Xử lý ảnh số - Các phép biến đổi part 2 pdf

H`ınh 3.1: A
˙’
nh gˆo
´
c, a
˙’
nh cu
˙’
a ln(1 + F [u, v])v`aa
˙’
nh cu
˙’
a g´oc pha ϕ(u, v).
3.3.1 T´ınh t´ach d¯u
.
o
.
.
c
X´et cˇa
.
pbiˆe
´
nd¯ˆo
˙’
i Fourier r`o
.
ira
.
ccu
˙’

a h`am a
˙’
nh f(x, y) c´o k´ıch thu
.
´o
.
c M × N :
F (u, v)=
1
MN
M−1

x=0
N−1

y=0
f(x, y)e
−2πi
(
ux
M
+
vy
N
)
, (3.3)
v`a
f(x, y)=
M−1


u=0
N−1

v=0
F (u, v)e
2πi
(
ux
M
+
vy
N
)
, (3.4)
trong d¯´o x, u =0,1, ,M − 1v`ay, v =0, 1, ,N − 1.
U
.
ud¯iˆe
˙’
ml`aF ( u, v) hoˇa
.
c f(x, y) c´o thˆe
˙’
nhˆa
.
nd¯u
.
o
.
.

c theo hai bu
.
´o
.
cbiˆe
´
nd¯ˆo
˙’
i Fourier
1D thuˆa
.
n hoˇa
.
c ngu
.
o
.
.
c. D
-
iˆe
`
u n`ay l`a hiˆe
˙’
n nhiˆen, v`ı t `u
.
(3.3) ta c´o
F (u, v)=
1
M

M−1

x=0
G(x, v)e
−2πi
ux
M
,
trong d¯´o
G(x, v):=
1
N
N−1

y=0
f(x, y)e
−2πi
vy
N
. (3.5)
D
-
ˆo
´
iv´o
.
imˆo
˜
i gi´a tri
.

x, vˆe
´
pha
˙’
icu
˙’
abiˆe
˙’
uth´u
.
c (3.5) l`a biˆe
´
nd¯ˆo
˙’
i Fourier 1D v´o
.
i c´ac
gi´a tri
.
tˆa
`
nsˆo
´
v =0, 1, ,N − 1. V`ıvˆa
.
y h`am hai chiˆe
`
u F (u, v) nhˆa
.
nd¯u

.
o
.
.
c theo c´ac
bu
.
´o
.
c sau
Bu
.
´o
.
c1. Biˆe
´
nd¯ˆo
˙’
i Fourier 1D theo t`u
.
ng h`ang cu
˙’
a f(x, y) ta d¯u
.
o
.
.
cma
˙’
ng trung gian

G(x, v);
Bu
.
´o
.
c2. Biˆe
´
nd¯ˆo
˙’
i Fourier 1D theo cˆo
.
tcu
˙’
a G(x, v).
48
C´o thˆe
˙’
nhˆa
.
nd¯u
.
o
.
.
ckˆe
´
t qua
˙’
giˆo
´

ng nhu
.
trˆen khi biˆe
´
nd¯ˆo
˙’
i theo c´ac cˆo
.
tcu
˙’
a f(x, y)
v`a sau d¯´o do
.
c theo c´ac h`ang.
3.3.2 Ti
.
nh tiˆe
´
n
V´o
.
imo
.
i x
0
,y
0
,u
0
,v

0
∈ C ta c´o
F

f(x, y)e
2πi
(
u
0
x
M
+
v
0
y
N
)

= F(u − u
0
,v− v
0
),
F [f(x − x
0
,y− y
0
)] = F ( u, v)e
−2πi
(

ux
0
M
+
vy
0
N
)
.
T`u
.
d¯´o suy ra
F

(−1)
x+y
f(x, y)

= F(u − M/2,v−N/2).
Ho
.
nn˜u
.
a, ti
.
nh tiˆe
´
n khˆong l`am thay d¯ˆo
˙’
i phˆo

˙’
Fourier cu
˙’
a F.
3.3.3 Chu k`y
Gia
˙’
su
.
˙’
h`am a
˙’
nh f tuˆa
`
n ho`an theo c´ac tru
.
c x v`a y tu
.
o
.
ng ´u
.
ng v´o
.
ichuk`y M v`a N;t´u
.
c
l`a
f(x, y)=f(x + M,y)=f(x, y + N)=f(x + M,y + N). (3.6)
Khi d¯´o

F (u, v)=F (u + M,v)=F(u, v + N)=F (u + M,v + N). (3.7)
Ngu
.
o
.
.
cla
.
i, nˆe
´
ubiˆe
´
nd¯ˆo
˙’
i Fourier cu
˙’
a f thoa
˙’
(3.7) th`ı h`am a
˙’
nh f thoa
˙’
m˜an (3.6). Ho
.
n
n˜u
.
a, nˆe
´
u h`am f thu

