Anti-Aliasing

1
Anti-Aliasing

• Jian Huang, CS594, Fall 2003
• This set of slides are revised from those used at
  Ohio State by Prof. Roger Crawfis.

2
Anti-aliasing?
3
Aliasing

• Aliasing comes from in-adequate sampling rates of
  the continuous signal
• The theoretical foundation of anti-aliasing has
  to do with frequency analysis
• Its always easier to look at 1D cases, so lets
  first look at a few of those.

4
Example of Sampling
5
Examples of Inadequate Sampling
6
Cosine Integrations

• Consider these formulas

7
Improper Cosine Integrals

• Evaluating from (-?,?)

8
Fourier Transform

• The Fourier Transform represents a periodic
  function as a continuous summation of sins and
  coss.

9
Fourier Transform

• Note
• This is also called the direct current or DC
  component.

10
Inverse Fourier Transform

• We can get back our original function f(x) from
  F(u) using the inverse transform

11
OK Why do this?

• Fourier space is a very good space for analyzing
  and understanding our signals.
• Rarely ever want to transform to Fourier space.
• There are some great theories developed in terms
  of sampling and convolution.

12
Fourier Analysis

• By looking at F(u), we get a feel for the
  frequencies of the signal.
• We also call this frequency space.
• Intuitively, you can envision, the sharper an
  edge, the higher the frequencies.
• From a numerical analysis standpoint, the sharper
  the edge the greater the tangent magnitude, and
  hence the interpolation errors.

13
Fourier Analysis

• Bandlimited
• We say a function is bandlimited, if F(u)0 for
  all frequencies ugtc and ult-c.
• Amplitude Spectrum
• The magnitude, F(u), is called the amplitude
  spectrum or simply the spectrum.
• Phase Spectrum or Phase

14
Fourier Properties

• Linearity
• Scaling

15
Convolution

• Definition

16
Convolution

• Pictorially

f(x)
h(x)
17
Convolution
h(t-x)
f(t)
18
Convolution

• Consider the function (box filter)

19
Convolution

• This function windows our function f(x).

f(t)
20
Convolution

• This function windows our function f(x).

f(t)
21
Convolution

• This function windows our function f(x).

f(t)
22
Convolution

• This function windows our function f(x).

f(t)
23
Convolution

• This function windows our function f(x).

f(t)
24
Convolution

• This function windows our function f(x).

f(t)
25
Convolution

• This function windows our function f(x).

f(t)
26
Convolution

• This function windows our function f(x).

f(t)
27
Convolution

• This function windows our function f(x).

f(t)
28
Convolution

• This function windows our function f(x).

f(t)
29
Convolution

• This function windows our function f(x).

f(t)
30
Convolution

• This function windows our function f(x).

f(t)
31
Convolution

• This function windows our function f(x).

f(t)
32
Convolution

• This function windows our function f(x).

f(t)
33
Convolution

• This function windows our function f(x).

f(t)
34
Convolution

• This function windows our function f(x).

f(t)
35
Convolution

• This function windows our function f(x).

f(t)
36
Convolution

• This function windows our function f(x).

f(t)
37
Convolution

• This function windows our function f(x).

f(t)
38
Convolution

• This function windows our function f(x).

f(t)
39
Convolution

• This function windows our function f(x).

f(t)
40
Convolution

• This function windows our function f(x).

f(t)
41
Convolution

• This particular convolution smooths out some of
  the high frequencies in f(x).

f(x)?g(x)
42
Impulse Function

• Consider the special function (called impulse)
• such that,

43
Impulse and Convolution

• Then, if we take the convolution of f(x) with
  d(x), we get
• f(x)?d(x) f(x)

44
Sampling Function

• A Sampling Function or Impulse Train is defined
  by
• where T is the sample spacing.

T
45
Sampling Function

• The Fourier Transform of the Sampling Function is
  itself a sampling function.
• The sample spacing is the inverse.

46
Sampling

• What we have in computer graphics is a point
  sampling of our scene, or
• I(x) f(x)ST(x)
• What we would like is more of an integration
  across the pixel (or larger area)
• I(x) f(x)? h(x)
• What should h(x) be?

47
Convolution Theorem

• The convolution theorem states that convolution
  in the spatial domain is equivalent to
  multiplication in the frequency domain, and vica
  versa.

48
Convolution Theorem

• This powerful theorem can illustrate the problems
  with our point sampling and provide guidance on
  avoiding aliasing.
• Consider f(x) ST(x)

f(t)
T
49
Convolution Theorem

• What does this look like in the Fourier domain?

F(u)
50
Convolution Theorem

• In Fourier domain we would convolve

F(u)
51
Aliasing

• What this says, is that any frequencies greater
  than a certain amount will appear intermixed with
  other frequencies.
• In particular, the higher frequencies for the
  copy at 1/T intermix with the low frequencies
  centered at the origin.

52
Aliasing and Sampling

• Note, that the sampling process introduces
  frequencies out to infinity.
• We have also lost the function f(x), and now have
  only the discrete samples.
• This brings us to our next powerful theory.

