
ly im
ages
are
2-D
arr
ays o
f int
ensi
ty v
alue
s of
loca
lly v
aryi
ng st
atis
tics (
resu
lting
from
edg
es,
hom
ogen
ous r
egio
ns, e
tc. –
see
Fig.
7.1
)
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
His
togr
ams c
ompu
ted
in lo
cal s
quar
esH
isto
gram
s com
pute
d in
loca
l squ
ares
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Imag
e Py
ram
ids
Imag
e Py
ram
ids
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atio
n by
iter
ated
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roxi
mat
ions
and
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rpol
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redi
ctio
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roxi
mat
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n le
vel
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umbe
r of l
evel
s P, t
wo
pyra
mid
s: a
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auss
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dual
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ted
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filte
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ould
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ragi
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s (G
auss
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10
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Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Gau
ssia
n a
nd L
apla
cian
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nd L
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cts (
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se, f
low
er, e
tc.)
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Subban
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Dig
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mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Brief
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Dig
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mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
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Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
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Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Fam
ilies
of
hal
f-ban
d P
R f
ilter
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mili
es o
f hal
f-ban
d P
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Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Sep
arab
le f
ilter
ing in 2
-D (
imag
es)
Sep
arab
le f
ilter
ing in 2
-D (
imag
es)
•Ver
tical
follo
wed
by
horiz
onta
l filt
erin
g
•Fou
r out
put s
ubba
nds:
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roxi
mat
ion,
and
ver
tical
, hor
izon
tal,
and
diag
onal
det
ail
subb
ands
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Exa
mple
: 8-t
ap o
rthonorm
alfilter
des
igned
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rthonorm
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s
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Exa
mple
: fo
ur
ban
d s
ubban
dfilter
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of
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age
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s due
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ring
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n st
age.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Haa
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nsf
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ldes
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d s
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st w
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Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Haa
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nct
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Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Multires
olu
tion E
xpan
sions
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of e
xpan
sion
func
tions
:
•Bas
is fu
nctio
ns a
nd fu
nctio
n sp
ace:
•Dua
l fun
ctio
ns:
•Cas
es: o
rthon
orm
alba
sis,
bior
thog
onal
bas
is, o
verc
ompl
ete
expa
nsio
ns
(fra
mes
and
tigh
t fra
mes
)
∑=
kk
kx
xf
)(
)(
ϕα
{} )
(sp
anx
Vk
kϕ
=
{}
dxx
fx
xf
xx
kk
k∫
∗=
=)
()
(~
)(
),(
~su
ch th
at
)(~
ϕϕ
αϕ
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
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© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Multires
olu
tion E
xpan
sions
Multires
olu
tion E
xpan
sions
•Sca
ling
Func
tions
•Set
of f
unct
ions
{ϕ
j,k(x
)} fo
rmed
by
inte
ger t
rans
latio
ns a
nd b
inar
y sc
alin
g of
a p
roto
type
func
tion
ϕ (x
):•t
rans
latio
n (p
ositi
on) p
aram
eter
k; s
cale
par
amet
er j
•sca
ling
func
tions
can
be
mad
e to
span
the
spac
e of
all
mea
sura
ble,
sq
uare
-inte
grab
lefu
nctio
ns, d
enot
ed b
y L2 (
R)
•by
fixin
g th
e sc
ale
to j=
j 0, th
e re
sulti
ng se
t spa
ns a
subs
pace
of L
2 (R
)
•if f
(x)∈
V j0 th
en
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2)
(2/
,k
xx
jj
kj
−=
ϕϕ
{} )
(sp
an,
00
xV
kj
kj
ϕ= ∑=
kk
jk
xx
f)
()
(,
0ϕ
α
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
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zale
z &
R. E
. Woo
ds
The
Haa
rSca
ling F
unct
ion
The
Haa
rSca
ling F
unct
ion
)(
21)
(21
)(
12,1
2,1,0
xx
xk
kk
++
=ϕ
ϕϕA
ny V
0ex
pans
ion
func
tion
can
be re
pres
ente
d by
V1
expa
nsio
n fu
nctio
ns, i
.e. V
0⊂
V 1.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
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© 2
002
R. C
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zale
z &
R. E
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ds
MRA R
equirem
ents
MRA R
equirem
ents
The
scal
ing
func
tion
is o
rthog
onal
to it
s int
eger
tra
nsla
tes
MR
A r
equi
rem
ent 2
:The
subs
pace
s spa
nned
by
the
scal
ing
func
tion
at lo
w sc
ales
are
nes
ted
with
in th
ose
span
ned
at h
ighe
r sca
les
∞−
∞−⊂
⊂⊂
⊂⊂
VV
VV
VV
......
