Sampling II, Anti-aliasing, Texture Filtering

Transcription

1 ME-C3100 Computer Graphics, Fall 2017 Jaakko Lehtinen Many slides from Frédo Durand Sampling II, Anti-aliasing, Texture Filtering 1

2 Sampling The process of mapping a function defined on a continuous domain to a discrete one is called sampling The process of mapping a continuous variable to a discrete one is called quantization To represent or render an image using a computer, we must both sample and quantize Today we focus on the effects of sampling continuous value discrete value continuous position discrete position 2

3 Sampling & Reconstruction The visual array of light is a continuous function 1/ We sample it... with a digital camera, or with our ray tracer This gives us a finite set of numbers (the samples) We are now in a discrete world 2/ We need to get this back to the physical world: we reconstruct a continuous function from the samples For example, the point spread of a pixel on a CRT or LCD Reconstruction also happens inside the rendering process (e.g. texture mapping) Both steps can create problems Insufficient sampling or pre-aliasing (our focus today) Poor reconstruction or post-aliasing 3

4 Sampling Example Continuous function 4

5 Sampling Example Sampled discrete function 5

6 Reconstruction Example Nearest neighbor reconstruction 6

7 Reconstruction Example Piecewise linear reconstruction 7

8 Reconstruction Example As you can see, there are many ways of reconstructing a function from the samples. Some are more accurate than others Piecewise cubic reconstruction 8

9 Wiggliness vs. Sampling Density

10 Wiggliness vs. Sampling Density 1.4 2x sampling density

11 Sampling Density If we sample too coarsely, the samples can be mistaken for something simpler during reconstruction This is why it s called aliasing The new (erroneous) low-frequency sine wave is an alias/ghost of the high-frequency one Input Reconstructed 11

12 Pre- vs. Post-Aliasing If the sampling density is too low, no reconstruction method can recover the original function This is pre-aliasing, information was lost in sampling Poor reconstruction is post-aliasing

13 Recap: Texture Aliasing 13

14 Aliasing What Does it Look Like? Nasty mess Nearest neighbor Mess gets blurred out Tri-linear Mip-mapping 14

15 Questions? 15

16 Sampling Density Revisited If we sample too coarsely, the samples can be mistaken for something simpler during reconstruction When does this happen? The signal must not contain frequencies higher than half the sampling density (Nyquist, Shannon, and others) Input Reconstructed 16

17 Sampling Density Revisited So, in order to prevent aliasing, we should either.. Filter the high frequencies out from the signal before sampling (this is called low-pass filtering ) Sample at a higher rate Input Reconstructed 17

18 Solution? Removing high freq s = low-pass filtering = blur In practice, we blur before sampling in order to stop high frequencies from messing our image up In audio, use an analog low-pass filter before sampling For ray tracing/rasterization: compute image at higher resolution, blur, then resample at lower resolution Not exactly a proper low-pass prefilter, because aliasing still happens at the higher resolution, but it s all we can do For textures, we blur the texture before doing the lookup (not necessary to compute in higher resolution first) To understand what really happens, we need semi-serious math (Fourier transforms) 18

19 Digital Cameras Most digital cameras have an optical low-pass filter in front of the sensor Yes, it s essentially a piece of frosted glass (maitolasi) Some cameras don t See comparison between Nikon D800 and D800e 19

20 Blurring Removes High Frequencies Original image High frequencies removed by blur 20

21 Types of Blur Blur = local weighted average Different set of weights => different blur e.g.: box, Gaussian, bilinear, bicubic Box filter: Equally-weighted average of N samples 21

22 Types of Blur Blur = local weighted average Different set of weights => different blur e.g.: box, Gaussian, bilinear, bicubic Bicubic filter: A wider filter with small negative lobes 22

