Image Filtering

1. 02/02/10 Image Filtering Computer Vision CS 543 / ECE 549 University of Illinois Derek Hoiem

2. Questions about HW 1?

3. Questions about class? • Room change starting thursday: Everitt 163, same time

4. Key ideas from last class • Lighting • Ambiguity between light source and albedo • Shading is a strong cue for shape • Interreflections, multiple sources, ambient light, etc. make lighting and shadows complicated • Color constancy • Color can be rebalanced by making assumptions (e.g., average pixel is gray)

5. Lightness Perception from Ted Adelson

6. Lightness Perception from Ted Adelson

7. By nickwheeleroz, on Flickr

8. By nickwheeleroz, on Flickr

9. Karsch et al. in review

10. Today’s class • How can we represent color? • What is image filtering and how do we do it? • What are some useful filters and what do they do? • What is linear separability? • Thinking in the frequency domain

11. Color spaces • How can we represent color? http://en.wikipedia.org/wiki/File:RGB_illumination.jpg

12. Color spaces: RGB 0,1,0 • Some drawbacks • Strongly correlated channels • Non-perceptual 1,0,0 0,0,1 Image from: http://en.wikipedia.org/wiki/File:RGB_color_solid_cube.png

13. Color spaces: HSV

14. Color spaces: L*a*b* “Perceptually uniform” color space

15. If you had to choose, would you rather go without luminance or chrominance?

16. If you had to choose, would you rather go without luminance or chrominance?

17. Most information in intensity Only color shown – constant intensity

18. Most information in intensity Only intensity shown – constant color

19. Most information in intensity Original image

20. The raster image (pixel matrix)

21. The raster image (pixel matrix)

22. Image filtering • Image filtering: compute function of local neighborhood at each position • Why bother? • Modify images • Denoise, resize, enhance contrast, etc. • Extract information from images • Edge detection, matching, find distinctive points, etc.

23. Example: box filter 1 1 1 1 1 1 1 1 1 Slide credit: David Lowe (UBC)

24. Image filtering 1 1 1 1 1 1 1 1 1 Credit: S. Seitz

25. Image filtering 1 1 1 1 1 1 1 1 1 Credit: S. Seitz

26. Image filtering 1 1 1 1 1 1 1 1 1 Credit: S. Seitz

27. Image filtering 1 1 1 1 1 1 1 1 1 Credit: S. Seitz

28. Image filtering 1 1 1 1 1 1 1 1 1 Credit: S. Seitz

29. Image filtering Q? 1 1 1 1 1 1 1 1 1 Credit: S. Seitz

30. Box Filter 1 1 1 1 1 1 1 1 1 • What does it do? • Replaces each pixel with an average of its neighborhood • Achieve smoothing effect (remove sharp features) Slide credit: David Lowe (UBC)

31. Smoothing with box filter

32. Practice with linear filters 0 0 0 0 1 0 0 0 0 ? Original Source: D. Lowe

33. Practice with linear filters 0 0 0 0 1 0 0 0 0 Original Filtered (no change) Source: D. Lowe

34. Practice with linear filters 0 0 0 0 0 1 0 0 0 ? Original Source: D. Lowe

35. Practice with linear filters 0 0 0 0 0 1 0 0 0 Original Shifted left By 1 pixel Source: D. Lowe

36. Practice with linear filters 0 1 0 1 0 1 1 0 2 1 0 1 1 0 1 0 0 1 - ? (Note that filter sums to 1) Original Source: D. Lowe

37. Practice with linear filters 0 1 0 1 0 1 1 0 2 1 0 1 1 0 1 0 0 1 - Original • Sharpening filter • Accentuates differences with local average Source: D. Lowe

38. Sharpening Source: D. Lowe

39. Other filters 1 0 -1 2 0 -2 1 0 -1 Sobel Vertical Edge (absolute value)

40. Other filters Q? 1 2 1 0 0 0 -1 -2 -1 Sobel Horizontal Edge (absolute value)

41. Filtering vs. Convolution • 2d filtering • h=filter2(f,g); or h=imfilter(f,g); • 2d convolution • h=conv2(f,g);

42. Key properties of linear filters • Linearity: filter(f1 + f2 ) = filter(f1) + filter(f2) • Shift invariance: same behavior regardless of pixel location: filter(shift(f)) = shift(filter(f)) • Any linear shift-invariant operator can be represented as a convolution Source: S. Lazebnik

43. More properties • Commutative: a * b = b * a • Conceptually no difference between filter and signal • Associative: a * (b * c) = (a * b) * c • Often apply several filters one after another: (((a * b1) * b2) * b3) • This is equivalent to applying one filter: a * (b1 * b2 * b3) • Distributes over addition: a * (b + c) = (a * b) + (a * c) • Scalars factor out: ka * b = a * kb = k (a * b) • Identity: unit impulse e = [0, 0, 1, 0, 0],a * e = a Source: S. Lazebnik

44. Important filter: Gaussian • Weight contributions of neighboring pixels by nearness 0.003 0.013 0.022 0.013 0.003 0.013 0.059 0.097 0.059 0.013 0.022 0.097 0.159 0.097 0.022 0.013 0.059 0.097 0.059 0.013 0.003 0.013 0.022 0.013 0.003 5 x 5,  = 1 Slide credit: Christopher Rasmussen

45. Gaussian filters • Remove “high-frequency” components from the image (low-pass filter) • Convolution with self is another Gaussian • So can smooth with small-width kernel, repeat, and get same result as larger-width kernel would have • Convolving two times with Gaussian kernel of width σ is same as convolving once with kernel of width σ√2 • Separable kernel • Factors into product of two 1D Gaussians Source: K. Grauman

46. Separability of the Gaussian filter Source: D. Lowe

47. Separability example * = = * 2D convolution(center location only) The filter factorsinto a product of 1Dfilters: Perform convolutionalong rows: Followed by convolutionalong the remaining column: Source: K. Grauman

48. Separability • Why is separability useful in practice?

49. Smoothing with Gaussian filter

50. Smoothing with box filter