21
01
MR
A r
equi
rem
ents
1:
MR
A r
equi
rem
ent 3
:The
onl
y fu
nctio
ns th
at is
com
mon
to a
ll V j
is
f (x)
= 0
MR
A r
equi
rem
ent 4
:Any
func
tion
can
be re
pres
ente
d w
ith a
rbitr
ary
prec
isio
n
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
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R. C
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ds
Dila
tation E
quat
ion
Dila
tation E
quat
ion
Und
er th
e M
RA
requ
irem
ents
, the
exp
ansi
on
func
tions
of s
ubsp
ace
V jca
n be
exp
ress
ed a
s a
wei
ghte
d su
m o
f the
exp
ansi
on fu
nctio
ns o
f su
bspa
ce V
j+1.
)2(
2)
()
(
equa
tion
di
lata
tion
ca
lled
-so
obta
in th
e
we
0 to
an
d
both
setti
ng
)2(
2)(
)(
)(
)(
12/)1
(,
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nx
nh
x
kj
nx
nh
x
xx
n
jn
jk
j
nj
nn
kj
−=
−== ∑∑∑
++
+
ϕϕ
ϕϕ
ϕα
ϕ
ϕ
ϕ
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
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zale
z &
R. E
. Woo
ds
Wav
elet
Funct
ions
Wav
elet
Funct
ions
{}
jj
j
kkj
kj
kjk
j
jj
kj
WV
V
xf
Wx
f
xW
kx
x
⊕=
=⇒
∈
=
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+
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,
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)(
)(
if
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ll W
jorth
ogon
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V j+1
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)(
),(
,,
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kjψ
ϕ
Def
ine
a w
avel
et fu
nctio
n ψ
(x) t
hat,
toge
ther
w
ith it
s int
eger
tran
slat
es a
nd b
inar
y sc
alin
gs,
span
s the
diff
eren
ce b
etw
een
any
two
adja
cent
sc
alin
g su
bspa
ces
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
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zale
z &
R. E
. Woo
ds
Wav
elet
Funct
ions
Wav
elet
Funct
ions
The
spac
e of
mea
sura
ble,
squa
re-in
tegr
able
func
tions
can
be
expr
esse
d as
......
)(
...)
(
...)
(
21
01
22
21
12
10
02
⊕⊕
⊕⊕
⊕=
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⊕=
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⊕=
−−
WW
WW
WL
WW
VL
WW
VL
RRR
Any
wav
elet
func
tion
can
be e
xpre
ssed
as a
wei
ghte
d su
m o
f shi
fted,
dou
ble-
reso
lutio
n sc
alin
gfu
nctio
ns
)1(
)1(
)(
w
here
)2(
2)
()
(
nh
nh
nx
nh
xn
n
−−
=
−=
∑ϕ
ψ
ψϕ
ψ
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
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zale
z &
R. E
. Woo
ds
Haa
rW
avel
et F
unct
ion
Coef
fici
ents
Haa
rW
avel
et F
unct
ion
Coef
fici
ents
)(
82)
(4
2)
(
)(
82)
(42
3)
(
)(
)(
)(
else
whe
re15.0
5.00
011)
( :
func
tion
et
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r wav
el
2/1
)1(;2
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ter
et
Haa
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el
2/1
)1()0(
:fil
ter
sc
alin
gH
aar
2,00,0
2,00,0
xx
xf
xx
xf
xf
xf
xf
xx
x
hh
hh
da
da
ψψ
ϕϕ
ψ
ψψ
ϕϕ −
−=
−=
+=
<
≤<
≤−
=
−=
=
==
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
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zale
z &
R. E
. Woo
ds
Wav
elet
Ser
ies
Exp
ansi
ons
Wav
elet
Ser
ies
Exp
ansi
ons )
(),
()
(
)(
),(
)(
)(
)(
)(
)(
)(
,,
,,
00
00
0
xx
fk
d
xx
fk
c
xk
dx
kc
xf
kj
j
kj
j
jj
kk
jj
kk
jj
ψϕ
ψϕ
==
+=
∑∑
∑∞ =
Con
side
r the
orth
ogon
al c
ase
c(k)
are
refe
rred
as a
ppro
xim
atio
n co
effic
ient
s and
d(k
) are
refe
rred
as d
etai
l co
effic
ient
s.