23 Filter Examples Notice that there are still sharp features in the result Original Box filter 23

24 Filter Examples Notice that there are still sharp features in the result Gaussian blur Box filter 24

25 Filter Examples No sharp edges like box, but overall sharper (good) Gaussian blur Bicubic filter 25

26 Blurring: (Discrete) Convolution Input Convolution sign Kernel Convolution of a function and a kernel gives another function. All values in the output function are locally weighted averages of the input function. 26

27 Blurring: (Discrete) Convolution Input Convolution sign Kernel Same shape, just reduced contrast!!! The sine wave is an eigenvector (output is the input multiplied by a constant) This means sines and cosines are particularly simple for convolution 27

28 Animated Convolution See Wikipedia 28

29 (Convolutional Neural Nets) Yes, the CNNs everyone is talking about now are based on precisely this operation. Many layers of convolutions where the kernels are learned Layers separated by point-wise nonlinearities (eg. ReLU) See wikipedia or Goodfellow s Deep Learning book 29

30 Types of Convolution That was discrete; the functions and the kernel were sampled representations defined on a grid There is also a continuous convolution input and output functions and the kernel are defined on a continuous domain weighted sum is replaced by an integral And actually, a semi-discrete one as well! Reconstructing a continuous signal from samples is a convolution as well: you convolve the spiky samples with a continuous reconstruction kernel 30

31 Example: Samples 31

32 Example: Reconstruction Kernel 32

33 1 Sample * Translated Reconstruction Kernel 33

34 All Samples * Translated Kernels 34

35 Final Reconstruction = sum of samples * translated kernels 35

36 Questions? 36

37 Signal Processing 101 Sampling and filtering are best understood in terms of Fourier analysis We already saw aliasing for sine waves: a high frequency sine wave turns into a low frequency one when undersampled 37

38 Remember Fourier Analysis? A signal in the spatial domain has a dual representation in the frequency domain spatial domain frequency domain This particular signal is band-limited, meaning it has no frequencies above some threshold 38

39 Remember Fourier Analysis? We can transform from one domain to the other using the Fourier Transform Just a change of basis: Represent the image in terms of sines and cosines of different frequencies and orientations An invertible transform frequency domain spatial domain Fourier Transform Inverse Fourier Transform 39

40 The Fourier basis functions are sines and cosines each with a certain orientation and frequency v u

41 The Fourier basis functions are sines and cosines each with a certain orientation and frequency v u

42 The Fourier basis functions are sines and cosines each with a certain orientation and frequency v u

43 Fourier Transforms and Series are bread-and-butter signal processing tools you learned in your introductory math classes You can refresh at Wikipedia, again If the complex exponentials feel weird to you, you can just think of it as doing sines and cosines at once through Euler s formula exp(ix) = cos(x) + i sin(x) 43

44 Great animated GIF that illustrates the decomposition 44

45 Questions? 45

46 Remember Convolution? Some operations that are difficult in the spatial domain are simpler in the frequency domain For example, convolution in the spatial domain (local weighted average) is the same as multiplication in the frequency domain. This is because sine waves are eigenfunctions! And, convolution in the frequency domain is the same as multiplication in the spatial domain 46

47 Low-Pass Filter black means 0, white means 1 => frequencies outside circle get set to zero

48 High-Pass Filter black means 0, white means 1 => frequencies inside circle get set to zero

49 Sampling in the Frequency Domain original signal Fourier Transform sampling grid ( impulse train ) (multiplication) Fourier Transform (convolution) sampled signal Fourier Transform Replicated spectra 49

50 Reconstruction Sampling gives us discrete samples at the cost of frequency replicas If we can get rid of the replicas, we can reconstruct the original signal in spatial domain How to get rid of replicas? Low-pass filter, i.e., blur the spiky sampled function! But there may be overlap between the copies This is aliasing Spectrum of sampled function 50

51 Guaranteeing Proper Reconstruction Separate by removing high frequencies from the original signal (low pass pre-filtering before sampling)......or separate replicas by increasing sampling density. If we can't separate the copies, we will have overlapping frequency spectrum during reconstruction aliasing. 51

