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Math 3360: Mathematical Imaging

Math 3360: Mathematical Imaging. Lecture 11: Types of noises. Prof. Ronald Lok Ming Lui Department of Mathematics, The Chinese University of Hong Kong. Lecture 1: Introduction to Image Processing Lecture 2: Basic idea of image transformation

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Math 3360: Mathematical Imaging

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

  2. 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 Class schedules 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 Lecture 15 to Lecture 17: Image deblurring Lecture 18 to Lecture 21: Image segmentation Lecture 22 to Lecture 24: Image registration

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

  4. 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:

  5. 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

  6. 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,

  7. Example of Gaussian noises: Gaussian noise

  8. Example of white noises: White noise

  9. Image components

  10. 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

  11. 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 Linear filter = Convolution

  12. Linear filter

  13. Type of filter

  14. In Photoshop

  15. Mean filter

  16. Mean filter Impulse noise After 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 image After mean filter

  21. Gaussian filter Real image After mean filter

  22. Gaussian filter Real image After Gaussian filter

  23. Gaussian filter After mean filter Real image

  24. Gaussian filter After Gaussian filter Real image

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

  26. Laplace filter Original Laplace filter

  27. Laplace filter Original Laplace filter

  28. Laplace filter Laplace filter Original

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

  30. Median filter Real image After mean filter

  31. Median filter Salt & Pepper Mean filter Median filter

  32. Median filter Noisy image Median filter

  33. Median filter

  34. Median filter Noisy image Median filter

  35. Median filter

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

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