1. The Chinese University of Hong Kong
Math 3360: Mathematical Imaging
Lecture 12:
Image Enhancement &
Types of noises
Prof. Ronald Lok Ming Lui Department of Mathematics,
The Chinese University of Hong Kong

2. Image Enhancement What is image enhancement? How?
Image enhancement is the process by which we improve an image so that it looks subjectively better.
How?
An image is enhanced when we:
remove additive noise and interference;
remove multiplicative interference;
increase its contrast;
decrease its blurring.
Some standard methods:
smoothing or low pass filtering;
sharpening or high pass filtering;
histogram manipulation and
algorithms that remove noise while avoid blurring the image.

3. Image Enhancement We will consider two image enhancement problems:
Image denoising
Image deblurring

4. Image Enhancement We will consider two image enhancement problems:
Image denoising
Image deblurring

5. Image Enhancement Linear filtering:
Modifying a pixel value (in the spatial domain) by a linear combination of neighborhood values.
Operations in spatial domain v.s. operations in frequency domains:
Linear filtering (matrix multiplication in spatial domain) = discrete convolution
In the frequency domain, it is equivalent to multiplying the Fourier transform of the image with a certain function that “kills” or modifies certain frequency components

6. Spatial transform v.s. frequency transform
Discrete convolution:
DFT of Discrete convolution: Product of fourier transform
DFT(convolution of f and w) = C*DFT(f)*DFT(w)
Multiplying the Fourier transform of the image with a certain function that “kills” or modifies certain frequency components

7. Spatial transform v.s. frequency transform

8. Spatial transform v.s. frequency transform

9. Type of noises Preliminary statistical knowledge: Random variables;
Random field;
Probability density function;
Expected value/Standard deviation;
Joint Probability density function;
Independence;
Uncorrelated;
Covariance;
Autocorrelation;
Cross-correlation;
Cross covariance;
Noise as random field etc…
Please refer to Lecture Note Chapter 3 for details.

10. Statistical Background
Please refer to Lecture Note Chapter 3 for details.

11. Statistical Background

12. Statistical Background

13. Statistical Background

14. Statistical Background

15. Statistical Background

16. Statistical Background
What is noise?

17. Statistical Background

18. Statistical Background

19. Statistical Background

20. Statistical Background

21. Type of noises Impulse noise: Gaussian noise:
Change value of an image pixel at random;
The randomness follows the Poisson distribution = Probability of having pixels affected by the noise in a window of certain size
Poisson distribution:
Gaussian noise:
Noise at each pixel follows the Gaussian probability density function:

22. Type of noises Additive noise: Multiplicative noise: Homogenous noise:
Noisy image = original (clean) image + noise
Multiplicative noise:
Noisy image = original (clean) image * noise
Homogenous noise:
Noise parameter for the probability density function at each pixel are the same (same mean and same standard derivation)
Zero-mean noise:
Mean at each pixel = 0
Biased noise:
Mean at some pixels are not zero

23. Type of noises Independent noise: Uncorrelated noise: White noise:
The noise at each pixel (as random variables) are independent.
Uncorrelated noise:
Let Xi = noise at pixel i (as random variable);
E(Xi Xj) = E(Xi) E(Xj) for all i and j.
White noise:
Zero mean + Uncorrelated + additive
iid noise:
Independent + identically distributed;
Noise component at every pixel follows the SAME probability density function (identically distributed)
For Gaussian distribution,

24. Gaussian noise
Example of Gaussian noises:

25. White noise
Example of white noises:

26. White noise
Example of white noises:

27. Noises as high frequency component
Why noises are often considered as high frequency component?
frequency
frequency
(a) Clean image spectrum and Noise spectrum (Noise dominates the high-frequency component);
(b) Filtering of high-frequency component