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Lecture 11 Image restoration and reconstruction II
1 Periodic noise1. Periodic noise2. Periodic noise reduction by frequency domain filter3 Introduction to degradation and filtering technique3. Introduction to degradation and filtering technique
• Linear, position-invariant degradation• Estimation of degradation function
I filt• Inverse filter• Wiener filter• Constrained least squares filtering
Periodic noise• Noise that appears periodically, typically from electronic or
electromechanical interference during image acquisition. • Example:• Example:
0 0( , ) sin(2 ( ) / 2 ( ) / )
( , ) ( , ) ( , )x yr x y A u x B M v y B N
g x y f x y r x y
π π= + + +
= +
• Periodic noise can be reduced significantly via frequency domain filteringdomain filtering
• The DFT of the r(x,y) is
0
0
2 /0 0
2 /0 0
( , ) [( ) ( , )2
( ) ( , )]
x
y
iu B M
iv B N
AR u v i e u u v v
e u u v v
= + +
− − −
π
π
δ
δ
Periodic Noise Reduction by Frequency Domain Filtering
• When the bandwidth of noise is known, bandreject filters can be used for filtering the noise
• Bandreject filters HBR(u, v) filtering that reject certain bandwidth– Idea, Butterworth, Gaussian
• Bandpass filters
( ) ( )HBP(u, v) = 1- HBR(u, v)
Obtained by Bandpass filtersHBP(u, v) = 1- HBR(u, v) The inverse FT
Notch Filters
• Reject (or pass) frequencies in a predefined neighborhood about the center of he frequency rectangle. Zero-phase-shift filters must be symmetric about the origin.must be symmetric about the origin.
• The Notch reject and pass filter can be represented as
( , ) ( , ) ( , )Q
N R k kH u v H u v H u v= ∏1
( , ) ( , ) ( , )
( , ) 1 ( , )
N R k kk
N P N R
H u v H u v H u v
H u v H u v
−=
= −
∏
where Hk(u, v), H-k(u, v) are the highpass filters whose centers are at (uk, vk) and (uk, vk), respectively. These centers are specified w.r.p.t the center of the frequency rectangle (M/2, N/2)
2 21 0 0
2 2 1 / 2
1 1( , ) [ ][ ]1 [ / ( , )] 1 [ / ( , )]
Q
N R n nk k k k k
H u vD D u v D D u v= −
=+ +∏
2 2 1 / 2
2 2 1 / 2
( , ) [( / 2 ) ( / 2 ) ]
( , ) [( / 2 ) ( / 2 ) ]k k k
k k k
D u v u M u v N v
D u v u M u v N v−
= − − + − −
= − + + − +
Linear, position-invariant degradation
• The image with degradation and noise:( , ) [ ( , )] ( , )g x y H f x y x y= +η
• Assume• H is linear if
( , ) 0x y =η
• Additivity:
1 2 1 2[ ( , ) ( , )] [ ( , )] [ ( , )]H af x y bf x y aH f x y bH f x y+ = +
Additivity:
• Homogenerity:1 2 1 2[ ( , ) ( , )] [ ( , )] [ ( , )]H f x y f x y H f x y H f x y+ = +
• Position invariant1 1[ ( , )] [ ( , )]H af x y aH f x y=
( ) [ ( )]g x y H f x y=( , ) [ ( , )][ ( , )] ( , )
g x y H f x yH f x y g x y
=− − = − −α β α β
Impulse response
• The image with degradation and noise:
( , ) ( , ) ( , )f x y f x y d d∞ ∞
