CSCE 441: Computer Graphics Image Filtering

1. CSCE 441: Computer GraphicsImage Filtering Jinxiang Chai

2. Outline Image Processing - Gaussian filtering - Median filtering - Bilateral filtering

3. Required Readings • Section 3.2 & 3.3.1 (Szeliski book)

4. Filtering • In signal processing, a filter is a process that removes from a signal some unwanted component or feature

5. 1D Signal Filtering

6. 2D Image Filtering

7. 2D Image Filtering

8. Image Filtering Image filtering: change range of image g(x) = h(f(x)) f f f h h x x x f x Image warping: change domain of image g(x) = f(h(x))

9. Image Filtering Image filtering: change range of image g(x) = h(f(x)) h h f g Image warping: change domain of image g(x) = f(h(x)) f g

10. Filtering in Spatial Domain = Filtered image Input image Filter function

11. Gaussian Filtering in Spatial Domain Discrete approximation to Gaussian function with σ=1.0

12. Filtering in Spatial Domain Discrete approximation to Gaussian function with σ=1.0

13. Filtering in Spatial Domain Discrete approximation to Gaussian function with σ=1.0

14. Filtering in Spatial Domain Discrete approximation to Gaussian function with σ=1.0

15. Filtering in Spatial Domain Discrete approximation to Gaussian function with σ=1.0 Filtered_I45 = X

16. Filtering in Spatial Domain Discrete approximation to Gaussian function with σ=1.0 Filtered_I45 = X

17. Filtering in Spatial Domain Discrete approximation to Gaussian function with σ=1.0 Filtered_I45 = X

18. Filtering input Gaussian filter

19. Median Filter • For each neighbor in image, sliding the window • Sort pixel values • Set the center pixel to the median

20. Median Filter input Gaussian filter Median filter

21. Median Filter Examples input Median 7X7

22. Median Filter Examples Median 11X11 Median 3X3

23. Median Filter Examples Straight edges kept Median 11X11 Median 3X3 Sharp features lost

24. Median Filter Properties Can remove outliers (peppers and salts) Window size controls size of structure Preserve some details but sharp corners and edges might get lost

25. Comparison of Mean, Gaussian, and Median original Mean with 6 pixels

26. Comparison of Mean, Gaussian, and Median original Gaussian with 6 pixels

27. Comparison of Mean, Gaussian, and Median original Median with 6 pixels

28. Common Problems Mean: blurs image, removes simple noise, no details are preserved

29. Common Problems Mean: blurs image, removes simple noise, no details are preserved Gaussian: blurs image, preserves details only for small σ.

30. Common Problems Mean: blurs image, removes simple noise, no details are preserved Gaussian: blurs image, preserves details only for small σ. Median: preserves some details, good at removing strong noise

31. Common Problems Mean: blurs image, removes simple noise, no details are preserved Gaussian: blurs image, preserves details only for small σ. Median: preserves some details, good at removing strong noise Can we find a filter that not only smooths regions but preserves edges?

32. Common Problems Mean: blurs image, removes simple noise, no details are preserved Gaussian: blurs image, preserves details only for small σ. Median: preserves some details, good at removing strong noise Can we find a filter that not only smooths regions but preserves edges? - yes, bilateral filter

33. Outline Image Processing - Gaussian filtering - Median filtering - Bilateral filtering

34. What Is Bilateral Filter? Bilateral - Affecting or undertaken by two sides equally Property: - Convolution filter - Smooth image but preserve edges - Operates in the domain and the range of image

35. Bilateral Filter Example Gaussian filter Bilateral filter

36. Bilateral Filter Example Gaussian filter Bilateral filter

37. 1D Graphical Example Center Sample u I(p) Neighborhood p It is clear that in weighting this neighborhood, we would like to preserve the step

38. The Weights I(p) p

39. Filtered Values I(p) Filtered value p

40. Edges Are Smoothed I(p) Filtered value p

41. What Causes the Problem? I(p) Filtered value p

42. What Causes the Problem? Same weights for these two pixels!! I(p) Filtered value p

43. The Weights I(p) p

44. Bilateral Filtering Denoise Feature preserving Bilateral filter Normalization

45. Kernel Properties • Per each sample, we can define a ‘Kernel’ that averages its neighborhood • This kernel changes from sample to sample! • The sum of the kernel entries is 1 due to the normalization, • The center entry in the kernel is the largest, • Subject to the above, the kernel can take any form (as opposed to filters which are monotonically decreasing).

46. Filter Parameters As proposed by Tomasi and Manduchi, the filter is controlled by 3 parameters: The filter can be applied for several iterations in order to further strengthen its edge-preserving smoothing N(u) – The neighbor size of the filter support, c – The variance of the spatial distances, s – The variance of the value distances,

47. Bilateral Filter input

48. Bilateral Filter input

49. Bilateral Filter input Wc

50. Bilateral Filter input Wc Ws