Aliasing

1
Aliasing Antialiasing

• Aaron Bloomfield
• CS 445 Introduction to Graphics
• Fall 2006
• (Slide set originally by David Luebke)

2
Overview

• Introduction
• Signal Processing
• Sampling Theorem
• Prefiltering
• Supersampling
• Continuous Antialiasing
• Catmull's Algorithm
• The A-Buffer
• Stochastic Sampling

3
Antialiasing

• Aliasing signal processing term with very
  specific meaning
• Aliasing computer graphics term for any unwanted
  visual artifact
• Antialiasing computer graphics term for avoiding
  unwanted artifacts
• Well tackle these in order

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No anti-aliasing
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5 sample anti-aliasing
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16 sample anti-aliasing
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Overview

• Introduction
• Signal Processing
• Sampling Theorem
• Prefiltering
• Supersampling
• Continuous Antialiasing
• Catmull's Algorithm
• The A-Buffer
• Stochastic Sampling

8
Signal Processing

• Raster display regular sampling of a continuous
  function (Really?)
• Think about sampling a 1-D function

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Signal Processing

• Sampling a 1-D function

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Signal Processing

• Sampling a 1-D function

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Signal Processing

• Sampling a 1-D function
• What do you notice?

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Signal Processing

• Sampling a 1-D function what do you notice?
• Jagged, not smooth

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Signal Processing

• Sampling a 1-D function what do you notice?
• Jagged, not smooth
• Loses information!

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Signal Processing

• Sampling a 1-D function what do you notice?
• Jagged, not smooth
• Loses information!
• What can we do about these?
• Use higher-order reconstruction
• Use more samples
• How many more samples?

15
Overview

• Introduction
• Signal Processing
• Sampling Theorem
• Prefiltering
• Supersampling
• Continuous Antialiasing
• Catmull's Algorithm
• The A-Buffer
• Stochastic Sampling

16
The Sampling Theorem

• Obviously, the more samples we take the better
  those samples approximate the original function
• The Nyquist sampling theorem
• A continuous bandlimited function can be
  completely represented by a set of equally spaced
  samples, if the samples occur at more than twice
  the frequency of the highest frequency component
  of the function

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The Sampling Theorem

• In other words, to adequately capture a function
  with maximum frequency F, we need to sample it at
  frequency N 2F.
• N is called the Nyquist limit.
• The Nyquist sampling theorem applied to CDs
• Most humans can hear to 20 kHz
• Some (like me!) to 22 kHz
• CDs are sampled at 44.1 kHz
• Although not twice the highest frequency
  component, it is twice the highest frequency
  component that can be (generally) heard

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The Sampling Theorem

• An example sinusoids

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The Sampling Theorem

• An example sinusoids

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Fourier Theory

• All our examples have been sinusoids
• Does this help with real world signals? Why?
• Fourier theory lets us decompose any signal into
  the sum of (a possibly infinite number of) sine
  waves

21
Overview

• Introduction
• Signal Processing
• Sampling Theorem
• Prefiltering
• Supersampling
• Continuous Antialiasing
• Catmull's Algorithm
• The A-Buffer
• Stochastic Sampling

22
Prefiltering

• Eliminate high frequencies before sampling (Foley
  van Dam p. 630)
• Convert I(x) to F(u)
• Apply a low-pass filter
• A low-pass filter allows low frequencies through,
  but attenuates (or reduces) high frequencies
• Then sample. Result no aliasing!

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Prefiltering
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Prefiltering
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Prefiltering

• So whats the problem?
• Problem most rendering algorithms generate
  sampled function directly
• e.g., Z-buffer, ray tracing

26
Overview

• Introduction
• Signal Processing
• Sampling Theorem
• Prefiltering
• Supersampling
• Continuous Antialiasing
• Catmull's Algorithm
• The A-Buffer
• Stochastic Sampling

27
Supersampling

• The simplest way to reduce aliasing artifacts is
  supersampling
• Increase the resolution of the samples
• Average the results down
• For now, well assume that the samples are evenly
  spaced
• In the Stochastic Sampling section, well see
  other ways of doing it
• Sometimes called postfiltering
• Create virtual image at higher resolution than
  the final image
• Apply a low-pass filter
• Resample filtered image

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Supersampling Limitations

• Q What practical consideration hampers
  super-sampling?
• A Storage goes up quadratically
• Q What theoretical problem does supersampling
  suffer?
• A Doesnt eliminate aliasing! Supersampling
  simply shifts the Nyquist limit higher

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Supersampling Worst Case

• Q Give a simple scene containing infinite
  frequencies
• A A checkered ground plane receding into
  infinity
• See next slide

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(No Transcript)
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Supersampling

• Despite these limitations, people still use
  super-sampling (why?)
• So how can we best perform it?

