Prof. Feng Liu. Winter /10/2019

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1 Prof. Feng Liu Winter 29 //29

2 Last Time Course overview Admin. Info Computer Vision Computer Vision at PSU Image representation Color 2

3 Today Filter 3

4 Today Filters and its applications noisy image naïve denoising Gaussian blur better denoising edge-preserving filter Slide credit: Sylvain Paris and Frédo Durand 4

5 Image Filtering Image filtering: compute function of local neighborhood at each pixel position One type of Local operator, Neighborhood operator, Window operator Useful for: Enhancing images Noise reduction, smooth, resize, increase contrast, etc. Extracting information from images Texture, edges, distinctive points, etc. Detecting patterns Template matching, e.g., eye template Slide credit: C. Dyer and D. Hoiem

6 Example: Box Filter g[, ] Slide credit: David Lowe

7 Image Correlation Filtering Center filter g at each pixel in image f Multiply weights by corresponding pixels and add them together Set resulting value in output image h g is called a filter, mask, kernel, or template Linear filtering is sum of dot product at each pixel position Filtering operation called cross-correlation, and denoted h = f g Slide credit: C. Dyer

8 Image Filtering g[, ] f [.,.] h[.,.] h[ m, n] g[ k, l] k, l f [ m k, n l] Credit: S. Seitz

9 Image Filtering g[, ] f [.,.] h[.,.] h[ m, n] g[ k, l] k, l f [ m k, n l] Credit: S. Seitz

10 Image Filtering g[, ] f [.,.] h[.,.] h[ m, n] g[ k, l] k, l f [ m k, n l] Credit: S. Seitz

11 Image Filtering g[, ] f [.,.] h[.,.] h[ m, n] g[ k, l] k, l f [ m k, n l] Credit: S. Seitz

12 Image Filtering g[, ] f [.,.] h[.,.] h[ m, n] g[ k, l] k, l f [ m k, n l] Credit: S. Seitz

13 Image Filtering g[, ] f [.,.] h[.,.] ? 9 h[ m, n] g[ k, l] k, l f [ m k, n l] Credit: S. Seitz

14 Image Filtering g[, ] f [.,.] h[.,.] ? h[ m, n] g[ k, l] k, l f [ m k, n l] Credit: S. Seitz

15 g[ Image Filtering, ] f [.,.] h[.,.] h[ m, n] g[ k, l] k, l f [ m k, n l] Credit: S. Seitz

16 Box Filter What does it do? Replaces each pixel with an average of its neighborhood g[, ] Achieves smoothing effect (i.e., removes sharp features) Weaknesses: Blocky results Axis-aligned streaks Slide credit: David Lowe

17 Smoothing with Box Filter Slide credit: C. Dyer

18 Properties of Smoothing Filters Smoothing Values all positive Sum to constant regions same as input Amount of smoothing proportional to mask size Removes high-frequency components low-pass filter Slide credit: C. Dyer

19 Practice with Linear Filters? Original Source: D. Lowe

20 Practice with Linear Filters Original Filtered (no change) Source: D. Lowe

21 Practice with Linear Filters? Original Source: D. Lowe

22 Practice with Linear Filters Original Shifted left by pixel Source: D. Lowe

23 Practice with Linear Filters 2 -? Original (Note that filter sums to ) Source: D. Lowe

24 Practice with Linear Filters + -? Original (Note that filter sums to ) Source: D. Lowe

25 Practice with Linear Filters 2 - Original Sharpening filter - Sharpen an out of focus image by subtracting a multiple of a blurred version Source: D. Lowe

26 Sharpening Source: D. Lowe

27 Sharpening by Unsharp Masking h = f + k(f * g) where k is a small positive constant and g = -4 called a Laplacian mask Called unsharp masking in photography

