Filtering Images in the Spatial Domain Chapter 3b G&W. Ross Whitaker (modified by Guido Gerig) School of Computing University of Utah

Transcription

1 Filtering Images in the Spatial Domain Chapter 3b G&W Ross Whitaker (modified by Guido Gerig) School of Computing University of Utah 1

2 Overview Correlation and convolution Linear filtering Smoothing, kernels, models Detection Derivatives Nonlinear filtering Median filtering Bilateral filtering Neighborhood statistics and nonlocal filtering 2

3 Cross Correlation Operation on image neighborhood and small mask, filter, stencil, kernel Linear operations within a moving window Input image Filter Output image 3

4 Cross Correlation Operation on image neighborhood and small mask, filter, stencil, kernel Linear operations within a moving window Input image Filter Output image 4

5 Cross Correlation Operation on image neighborhood and small mask, filter, stencil, kernel Linear operations within a moving window Input image Filter Output image 5

6 Cross Correlation Operation on image neighborhood and small mask, filter, stencil, kernel Linear operations within a moving window Input image Filter Output image 0.0* * * * * * * * *205 =

7 Cross Correlation Operation on image neighborhood and small mask, filter, stencil, kernel Linear operations within a moving window Input image Filter Output image * * * * * * * * *205 =

8 Cross Correlation Operation on image neighborhood and small mask, filter, stencil, kernel Linear operations within a moving window Input image Filter Output image * * * * * * * * *205 =

9 Cross Correlation Operation on image neighborhood and small mask, filter, stencil, kernel Linear operations within a moving window Input image Filter Output image

10 Cross Correlation Operation on image neighborhood and small mask, filter, stencil, kernel Linear operations within a moving window Input image Filter Output image 10

11 Cross Correlation Operation on image neighborhood and small mask, filter, stencil, kernel Linear operations within a moving window Input image Filter Output image 11

12 Cross Correlation Operation on image neighborhood and small mask, filter, stencil, kernel Linear operations within a moving window Input image Filter Output image 0.0* * * * * * * * *77 =

13 Cross Correlation Operation on image neighborhood and small mask, filter, stencil, kernel Linear operations within a moving window Input image Filter Output image * * * * * * * * *77 =

14 Cross Correlation 1D 2D 14

15 Correlation: Technical Details How to filter boundary?????????? 15

16 Correlation: Technical Details Boundary conditions Boundary not filtered (keep it 0) Pad image with amount (a,b) Constant value or repeat edge values Cyclical boundary conditions Wrap or mirroring 16

17 Correlation: Technical Details Boundaries Can also modify kernel no longer correlation For analysis Image domains infinite Data compact (goes to zero far away from origin) 17

18 Correlation: Properties Shift invariant 18

19 Correlation: Properties Shift invariant Linear Compact notation 19

20 Filters: Considerations Normalize Sums to one Sums to zero (some cases, see later) Symmetry Left, right, up, down Rotational Special case: auto correlation 20

21 Examples

22 Examples

23 Examples /9 *

24 Examples /9 *

25 Examples 1 1/9 *

26 Examples 2 1/9 * /25 *

27 Smoothing and Noise Noisy image 5x5 box filter 27

28 Noise Analysis Consider an a simple image I() with additive, uncorrelated, zero-mean noise of variance s What is the expected rms error of the corrupted image? If we process the image with a box filter of size 2a+1 what is the expected error of the filtered image? 28

29 Other Filters Disk Circularly symmetric, jagged in discrete case Gaussians Circularly symmetric, smooth for large enough stdev Must normalize in order to sum to one Derivatives discrete/finite differences Operators 30

30 Gaussian Kernel 31

31 Gaussian Kernel Normalization to

32 Box versus Gaussian 33

33 Discrete Convolution Java demo: Continuous Same as cross correlation with kernel transposed around each axis The two operations (correlation and convolution) are the same if the kernel is symmetric about axes reflection of w 34

34 Convolution: Properties Shift invariant, linear Commutative Associative Others (discussed later): Derivatives, convolution theorem, spectrum 35

35 Computing Convolution 36

36 Computing Convolution Compute time MxM mask NxN image 37

37 Computing Convolution Compute time MxM mask NxN image O(M 2 N 2 ) for loops are nested 4 deep 38

38 Computing Convolution Compute time MxM mask NxN image O(M 2 N 2 ) Special case: separable for loops are nested 4 deep Two 1D kernels = * O(M 2 N 2 ) O(MN 2 ) 39

