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⎢ ⎥−⎣ ⎦