53
Sampling Theorem

• The Shannon Sampling Theorem
• A band-limited signal f(x), with a cutoff
  frequency of l, that is sampled with a sampling
  spacing of T may be perfectly reconstructed from
  the discrete values fnT by convolution with the
  sinc(x) function, provided
• l is called the Nyquist limit.

54
Sampling Theory

• Why is this?
• The Nyquist limit will ensure that the copies of
  F(u) do not overlap in the frequency domain.
• I can completely reconstruct or determine f(x)
  from F(u) using the Inverse Fourier Transform.

55
Sampling Theory

• In order to do this, I need to remove all of the
  shifted copies of F(u) first.
• This is done by simply multiplying F(u) by a box
  function of width 2l.

S(u)
F(u)
l
-l
56
Sampling Theory

• In order to do this, I need to remove all of the
  shifted copies of F(u) first.
• This is done by simply multiplying F(u) by a box
  function of width 2l.

S(u)
F(u)
l
-l
57
General Process
58
How? - Convolution
Spatial Domain
Frequency Domain
Mathematicallyf(x)h(x)

• Multiplication

• Convolution

Evaluated at discrete points (sum)
59
Reconstruction
Mathematically f(x)h(x) (Sfi)h(x)
Sfi
h(x)
60
General Process - Frequency Domain
61
Pre-Filtering
62
Ideal Reconstruction with Sinc function

• Spatial Domain
• convolution is exact

• Frequency Domain
• cut off freq. replica

63
Reconstructing Derivatives

• Spatial Domain
• convolution is exact

• Frequency Domain
• cut off freq. replica

64
Possible Errors

• Post-aliasing
• reconstruction filter passes frequencies beyond
  the Nyquist frequency (of duplicated frequency
  spectrum) gt frequency components of the original
  signal appear in the reconstructed signal at
  different frequencies
• Smoothing
• frequencies below the Nyquist frequency are
  attenuated
• Ringing (overshoot)
• occurs when trying to sample/reconstruct
  discontinuity
• Anisotropy
• caused by not spherically symmetric filters

65
How Good? Error

• Spatial Domain
• local error
• asymptotic error
• numerical error

• Frequency Domain
• global error
• visual appearance
• blurring
• aliasing
• smoothing

Approximation Theory/Analysis
Signal Processing
66
Sources of Aliasing

• Non-bandlimited signal

• Low sampling rate (below Nyquist)

• Non perfect reconstruction

67
Reconstruction Kernels
stop band
pass band
The spatial extent of reconstruction kernels, or
interpolation basis functions, depend on the
cut-off frequency as well.
filter
68
Reconstruction Kernels

• Nearest Neighbor (Box)

• Triangular func

• Sinc

• Gaussian

many others
Spatial d.
Frequency d.
69
Higher Dimensions

• An-isotropic Filters
• (radially symmetric)

• separable filters

70
Interpolation (an example)

• Very important regardless of algorithm
• expensive gt done very often for one image
• Requirements for good reconstruction
• performance
• stability of the numerical algorithm
• accuracy

Linear
Nearest neighbor
71
Put Things in Perspective

• In graphics, need to use continuous space
  functions. But can only work with discrete data.
  So, lets reconstruct from discrete data to
  continuous space (convolution) and resample
• Interpolation is doing the same thing. Computing
  one data point in the resulting function, say, at
  x1.

• So, which reconstruction kernel (basis function)
  does linear/bilinear/tri-linear interpolations
  use?

72
A Cosine Example

• Consider the function f(x)cos(2px).

f(x)
x
F(u)
u
73
Sampling Theory

• So, given fnT and an assumption that f(x) does
  not have frequencies greater than 1/2T, we can
  write the formula
• fnT f(x) ST(x) ? F(u)? ST(u)
• F(u) (F(u)? ST(u)) Box1/2T(u)
• therefore,
• f(x) fnT ? sinc(x)

74
A Cosine Example

• Now sample it at T½

f(x)
x
F(u)
u
1
75
A Cosine Example

• Problem
• The amplitude is now wrong or undefined.
• Note however, that there is one and only one
  cosine with a frequency less than or equal to 1
  that goes through the sample pts.

F(u)
u
1
76
A Cosine Example

• What if we sample at T¾?

f(x)
x
F(u)
u
1
77
Supersampling

• Supersampling increases the sampling rate, and
  then integrates or convolves with a box filter,
  which is finally followed by the output sampling
  function.

f(x)
x
78
Sampling and Anti-aliasing

• If you can not get rid of it, convert it to noise.

Basic checkerboard
Checkerboard with noise
79
Sampling and Anti-aliasing

• The images were calculated as follows
• A 2Kx2K image was constructed and smoothly
  rotated into 3D.
• For Uniform Sampling, it was downsampled toa
  512x512 image.
• Noise was added to the image, sharpened and
  thendownsampled for the other one.
• Both were converted to BW.

80
Sampling and Anti-aliasing

• The problem
• The signal is not band-limited.
• Uniform sampling can pick-up higher frequency
  patterns and represent them as low-frequency
  patterns.