= − −∫ ∫ α β δ α β α β( , ) ( , ) ( , )
( , ) [ ( , )]
[ ( , ) ( , ) ]
f x y f x y d d
g x y H f x y
H f x y d d
−∞ −∞
∞ ∞
=
= − −
∫ ∫
∫ ∫
α β δ α β α β
α β δ α β α β[ ( , ) ( , ) ]
[ ( , ) ( , )]
H f x y d d
H f x y d d
−∞ −∞
∞ ∞
−∞ −∞
∞ ∞
= − −
∫ ∫∫ ∫∫ ∫
α β δ α β α β
α β δ α β α β
• Impulse response of H (point spread)
( , ) [ ( , )]f H x y d d−∞ −∞
= − −∫ ∫ α β δ α β α β
• Superposition integral of the first kind ( , , , ) [ ( , )]h x y H x y= − −α β δ α β
( , ) ( , ) ( , , , )g x y f h x y d d∞ ∞
−∞ −∞= ∫ ∫ α β α β α β
Convolution
• If H is position invariant[ ( , )] ( , )H x y h x y− − = − −δ α β α β
• The convolution integral
With i
( , ) ( , ) ( , )g x y f h x y d d∞ ∞
−∞ −∞= − −∫ ∫ α β α β α β
• With noise
( , ) ( , ) ( , ) ( , )g x y f h x y d d x y∞ ∞
−∞ −∞= − − +∫ ∫ α β α β α β η
( , ) ( , ) ( , ) ( , )( , ) ( , ) ( , ) ( , )
g x y h x y f x y x yG u v H u v F u v N u v
∞ ∞
= ∗ += +
∫ ∫η
Summary
• A linear spatially-invariant degradation system with additive noise can be modelled in spatial domain as the
l ti f th d d ti ( i t d) f ticonvolution of the degradation (point spread) function with an image, followed by the addition of the noise
Or equivalently in frequency domain: the product of the FTs of the image and degradation, followed by the addition of Ft of the noiseaddition of Ft of the noise
• Solution is available for linear spatially-invariant p ydegradation model.
Inverse filter
• Restoration image degraded by a degradation function H, where H is given or obtained by estimation
( , )ˆ ( , ) G u vF u v =( , )( , )
( , )ˆ ( , ) ( , )( , )
F u vH u v
N u vF u v F u vH u v
= +( , )H u v
Minimum Mean Square Error (Wiener) Filtering
• Incorporate both the degradation function and statistical characterization of the noise into restoration process.
• Find the estimate image which minimize the error2 2ˆ{( ) }e E f f= −
2
( , ) ( , )ˆ ( , ) ( , )( , ) | ( , ) | ( , )
( )
f
f
H u v S u vF u v G u v
S u v H u v S u v
H u v
∗
∗
⎡ ⎤= ⎢ ⎥
+⎢ ⎥⎣ ⎦⎡ ⎤
η
2
2
2
( , ) ( , )| ( , ) | ( , ) / ( , )
1 ( , ) ( , )
f
H u v G u vH u v S u v S u v
H u v G u v
⎡ ⎤= ⎢ ⎥
+⎢ ⎥⎣ ⎦⎡ ⎤
= ⎢ ⎥
η
2 ( , )( , ) | ( , ) | ( , ) / ( , )f
G u vH u v H u v S u v S u v⎢ ⎥
+⎢ ⎥⎣ ⎦η
2( , ) | ( , ) |S u v N u v=η2
2
1 ( , )ˆ ( , ) ( , )H u vF u v G u v⎡ ⎤
= ⎢ ⎥2( , ) | ( , ) |fS u v F u v=2( , ) ( , )
( , ) | ( , ) |F u v G u v
H u v H u v K⎢ ⎥+⎣ ⎦
Constrained Least Square Filtering
• When H and the mean and variance of the noise are known.
1 1M N− −
• To minimize
Subject to
2 2
0 0[ ( , )]
x yC f x y
= =
= ∇∑∑2 2ˆ|| || || ||g Hf− = ηSubject to
Then
|| || || ||g Hf η
2 2
( , )ˆ ( , ) ( , )| ( ) | | ( ) |
H u vF u v G u vH u v P u v
∗⎡ ⎤= ⎢ ⎥+⎣ ⎦γ
• Where P(u,v) is the FT of
⎡ ⎤
| ( , ) | | ( , ) |H u v P u v+⎣ ⎦γ
0 1 0( , ) 1 4 1p x y
−⎡ ⎤⎢ ⎥= − −⎢ ⎥⎢ ⎥0 1 0⎢ ⎥−⎣ ⎦