Xử lý ảnh số - Các phép biến đổi part 1 ppsx

Chu
.
o
.
ng 3
C
´
AC PH
´
EP BI
ˆ
E
´
ND
-
ˆ
O
˙’
I
Mu
.
cd¯´ıch ch´ınh cu
˙’
a chu
.
o
.
ng n`ay l`a tr`ınh b`ay c´ac ph´ep biˆe
´
nd¯ˆo
˙’

i hai chiˆe
`
u v`a c´ac t´ınh
chˆa
´
tcu
˙’
ach´ung. C´ac ph´ep biˆe
´
nd¯ˆo
˙’
i d¯´ong vai tr`o quan tro
.
ng trong xu
.
˙’
l´y a
˙’
nh ca
˙’
vˆe
`
mˇa
.
t
l´y thuyˆe
´
tc˜ung nhu
.
´u

.
ng du
.
ng. C´ac ph´ep biˆe
´
nd¯ˆo
˙’
i hai chiˆe
`
us˜ed¯u
.
o
.
.
csu
.
˙’
du
.
ng trong
nh˜u
.
ng chu
.
o
.
ng sau d¯ˆe
˙’
nˆang cao chˆa
´

tlu
.
o
.
.
ng a
˙’
nh, phu
.
chˆo
`
ia
˙’
nh, m˜a ho´a v`a miˆeu ta
˙’
a
˙’
nh.
Mˇa
.
cd`u c´ac ph´ep biˆe
´
nd¯ˆo
˙’
i kh´ac c˜ung d¯u
.
o
.
.
cd¯ˆe

`
cˆa
.
p, tuy nhiˆen ch´ung ta vˆa
˜
n nhˆa
´
n
ma
.
nh v`ao ph´ep biˆe
´
nd¯ˆo
˙’
i Fourier v`ın´od¯u
.
o
.
.
csu
.
˙’
du
.
ng rˆo
.
ng r˜ai trong c´ac b`ai to´an xu
.
˙’
l´y

a
˙’
nh.
3.1 Biˆe
´
nd¯ˆo
˙’
i Fourier liˆen tu
.
c
3.1.1 Biˆe
´
nd¯ˆo
˙’
i Fourier mˆo
.
tchiˆe
`
u
Gia
˙’
su
.
˙’
f(x) l`a h`am liˆen tu
.
c theo biˆe
´
n thu
.

.
c x. Biˆe
´
nd¯ˆo
˙’
i Fourier cu
˙’
a f(x), k´yhiˆe
.
u F(f)
hoˇa
.
c F, x´ac d¯i
.
nh bo
.
˙’
i
F(f)(u)=F (u):=

+∞
−∞
f(x)e
−2πiux
dx,
trong d¯´o i :=
√
−1.
Cho tru
.

´o
.
c F (u) ta c´o thˆe
˙’
nhˆa
.
nd¯u
.
o
.
.
c f(x)bˇa
`
ng c´ach su
.
˙’
du
.
ng biˆe
´
nd¯ˆo
˙’
i Fourier
43
ngu
.
o
.
.
c

F
−1
(F )(x)=f(x)=

+∞
−∞
F (u)e
2πiux
du.
C´o thˆe
˙’
ch´u
.
ng minh su
.
.
tˆo
`
nta
.
icu
˙’
a cˇa
.
pbiˆe
´
nd¯ˆo
˙’
i Fourier, nˆe
´

u f(x)liˆen tu
.
c, kha
˙’
t´ıch v`a F (u) kha
˙’
t´ıch. C´ac d¯iˆe
`
ukiˆe
.
n n`ay thu
.
`o
.
ng thoa
˙’
m˜an trong thu
.
.
ctˆe
´
.
Trong gi´ao tr`ınh n`ay, ta luˆon gia
˙’
su
.
˙’
f l`a h`am thu
.
.

c. N´oi chung biˆe
´
nd¯ˆo
˙’
i Fourier
cu
˙’
a h`am thu
.
.
c f l`a mˆo
.
t h`am ph´u
.
c; t´u
.
cl`a
F (u)=R( u)+iI(u),
trong d¯´o R(u) (tu
.
o
.
ng ´u
.
ng, I(u)) l`a phˆa
`
n thu
.
.
c (tu

.
o
.
ng ´u
.
ng, phˆa
`
na
˙’
o) cu
˙’
a F(u):
R(u)=

+∞
−∞
f(x) cos[2πux]dx,
I(u)=

+∞
−∞
f(x) sin[2 πux]dx.
C´ac h`am n`ay d¯ˆoi khi c`on go
.
il`abiˆe
´
nd¯ˆo
˙’
i Fourier cosin v`a biˆe
´

nd¯ˆo
˙’
i Fourier sin cu
˙’
a f.
Ta thu
.
`o
.
ng biˆe
˙’
udiˆe
˜
n h`am F(u)du
.
´o
.
ida
.
ng
F (u)=F (u)e
iϕ(u)
,
trong d¯´o
F (u) :=