52 Recap: Sampling/Reconstruction 1/ Prefilter Blur signal with given low-pass filter to remove frequencies that would alias when sampled 2/ Sample From continuous to discrete domain Creates replicas in frequency domain 3/ Reconstruct Get rid of the replicas Another blur: Apply a low-pass filter to the spiky discrete sampled function 52

53 The Sampling Theorem When sampling a signal at discrete intervals, the sampling frequency must be greater than twice the highest frequency of the input signal in order to be able to reconstruct the original perfectly from the sampled version (Shannon, Nyquist, Whittaker, Kotelnikov) 53

54 Questions? 54

55 Supersampling in Graphics Pre-filtering (blurring before sampling) is hard Our primitives have infinite frequencies (e.g., a mathematical triangle has perfectly sharp edges) And we can usually only take point samples of the image Think of your ray tracer! And still difficult to integrate analytically with prefilter Possible for lines, or if visibility is ignored Not very interesting... Usually, fall back to supersampling Just bite the bullet and sample more 55

56 In Practice: Supersampling Your intuitive solution is to compute multiple color values per pixel (at slightly different locations) and average them jaggies w/ antialiasing 56

57 In Practice: Supersampling Your intuitive solution is to compute multiple color values per pixel and average them A better interpretation of the same idea is that You first render a high resolution image You blur it (low-pass, prefilter) You resample it at a lower resolution 57

58 Uniform Supersampling Compute image at resolution k*width, k*height Downsample using low-pass filter (e.g. Gaussian, bicubic, etc.) 58

59 Uniform Supersampling Advantage The first (super)sampling captures more high frequencies that are not aliased Downsampling can use a good filter Issue Frequencies above the (super)sampling limit are still aliased Works well for edges Not as well for repetitive textures We only pushed the problem farther But solutions exist 59

60 Related Idea: Multisampling Problem Shading is very expensive today (complicated shaders) Full supersampling has linear cost in #samples (k*k) Goal: High-quality edge antialiasing at lower cost Solution Compute shading only once per pixel for each primitive, but resolve visibility at sub-pixel level Store (k*width, k*height) frame and z buffers, but share shading results between sub-pixels within a real pixel When visibility samples within a pixel hit different primitives, we get an average of their colors Edges get antialiased without large shading cost 60

61 Multisampling, Visually = sub-pixel visibility sample One pixel 61

62 Multisampling, Visually = sub-pixel visibility sample One pixel 62

63 Multisampling, Visually = sub-pixel visibility sample The color is only computed once per pixel per triangle and reused for all the visibility samples that are covered by the triangle. One pixel 63

64 Supersampling, Visually = sub-pixel visibility sample When supersampling, we compute colors independently for all the visibility samples. One pixel 64

65 Multisampling Pseudocode For each triangle For each pixel if pixel overlaps triangle color=shade() // only once per pixel! for each sub-pixel sample compute edge equations & z if subsample passes edge equations && z < zbuffer[subsample] zbuffer[subsample]=z framebuffer[subsample]=color 65

66 Multisampling Pseudocode For each triangle For each pixel if pixel overlaps triangle color=shade() // only once per pixel! for each sub-pixel sample compute edge equations & z if subsample passes edge equations && z < zbuffer[subsample] zbuffer[subsample]=z framebuffer[subsample]=color At display time: //this is called resolving For each pixel color = average CS-C3100 of Fall 2017 subsamples Lehtinen 66

67 Multisampling vs. Supersampling Supersampling Compute an entire image at a higher resolution, then downsample (blur + resample at lower res) Multisampling Supersample visibility, compute expensive shading only once per pixel, reuse shading across visibility samples But Why? Visibility edges are where supersampling really works Shading can be prefiltered more easily than visibility This is how GPUs perform antialiasing these days 67

