Math 3360: Mathematical Imaging

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

2. 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 Image Enhancement

3. Discrete convolution: Spatial transform v.s. frequency transform (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. LP = Low Pass; HP = High Pass Image components

7. Image components

8. 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… Type of noises • Please refer to Supplemental note 6 for details.

9. Impulse 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: Type of noises • Gaussian noise: • Noise at each pixel follows the Gaussian probability density function:

10. Additive noise: • Noisy image = original (clean) image + noise • Multiplicative noise: • Noisy image = original (clean) image * noise Type of noises • 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. Independent 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. Type of noises • 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. Example of Gaussian noises: Gaussian noise

13. Example of white noises: White noise

14. Image components

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