R
2
(u)+I
2

(u),
tan[ϕ( u)] :=

I(u)
R(u)

.
H`am F (u) d¯ u
.
o
.
.
cgo
.
il`aphˆo
˙’
Fourier cu
˙’
a f, v`a ϕ(u)l`ag´oc pha.B`ınh phu
.
o
.
ng cu
˙’
a phˆo
˙’
Fourier go
.
il`aphˆo
˙’

cˆong suˆa
´
t.Biˆe
´
n u thu
.
`o
.
ng d¯u
.
o
.
.
cgo
.
il`abiˆe
´
n tˆa
`
nsˆo
´
.
V´ı du
.
3.1.1 Biˆe
´
nd¯ˆo
˙’
i Fourier cu
˙’

a h`am
f(x):=



A nˆe
´
u0≤ x ≤ α,
0nˆe
´
u ngu
.
o
.
.
cla
.
i,
l`a
F (u)=

+∞
−∞
f(x)e
−2πixu
dx
=

α
0

Ae
−2πixu
dx
44
=
A
πu
sin(πuα)e
−πiαu
.
Suy ra phˆo
˙’
Fourier
F (u) =




A
πu




|sin(πuα)||e
−πiαu
|
= |A|α





sin(πuα)
(πuα)




.
3.1.2 Biˆe
´
nd¯ˆo
˙’
i Fourier hai chiˆe
`
u
Dˆe
˜
d`ang mo
.
˙’
rˆo
.
ng biˆe
´
nd¯ˆo
˙’
i Fourier trong tru
.
`o

.
ng ho
.
.
p hai chiˆe
`
u. Nˆe
´
u f(x, y) l`a h`am
liˆen tu
.
c v`a kha
˙’
t´ıch, v`a F(u, v) kha
˙’
t´ıch, th`ı tˆo
`
nta
.
icˇa
.
pbiˆe
´
nd¯ˆo
˙’
i Fourier:
F(f)(u, v)=F (u, v):=

+∞
−∞


+∞
−∞
f(x, y)e
−2πi(ux+vy)
dxdy
v`a
F
−1
(F )(x, y)=f(x, y)=

+∞
−∞

+∞
−∞
F (u, v)e
2πi(ux+vy)
dudv,
trong d¯´o u, v l`a c´ac biˆe
´
ntˆa
`
nsˆo
´
.
Tu
.
o
.

ng tu
.
.
trong tru
.
`o
.
ng ho
.
.
pmˆo
.
tchiˆe
`
u, phˆo
˙’
Fourier, g´oc pha v`a phˆo
˙’
cˆong suˆa
´
t
x´ac d¯i
.
nh tu
.
o
.
ng ´u
.
ng bo

.
˙’
i
F (u, v) :=

R
2
(u, v)+I
2
(u, v),
ϕ(u, v) := tan
−1

I(u, v)
R(u, v)

v`a
P (u, v):=F (u, v)
2
= R
2
(u, v)+I
2
(u, v).
V´ı du
.
3.1.2 Biˆe
´
nd¯ˆo
˙’

i Fourier cu
˙’
a h`am
f(x, y):=



A nˆe
´
u0≤ x ≤ α, 0 ≤ y ≤ β,
0nˆe
´
u ngu
.
o
.
.
cla
.
i,
l`a
F (u, v)=

+∞
−∞

+∞
−∞
f(x, y)e
−2πi(ux+vy)

dxdy
= Aαβ

sin(πuα)e
−πiαu
(πuα)

sin(πuβ)e
−πiβu
(πuβ)

.
Do d¯´o phˆo
˙’
Fourier l`a
F (u, v) = |A|αβ




sin(πuα)
(πuα)









sin(πuβ)
(πuβ)




.
45
3.2 Biˆe
´
nd¯ˆo
˙’
i Fourier r`o
.
ira
.
c
X´et d˜ay f(x),x=0, 1, ,N−1. Biˆe
´
nd¯ˆo
˙’
i Fourier r`o
.
ira
.
c thuˆa
.
n ngu
.
o

