Xử lý ảnh số - Khôi phục ảnh part 2 doc

v`a g l`a c´ac vector cˆo
.
t trong R
6
v`a H l`a ma trˆa
.
n vuˆong cˆa
´
p6:
H =









h
e
(0) h
e
(5) h
e
(4) ··· h
e
(1)
h
e
(1) h

e
(0) h
e
(5) ··· h
e
(2)
h
e
(2) h
e
(1) h
e
(0) ··· h
e
(3)
.
.
.
h
e
(5) h
e
(4) h
e
(3) ··· h
e
(0)










.
Nhu
.
ng h
e
(x)=0v´o
.
i x =3, 4, 5v`ah
e
(x)=h(x)v´o
.
i x =0, 1, 2nˆen
H =












h(0) h(2) h(1)
h(1) h(0) h(2)
h(2) h(1) h(0)
h(2) h(1) h(0)
h(2) h(1) h(0)
h(2) h(1) h(0)











trong d¯´o tˆa
´
tca
˙’
c´ac phˆa
`
ntu
.
˙’
khˆong d¯u
.
o
.

.
cviˆe
´
t ra bˇa
`
ng khˆong.
Bˆay gi`o
.
x´et tru
.
`o
.
ng ho
.
.
p hai chiˆe
`
u: hai a
˙’
nh sˆo
´
f(x, y)v`ah(x, y)c´ok´ıch thu
.
´o
.
c
A ×B v`a C × D tu
.
o
.

ng ´u
.
ng. Cho
.
n M,N thoa
˙’
m˜an
M ≥ A + B −1,N≥ C + D − 1.
Ta mo
.
˙’
rˆo
.
ng k´ıch thu
.
´o
.
c c´ac a
˙’
nh bˇa
`
ng c´ach d¯ˇa
.
t
f
e
(x, y):=




f(x, y)nˆe
´
u0≤ x ≤ A − 1, v`a 0 ≤ y ≤ B −1,
0nˆe
´
u A ≤ x ≤ M −1, hoˇa
.
c B ≤ y ≤ N − 1,
v`a
h
e
(x, y):=



h(x, y)nˆe
´
u0≤ x ≤ C − 1, v`a 0 ≤ y ≤ D −1,
0nˆe
´
u C ≤ x ≤ M −1, hoˇa
.
c D ≤ y ≤ N − 1,
Ch´ung ta c˜ung gia
˙’
thiˆe
´
t c´ac h`am f
e
(x, y)v`ah

e
(x, y) tuˆa
`
n ho`an v´o
.
ichuk`yM v`a N
theo hai hu
.
´o
.
ng x v`a y tu
.
o
.
ng ´u
.
ng. Nhˇa
´
cla
.
i
g
e
(x, y)=
M−1

m=0
N−1

n=0

f
e
(m, n)h
e
(x −m, y −n),
v´o
.
i x =0, 1, ,M − 1, v`a y =0, 1, ,N − 1. H`am g
e
(x, y) tuˆa
`
n ho`an v´o
.
ic`ung chu
k`ynhu
.
f
e
(x, y)v`ah
e
(x, y) . D
-
ˆe
˙’
ho`an thiˆe
.
nmˆoh`ınh suy gia
˙’
mchˆa
´

tlu
.
o
.
.
ng r`o
.
ira
.
c, cˆa
`
n
thˆem nhiˆe
˜
u η
e
(x, y) v`ao h`am g
e
(x, y); t ´u
.
cl`a
g
e
(x, y)=
M−1

m=0
N−1

n=0

f
e
(m, n)h
e
(x −m, y −n)+η
e
(x, y) , (5.4)
116
v´o
.
i x =0, 1, ,M −1, v`a y =0, 1, ,N − 1.
K´yhiˆe
.
u f, g v`a n l`a c´ac vector cˆo
.
t k´ıch thu
.
´o
.
c MN nhˆa
.
nd¯u
.
o
.
.
cbˇa
`
ng c´ach sˇa
´

pxˆe
´
p
la
.
i c´ac h`ang cu
˙’
a c´ac ma
˙’
ng tu
.
o
.
ng ´u
.
ng f
e
,g
e
v`a η
e
v´o
.
i k´ıch thu
.
´o
.
c M × N. Chˇa
˙’
ng ha