39 Separable Kernels Examples Box/rectangle Bilinear interpolation Combinations of partial derivatives d 2 f/dxdy Gaussian Only filter that is both circularly symmetric and separable Counter examples Disk Cone Pyramid 40

40 Separability 41

41 Digital Images: Boundaries are Lines or Discontinuities Example: Characterization of discontinuities? Source: 42

42 Digital Images: Boundaries are Lines or Discontinuities Example: Characterization of discontinuities? Source: 43

43 Source: 44

44 Source: 45

45 Derivatives: Finite Differences 46

46 Derivative Example

47 Source: 48

48 Source: 49

49 Source: 50

50 Pattern Matching 51

51 Pattern Matching/Detection The optimal (highest) response from a filter is the autocorrelation evaluated at position zero A filter responds best when it matches a pattern that looks itself Strategy Detect objects in images by correlation with matched filter 52

52 Matched Filter Example 53

53 Matched Filter Example: Correlation of template with image 54

54 Matched Filter Example: Thresholding of correlation results 55

55 Matched Filter Example: High correlation template found 56

56 Summary of 9/15 Spatial Filtering Consider neighborhood information Special consideration at boundary????????? 57

57 Summary of 9/15 Common Filters Box Gaussian Derivative 1/9 *

58 Box versus Gaussian 59

59 Gaussian Filtering 60

60 Gaussian Filtering 61

61 Source: 62

62 Cross-correlation and Convolution Cross-correlation Convolution 63

63 Separability Key Concepts = O(M 2 N 2 ) O(MN 2 ) 64

64 Nonlinear Methods For Median filtering Bilateral filtering Filtering Neighborhood statistics and nonlocal filtering 65

65 Median Filtering For each neighborhood in image Sliding window Usually odd size (symmetric) 5x5, 7x7, Sort the greyscale values Set the center pixel to the median Important: use Jacobi updates Separate input and output buffers All statistics on the original image old new 66

66 Median vs Gaussian Original + Gaussian Noise 3x3 Median 3x3 Box 67

67 Median Filter Issues Boundaries Compute on pixels that fall within window Computational efficiency What is the best algorithm? Properties Removes outliers (replacement noise salt and pepper) Window size controls size of structures Preserves straight edges, but rounds corners and features 68

68 Replacement Noise Also: shot noise, salt&pepper Replace certain % of pixels with samples from pdf Best filtering strategy: filter to avoid outliers 69

69 Smoothing of S&P Noise It s not zero mean (locally) Averaging produces local biases 70

70 Smoothing of S&P Noise It s not zero mean (locally) Averaging produces local biases 71

71 Median Filtering Median 3x3 Median 5x5 72

72 Median Filtering Median 3x3 Median 5x5 73

73 Median Filtering Iterate Median 3x3 2x Median 3x3 74

74 Median Filtering Image model: piecewise constant (flat) 75

75 Median Filtering Image model: piecewise constant (flat) Ordering Output 76

76 Median Filtering Image model: piecewise constant (flat) 77

77 Median Filtering Image model: piecewise constant (flat) Ordering Output 78

78 Order Statistics Median is special case of order-statistics filters Instead of weights based on neighborhoods, weights are based on ordering of data Neighborhood Ordering Filter Neighborhood average (box) Median filter Trimmed average (outlier removal) 79

79 Source: 80

80 Piecewise Flat Image Models Image piecewise flat -> average only within similar regions Problem: don t know region boundaries 81

81 Piecewise-Flat Image Models Assign probabilities to other pixels in the image belonging to the same region Two considerations Distance: far away pixels are less likely to be same region Intensity: pixels with different intensities are less likely to be same region 82

82 Gaussian: Blur Comes from Averaging across Edges input * output * * Same Gaussian kernel everywhere. Source:

83 Bilateral Filter No Averaging across Edges [Aurich 95, Smith 97, Tomasi 98] input * output * * The kernel shape depends on the image content. Source:

84 Main Idea Distance (kernel/pdf) Distance (pdf) Prob pixel belongs to same region as i position Prob pixel belongs to same region as i intensity 85

85 BF Bilateral Filter Definition: an Additional Edge Term Same idea: weighted average of pixels. new q S not new 1 [ I] p G r W s p new p q G I I p q I q normalization factor space weight range weight Source:

86 Illustration a 1D Image 1D image = line of pixels Better visualized as a plot pixel intensity pixel position Source:

87 Gaussian Blur and Bilateral Filter Gaussian blur p q GB I] G p q [ p q S space I q space Source:

88 Gaussian Blur and Bilateral Filter Gaussian blur p q GB [ I] p G q S p q space I q space Bilateral filter [Aurich 95, Smith 97, Tomasi 98] q p range BF 1 [ I ] p G W s r p q S space normalization p q G I I p range q I q space Source:

89 Bilateral Filter Neighborhood sliding window Weight contribution of neighbors according to: normalization: all weights add up to 1 G is a Gaussian (or lowpass), as is H, N is neighborhood, Often use G(r ij ) where r ij is distance between pixels Update must be normalized for the samples used in this (particular) summation Spatial Gaussian with extra weighting for intensity Weighted average in neighborhood with downgrading of intensity outliers Tomasi, Manduchi:

90 Bilateral Filter Replaces the pixel value at x with an average of similar and nearby pixel values. When the bilateral filter is centered, say, on a pixel on the bright side of the boundary, the similarity function s assumes values close to one for pixels on the same side, and values close to zero for pixels on the dark side. The similarity function is shown in figure 1(b) for a 23x23 filter support centered two pixels to the right of the step in figure 1(a). 91

91 Bilateral Filtering Replaces the pixel value at x with an average of similar and nearby pixel values. Gaussian Blurring Bilateral 92

92 Bilateral Filtering Gaussian Blurring Bilateral 93

93 Nonlocal Averaging Recent algorithm NL-means, Baudes et al., 2005 UINTA, Awate & Whitaker, 2005 Different model No need for piecewise-flat Images consist of some set of pixels with similar neighborhoods average several of those Scattered around General area of a pixel All around Idea Average sets of pixels with similar neighborhoods 94

94 UINTA: Unsupervised Information-Theoretic Adaptive Filtering : Excellent Introduction and Additional Readings (Suyash P. Awate) /~suyash/pubs/uinta/ Suyash P. Awate, Ross T. Whitaker Unsupervised, Information-Theoretic, Adaptive Image Filtering with Applications to Image Restoration IEEE Trans. Pattern Analysis & Machine Intelligence (TPAMI) 2006, Vol. 28, Num. 3, pp

95 Nonlocal Averaging Strategy: Average pixels to alleviate noise Combine pixels with similar neighborhoods Formulation n i,j vector of pixels values, indexed by j, from neighborhood around pixel i vector = 96

96 Nonlocal Averaging Formulation Distance between neighborhoods Kernel weights based on distances w i, k K ( di, k ) e d 2 i, k 2 2 Pixel values of k neighborhoods: f k 97

97 Averaging Pixels Based on Weights For each pixel, i, choose a set of pixel locations k: k = 1,., M Average them together based on neighborhood weights (prop. to intensity pattern difference) g i 1 M M w k 1 i, k k 1 w i, k f k 98

98 Nonlocal Averaging 99

99 Some Details Window sizes: good range is 5x5- >11x11 How to choose samples: Random samples from around the image UINTA, Awate&Whitaker Block around pixel (bigger than window, e.g. 51x51) NL-means Iterate UNITA: smaller updates and iterate 100

100 NL-Means Algorithm For each pixel, p Loop over set of pixels nearby Compare the neighorhoods of those pixels to the neighborhood of p and construct a set of weights Replace the value of p with a weighted combination of values of other pixels Repeat but 1 iteration is pretty good 101

101 Results Noisy image (range ) Bilateral filter (3.0, 0.1) 102

102 Results Bilateral filter (3.0, 0.1) NL means (7, 31, 1.0) 103

103 Results Bilateral filter (3.0, 0.1) NL means (7, 31, 1.0) 104

104 Less Noisy Example 105

105 Less Noisy Example 106

106 Results Original Noisy Filtered 107

107 Checkerboard With Noise Original Noisy Filtered 108

108 Quality of Denoising, joint entropy, and RMS- error vs. number of iterations 109

109 MRI Head 110

110 MRI Head 111

111 Fingerprint 112

112 Fingerprint 113

113 Results Original Noisy Filtered 114

114 Results Original Noisy Filtered 115

115 Results Original Noisy Filtered 116

116 Fractal Original Noisy Filtered 117

117 Piecewise Constant Several 10s of Iterations Tends to obliterate rare events (e.g. corners) 118

118 Texture, Structure 119

119 120