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Supersampling

• The process
• Create virtual image at higher resolution than
  the final image
• Apply a low-pass filter
• Resample filtered image

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Supersampling Step 1

• Create virtual image at higher resolution than
  the final image
• This is easy

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Supersampling Step 2

• Apply a low-pass filter
• Convert to frequency domain
• Multiply by a box function
• Expensive! Can we simplify this?
• Recall multiplication in frequency equals
  convolution in space, so
• Just convolve initial sampled image with the FT
  of a box function
• Isnt this a lot of work?

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Supersampling Digital Convolution

• In practice, we combine steps 2 3
• Create virtual image at higher resolution than
  the final image
• Apply a low-pass filter
• Resample filtered image
• Idea only create filtered image at new sample
  points
• I.e., only convolve filter with image at new
  points

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Supersampling Digital Convolution

• Q What does convolving a filter with an image
  entail at each sample point?
• A Multiplying and summing values
• Example

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Supersampling

• Typical supersampling algorithm
• Compute multiple samples per pixel
• Combine sample values for pixels value using
  simple average
• Q What filter does this equate to?
• A Box filter -- one of the worst!
• Q Whats wrong with box filters?
• Passes infinitely high frequencies
• Attenuates desired frequencies

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Supersampling

• Common filters
• Truncated sinc
• sinc(x) sin(x)/x
• Box
• Triangle
• Guassian

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Supersampling In Practice

• Sinc function ideal but impractical
• One approximation sinc2
• Another Gaussian falloff
• Q How wide (what res) should filter be?
• A As wide as possible (duh)
• In practice 3x3, 5x5, at most 7x7
• In other words, the filter is larger than the
  final pixel
• So we are using the values of neighboring pixels
• This creates a better visual effect

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Supersampling Summary

• Supersampling improves aliasing artifacts by
  shifting the Nyquist limit
• It works by calculating a high-res image and
  filtering down to final res
• Filtering down means simultaneous convolution
  and resampling
• This equates to a weighted average
• Wider filter ? better results ? more work

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Summary So Far

• Prefiltering
• Before sampling the image, use a low-pass filter
  to eliminate frequencies above the Nyquist limit
• This blurs the image
• But ensures that no high frequencies will be
  misrepresented as low frequencies

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Summary So Far

• Supersampling
• Sample image at higher resolution than final
  image, then average down
• Average down means multiply by low-pass
  function in frequency domain
• Which means convolving by that functions FT in
  space domain
• Which equates to a weighted average of nearby
  samples at each pixel

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Summary So Far

• Supersampling cons
• Doesnt eliminate aliasing, just shifts the
  Nyquist limit higher
• Cant fix some scenes (e.g., checkerboard)
• Badly inflates storage requirements
• Supersampling pros
• Relatively easy
• Often works all right in practice
• Can be added to a standard renderer

44
Overview

• Introduction
• Signal Processing
• Sampling Theorem
• Prefiltering
• Supersampling
• Continuous Antialiasing
• Catmull's Algorithm
• The A-Buffer
• Stochastic Sampling

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Antialiasing in the Continuous Domain

• Problem with prefiltering
• Sampling and image generation inextricably linked
  in most renderers
• Z-buffer algorithm
• Ray tracing
• Why?
• Still, some approaches try to approximate effect
  of convolution in the continuous domain

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Antialiasing in the Continuous Domain
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Antialiasing in the Continuous Domain

• The good news
• Exact polygon coverage of the filter kernel can
  be evaluated
• What does this entail?
• Clipping
• Hidden surface determination

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Antialiasing in the Continuous Domain

• The bad news
• Evaluating coverage is very expensive
• The intensity variation is too complex to
  integrate over the area of the filter
• Q Why does intensity make it harder?
• A Because polygons might not be flat- shaded
• Q How bad a problem is this?
• A Intensity varies slowly within a pixel, so
  shape changes are more important

49
Overview

• Introduction
• Signal Processing
• Sampling Theorem
• Prefiltering
• Supersampling
• Continuous Antialiasing
• Catmull's Algorithm
• The A-Buffer
• Stochastic Sampling

50
Catmulls Algorithm
A2
A1

• Find fragment areas
• Multiply by fragment colors
• Sum for final pixel color

AB
A3
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Catmulls Algorithm

• First real attempt to filter in continuous domain
• Very expensive
• Clipping polygons to fragments
• Sorting polygon fragments by depth (Whats wrong
  with this as a hidden surface algorithm?)
• Equates to box filter (Is that good?)