F(u)
81
Sampling and Anti-aliasing

• Turning the high-frequencies into noise.
• Recall, multiplication by the sampling function
  is equivalent to convolution in the Fourier
  domain.
• Fourier transform of noise.

82
Non-uniform Sampling
83
Quality considerations

• So far we just mapped one point
• results in bad aliasing (resampling problems)
• we really need to integrate over polygon
• super-sampling is not a very good solution
  (slow!)
• most popular (easiest) - mipmaps

84
Quality considerations

• Pixel area maps to weird (warped) shape in
  texture space

v
ys
pixel
u
xs
85
Quality considerations

• We need to
• Calculate (or approximate) the integral of the
  texture function under this area
• Approximate
• Convolve with a wide filter around the center of
  this area
• Calculate the integral for a similar (but
  simpler) area.

86
Quality considerations

• the area is typically approxiated by a
  rectangular region (found to be good enough for
  most applications)
• filter is typically a box/averaging filter -
  other possibilities
• how can we pre-compute this?

87
Mip-maps

• An image-pyramid is built.

256 pixels 128 64 32 16 8 4 2 1
88
Mip-maps

• Find level of the mip-map where the area of each
  mip-map pixel is closest to the area of the
  mapped pixel.

2x2 pixels level selected
89
Mip-maps

• Pros
• Easy to calculate
• Calculate pixels area in texture space
• Determine mip-map level
• Sample or interpolate to get color
• Cons
• Area not very close restricted to square shapes
  (64x64 is far away from 128x128).
• Location of area is not very tight.

90
Summed Area Table (SAT)

• Use an axis aligned rectangle, rather than a
  square
• Precompute the sum of all texels to the left and
  below for each texel location
• For texel (u,v), replace it with sum
  (texels(i0u,j0v))

91
Summed Area Table (SAT)

• Determining the rectangle
• Find bounding box and calculate its aspect ratio

92
Summed Area Table (SAT)

• Determine the rectangle with the same aspect
  ratio as the bounding box and the same area as
  the pixel mapping.

93
Summed Area Table (SAT)

• Center this rectangle around the bounding box
  center.
• Formula
• Area aspect_ratioxx
• Solve for x the width of the rectangle
• Other derivations are also possible using the
  aspects of the diagonals,

94
Summed Area Table (SAT)

• Calculating the color
• We want the average of the texel colors within
  this rectangle

(u4,v4)
(u3,v3)

-

-
(u2,v2)
(u1,v1)

-

-
95
Summed Area Table (SAT)

• To get the average, we need to divide by the
  number of texels falling in the rectangle.
• Color SAT(u3,v3)-SAT(u4,v4)-SAT(u2,v2)SAT(u1,v1
  )
• Color Color / ( (u3-u1)(v3-v1) )
• This implies that the values for each texel may
  be very large
• For 8-bit colors, we could have a maximum SAT
  value of 255nxny
• 32-bit pixels would handle a 4kx4k texture with
  8-bit values.
• RGB images imply 12-bytes per pixel.

96
Summed Area Table (SAT)

• Pros
• Still relatively simple
• Calculate four corners of rectangle
• 4 look-ups, 5 additions, 1 mult and 1 divide.
• Better fit to area shape
• Better overlap
• Cons
• Large texel SAT values needed
• Still not a perfect fit to the mapped pixel.

97
Elliptical Weighted Average (EWA) Filter

• Treat each pixel as circular, rather than square.
• Mapping of a circle is elliptical in texel space.

ys
v
pixel
xs
u
98
EWA Filter

• Precompute?
• Can use a better filter than a box filter.
• Heckbert chooses a Gaussian filter.

99
EWA Filter

• Calculating the Ellipse
• Scan converting the Ellipse
• Determining the final color (normalizing the
  value or dividing by the weighted area).

100
EWA Filter

• Calculating the ellipse
• We have a circular function defined in (x,y).
• Filtering that in texture space h(u,v).
• (u,v) T(x,y)
• Filter h(T(x,y))

101
EWA Filter

• Ellipse
• ?(u,v) Au2 Buv Cv2 F
• (u,v) (0,0) at center of the ellipse
• A vx2 vy2
• B -2(uxvy uyvx)
• C ux2 uy2
• F uxvy uyvx

102
EWA Filter

• Scan converting the ellipse
• Determine the bounding box
• Scan convert the pixels within it, calculating
  ?(u,v).
• If ?(u,v) lt F, weight the underlying texture
  value by the filter kernel and add to the sum.
• Also, sum up the filter kernel values within the
  ellipse.

103
EWA Filter

• Determining the final color
• Divide the weighted sum of texture values by the
  sum of the filter weights.

104
EWA Filter

• What about large areas?
• If m pixels fall within the bounding box of the
  ellipse, then we have O(n2m) algorithm for an nxn
  image.
• m maybe rather large.
• We can apply this on a mip-map pyramid,rather
  than the full detailed image.
• Tighter-fit of the mapped pixel
• Cross between a box filter and gaussian filter.
• Constant complexity - O(n2)