This presentation contains concepts of different image restoration and reconstruction techniques used nowadays in the field of digital image processing. Slides are prepared from Gonzalez book and Pratt book.

1. Image
Restoration
and Recon-
struction
Upendra Image Restoration and Reconstruction
Upendra
Indian Institute of Information Technology
February 26, 2017

2. Image
Restoration
and Recon-
struction
Upendra
Image Restoration and Reconstruction
Introduction
1 Image enhancement as a subjective process
2 Image restoration, an objective version
3 Goal remains more or less the same: to improve the image
in some aspect

3. Image
Restoration
and Recon-
struction
Upendra
Image Restoration and Reconstruction
Image Enhancement
1 based on heuristic measures to manipulate the image
2 Goal: to get the most of psychophysical aspects of human
eye. For example, persistence of vision, resolution etc.
3 Example- constrast stretching , histogram equalization etc.

4. Image
Restoration
and Recon-
struction
Upendra
Image Restoration and Reconstruction
Image Restoration
1 Restoration tries to recover the image given some apriori
knowledge of the degradation phenomenon.
2 Restoration = detect,model and invert degradation
3 Output, a restored image in some sense.

5. Image
Restoration
and Recon-
struction
Upendra
Image Restoration and Reconstruction
Image Restoration
1 Not all restoration techniques amenable in any domain.
2 Additive components best dealt in spatial domain while
multiplication terms in frequency domain.
3 In other case, go with the experience

6. Image
Restoration
and Recon-
struction
Upendra
A Model for Image Degradation/Restoration
Description
Figure: A model of the image degradation/restoration process

7. Image
Restoration
and Recon-
struction
Upendra
A Model for Image Degradation/Restoration
Description
Assuming H to be a linear, position-invariant process, the
degraded image is given by
g(x, y) = h(x, y) f (x, y) + η(x, y) (1)
Alternatively,
G(u, v) = H(u, v)F(u, v) + N(u, v) (2)
Here, we assume H to be an identity operator and as such we
discuss the degradation due to noise only.

8. Image
Restoration
and Recon-
struction
Upendra
A Model for Image Degradation/Restoration
Noise Models
1 Noise sources: image acquisition, transmission,
environmental factors, sensing devices etc.
2 Parameters of noise model, correlation with the image etc.
are prerequisites to image restoration
3 Concept of white noise.
4 Our assumptions:
noise independent of spatial coordinates
noise uncorrelated with the image
5 Assumptions not so strong as in X-ray, nuclear imaging
etc.

9. Image
Restoration
and Recon-
struction
Upendra
Noise Probability Density Function Models
Gaussian Noise
1 Mathematically tractible, hence, widely used
2 Better known as normal noise
3 factors like electronic circuit noise, sensor noise due to
poor illumination and/or high temperature
p(z) =
1
√
2πσ
e
−(z−z)2
2σ2 (3)
where z represents the intensity, z, the mean value.

10. Image
Restoration
and Recon-
struction
Upendra
Noise Probability Density Function Models
Gaussian Noise
Figure: Plot of Gaussian probability density function

11. Image
Restoration
and Recon-
struction
Upendra
Noise Probability Density Function Models
Rayleigh Noise
1 Evident in image ranging
2 The PDF for a Rayleigh noise is given by
p(z) =
2
b (z − a)e
−(z−a)2
b , for z ≥ a
0, for z < a
(4)
The mean and variance for this density are given by
z = a +
πb
4
(5)
σ2
=
b(4 − π)
4
(6)

12. Image
Restoration
and Recon-
struction
Upendra
Noise Probability Density Function Models
Rayleigh Noise
Figure: Rayleigh probability density function

13. Image
Restoration
and Recon-
struction
Upendra
Noise Probability Density Function Models
Erlang Noise
The PDF for Erlang noise is given by
p(z) =
abzb−1
(b−1)! e−az, for z ≥ a
0, for z < 0
(7)
where the parameters are such that a > 0, b is a positive
integer
The mean and variance of this density are given by
z =
b
a
(8)
σ2
=
b
a2
(9)

14. Image
Restoration
and Recon-
struction
Upendra
Noise Probability Density Function Models
Erlang Noise
Figure: Erlang probability density function

15. Image
Restoration
and Recon-
struction
Upendra
Noise Probability Density Function Models
Exponential Noise
The PDF of exponential noise is given by,
p(z) =
ae−az, for z ≥ 0
0, for z < 0
(10)
where a > 0. The mean and variance of this density function
are
z =
1
a
(11)
σ2
=
1
a2
(12)