52
Overview

• Introduction
• Signal Processing
• Sampling Theorem
• Prefiltering
• Supersampling
• Continuous Antialiasing
• Catmull's Algorithm
• The A-Buffer
• Stochastic Sampling

53
The A-Buffer

• Idea approximate continuous filtering by
  subpixel sampling
• Summing areas now becomes simple

54
The A-Buffer

• Advantages
• Incorporating into scanline renderer reduces
  storage costs dramatically
• Processing per pixel depends only on number of
  visible fragments
• Can be implemented efficiently using bitwise
  logical ops on subpixel masks

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The A-Buffer

• Disadvantages
• Still basically a supersampling algorithm
• Not a hardware-friendly algorithm
• Lists of potentially visible polygons can grow
  without limit
• Work per-pixel non-deterministic

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Recap Antialiasing Strategies

• Supersampling sample at higher resolution, then
  filter down
• Pros
• Conceptually simple
• Easy to retrofit existing renderers
• Works well most of the time
• Cons
• High storage costs
• Doesnt eliminate aliasing, just shifts Nyquist
  limit upwards

• A-Buffer approximate pre-filtering of continuous
  signal by sampling
• Pros
• Integrating with scan-line renderer keeps storage
  costs low
• Can be efficiently implemented with clever
  bitwise operations
• Cons
• Still basically a super-sampling approach
• Doesnt integrate with ray-tracing

57
Overview

• Introduction
• Signal Processing
• Sampling Theorem
• Prefiltering
• Supersampling
• Continuous Antialiasing
• Catmull's Algorithm
• The A-Buffer
• Stochastic Sampling

58
Stochastic Sampling

• Stochastic involving or containing a random
  variable
• Sampling theory tells us that with a regular
  sampling grid, frequencies higher than the
  Nyquist limit will alias
• Q What about irregular sampling?
• A High frequencies appear as noise, not aliases
• This turns out to bother our visual system less!

59
Stochastic Sampling

• An intuitive argument
• In stochastic sampling, every region of the image
  has a finite probability of being sampled
• Thus small features that fall between uniform
  sample points tend to be detected by non-uniform
  samples

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Stochastic Sampling

• Integrating with different renderers
• Ray tracing
• It is just as easy to fire a ray one direction as
  another
• Z-buffer hard, but possible
• Notable example REYES system (?)
• Using image jittering is easier (more later)
• A-buffer nope
• Totally built around square pixel filter and
  primitive-to-sample coherence

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Stochastic Sampling

• Idea randomizing distribution of samples
  scatters aliases into noise
• Problem what type of random distribution to
  adopt?
• Reason type of randomness used affects spectral
  characteristics of noise into which high
  frequencies are converted

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Stochastic Sampling

• Problem given a pixel, how to distribute points
  (samples) within it?
• Grid Random Poisson Disc
  Jitter

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Stochastic Sampling

• Poisson distribution
• Completely random
• Add points at random until area is full.
• Uniform distribution some neighboringsamples
  close together, some distant

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Stochastic Sampling

• Poisson disc distribution
• Poisson distribution, with minimum-distance
  constraint between samples
• Add points at random, removing again if they are
  too close to any previous points
• Very even-looking distribution

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Stochastic Sampling

• Jittered distribution
• Start with regular grid of samples
• Perturb each sample slightly in a random
  direction
• More clumpy or granular in appearance

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Stochastic Sampling

• Spectral characteristics of these distributions
• Poisson completely uniform (white noise). High
  and low frequencies equally present
• Poisson disc Pulse at origin (DC component of
  image), surrounded by empty ring (no low
  frequencies), surrounded by white noise
• Jitter Approximates Poisson disc spectrum, but
  with a smaller empty disc.

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Stochastic Sampling

• Watt Watt, p. 134
• See Foley van Dam, p 644-645
• Should have images next time