Digital image processing: Image Restoration, Slides of Digital Image Processing

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Digital Image Processing

Image Restoration

H.R. Pourreza

• Image restoration vs. image enhancement

 Enhancement:

 largely a subjective process

 Priori knowledge about the degradation is not a must

(sometimes no degradation is involved)

 Procedures are heuristic and take advantage of the

psychophysical aspects of human visual system

 Restoration:

 more an objective process

 Images are degraded

 Tries to recover the images by using the knowledge

about the degradation

Image Restoration

• Two types of degradation

 Additive noise  Spatial domain restoration (denoising) techniques are preferred

 Image blur

 Frequency domain methods are preferred

• We model the degradation process by a degradation function h(x,y),

an additive noise term, (x,y), as g(x,y)=h(x,y)*f(x,y)+ (x,y)

 f(x,y) is the (input) image free from any degradation

 g(x,y) is the degraded image

 (^) * is the convolution operator

 The goal is to obtain an estimate of f(x,y) according to the knowledge about the degradation function h and the additive noise 

 In frequency domain: G(u,v)=H(u,v)F(u,v)+N(u,v)

• Three cases are considered in this Chapter

 g(x,y)=f(x,y)+ (x,y) (5-2~5-4)

 g(x,y)=h(x,y)*f(x,y) (5-5~5-6)

 g(x,y)=h(x,y)*f(x,y)+ (x,y) (5-7~5-9)

An Image Degradation Model

A Model of the Image

Degradation/Restoration Process

Noise

• We first consider the degradation due to noise only

 h is an impulse for now ( H is a constant)

• White noise

 Autocorrelation function is an impulse function multiplied by

a constant



 It means there is no correlation between any two pixels in the

noise image

 There is no way to predict the next noise value

 The spectrum of the autocorrelation function is a constant

(white) (the statement in page 222 about white noise is

wrong)

( , ) ( , ) ( , ) 0 ( , )

1

0

1

0

a x y s t s x t y N x y

N

t

M

s

 (^)      









Noise Model

Gaussian Noise

Gaussian Noise

• Rayleigh noise

 The mean and variance of this

density are given by

 a and b can be obtained through

mean and variance

 

z a

z a e z a p z b

z a b

0 for

( ) for

( )^2 /

/ 4 and

b a b

Other Noise Models

• Erlang (Gamma) noise

 The mean and variance of this

density are given by

 a and b can be obtained

through mean and variance





0 for 0

for 0

1

z

e z

b

a z

p z

az

b b

2

2 / and a

b

  b a  

Other Noise Models

• Uniform noise



 The mean and variance

of this density are given

by

0 otherwise

if

a z b

p z b a

( )/ 2 and

2 2 b a a b

Other Noise Models

• Impulse (salt-and-pepper)

noise



 If either Pa or Pb is zero,

the impulse noise is called

unipolar

 a and b usually are

extreme values because

impulse corruption is

usually large compared

with the strength of the

image signal

 It is the only type of noise

that can be distinguished

from others visually

0 otherwise

for

for

( ) P z b

P z a

p z b

a

Other Noise Models

Salt-and-pepper Noise

A Sample Image