1. Digital Image Processing
Chapter 5 Image Restoration

2. Introduction Image enhancement Image restoration Subjective process
Objective process to recover the original image.
Involving formula or criteria that will yield an optimal result.

3. 5. 1 Model of Image degradation and restoration
g(x,y)=h(x,y)f(x,y)+(x,y)
Note: H is a linear, position-invariant process.

4. 5.2 Noise Model Spatial and frequency property of noise
White noise (random noise)
A sequence of random positive/negative numbers whose mean is zero.
Independent of spatial coordinates and the image itself.
In the frequency domain, all the frequencies are the same.
All the frequencies are corrupted by an additional constant frequency.
Periodic noise

5. Noise Probability Density Functions (1)
Gaussian noise
 : mean;  : variance
70% [(-), (+)]
95 % [(-2), (+2)]
Because of its tractability, Gaussian (normal) noise model is often applicable at best.

6. Implementation
The problem of adding noise to an image is identical to that of adding a random number to the gray level of each pixel.
Noise models describe the distribution (probability density function, PDF) of these random numbers.
How to match the PDF of a group of random numbers to a specific noise model?
Histogram matching.

7. Noise Probability Density Functions (2)
Rayleigh noise
 = a+(b/4)1/2
 = b(4-)/4

8. Noise Probability Density Functions (3)
Erlang (gamma) noise
=b/a;  =b/a2

9. Noise Probability Density Functions (4)
Exponential noise
=1/a;  =1/a2
A special case Erlang noise model when b = 1.

10. Noise Probability Density Functions (5)
Uniform noise
=(a+b)/2;  =(b-a)2/12

11. Noise Probability Density Functions (6)
Impulse noise (salt and pepper noise)

12. Example

13. Results of adding noise

14. Results of adding noise

15. The Principle Use of Noise Models
Gaussian noise:
electronic circuit noise and sensor noise due to poor illumination or high temperature.
Rayleigh noise:
Noise in range imaging.
Erlang noise:
Noise in laser imaging.
Impulse noise:
Quick transients take place during imaging.
Uniform noise:
Used in simulations.

16. 5.2.4 Estimation of Noise Parameters
How do we know which noise model adaptive to the currently available imaging tool?
Image a solid gray board that is illuminated uniformly.
Crop a small patch of constant grey level and analyze its histogram to see which model matches.

17. 5.2.4 Estimation of Noise Parameters
Gaussian n:
Find the mean  and standard deviation  of the histogram (Gaussian noise).
Rayleigh, Erlang, and uniform noise:
Calculate the a and b from  and .
Impulse noise:
Compute the height of peaks at gray levels 0 and 255 to find Pa and Pb.

18. Estimation of Noise Parameters

19. 5.3 Restoration using spatial filter
Used when only additive noise is present.
Mean filters
Arithmetic mean filter:
The new pixel value is resulted from averaging the pixels in the area by Sxy.
Geometric mean filter:
Achieves smoothing comparable to arithmetic approach, but tends to lose less image detail.

20. Order-Statistics filters
Median filter
Max filter: find the brightest points to reduce the pepper noise
Min filter: find the darkest point to reduce the salt noise
Midpoint filter: combining statistics and averaging.

21. Example 5.3
Iteratively applying median filter to an image corrupted by impulse noise.

22. Alpha-Trimmed Mean Filter
Combining the advantages of mean filter and order-statistics filter.
Suppose delete d/2 lowest and d/2 highest gray- level value in the neighborhood of Sxy and average the remaining mn-d pixel, denoted by gr(s, t).
where d=0 ~ mn-1

24. Adaptive Filter
Filters whose behavior changes based on statistical characteristics of the image.
Two adaptive filters are considered:
Adaptive, local noise reduction filter.
Adaptive median filter.

25. Adaptive, Local Noise Reduction Filter
Two parameters are considered:
Mean: measure of average gray level.
Variance: measure of average contrast.
Four measurements:
noisy image at (x, y ): g(x, y )
The variance of noise 2
The local mean mL in Sxy
The local variance 2L

26. Adaptive, Local Noise Reduction Filter
Given the corrupted image g(x, y), find f(x, y).
Conditions:
2 is zero (Zero-noise case)
Simply return the value of g(x, y).
If 2L is higher than 2
Could be edge and should be preserved.
Return value close to g(x, y).
If 2L = 2
when the local area has similar properties with the overall image.
Return arithmetic mean value of the pixels in Sxy.
General expression:

27. Adaptive, Local Noise Reduction Filter

28. Adaptive Median Filter
Adaptive median filter can handle impulse noise with larger probability (Pa and Pb are large).
This approach changes window size during operation (according to certain criteria).
First, define the following notations:
zmin=minimum gray-level value in Sxy
zmax=maximum gray-level value in Sxy
zmed=median gray-level value in Sxy
zxy= gray-level at (x,y)
Smax=maximum allowed size of Sxy

29. Adaptive Median Filter
The adaptive median filter algorithm works in two levels: A and B
Level A: A1=zmed-zmin
A2=zmed-zmax
If A1>0 and A2<0 goto level B
else increase the window size
If window size  Smax repeat level A
else output zxy
Level B: B1=zxy-zmin
B2=zxy-zmax
If B1>0 AND B2<0, output zxy
Else output zmed.

30. Example 5.5