68 Questions? 68

69 BUT: Supersampling has its limits 69

70 1 Sample / Pixel 70

71 Supersampling, 16 Samples / Pixel 71

72 100 Samples / Pixel Even this sampling rate cannot get rid of all aliasing artifacts! We are really only pushing the problem farther. 72

73 Questions? 73

74 Texture Filtering Problem: Prefiltering is impossible when you can only take point samples This is why visibility (edges) need supersampling Texture mapping is simpler Imagine again we are looking at an infinite textured plane textured plane 74

75 Texture Filtering We should pre-filter image function before sampling That means blurring the image function with a low-pass filter (convolution of image function and filter) textured plane Low-pass filter 75

76 Texture Filtering We can combine low-pass and sampling The value of a sample is the integral of the product of the image f and the filter h centered at the sample location A local average of the image f weighted by the filter h ˆf i = f(x) h(x)dx image textured plane Low-pass filter 76

77 Texture Filtering Well, we can just as well change variables and compute this integral on the textured plane instead In effect, we are projecting the pre-filter onto the plane textured plane Low-pass filter 77

78 Texture Filtering Well, we can just as well change variables and compute this integral on the textured plane instead In effect, we are projecting the pre-filter onto the plane It s still a weighted average of the texture under filter ˆf i = plane f(x ) h(x ) J(x, x ) dx textured plane Just the usual change of variables formula, you learned in math class, see here Low-pass filter 78

79 Texture Pre-Filtering, Visually textured surface (texture map) image plane Image adapted from McCormack et al. image-space filter image-space filter projected onto plane Must still integrate product of projected filter and texture That doesn t sound any easier... 79

80 Solution: Precomputation We ll precompute and store a set of prefiltered results from each texture with different sizes of prefilters 80

81 Solution: Precomputation We ll precompute and store a set of prefiltered results from each texture with different sizes of prefilters 81

82 Solution: Precomputation We ll precompute and store a set of prefiltered results from each texture with different sizes of prefilters Because it s low-passed, we can also subsample 82

83 Solution: Precomputation We ll precompute and store a set of prefiltered results from each texture with different sizes of prefilters Because it s low-passed, we can also subsample 83

84 This is Called MIP-Mapping Construct a pyramid of images that are pre-filtered and re-sampled at 1/2, 1/4, 1/8, etc., of the original image's sampling During rasterization we compute the index of the decimated image that is sampled at a rate closest to the density of our desired sampling rate MIP stands for multum in parvo which means many in a small place 84

85 MIP-Mapping When a pixel wants an integral of the pre-filtered texture, we must find the closest results from the precomputed MIP-map pyramid Must compute the size of the projected pre-filter in the texture UV domain Projected pre-filter 85

86 MIP-Mapping When a pixel wants an integral of the pre-filtered texture, we must find the closest results from the precomputed MIP-map pyramid Must compute the size of the projected pre-filter in the texture UV domain Simplest method: Just pick the one scale that is closest (in some sense), then perform usual reconstruction on that level (e.g. bilinear) Projected pre-filter 86

87 Tri-Linear MIP-Mapping Next-simplest method: See which two scales are closest, compute reconstruction results from both, and linearly interpolate between them When using bilinear reconstruction on each scale, this is called tri-linear MIP-mapping Projected pre-filter Note that we are not getting the correct answer because precomputed filters are not the right shape 87

88 Anisotropic MIP-Mapping Anisotropic MIP-mapping approximates the true prefilter with a number of samples from different levels of the pyramid Mostly black magic Projected pre-filter 88