.
n,
N phˆa
`
ntu
.
˙’
d¯ ˆa
`
u tiˆen cu
˙’
a f tu
.
o
.
ng ´u
.
ng c´ac phˆa
`
ntu
.
˙’
trong h`ang d¯ˆa
`
u tiˆen cu
˙’
a f
e
(x, y); N
phˆa

`
ntu
.
˙’
kˆe
´
tiˆe
´
ptu
.
o
.
ng ´u
.
ng h`ang th´u
.
hai, v`a vˆan vˆan. Khi d¯´o biˆe
˙’
uth´u
.
c (5.4) c´o thˆe
˙’
viˆe
´
tdu
.
´o
.
ida
.

ng ma trˆa
.
n
g=Hf+n, (5.5)
trong d¯´o H l`a ma trˆa
.
n vuˆong k´ıch thu
.
´o
.
c MN. Ma trˆa
.
n n`ay c´o thˆe
˙’
phˆan hoa
.
ch th`anh
M
2
ma trˆa
.
n con v´o
.
imˆo
˜
i ma trˆa
.
nconk´ıch thu
.
´o

.
c N × N v`a d¯u
.
o
.
.
csˇa
´
p theo th´u
.
tu
.
.
H =











H
0
H
M−1
H

M−2
H
1
H
1
H
0
H
M−1
H
2



H
M−1
H
M−2
H
M−3
H
0












,
v´o
.
imˆo
˜
i phˆa
`
ntu
.
˙’
H
j
cu
˙’
a h`ang th´u
.
j x´ac d¯i
.
nh bo
.
˙’
i h`am mo
.
˙’
rˆo
.
ng h

e
(x, y):
H
j
=













h
e
(j, 0) h
e
(j, N − 1) h
e
(j, N − 2) h
e
(j, 1)
h
e
(j, 1) h

e
(j, 0) h
e
(j, N − 1) h
e
(j, 2)
h
e
(j, 2) h
e
(j, 1) h
e
(j, N − 2) h
e
(j, 3)



h
e
(j, N −1) h
e
(j, N − 2) h
e
(j, N − 3) h
e
(j, 0)














.
Nhˆa
.
n x´et rˇa
`
ng, H
j
l`a ma trˆa
.
n chu tr`ınh v`a c´ac khˆo
´
icu
˙’
a H d¯ u
.
o
.
.
c d¯´anh chı
˙’

sˆo
´
theo ´y
ngh˜ıa chu tr`ınh. Do d¯´o ma trˆa
.
n H go
.
il`ama trˆa
.
n chu tr`ınh khˆo
´
i.
Hˆa
`
uhˆe
´
t c´ac kˆe
´
t qua
˙’
trong phˆa
`
n sau tˆa
.
p trung v`ao mˆo h`ınh suy gia
˙’
mchˆa
´
tlu
.

o
.
.
ng
da
.
ng (5.5). Ch´u´yrˇa
`
ng biˆe
˙’
uth´u
.
c n`ay du
.
.
a trˆen gia
˙’
thiˆe
´
t tuyˆe
´
n t´ınh v`a bˆa
´
tbiˆe
´
nvi
.
tr´ı
cu
˙’

amˆoh`ınh xu
.
˙’
l´y. Mu
.
c tiˆeu l`a t`ım u
.
´o
.
clu
.
o
.
.
ng f(x, y)du
.
.
a trˆen h`am g(x, y)v´o
.
isu
.
.
hiˆe
˙’
ubiˆe
´
tvˆe
`
h(x, y)v`aη(x, y). N´oi c´ach kh´ac cˆa
`

nt`ımu
.
´o
.
clu
.
o
.
.
ng f du
.
.
a trˆen g, n v`a H.
Mˇa
.
cd`ud¯o
.
n gia
˙’
n, nhu
.
ng viˆe
.
c x´ac d¯i
.
nh f t`u
.
Phu
.
o

.
ng tr`ınh (5.5) l`a rˆa
´
tph´u
.
cta
.
p
trong tru
.
`o
.
ng ho
.
.
p k´ıch thu
.
´o
.
cl´o
.
n. Chˇa
˙’
ng ha
.
n, nˆe
´
u M = N = 512 th`ı H c´o k´ıch thu
.
´o

.
c
262144. Do d¯´o d¯ˆe
˙’
x´ac d¯i
.
nh f ta cˆa
`
n gia
˙’
ihˆe
.
262144 phu
.
o
.
ng tr`ınh tuyˆe
´
nt´ınh. Tuy
nhiˆen, su
.
˙’
du
.
ng t´ınh chˆa
´
t chu tr`ınh cu
˙’
a ma trˆa
.