.
.
c mˆo
.
tchiˆe
`
u x´ac
d¯ i
.
nh bo
.
˙’
i
F(f)(u)=F (u):=
1
N
N−1

x=0
f(x)e
−2πi
ux
N
, (3.1)
v´o
.
i u =0, 1, ,N − 1, v`a
F
−1
(F )(x)=f(x):=

N−1

u=0
F (u)e
2πi
ux
N
, (3.2)
trong d¯´o x =0, 1, ,N − 1.
V´ı du
.
3.2.1 Gia
˙’
su
.
˙’
f(0) = 2,f(1) = 3,f(2) = f(3) = 4. Tac´obiˆe
´
nd¯ˆo
˙’
i Fourier cu
˙’
a
f l`a
F (0) =
1
4
3

x=0

f(x)e
0
=3.25;
F (1) =
1
4
3

x=0
f(x)e
−2πix/4
=
1
4
(−2+i);
F (2) =
1
4
3

x=0
f(x)e
−2πi2x/4
= −
1
4
;
F (3) =
1
4

3

x=0
f(x)e
−2πi3x/4
= −
1
4
(2 + i).
Trong tru
.
`o
.
ng ho
.
.
p hai biˆe
´
n, cˇa
.
pbiˆe
´
nd¯ˆo
˙’
i Fourier r`o
.
ira
.
cchobo
.

˙’
i
F(f)(u, v):=
1
MN
M−1

x=0
N−1

y=0
f(x, y)e
−2πi
(
ux
M
+
vy
N
)
,
v´o
.
i u =0, 1, ,M − 1, v`a v =0, 1, ,N − 1, v`a
F
−1
(F )(x, y):=
M−1

u=0

N−1

v=0
F (u, v)e
2πi
(
ux
M
+
vy
N
)
,
trong d¯´o x =0, 1, ,M − 1, v`a y =0, 1, ,N − 1.
46
Khi c´ac a
˙’
nh c´o k´ıch thu
.
´o
.
c vuˆong, t´u
.
cl`aM = N, d¯ ˆe
˙’
thuˆa
.
ntiˆe
.
n trong c´ac t´ınh

to´an ta thu
.
`o
.
ng su
.
˙’
du
.
ng cˆong th´u
.
cbiˆe
´
nd¯ˆo
˙’
i Fourier thuˆa
.
n ngu
.
o
.
.
c sau














F (u, v)=
1
N
N−1

x=0
N−1

y=0
f(x, y)e
−2πi
(
ux
N
+
vy
N
)
,
f(x, y)=
1
N
N−1


u=0
N−1

v=0
F (u, v)e
2πi
(
ux
N
+
v
y
N
)
,
trong d¯´o u, x =0, 1, ,N − 1, v`a y,v =0, 1, ,N − 1.
Ho`an to`an tu
.
o
.
ng tu
.
.
ta c ˜ung c´o c´ac kh´ai niˆe
.
m phˆo
˙’
Fourier, g´oc pha, phˆo
˙’
cˆong

suˆa
´
tcu
˙’
a h`am r`o
.
ira
.
c f.
Kh´ac v´o
.
i tru
.
`o
.
ng ho
.
.
p liˆen tu
.
c, dˆe
˜
d`ang ch ´u
.
ng minh tˆo
`
nta
.
icu
˙’

abiˆe
´
nd¯ˆo
˙’
i Fourier
r`o
.
ira
.
c. Chˇa
˙’
ng ha
.
n trong tru
.
`o
.
ng ho
.
.
pmˆo
.
t chiˆe
`
u, ta c´o thˆe
˙’
ch´u
.
ng minh bˇa
`

ng c´ach thay
tru
.
.
ctiˆe
´
p (3.2) v`ao (3.1):
F (u)=
1
N
N−1

x=0

N−1

r=0
F (r)e
2πirx/N
e
−2πiux/N

=
1
N
N−1

r=0
F (r)


N−1

x=0
e
2πirx/N
e
−2πiux/N

= F(u).
D
-
ˆo
`
ng nhˆa
´
tth´u
.
c trˆen suy t `u
.
d¯ i ˆe
`
ukiˆe
.
n tru
.
.
c giao
N−1

x=0

e
2πirx/N
e
−2πiux/N
=



N nˆe
´
u r = u,
0nˆe
´
u ngu
.
o
.
.
cla
.
i.
V´ı du
.
3.2.2 H`ınh 3.1 l`a a
˙’
nh gˆo
´
c, a
˙’
nh cu

˙’
a ln(1+F[u, v])v`aa
˙’
nh cu
˙’
a g´oc pha ϕ(u, v).
3.3 C´ac t´ınh chˆa
´
t
Tru
.
´o
.
chˆe
´
t ta c´o
D
-
i
.
nh l´y 3.3.1 [Raleigh] Gia
˙’
su
.
˙’
F l`a biˆe
´
nd¯ˆo
˙’
i Fourier cu

˙’
a f. Khi d¯´o

+∞
−∞

+∞
−∞
f
2
(x, y)dxdy =

+∞
−∞

+∞
−∞
F (u, v)
2
dudv.
47