89 MIP Mapping Example Nearest Neighbor MIP Mapped (Tri-Linear) 89

90 MIP Mapping Example Small details may "pop" in and out of view Nearest Neighbor MIP Mapped (Tri-Linear) 90

91 MIP Mapping Example nearest neighbor/ point sampling mipmaps & linear interpolation (tri-linear) 91

92 Storing MIP Maps Can be stored compactly: Only 1/3 more space! 92

93 Finding the MIP Level Often we think of the pre-filter as a box What is the projection of the rectangular pixel window in texture space? MIP-map prefilters are symmetric (isotropic) so there is no one right way to pick the level Projected pre-filter 93

94 Finding the MIP Level Often we think of the pre-filter as a box What is the projection of the rectangular pixel window in texture space? MIP-map prefilters are symmetric (isotropic) so there is no one right way to pick the level Answer is in the partial derivatives px and py of (u,v) w.r.t. screen (x,y) Projection of pixel center px = (du/dx, dv/dx) py = (du/dy, dv/dy) Projected pre-filter 94

95 Finding the MIP Level Two most common approaches are Pick level according to the length (in texels) of the longer partial log 2 max {w p x,h p y } Projection of pixel center Projected pre-filter px = (du/dx, dv/dx) py = (du/dy, dv/dy) Pick level according to the length of their sum log 2 (w p x ) 2 +(h p y ) 2 w x h 95

96 Questions? 96

97 How Are Partials Computed? You can derive closed form formulas based on the uv and xyw coordinates of the vertices... This is what used to be done..but shaders may compute texture coordinates programmatically, not necessarily interpolated No way of getting analytic derivatives! In practice, use finite differences GPUs process pixels in blocks of (at least) 4 anyway These 2x2 blocks are called quads 97

98 Texture Filtering: Magnification Pre-aliasing problems happen when function varies too fast for samples to capture But when we are looking at a texture closer than the sampling rate, poor reconstruction (post-aliasing) is apparent Must also use proper filtering... If not enough texture resolution, result is blurry Oh well, not much you can do there! Nearest neighbor Bilinear 98

99 Elliptical Weighted Average (EWA) Isotropic filter w.r.t. screen space Becomes anisotropic in texture space Use e.g. anisotropic Gaussian Very, very high quality, but slow No hardware support 99

100 EWA Filtering in Action 100

101 Image Quality Comparison EWA vs. tri-linear MIP-mapping EWA trilinear mipmapping 101

102 Further Reading Paul Heckbert published seminal work on texture mapping and filtering in his master s thesis (!) Including EWA Highly recommended reading! See More reading Feline: Fast Elliptical Lines for Anisotropic Texture Mapping, McCormack, Perry, Farkas, Jouppi SIGGRAPH 1999 Texram: A Smart Memory for Texturing Schilling, Knittel, Strasser,. IEEE CG&A, 16(3): Arf! 102

103 That s All for Today! Let s see a cool real-time video to get back from mathland! 103

104 Sampling Texture Maps textured surface (texture map) image plane circular pixel window How to map the texture area seen through the pixel window to a single pixel value? 104

105 Sampling Texture Maps When texture mapping it is rare that the screen-space sampling density matches the sampling density of the texture. 64x64 pixels Original Texture Magnification for Display Minification for Display 105

106 Bilinear Texture Filtering Tell OpenGL to use a tent filter instead of a box filter Magnification looks better, but blurry (texture is under-sampled for this resolution) Oh well

107 Questions? 107

108 Minification: Examples of Aliasing point sampling 108

109 Potential Solutions 1/ We can supersample the final image But only offsets the Nyquist limit 2/ We can blur the texture (prefilter) We ll treat the projected texture as a 2D function of screen coordinates, and project the image-space prefilter back onto the texture map The right way to go textured surface (texture map) image circular pixel window 109

110 Spatial Filtering Remove the high frequencies which cause artifacts in texture minification Compute a spatial integration over the extent of the pixel This is equivalent to convolving the texture with a filter kernel centered at the sample (i.e., pixel center)! Expensive to do during rasterization, but an approximation it can be precomputed projected texture in image plane pixels projected in texture plane 110

