Introduction Bilateral Filtering Results. Bilateral Filtering. Mathias Eitz. TU Berlin. November, 21st 2006

Transcription

1 Introduction TU Berlin November, 21st 2006

2 About Me Introduction Student at TU Berlin since 2002

3 Outline Introduction 1 Introduction Smoothing Filters Comparison 2 Intuition Mathematical 3 Pictures

4 Outline Introduction Smoothing Filters Comparison 1 Introduction Smoothing Filters Comparison 2 Intuition Mathematical 3 Pictures

5 Introduction Image Smoothing Filters Smoothing Filters Comparison Aim: Smooth image to reduce noise Problem: Details are reduced as well Examples Mean (average) filter Median filter Gaussian filter

6 Introduction Smoothing Filters - Mean Filter Smoothing Filters Comparison Replace pixel value by mean (average) of its neigbours /9 1/9 1/9 * 1/9 1/9 1/9 = 1/9 1/9 1/ Figure: Computing mean of central pixel

7 Introduction Smoothing Filters - Median Filter Smoothing Filters Comparison Replace pixel value by median of its neigbours Variance in neigbouring values does not influence mean Sort values, find the middle one Figure: Computing median of central pixel

8 Introduction Smoothing Filters - Gaussian Smoothing Filters Comparison Replace pixel value by weighted average Pixels near center of kernel are weighted higher Pixels near border of kernel are weighted lower Weighting function G (x, y) = 1 x 2 +y 2 2πσ 2 e 2σ 2

9 Gaussian Function Introduction Smoothing Filters Comparison Figure: Gaussian, σ = 0.25 Figure: Filtered, σ = 0.25

10 Gaussian Function Introduction Smoothing Filters Comparison Figure: Gaussian, σ = 0.5 Figure: Filtered, σ = 0.5

11 Gaussian Function Introduction Smoothing Filters Comparison Figure: Gaussian, σ = 1.0 Figure: Filtered, σ = 1.0

12 Gaussian Function Introduction Smoothing Filters Comparison Figure: Gaussian, σ = 2.0 Figure: Filtered, σ = 2.0

13 Gaussian Function Introduction Smoothing Filters Comparison Figure: Gaussian, σ = 4.0 Figure: Filtered, σ = 4.0

14 Gaussian Function Introduction Smoothing Filters Comparison Figure: Gaussian, σ = 8.0 Figure: Filtered, σ = 8.0

15 Outline Introduction Smoothing Filters Comparison 1 Introduction Smoothing Filters Comparison 2 Intuition Mathematical 3 Pictures

16 Introduction Smoothing Filters Comparison Comparison of Mean, Median and Gaussian Figure: Original Figure: Mean, radius 6px

17 Introduction Smoothing Filters Comparison Comparison of Mean, Median and Gaussian Figure: Original Figure: Gaussian, σ = 4.0

18 Introduction Smoothing Filters Comparison Comparison of Mean, Median and Gaussian Figure: Original Figure: Median, radius 6px

19 Common Problems Introduction Smoothing Filters Comparison Aim Mean Blurs the image, removes simple noise, no details are preserved Gaussian Blurs the image, results related to the mean filter, preserves details only for small σ Median Preserves some details, good at removing strong noise We need a filter that only smooths regions but does not smooth edges

20 Outline Introduction Intuition Mathematical 1 Introduction Smoothing Filters Comparison 2 Intuition Mathematical 3 Pictures

21 Introduction Intuition Mathematical What Is a Bilateral Filter - Definition Some properties Convolution filter Smooth image but preserve edges Operates both in the domain and the range of the image Definition Bilateral Affecting or undertaken by two sides equally

22 Introduction Intuition Mathematical What Is a Bilateral Filter - Definition Some properties Convolution filter Smooth image but preserve edges Operates both in the domain and the range of the image Definition Bilateral Affecting or undertaken by two sides equally

23 Introduction - Basic Idea Intuition Mathematical Algorithm Idea Smooth as usual in the domain of the image (e.g. Gaussian) Do not smooth when pixels are not similar (edge) Similarity Function Determines the amount of smoothing Similar pixels: Strong smoothing Otherwise (edges): No smoothing Similarity based on human perception Simplest example: based on intensity values of pixel, two pixels considered similar if they have the same value

24 Introduction - Basic Idea Intuition Mathematical Algorithm Idea Smooth as usual in the domain of the image (e.g. Gaussian) Do not smooth when pixels are not similar (edge) Similarity Function Determines the amount of smoothing Similar pixels: Strong smoothing Otherwise (edges): No smoothing Similarity based on human perception Simplest example: based on intensity values of pixel, two pixels considered similar if they have the same value

25 Introduction of a Picture Intuition Mathematical Smoothing Weight: Similarity Weight: * 0.9 * 1.0 * 0.9 Resulting Weight: Weight * PixelValue: 0.30*0 * *21 sum() normalize()

26 Introduction of a Picture Intuition Mathematical Smoothing Weight: Similarity Weight: * 0.9 * 1.0 * 0.9 Resulting Weight: Weight * PixelValue: 0.30*15 * *7 sum() normalize()