Exam
ple:
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=ot
herw
ise1
0
0)
(2
xx
xf
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
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zale
z &
R. E
. Woo
ds
Dis
cret
e W
avel
et T
ransf
orm
(D
WT)
Dis
cret
e W
avel
et T
ransf
orm
(D
WT)
Tran
sfor
m p
air (
forw
ard
and
inve
rse
trans
form
) app
licab
le fo
r dis
cret
e si
gnal
s (or
sam
pled
sign
als)
∑∑
∑
∑∑
∞ =+
=
==
000
)(
),
(1
)(
),
(1
)(
)(
)(
1)
,(
)(
)(
1)
,(
,,
0
,
,0
jj
kk
jk
kj
xk
j
xk
j
xk
jW
Mx
kj
WM
xf
xx
fM
kj
W
xx
fM
kj
W
ψϕψϕ
ψϕ
ψϕ
Her
e x
is a
dis
cret
eva
riabl
e: x
= 0
,1,2
,…M
-1
N
orm
ally
, M =
2J ;
j 0 =
0;
j =
0,1,
2,…
J-1
and
k =
0,1,
2,..2
j -1W
ϕan
d W
ψar
e re
ferr
ed a
s app
roxi
mat
ion
and
deta
il co
effic
ient
s re
spec
tivel
y.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
dsContinuous
Wav
elet
Tra
nsf
orm
(CW
T)
Continuous
Wav
elet
Tra
nsf
orm
(CW
T)
Tran
sfor
m a
con
tinuo
us fu
nctio
n in
to a
hig
hly
redu
ndan
t fun
ctio
nof
tw
o va
riabl
es –
trans
latio
n ( τ
) and
scal
e (s
).
ωωω
τψ
τ
τψ
ψ
ψτ
ψ
τψ
ψ
τ
τψ
dC
dsd
sx
sW
Cx
f
sx
sx
xx
fs
W
s
s
s
∫∫∫
∫
∞ ∞−
∞∞ ∞−
∞ ∞−
Ψ=
=
−=
=
2
02
,
,
, )(
w
here
)(
),
(1
)(
)(
1)
(
whe
re
)(
)(
),
(
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
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002
R. C
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zale
z &
R. E
. Woo
ds
Com
par
ison b
etw
een F
ourier
Tra
nsf
orm
an
d C
ontinuous
WT
Com
par
ison b
etw
een F
ourier
Tra
nsf
orm
an
d C
ontinuous
WT
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Fast
Wav
elet
Tra
nsf
orm
(FW
T)
In b
rief
Fast
Wav
elet
Tra
nsf
orm
(FW
T)
In b
rief
Synt
hesi
s filt
er b
ank
Ana
lysi
s filt
er b
ank
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
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© 2
002
R. C
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zale
z &
R. E
. Woo
ds
A T
wo-s
tage
(Tw
o-s
cale
) FW
T:
Anal
ysis
A T
wo-s
tage
(Tw
o-s
cale
) FW
T:
Anal
ysis
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
dsA T
wo-s
tage
(Tw
o-s
cale
) FW
T:
Syn
thes
isA T
wo-s
tage
(Tw
o-s
cale
) FW
T:
Syn
thes
is
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
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zale
z &
R. E
. Woo
ds
Exa
mple
: Tw
o-s
cale
FW
T b
y H
aar
Exa
mple
: Tw
o-s
cale
FW
T b
y H
aar
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
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002
R. C
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zale
z &
R. E
. Woo
ds
Exa
mple
: Tw
o-s
cale
FW
T b
y H
aar
Exa
mple
: Tw
o-s
cale
FW
T b
y H
aar
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Wav
elet
Pac
kets
Wav
elet
Pac
kets
The
Wav
elet
Tra
nsfo
rm d
ecom
pose
s a fu
nctio
n in
to a
serie
s of l
ogar
ithm
ical
ly re
late
d fr
eque
ncy
band
s. Lo
w fr
eque
ncie
s are
gro
uped
into
nar
row
ban
ds, w
hile
the
high
freq
uenc
ies
are
grou
ped
into
wid
er b
ands
. W