111 MIP Mapping Construct a pyramid of images that are pre-filtered and re-sampled at 1/2, 1/4, 1/8, etc., of the original image's sampling During rasterization we compute the index of the decimated image that is sampled at a rate closest to the density of our desired sampling rate MIP stands for multum in parvo which means many in a small place 111

112 MIP Mapping Example Nearest Neighbor MIP Mapped (Bi-Linear) 112

113 MIP Mapping Example Small details may "pop" in and out of view Nearest Neighbor MIP Mapped (Bi-Linear) 113

114 Examples of Aliasing Texture Errors nearest neighbor/ point sampling mipmaps & linear interpolation 114

115 Storing MIP Maps Can be stored compactly Illustrates the 1/3 overhead of maintaining the MIP map 115

116 Finding the MIP Level textured surface (texture map) area pre- filtered in MIPmap circular pixel window image plane Square MIP-map area is a bad approximation 116

117 Finding the MIP level How does a screen-space change dt relates to a texture-space change du,dv. => derivatives, ( du/dt, dv/dt ). e.g. computed by hardware during rasterization often: finite difference (pixels are handled by quads) du, dv dt 117

118 Finding the MIP level How does a screen-space change dt relates to a texture-space change du,dv. This sounds like the derivatives, ( du/dt, dv/dt ). They can be computed simply using the chain rule: du, dv dt 118

119 Finding the MIP level What we'd like to find is the step size that a uniform step in screen-space causes in three-space, or, in other words how a screen-space change relates to a 3-space change. This sounds like the derivatives, ( du/dt, dv/dt ). They can be computed simply using the chain rule: Notice that the term being squared under the numerator is just the w plane equation that we are already computing. The remaining terms are constant for a given rasterization. Thus all we need to do to compute the derivative is a square the w accumulator and multiply it by a couple of constants. Now, we know how a step in screen-space relates to a step in 3-space. So how do we translate this to an index into our MIP table? 119

120 MIP Indices Actually, you have a choice of ways to translate this derivative value into a MIP level. Because we have two derivatives, for u and for v (anisotropy) This also brings up one of the shortcomings of MIP mapping. MIP mapping assumes that both the u and v components of the texture index are undergoing a uniform scaling, while in fact the terms du/dt and dv/dt are relatively independent. Thus, we must make some sort of compromise. Two of the most common approaches are given below: 120

121 MIP Indices Actually, you have a choice of ways to translate this derivative value into a MIP level. Because we have two derivatives, for u and for v (anisotropy) This also brings up one of the shortcomings of MIP mapping. MIP mapping assumes that both the u and v components of the texture index are undergoing a uniform scaling, while in fact the terms du/dt and dv/dt are relatively independent. Thus, we must make some sort of compromise. Two of the most common approaches are given below: The differences between these level selection methods is illustrated by the accompanying figure. 121

122 Anisotropy & MIP-Mapping What happens when the surface is tilted? Nearest Neighbor MIP Mapped (Bi-Linear) 122

123 Questions? 123

124 Elliptical weighted average Isotropic filter wrt screen space Becomes anisotropic in texture space e.g. use anisotropic Gaussian Called Elliptical Weighted Average (EWA) 124

125 EWA resampling 125

126 Image Quality Comparison Trilinear mipmapping EWA trilinear mipmapping 126

127 Approximation of anisotropic Feline: Fast Elliptical Lines for Anisotropic Texture Mapping Joel McCormack, Ronald Perry, Keith I. Farkas, and Norman P. Jouppi SIGGRAPH 1999 Andreas Schilling, Gunter Knittel & Wolfgang Strasser. Texram: A Smart Memory for Texturing. IEEE Computer Graphics and Applications, 16(3): 32-41, May Aproximate Anisotropic Gaussian by a set of isotropic probes 127

128 FELINE results 128

129 Questions? 129

130 Jittering Uniform sample + random perturbation Sampling is now non-uniform Signal processing gets more complex In practice, adds noise to image But noise is better than aliasing Moiré patterns 130