n H c´o thˆe
˙’
gia
˙’
m d¯´ang kˆe
˙’
khˆo
´
ilu
.
o
.
.
ng
t´ınh to´an.
117
5.2 Ch´eo ho´a ma trˆa
.
n chu tr`ınh v`a ma trˆa
.
n khˆo
´
i
chu tr`ınh
Phˆa
`
n n`ay tr`ınh b`ay phu
.
o
.

ng ph´ap hiˆe
.
u qua
˙’
gia
˙’
iPhu
.
o
.
ng tr`ınh (5.5) bˇa
`
ng c´ach ch´eo
ho´a ma trˆa
.
n H.D
-
ˆe
˙’
d¯ o
.
n gia
˙’
n, ch´ung ta bˇa
´
td¯ˆa
`
uv´o
.
i ma trˆa

.
n chu tr`ınh v`a sau d¯´o s˜e x´et
ma trˆa
.
n khˆo
´
i chu tr`ınh.
5.2.1 Ma trˆa
.
n chu tr`ınh
X´et ma trˆa
.
n chu tr`ınh cˆa
´
p M × M da
.
ng
H =














h
e
(0) h
e
(M − 1) h
e
(M − 2) ··· h
e
(1)
h
e
(1) h
e
(0) h
e
(M − 1) ··· h
e
(2)
h
e
(2) h
e
(1) h
e
(0) ··· h
e
(3)
. . . .
. . . .

. . . .
h
e
(M − 1) h
e
(M − 2) h
e
(M − 3) ··· h
e
(0)













.
D
-
ˇa
.
t
λ(k):=

M

j=1
h
e
(M − j) exp

2πi
M
jk

v`a
w(k):=









1
exp

2πi
M
k

exp


2πi
M
2k

.
.
.
exp

2πi
M
(M − 1)k










v´o
.
i k =0, 1, ,M − 1. Dˆe
˜
d`ang kiˆe
˙’
m tra rˇa

`
ng
Hw(k)=λ(k)w(k),k=0, 1, ,M −1.
N´oi c´ach kh´ac, ma trˆa
.
n chu tr`ınh H c´o M gi´a tri
.
riˆeng λ(k)tu
.
o
.
ng ´u
.
ng v´o
.
i c´ac vector
riˆeng w(k),k =0, 1, ,M − 1. Ho
.
nn˜u
.
a, c´ac vector riˆeng n`ay l`a tru
.
.
c giao, t´u
.
cl`a
w(k),w(k

) =0, v´o
.

imo
.
i k = k

,k,k

=0, 1, ,M − 1.
118
Gia
˙’
su
.
˙’
W := (W (k,j)) l`a ma trˆa
.
n vuˆong cˆa
´
p M v´o
.
i c´ac cˆo
.
t l`a c´ac vector riˆeng
cu
˙’
a ma trˆa
.
n chu tr`ınh H;t´u
.
cl`a
W(k,j) := exp


2πi
M
kj

,
v´o
.
i k,j =0, 1, ,M − 1. Dˆe
˜
thˆa
´
yrˇa
`
ng W l`a ma trˆa
.
n tru
.
.
c giao v`a do d¯´o tˆo
`
nta
.
ima
trˆa
.
n nghi
.
ch d¯a
˙’

o W
−1
:= (W
−1
(k,j)) v´o
.
i
W
−1
(k,j)=
1
M
exp

−
2πi
M
kj

.
Suy ra
H = WDW
−1
, (5.6)
trong d¯´o D l`a ma trˆa
.
n vuˆong cˆa
´
p M da
.

ng d¯u
.
`o
.
ng ch´eo, c´o c´ac phˆa
`
ntu
.
˙’
trˆen d¯u
.
`o
.
ng
ch´eo
D(k, k)=λ(k) .
5.2.2 Ma trˆa
.
n chu tr`ınh khˆo
´
i
Ma trˆa
.
nbiˆe
´
nd¯ˆo
˙’
id¯ˆe
˙’
ch´eo ho´a c´ac khˆo

´
i chu tr`ınh d¯u
.
o
.
.
c xˆay du
.
.
ng nhu
.
sau. D
-
ˇa
.
t
w
M
(k,j) := exp