16. Image
Restoration
and Recon-
struction
Upendra
Noise Probability Density Function Models
Exponential Noise
Figure: Exponential probability density function

17. Image
Restoration
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Upendra
Noise Probability Density Function Models
Uniform Noise
The PDF of uniform noise is given by,
p(z) =
ae−az, if a ≤ z ≤ b
0, otherwise
(13)
The mean and variance of this density function are given by
z =
a + b
2
(14)
σ2
=
(b − a)2
12
(15)

18. Image
Restoration
and Recon-
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Upendra
Noise Probability Density Function Models
Uniform Noise
Figure: Uniform probability density function

19. Image
Restoration
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Upendra
Noise Probability Density Function Models
Impulse (salt-and-pepper) Noise
The PDF of bipolar impulse noise is given by,
p(z) =



Pa, for z = a
Pb, for z = b
0, otherwise
(16)
If either Pa or Pb is 0, then the noise assumes unipolar nature.
If, however, they are almost same, then the impulse assumes
salt and pepper granules.

20. Image
Restoration
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struction
Upendra
Noise Probability Density Function Models
Impulse (salt-and-pepper) Noise
1 Noise impulses can be negative or positive as is evident in
sharp transitions
2 Scaling done since impulse corruption exceeds the image
signal values
3 ’a’ and ’b’ thus remains saturated values
4 As a consequence, negative extremes appear as black dots
while positive extremes appear as white dots.

21. Image
Restoration
and Recon-
struction
Upendra
Noise Probability Density Function Models
Impulse (salt-and-pepper) Noise
Figure: Impulse probability density function

22. Image
Restoration
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Upendra
Noise Probability Density Function Models
Comparative Study of Characteristics of Different Noise Models
Figure: Original Image

23. Image
Restoration
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Upendra
Noise Probability Density Function Models
Comparative Study of Characteristics of Different Noise Models
Figure: Images and histograms resulting from adding Gaussian,
Rayleigh and Erlang noise to the image

24. Image
Restoration
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struction
Upendra
Noise Probability Density Function Models
Comparative Study of Characteristics of Different Noise Models
Figure: Images and histograms resulting from adding exponential,
uniform and impulse noise to the image

25. Image
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Upendra
Noise Probability Density Function Models
Periodic Noise
1 Arises typically from electrical or electromechanical
interference during image acquisition
2 Spatially dependent noise
3 Can be reduced significantly by frequency domain filtering

26. Image
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Noise Probability density function models
Periodic Noise
Figure: (a) Image corrupted by sinusoidal noise. (b) Spectrum (each
pair of conjugate impulses corresponds to one sine wave). Original
image courtsey of NASA

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Noise Probability Density Function Models
Estimation of Noise Parameters
1 Typically estimated by inspecting the Fourier Transform.
Why?
2 In simple cases, from the image itself.
3 Automated analysis possible only when either noise spikes
are either exceptionally pronounced, or when knowledge is
available about the general location of the frequency
components of the interference.
4 Guess the noise model, estimate the parameters and model
the same

28. Image
Restoration
and Recon-
struction
Upendra
Noise Probability Density Function Models
Estimation of Noise Parameters
1 Small strips of images used for calculating the mean and
variance of intensity levels
2 Consider a strip, S, and let pS (zi ) i = 0, 1, 2....L − 1,
denote the probability estimates of the intensities of the
pixels in S, for L possible intensities in the image.
3 Given this information, the mean and variance of the
pixels in S is as follows
z =
L−1
i=0
zi pS (zi ) (17)
σ2
=
L−1
i=0
(zi − z)2
pS (zi ) (18)

29. Image
Restoration
and Recon-
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Upendra
Noise Probability Density Function Models
Estimation of Noise Parameters
Figure: Historams computed using small strips (shown as inserts)
from (a) the Gaussian, (b) the Rayleigh, and (c) the uniform noisy
images

30. Image
Restoration
and Recon-
struction
Upendra
Noise Probability density function models
Estimation of Noise Parameters
1 The shape of the histogram identifies the closest match
2 Find the best match, then get parameters given the mean
and variance values
3 Use the inverse model for image restoration.

31. Image
Restoration
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Upendra
References
1 Digital Image Processing (3rd Edition) by Gonzalez, R.C.
and Woods, R.R.,PHI
2 Digital Image Processing (3rd Edition) by Willian k. Pratt,
John Wiley and Sons