28 Sharpening using Unsharp Mask Filter Original Filtered result Slide credit: C. Dyer

29 Other Filters: Edge Detection Sobel Slide credit: C. Dyer Vertical Edge (absolute value)

30 Other Filters: Edge Detection Sobel Slide credit: C. Dyer Horizontal Edge (absolute value)

31 Key Properties of Linear Filters Linearity: filter(f + f 2 ) = filter(f ) + filter(f 2 ) Shift invariance: same behavior regardless of pixel location filter(shift(f)) = shift(filter(f)) Source: S. Lazebnik

32 Gaussian Filtering Weight contributions of neighboring pixels by nearness x 5, = Constant factor at front makes volume sum to Convolve each row of image with D kernel to produce new image; then convolve each column of new image with same D kernel to yield output image Slide credit: Christopher Rasmussen

33 Smoothing with a Gaussian Smoothing with a box actually doesn t compare at all well with a defocused lens Most obvious difference is that a single point of light viewed in a defocused lens looks like a fuzzy blob; but the averaging process would give a little square Gaussian is isotropic (i.e., rotationally symmetric) A Gaussian gives a good model of a fuzzy blob It closely models many physical processes (the sum of many small effects) Slide by D.A. Forsyth

34 What does Blurring take away? Slide credit: C. Dyer original

35 What does Blurring take away? Slide credit: C. Dyer smoothed (5x5 Gaussian)

36 Smoothing with Gaussian Filter

37 Smoothing with Box Filter

38 input Slide by S. Paris

39 box average Slide by S. Paris

40 Gaussian blur Slide by S. Paris

41 Gaussian Filters What parameters matter here? Standard deviation,, of Gaussian: determines extent of smoothing σ = 2 with 3 x 3 kernel σ = 5 with 3 x 3 kernel Source: D. Hoiem

42 Slide credit: C. Dyer

43 Smoothing with a Gaussian Parameter σ is the scale / width / spread of the Gaussian kernel, and controls the amount of smoothing Slide credit: C. Dyer

44 Small Slide by D.A. Forsyth

45 Medium Slide by D.A. Forsyth

46 Large Slide by D.A. Forsyth

47 Gaussian filters = pixel = 5 pixels = pixels = 3 pixels Slide credit: C. Dyer

48 Cascading Gaussian Filters Remove high-frequency components from the image (low-pass filter) Images become smoother / more blurred Convolution with self is another Gaussian * = Convolving two times with Gaussian kernel of width σ is same as convolving once with kernel of width σ 2 Source: K. Grauman

49 Separability A function g(x, y) is separable if g(x, y) = g (x) g 2 (y) The Gaussian function is separable: e -[(x2 + y 2 )/2 2 ] = e -[x2 /2 2 ] * e -[y2 /2 2 ] First convolve the image with a D horizontal filter Then convolve the result of the first convolution with a D vertical filter For a k x k Gaussian filter, 2D convolution requires k 2 multiplications and k 2 - additions per pixel But using the separable filters, we reduce this to 2k multiplications and 2k- additions per pixel Slide credit: C. Dyer

50 Separability f = g = g 2 = x = = 2 = = = = = 8 65 Slide credit: C. Dyer

51 Gaussian Filters What parameters matter here? Size of kernel or mask σ = 5 with x kernel σ = 5 with 3 x 3 kernel Slide credit: C. Dyer

52 How big should the filter be? Gaussian function has infinite support but need a finite-size kernel Values at edges should be near 98.8% of area under Gaussian in mask of size 5σ x 5σ In practice, use mask of size 2k+ x 2k+ where k 3 Normalize output by dividing by sum of all weights Slide credit: C. Dyer

53 Practical Matters What about near the edge? the filter window falls off the edge of the image need to extrapolate methods: clip filter (black) wrap around copy edge reflect across edge Source: S. Marschner

54 Filter Re-cap noisy image naïve denoising Gaussian blur better denoising edge-preserving filter Slide credit: Sylvain Paris and Frédo Durand 56

55 Median Filter Replace pixel by the median value of its neighbors No new pixel values introduced Removes spikes: good for impulse, salt & pepper noise Nonlinear filter Slide credit: C. Dyer