2πi
M
kj

v`a
w
N
(k,j) := exp

2πi

N
kj

.
Ta d¯i
.
nh ngh˜ıa W l`a mˆo
.
t ma trˆa
.
n vuˆong cˆa
´
p MN × MN gˆo
`
m M
2
khˆo
´
i, mˆo
˜
i khˆo
´
il`a
mˆo
.
t ma trˆa
.
n vuˆong cˆa
´
p N, khˆo

´
inˇa
`
m trˆen h`ang m cˆo
.
t n x´ac d¯i
.
nh bo
.
˙’
i
W(k,m):=w
M
(k,m)W
N
,
v´o
.
i k,m =0, 1, ,M − 1v`aW
N
l`a ma trˆa
.
n vuˆong cˆa
´
p N v´o
.
i c´ac phˆa
`
ntu
.

˙’
W
N
(k,n)=w
N
(k,n)
v´o
.
i k,n =0, 1, 2, ,N − 1.
Ma trˆa
.
n nghi
.
ch d¯a
˙’
o W
−1
c˜ung l`a ma trˆa
.
n vuˆong k´ıch thu
.
´o
.
c MN ×MN gˆo
`
m M
2
khˆo
´
i, mˆo

˜
i khˆo
´
i l`a ma trˆa
.
n vuˆong cˆa
´
p N. Khˆo
´
io
.
˙’
h`ang m cˆo
.
t n cu
˙’
a W
−1
=(W
−1
(m, n))
x´ac d¯i
.
nh bo
.
˙’
i
W
−1
(m, n):=

1
M
w
−1
M
(m, n)W
−1
N
,
119
trong d¯´o
w
−1
M
(m, n) = exp

−
2πi
M
mn

,
v`a
W
−1
N
:=
1
N


w
−1
N
(k,j)

k,j=0,1, ,N−1
l`a ma trˆa
.
n vuˆong cˆa
´
p N v´o
.
i
w
−1
N
(k,j) = exp

−
2πi
N
kj

.
T`u
.
c´ac kˆe
´
t qua
˙’

cu
˙’
a Phˆa
`
n 5.2.1 v`a nˆe
´
u H l`a ma trˆa
.
n khˆo
´
i chu tr`ınh th`ı c´o thˆe
˙’
chı
˙’
ra
rˇa
`
ng
D = W
−1
HW
l`a ma trˆa
.
n ch´eo ho´a cu
˙’
a H trong d¯´o c´ac phˆa
`
ntu
.
˙’

trˆen d¯u
.
`o
.
ng ch´eo D(k,k) c´o liˆen quan
d¯ ˆe
´
nbiˆe
´
nd¯ˆo
˙’
i Fourier r`o
.
ira
.
ccu
˙’
a h`am th´ac triˆe
˙’
n h
e
(x, y) trong Phˆa
`
n 5.1.3. Suy ra
H = WDW
−1
. (5.7)
Ho
.
nn˜u

.
a, ma trˆa
.
n chuyˆe
˙’
nvi
.
cu
˙’
a H l`a
H
t
= W
¯
DW
−1
trong d¯´o
¯
D l`a ma trˆa
.
n liˆen ho
.
.
pph´u
.
ccu
˙’
a D.
5.2.3 Hiˆe
.

u qua
˙’
cu
˙’
a ch´eo ho´a ma trˆa
.
n trong mˆo h`ınh suy gia
˙’
m
chˆa
´
tlu
.
o
.
.
ng
Ma trˆa
.
n H trong mˆo h`ınh 1D r`o
.
ira
.
c (5.3) l`a ma trˆa
.
n chu tr`ınh. V`ıvˆa
.
y n´o c´o thˆe
˙’
biˆe

˙’
u
diˆe
˜
nda
.
ng (5.6). Khi d¯´o (5.3) tro
.
˙’
th`anh
g = WDW
−1
f.
Suy ra
W
−1
g = DW
−1
f. (5.8)
Nhu
.
ng dˆe
˜
d`ang kiˆe
˙’
m tra rˇa
`
ng vector cˆo
.
t W

−1
f thuˆo
.
c R
M
c´o phˆa
`
ntu
.
˙’
o
.
˙’
h`ang th´u
.
k
F (k):=
1
M
M−1

j=0
f
e
(j) exp

2πi
M
kj


,k=0, 1, ,M − 1,
120