.
.
c, th`ı F (u, v)=
¯
F(−u, −v), trong d¯´o
¯
F (u, v) l`a sˆo
´
ph´u
.
c liˆen ho
.
.
p
cu
˙’
a F (u, v). Suy ra F (u, v) = F (−u, −v).
3.3.4 Ph´ep quay
X´et ph´ep biˆe
´
nd¯ˆo
˙’
ito
.
ad¯ˆo
.
cu
.
.
c

x(r, θ)=r cos θ, y(r, θ)=r sin θ,
u(ω,ϕ)=ω cos ϕ, v(ω,ϕ)=ω sin ϕ.
D
-
ˇa
.
t
g(r, θ):=f(x(r, θ),y(r, θ)),
G(ω,ϕ):=F(u(ω,ϕ),v(ω, ϕ)).
49
Khi d¯´o v´o
.
imo
.
i θ
0
∈ R ta c´o
F[g(r, θ + θ
0
)] = G(ω, ϕ + θ
0
),
N´oi c´ach kh´ac, quay f(x, y)mˆo
.
t g´oc θ
0
s˜e l`am quay F(u, v)c`ung mˆo
.
t g´oc. Tu
.

o
.
ng tu
.
.
,
ta quay F(u, v)s˜e l`am quay f(x, y)v´o
.
ic`ung mˆo
.
t g´oc.
3.3.5 Tuyˆe
´
n t´ınh v`a co gi˜an
Biˆe
´
nd¯ˆo
˙’
i Fourier l`a ´anh xa
.
tuyˆe
´
n t´ınh, t´u
.
cl`a
F(af + bg)=aF(f)+bF(g)v´o
.
imo
.
i a, b ∈ C.

Tuy nhiˆen, n´oi chung
F(fg) = F(f) F(g).
Ngo`ai ra, dˆe
˜
d`ang ch ´u
.
ng minh rˇa
`
ng v´o
.
imo
.
i a, b ∈ C v´o
.
i a, b = 0 ta c´o
F[f(ax, by)] =
1
ab
F

u
a
,
v
b

.
3.3.6 Gi´a tri
.
trung b`ınh

Gi´a tri
.
trung b`ınh cu
˙’
a h`am r`o
.
ira
.
c hai chiˆe
`
u f l`a
1
MN
M−1

x=0
N−1

y=0
f(x, y)=F (0, 0).
3.3.7 Biˆe
´
nd¯ˆo
˙’
i Laplace
Biˆe
´
nd¯ˆo
˙’
i Laplace cu

˙’
a f x´ac d¯i
.
nh bo
.
˙’
i
∆f(x, y):=
∂
2
f
∂x
2
+
∂
2
f
∂y
2
.
Dˆe
˜
d`ang ch ´u
.
ng minh rˇa
`
ng
F (∆f)=−(2π)
2
(u

2
+ v
2
)F (u, v).
Ph´ep biˆe
´
nd¯ˆo
˙’
i Laplace thu
.
`o
.
ng d¯u
.
o
.
.
cd`ung trong k˜y thuˆa
.
t t´ach biˆen cu
˙’
aa
˙’
nh.
50
3.3.8 T´ıch chˆa
.
p v`a tu
.
o

.
ng quan
Nhˇa
´
cla
.
il`at´ıch chˆa
.
p (liˆen tu
.
c) cu
˙’
a f v`a g, k´yhiˆe
.
u(f ∗ g), x´ac d¯i
.
nh bo
.
˙’
i
(f ∗ g)(x, y):=

+∞
−∞

+∞
−∞
g(x −α, y −β)f(α, β)dαdβ.
V´ı du
.

3.3.2 Gia
˙’
su
.
˙’
f(x):=



1nˆe
´
u0≤ x ≤ 1,
0nˆe
´
u ngu
.
o
.
.
cla
.
i,
g(x):=



1/2nˆe
´
u0≤ x ≤ 1,
0nˆe

´
u ngu
.
o
.
.
cla
.
i.
Khi d¯´o dˆe
˜
d`ang kiˆe
˙’
m tra rˇa
`
ng
(f ∗ g)(x)=









x/2nˆe
´
u0≤ x ≤ 1,
1 − x/2nˆe

´
u1≤ x ≤ 2,
0nˆe
´
u ngu
.
o
.
.
cla
.
i.
D
-
ˆe
˙’
d¯ i
.
nh ngh˜ıa t´ıch chˆa
.
pr`o
.
ira
.
c cu
˙’
a hai h`am a
˙’
nh f v`a g tu
.

o
.
ng ´u
.
ng c´ac ma
˙’
ng hai
chiˆe
`
uv´o
.
i k´ıch thu
.
´o
.
c A ×B v`a C ×D ta cˆa
`
nmo
.
˙’
rˆo
.
ng k´ıch thu
.
´o
.
ca
˙’
nh lˆen M ×N. D
-