avel
et P
acke
tsar
e ge
nera
lizat
ion
that
allo
ws g
reat
er c
ontro
l ove
r the
tim
e-fr
eque
ncy
plan
e pa
rtitio
ning
.C
onsi
der t
he tw
o-sc
ale
filte
r ban
k as
a b
inar
y tr
ee. T
he w
avel
et c
oeff
icie
nts a
re a
t the
nod
es o
f th
e tre
e. T
he ro
ot n
ode
cont
ains
the
high
est-s
cale
app
roxi
mat
ion
coef
ficie
nts (
i.e. t
he sa
mpl
ed
sign
al it
self)
. Eac
h no
de c
onta
ins c
oeff
icie
nts r
epre
sent
ing
diff
eren
t sub
spac
es –
subs
pace
an
alys
is tr
ee.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
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zale
z &
R. E
. Woo
ds
WT:
Filter
Ban
k, A
nal
ysis
Tre
e,
and S
pec
trum
Split
ting
WT:
Filter
Ban
k, A
nal
ysis
Tre
e,
and S
pec
trum
Split
ting
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Wav
elet
Pac
kets
Wav
elet
Pac
kets
Wav
elet
pac
kets
are
con
vent
iona
l wav
elet
tran
sfor
ms i
n w
hich
the
deta
ilsar
e al
so it
erat
ivel
y fil
tere
d.
Subs
crip
ts in
the
figur
e sh
ow th
e sc
ale
and
a st
ring
of A
’s a
nd D
’s e
ncod
ing
the
path
from
the
pare
nt to
the
node
.
The
pack
et tr
ee a
lmos
t trip
les t
he n
umbe
r of d
ecom
posi
tions
.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Wav
elet
Pac
kets
: Fi
lter
Str
uct
ure
an
d S
pec
trum
Split
ting
Wav
elet
Pac
kets
: Fi
lter
Str
uct
ure
an
d S
pec
trum
Split
ting
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Wav
elet
Pac
kets
Wav
elet
Pac
kets
The
freq
uenc
y se
lect
ivity
can
be
incr
ease
d to
war
d hi
gh fr
eque
ncie
s
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
2-D
Wav
elet
Pac
kets
2-D
Wav
elet
Pac
kets
In 2
-D, s
epar
able
impl
emen
tatio
n of
wav
elet
pac
kets
lead
s to
a qu
ad-tr
ee
stru
ctur
e.Th
e fr
eque
ncy
spec
trum
is d
ivid
ed in
to fo
ur a
reas
. The
low
-fre
quen
cy a
rea
is in
th
e ce
ntre
.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
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R. C
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Port
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Dig
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Dig
ital I
mag
e Pro
cess
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2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Wav
elet
Pac
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Dec
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tion
Wav
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Dig
ital I
mag
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cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Optim
al W
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Optim
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Dig
ital I
mag
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cess
ing,
2nd
ed.
Dig
ital I
mag
e Pro
cess
ing,
2nd
ed.
www.imageprocessingbook.com
© 2
002
R. C
. Gon
zale
z &
R. E
. Woo
ds
Optim
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Wav
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