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Math 3360: Mathematical Imaging

Understand the principles of linear filtering in image processing, including mean, Gaussian, Laplacian, and median filters. Learn how each filter type affects image quality and noise reduction. Explore nonlinear filtering methods for advanced applications.

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Math 3360: Mathematical Imaging

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  1. Math 3360: Mathematical Imaging Lecture 12: Linear filtering Prof. Ronald Lok Ming LuiDepartment of Mathematics, The Chinese University of Hong Kong

  2. 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

  3. Linear filter

  4. Type of filter

  5. In Photoshop

  6. Mean filter

  7. Mean filter Impulse noise After mean filter

  8. Mean filter Gaussian noise After mean filter

  9. Mean filter Real image After mean filter

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

  11. Gaussian filter Real image After mean filter

  12. Gaussian filter Real image After mean filter

  13. Gaussian filter Real image After Gaussian filter

  14. Gaussian filter After mean filter Real image

  15. Gaussian filter After Gaussian filter Real image

  16. ) Laplace filter Laplace filter (High pass filter)

  17. Laplace filter Original Laplace filter

  18. Laplace filter Original Laplace filter

  19. Laplace filter Laplace filter Original

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

  21. Median filter Real image After mean filter

  22. Median filter Salt & Pepper Mean filter Median filter

  23. Median filter Noisy image Median filter

  24. Median filter

  25. Median filter Noisy image Median filter

  26. Median filter

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

  28. Step 1: Consider all windows of fixed size around a pixel (not necessarily centered at that pixel) Step 2: Find a window with the least variance Step 3: Do a linear filter in that window. Median preserving filtering

  29. Median preserving filtering

  30. Median preserving filtering

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