ˆe
˙’
tr´anh hiˆe
.
ntu
.
o
.
.
ng lˆo
˜
ibo
.
c, ta cho
.
n M,N sao cho
M ≥ A + C − 1,N≥ B + D − 1. (3.8)
X´et c´ac mo
.
˙’
rˆo
.
ng cu
˙’
a f(x, y)l`a
f
r
(x, y):=




f(x, y)nˆe
´
u0≤ x ≤ A −1, 0 ≤ y ≤ B − 1,
0nˆe
´
u A ≤ x ≤ M − 1 hoˇa
.
c B ≤ y ≤ N − 1,
v`a mo
.
˙’
rˆo
.
ng cu
˙’
a g(x, y)l`a
g
r
(x, y):=



g(x, y)nˆe
´
u0≤ x ≤ C −1, 0 ≤ y ≤ D −1,
0nˆe
´
u C ≤ x ≤ M − 1 hoˇa
.

c D ≤ y ≤ N − 1.
T´ıch chˆa
.
p hai chiˆe
`
u (r`o
.
ira
.
c) cu
˙’
a f
r
v`a g
r
d¯ i
.
nh ngh˜ıa bo
.
˙’
i
(f
r
∗g
r
)(x, y):=
M−1

α=0
N−1


β=0
g
r
(x − α, y − β)f
r
(α, β), (3.9)
51
v´o
.
i x =0, 1, ,M −1,y=0, 1, ,N − 1.
Trong thu
.
.
ctˆe
´
,viˆe
.
c t´ınh to´an t´ıch chˆa
.
pr`o
.
ira
.
c trong miˆe
`
ntˆa
`
nsˆo
´

hiˆe
.
u qua
˙’
ho
.
n
khi ´ap du
.
ng cˆong th´u
.
c (3.9).
D
-
i
.
nh l ´y 3.3.3 Gia
˙’
su
.
˙’
F v`a G l`a c´ac biˆe
´
nd¯ˆo
˙’
i Fourier cu
˙’
a f v`a g. Khi d¯´o biˆe
´
nd¯ˆo

˙’
i
Fourier ngu
.
o
.
.
ccu
˙’
a FG ch´ınh l`a f ∗ g.
Ch´u
.
ng minh. Gia
˙’
su
.
˙’
H l`a biˆe
´
nd¯ˆo
˙’
i Fourier cu
˙’
a f ∗g. Ta cˆa
`
nch´u
.
ng minh rˇa
`
ng H = FG.

Thˆa
.
tvˆa
.
y
H(u, v)=

+∞
−∞

+∞
−∞
e
−2πi(ux+vy)


+∞
−∞

+∞
−∞
g(x −α, y −β)f(α, β)dαdβ

dxdy
=

+∞
−∞

+∞

−∞
f(α, β)


+∞
−∞

+∞
−∞
e
−2πi(ux+vy)
g(x − α, y −β)dxdy

dαdβ
=

+∞
−∞

+∞
−∞
f(α, β)e
−2πi(uα+vβ)
G(u, v)dαdβ
= G(u, v)

+∞
−∞

+∞

−∞
f(α, β)e
−2πi(uα+vβ)
dαdβ
= F(u, v)G(u, v).
D
-
i
.
nh l´y d¯u
.
o
.
.
cch´u
.
ng minh. ✷
Tu
.
o
.
ng quan cu
˙’
a hai h`am liˆen tu
.
c f v`a g, k´yhiˆe
.
u f ⊗ g, x´ac d¯i
.
nh bo

.
˙’
i
(f ⊗ g)(x, y):=

+∞
−∞

+∞
−∞
g(x + α, y + β)
¯
f(α, β)dαdβ.
Trong tru
.
`o
.
ng ho
.
.
pr`o
.
ira
.
c
(f
r
⊗ g
r
)(x, y):=

M−1

α=0
N−1

β=0
g
r
(x + α, y + β)
¯
f
r
(α, β), (3.10)
v´o
.
i x =0, 1, ,M − 1,y=0, 1, ,N − 1. Nhu
.
trong tru
.
`o
.
ng ho
.
.
pcu
˙’
at´ıchchˆa
.
pr`o
.

i
ra
.
c, f
r
(x, y)v`ag
r
(x, y) l`a nh˜u
.
ng h`am d¯u
.
o
.
.
cmo
.
˙’
rˆo
.
ng v`a M,N d¯ u
.
o
.
.
ccho
.
n theo (3.8)
d¯ ˆe
˙’
tr´anh hiˆe

.
ntu
.
o
.
.
ng lˆo
˜
ibo
.
c.
Nhˆa
.
nx´et 3.3.4 (i) D
-
ˆo
´
iv´o
.
ica
˙’
hai tru
.
`o
.
ng ho
.
.
pr`o
.

ira
.
c v`a liˆen tu
.
c, ta dˆe
˜
d`ang ch´u
.
ng
minh c´ac quan hˆe
.
sau:
F(f ⊗g)=
¯
FG,
F(
¯
f ⊗ g)=FG.
52