56 Median Filter Salt and pepper noise Median filtered Plots of a row of the image Slide credit: M. Hebert, C. Dyer

57 Median Filter Median filter is edge preserving Slide credit: C. Dyer

58 Slide credit: C. Dyer

59 Median Filter (9 x 9 window) Slide credit: C. Dyer input output images by J. Plush

60 Bilateral filter Tomasi and Manduci CCV98.pdf Related to SUSAN filter [Smith and Brady 95] Digital-TV [Chan, Osher and Chen 2] sigma filter Slide credit: F. Durand

61 Start with Gaussian filtering Here, input is a step function + noise J f I output Slide credit: F. Durand input

62 Gaussian filter as weighted average Weight of x depends on distance to x J(x) x f (x,x) I(x) x x x x output input Slide credit: F. Durand

63 The problem of edges Here, It is too different J(x) x x pollutes our estimate J(x) f (x,x) I(x) x x x x output input Slide credit: F. Durand

64 Principle of Bilateral filtering [Tomasi and Manduchi 998] Penalty g on the intensity difference J(x) k(x) x f (x,x) g(i(x) I(x)) I(x) x x I(x) output Slide credit: F. Durand input

65 Bilateral filtering [Tomasi and Manduchi 998] Spatial Gaussian f J(x) k(x) x f (x,x) g(i(x) I(x)) I(x) x x x output input Slide credit: F. Durand

66 Bilateral filtering Spatial Gaussian f Gaussian g on the intensity difference J(x) k(x) x f (x,x) [Tomasi and Manduchi 998] g(i(x) I(x)) I(x) x x I(x) output input Slide credit: F. Durand

67 Normalization factor [Tomasi and Manduchi 998] k(x)= J(x) x k(x) x f (x,x) f (x,x) g(i(x) I(x)) g(i(x) I(x)) I(x) output input Slide credit: F. Durand

68 Blur comes from averaging across edges input * output * * Same Gaussian kernel everywhere. Slide credit: P. Sylvain

69 Bilateral filter: no averaging across edges input * output * * The kernel shape depends on the image content. Slide credit: P. Sylvain

70 Parameter for intensity difference Gaussian g r =. r =.25 r = (Gaussian blur) input s = 2 Parameter for spatial distance Gaussian f s = 6 s = 8 Slide credit: P. Sylvain

71 Parameter for intensity difference Gaussian g r =. r =.25 r = (Gaussian blur) input s = 2 Parameter for spatial distance Gaussian f s = 6 s = 8 Slide credit: P. Sylvain

72 Result Input Output Reprint from Tomasi and Manduchi

73 Why do we say it is non-linear? It does not respect bila(f+g)=bila(f)+bila(g) Slide credit: F. Durand

74 Bilateral filtering is non-linear The weights are different for each output pixel J(x) k(x) x f (x,x) g(i(x) I(x)) I(x) x x output input Slide credit: F. Durand

75 Other view The bilateral filter uses the 3D distance Slide credit: F. Durand

76 Speed Direct bilateral filtering is slow (minutes) Accelerations exist: Subsampling in space & range Durand & Dorsey 22 Paris & Durand 26 Limit to box kernel & intelligent maintenance of histogram Weiss 26 Slide credit: F. Durand

77 Application: Filter Banks for Feature Detection LM Filter Bank Code for filter banks: Slide credit: C. Dyer

78 Filter Banks Process image with each filter and keep responses (or squared/abs responses) Slide credit: C. Dyer

79 Application: Hybrid Images I Gaussian Filter I G A. Oliva, A. Torralba, P.G. Schyns, Hybrid Images, SIGGRAPH 26 G I 2 (-G 2 ) Laplacian Filter Slide credit: C. Dyer Project Instructions: unit impulse GaussianLaplacian of Gaussian ComputationalPhotography_ProjectHybrid.html

80 Slide credit: C. Dyer

81

82 Next Time Feature detection 84

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