1. Math 3360: Mathematical Imaging Prof. Ronald Lok Ming Lui Department of Mathematics, The Chinese University of Hong Kong Lecture 11: Types of noises

2. Class schedules Lecture 1: Introduction to Image Processing Lecture 2: Basic idea of image transformation Lecture 3: Image decomposition & Stacking operator Lecture 4: Singular Value Decomposition for Image decomposition & Error analysis Lecture 5: Haar & Walsh Transform Lecture 6: Examples of Haar & Walsh Transform; R-Walsh transform Lecture 7: Discrete Fourier transform Lecture 8: Even Discrete Cosine Transform (JPEG) Lecture 9: EDCT + ODCT+ EDST + ODST; Introduction to Image enhancement Lecture 10: Introduction to Linear filtering & Statistical images Lecture 15 to Lecture 17: Image deblurring Lecture 18 to Lecture 21: Image segmentation Lecture 22 to Lecture 24: Image registration Lecture 11: Image denoising: Linear filtering model in the spatial domain; Image denoising: Nonlinear filtering model in the spatial domain; Relationship with the convolution Lecture 12: Image denoising: Linear filtering in the frequency domain Image denoising: Anisotropic diffusion Lecture 13: Image denoising: Total variation (TV) or ROF model Lecture 14: Image denoising: ROF model part 2

3. Type of noises Recap: 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.

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

5. Type of noises Additive 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

6. Type of noises 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. 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,

7. Gaussian noise Example of Gaussian noises:

8. White noise Example of white noises:

9. Image components

10. 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

11. Linear filter = Convolution Linear filtering of a (2M+1)x(2N+1) image I (defined on [-M,M]x[-N,N]) = CONVOLUTION OF I and H H is called the filter. Different filter can be used: Mean filter Gaussian filter Laplcian filter Variation of these filters (Non-linear) Median filter Edge preserving mean filter

12. Linear filter

13. Type of filter

14. In Photoshop

15. Mean filter

16. Impulse noiseAfter mean filter

17. Mean filter Gaussian noise After mean filter

18. Mean filter Real image After mean filter

19. Gaussian filter Define a function using Gaussian function Definition of H

20. Gaussian filter Real imageAfter mean filter

21. Gaussian filter Real image After mean filter

22. Gaussian filter Real image After Gaussian filter

23. Gaussian filter Real image After mean filter

24. Gaussian filter Real image After Gaussian filter

25. Laplace filter ) Laplace filter (High pass filter)

26. Laplace filter Original

27. Laplace filter Original

28. Laplace filter Original

29. Median filter Median Nonlinear filter Take median within a local window

30. Median filter Real image After mean filter

31. Median filter Salt & Pepper Mean filterMedian filter

32. Noisy imageMedian filter

34. Noisy imageMedian filter

36. Noisy image Can you guess what it is? Median filter