1. The Chinese University of Hong Kong
Math 3360: Mathematical Imaging
Lecture 10:
Types of noises
Prof. Ronald Lok Ming Lui Department of Mathematics,
The Chinese University of Hong Kong

2. 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

3. Spatial transform v.s. frequency transform
Discrete convolution:
(Matrix multiplication, which define output value as linear combination of its neighborhood)
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

4. Spatial transform v.s. frequency transform

5. Spatial transform v.s. frequency transform

6. Image components
LP = Low Pass; HP = High Pass

7. Image components

8. Type of noises Preliminary statistical knowledge: Random variables;
Random field;
Probability density function;
Expected value/Standard deviation;
Joint Probability density function;
Linear independence;
Uncorrelated;
Covariance;
Autocorrelation;
Cross-correlation;
Cross covariance;
Noise as random field etc…
Please refer to Supplemental note 6 for details.

9. 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:

10. 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

11. Type of noises Independent noise: Uncorrected noise: White noise:
The noise at each pixel (as random variables) are linearly independent
Uncorrected 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
idd noise:
Independent + identically distributed;
Noise component at every pixel follows the SAME probability density function (identically distributed)
For Gaussian distribution,

12. Gaussian noise
Example of Gaussian noises:

13. White noise
Example of white noises:

14. Image components

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