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ELEC317
Digital Image Processing
Lecture 6
Image Restoration
– Due to distortions in imaging process, data captured is different from the true image.
e.g. –Relative motion between subject & camera
–Band limited image due to diffraction.
–Corruption by noise
–Missing part of data
–Let true image be
u(m,n)
, captured image be
v(m,n)
, then objective is to find an
operator
H
such that
()
[]()
n
m
u
n
m
v
H
,
,
≈
–To measure “Goodness” of operator, we use mean squared error:
[]
∑∑
==
−
=
N
m
N
n
n
m
v
H
n
m
u
N
MSE
00
2
2
,
,
1
–image restoration in general consists of two stages:
A.
Restoration Models
This is physical understanding & modeling of the relationship between
received
image & real image.
For this purpose, we need to
A.1. Model the way image is collected.
A.2. Model/Measure response characteristics of detector and recorder
A.3. Model/Measure statistical characteristics of noise.
B.
Design and implementation of
restoration algorithms
There are two big classes of restoration algorithms:
B.1.Linear filtering method;
B.2. Nonlinear methods.
Image Observation Model
Before useful restoration algorithm can be designed, need to understand how
images are formed. A generally useful model is as follows:
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()
[]()
y
x
y
x
w
g
y
x
v
,
,
,
η
+
=
(
)
∫∫
∞
∞
−
=
'
'
'
'
'
'
,
,
,
,
dy
dx
y
x
u
y
x
y
x
h
y
x
w
i
[]
( )
y
x
y
x
y
x
w
g
f
y
x
,
,
,
,
2
1
+
⋅
=
Thus, the image observation modeling refers to characterization of:
1.
impulse response of linear system:
h(x,
,
y,
x’,y’)
normally, system is shift invariant
Æ
only need
h(x,y)
And
()()
y
x
u
y
x
h
y
x
w
,
*
,
,
=
2.
Characterize nonlinear functions
g(.)
and
f(.)
3.
Characterize noise characteristics
y
x
,
1
and
y
x
,
2
.
Image Formation Model
The impulse response
h(x,y)
is related to the way image is taken.
(i)
Coherent Diffraction Limited Model
–Here the aperture is rectangular
–Image obtained is (in frequency domain)
⋅
=
b
a
rect
u
W
2
1
2
1
2
1
,
,
,
ξ
↓
( ) ( ) ( )
by
c
ax
c
ab
y
x
u
y
x
w
sin
sin
*
,
,
=
(ii)
Incoherent diffraction limited model
Linear system
h(x,y;
x’,y’)
Point
nonlinearity
g(.)
w(x,y)
System may not
be shift invariant
u(x,y)
image
v(x,y)
(x,y)
1
(x,y)
Noise
2
(x,y)
Noise
f(.)
3
–Here aperture is rectangle
–But phase is lost
()
⋅
=
b
a
tri
u
W
2
1
2
1
2
1
,
,
,
ξ
↓
(
)(
)
( )
[ ]
by
c
ax
c
y
x
u
y
x
w
2
2
sin
sin
*
,
,
=
(iii)
Horizontal motion
–Here, while take the picture, the object move horizontally with uniform
velocity.
ds
y
s
x
u
y
x
w
∫
+
−
−
=
2
0
2
0
,
1
,
0
α
−
=
y
x
rect
y
x
u
δ
2
1
1
*
,
0
0
(iv)
Rectangular scanning aperture
–here due to finite aperture size, we measure the
average
of
u(x,y)
only a
square.
(
)
∫∫
−−
−
−
=
2
2
2
2
,
,
β
dudv
v
y
u
x
u
y
x
w
=
y
x
rect
y
x
u
,
*
,
↓
( ) ( ) ( )
[]
2
1
2
1
2
1
sin
sin
,
,
βξ
αξ
αβ
c
c
u
W
⋅
=
(v)
CCD interaction
–Here due to interaction between the CCD cells, value at
(x,y)
also depends on
the value of its neighbours:
(
)
l
y
k
x
u
y
x
w
kl
kl
∆
−
∆
−
=
∑∑
−
=−
=
,
,
1
1
1
1
(
)
∆
−
∆
−
=
−
=
l
y
k
x
y
x
u
kl
,
*
,
1
1
1
1
(vi)
Atmospheric turbulence
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This note was uploaded on 04/14/2010 for the course ELEC 317 taught by Professor Nil during the Spring '02 term at HKUST.
 Spring '02
 Nil

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