131 Jittered supersampling Regular, Jittered Supersampling 131

132 Jittering Displaced by a vector a fraction of the size of the subpixel distance Low-frequency Moire (aliasing) pattern replaced by noise Extremely effective Patented by Pixar, but expired a few years ago When jittering amount is 1, equivalent to stratified sampling (cf. later) 132

133 Questions? 133

134 Adaptive supersampling Use more sub-pixel samples around edges 134

135 Adaptive supersampling Use more sub-pixel samples around edges Compute color at small number of sample If their variance is high Compute larger number of samples 135

136 Adaptive supersampling Use more sub-pixel samples around edges Compute color at small number of sample If variance with neighbor pixels is high Compute larger number of samples 136

137 Problem with non-uniform distribution Reconstruction can be complicated 80% of the samples are black Yet the pixel should be light grey 137

138 Problem with non-uniform distribution Reconstruction can be complicated Solution: do a multi-level reconstruction Reconstruct uniform sub-pixels Filter those uniform sub-pixels 138

139 Filters (a.k.a convolution kernel) Weighting function or a convolution kernel Area of influence often bigger than "pixel" Sum of weights = 1 Each sample contributes the same total to image Constant brightness as object moves across the screen. No negative weights/colors (optional) 139

140 Filters Filters are used to reconstruct a continuous signal from a sampled signal (reconstruction filters) band-limit continuous signals to avoid aliasing during sampling (low-pass filters) Desired frequency domain properties are the same for both types of filters Often, the same filters are used as reconstruction and low-pass filters 140

141 Pre-Filtering Filter continuous primitives Treat a pixel as an area Compute weighted amount of object overlap What weighting function should we use? 141

142 Post-Filtering Filter samples Compute the weighted average of many samples Regular or jittered sampling (better) 142

143 The Ideal Filter Unfortunately it has infinite spatial extent Every sample contributes to every interpolated point Expensive/impossible to compute spatial frequenc y 143

144 Problems with Practical Filters Many visible artifacts in re-sampled images are caused by poor reconstruction filters Excessive pass-band attenuation results in blurry images Excessive high-frequency leakage causes "ringing" and can accentuate the sampling grid (anisotropy) frequency 144

145 Gaussian Filter This is more or less what a physical display does for free! spatial frequenc y 145

146 Box Filter / Nearest Neighbor Pretending pixels are little squares. spatial frequenc y 146

147 Tent Filter / Bi-Linear Interpolation Simple to implement Reasonably smooth spatial frequenc y 147

148 Bi-Cubic Interpolation Begins to approximate the ideal spatial filter, the sinc function spatial frequenc 148 y

149 Why is the Box filter bad? (Why is it bad to think of pixels as squares) Original high- resolution image Down-sampled with a 5x5 box filter (uniform weights) Down-sampled with a 5x5 Gaussian filter (non-uniform weights) notice the ugly horizontal banding 149

150 Questions? 150

151 Difficulties with perfect sampling Hard to prefilter Perfect filter has infinite support Fourier analysis assumes infinite signal and complete knowledge Not enough focus on local effects And negative lobes Emphasizes the two problems above Negative light is bad Ringing artifacts if prefiltering or supports are not perfect 151

152 At the end of the day Fourier analysis is great to understand aliasing But practical problems kick in As a result there is no perfect solution Compromises between Finite support Avoid negative lobes Avoid high-frequency leakage Avoid low-frequency attenuation Everyone has their favorite cookbook recipe Gaussian, tent, Mitchell bicubic 152

153 Recap: Ideal sampling/reconstruction Pre-filter with a perfect low-pass filter Box in frequency Sinc in time Sample at Nyquist limit Twice the frequency cutoff Reconstruct with perfect filter Box in frequency, sinc in time And everything is great! 153