27 Introduction of a Picture Intuition Mathematical Smoothing Weight: Similarity Weight: * 0.9 * 1.0 * 0.9 Resulting Weight: Weight * PixelValue: 0.30*21 *7 0.30*0 sum() normalize()

28 Introduction of a Picture Intuition Mathematical Smoothing Weight: Similarity Weight: * 0.9 * 1.0 * 0.0 Resulting Weight: Weight * PixelValue: 0.30*7 *0 0.0*255 sum() normalize()

29 Introduction of a Picture Intuition Mathematical Smoothing Weight: Similarity Weight: * 0.0 * 1.0 * 0.9 Resulting Weight: Weight * PixelValue: 0.0*0 * *240 sum() normalize()

30 Introduction of a Picture Intuition Mathematical Smoothing Weight: Similarity Weight: * 0.9 * 1.0 * 0.9 Resulting Weight: Weight * PixelValue: 0.30*255 * *231 sum() normalize()

31 Introduction of a Picture Intuition Mathematical Smoothing Weight: Similarity Weight: * 0.9 * 1.0 * 0.9 Resulting Weight: Weight * PixelValue: 0.30*240 * *242 sum() normalize()

32 Introduction of a Picture Intuition Mathematical Smoothing Weight: Similarity Weight: * 0.9 * 1.0 * 0.9 Resulting Weight: Weight * PixelValue: 0.30*231 * *251 sum() normalize()

33 Outline Introduction Intuition Mathematical 1 Introduction Smoothing Filters Comparison 2 Intuition Mathematical 3 Pictures

34 Introduction - Math Intuition Mathematical Formula J s = 1 f (p s) g (I p I s ) I p k s p Ω k s = p Ω f (p s) g (I p I s ) s coord. of center pixel, p coord. of current pixel, Ω set of all pixel coord. in the local neighbourhood (under kernel) J s resulting pixel intensity. I s, I p intensities of p and s f (p s) measures geometric distance between p and s g (I p I s ) measures photometric similarity betw. I p and I s

35 Introduction Domain Weighting Functions Intuition Mathematical Domain weighting Usually a standard Gaussian Filter f (p s) = e d(p s) 2 2σ d 2 d (p s) = p s = p 2 + s 2 d (p s) is the Euclidean distance between p and s

36 Introduction Range Weighting Function Intuition Mathematical Range weighting Usually Gaussian of the intensity difference g (I p I s ) = e δ(ip Is) 2 2σr 2 δ (I p I s ) = I p I s With I p I s suitable measure of the difference between two pixel values Simplest approach: Difference in intensity values I p I s = I p I s

37 Outline Introduction Pictures References 1 Introduction Smoothing Filters Comparison 2 Intuition Mathematical 3 Pictures

38 Lena Figure: Original Figure: σ d = 3.0, σ r = 3.0

39 Lena Figure: Original Figure: σ d = 6.0, σ r = 3.0

40 Lena Figure: Original Figure: σ d = 12.0, σ r = 3.0

41 Lena Figure: Original Figure: σ d = 12.0, σ r = 6.0

42 Lena Figure: Original Figure: σ d = 15.0, σ r = 8.0

43 Water Lily Figure: Original Figure: σ d = 3.0, σ r = 3.0

44 Water Lily Figure: Original Figure: σ d = 6.0, σ r = 3.0

45 Water Lily Figure: Original Figure: σ d = 12.0, σ r = 3.0

46 Water Lily Figure: Original Figure: σ d = 12.0, σ r = 6.0

47 Water Lily Figure: Original Figure: σ d = 15.0, σ r = 8.0

48 Chinese Handle

49 Chinese Handle

50 Chinese Handle

51 Chinese Handle

52 Chinese Handle

53 Peppers Figure: Original Figure: σ d = 3.0, σ r = 3.0

54 Peppers Figure: Original Figure: σ d = 6.0, σ r = 3.0

55 Peppers Figure: Original Figure: σ d = 12.0, σ r = 3.0

56 Peppers Figure: Original Figure: σ d = 12.0, σ r = 6.0

57 Peppers Figure: Original Figure: σ d = 15.0, σ r = 8.0

58 Questions Introduction Pictures References Any questions? Slides can be downloaded at eitz References: [1, 2, 3, 4, 5]

59 Bibliography I Introduction Pictures References Barash, D. and Anisotropic Diffusion: Towards a Unified Viewpoint. Springer, Barash, D. Fundamental relationship between bilateral filtering, adaptivesmoothing, and the nonlinear diffusion equation. Pattern Analysis and Machine Intelligence, IEEE Transactions on 24, 6 (2002),

60 Bibliography II Introduction Pictures References Durand, F., and Dorsey, J. Fast bilateral filtering for the display of high-dynamic-range images. Proceedings of the 29th annual conference on Computer graphics and interactive techniques (2002), Fleishman, S., Drori, I., and Cohen-Or, D. Bilateral mesh denoising. ACM Transactions on Graphics (TOG) 22, 3 (2003), Tomasi, C., and Manduchi, R. Bilateral filtering for gray and color images. Proceedings of the Sixth International Conference on Computer Vision (1998),