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
We’ll tackle these in order

4. No anti-aliasing

5. 5 sample anti-aliasing

6. 16 sample anti-aliasing

7. 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:

9. Signal Processing
Sampling a 1-D function:

10. Signal Processing
Sampling a 1-D function:

11. Signal Processing
Sampling a 1-D function:
What do you notice?

12. Signal Processing Sampling a 1-D function: what do you notice?
Jagged, not smooth

13. Signal Processing Sampling a 1-D function: what do you notice?
Jagged, not smooth
Loses information!

14. 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

17. 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

18. The Sampling Theorem
An example: sinusoids

19. The Sampling Theorem
An example: sinusoids

20. 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!

23. Prefiltering

24. Prefiltering

25. Prefiltering So what’s 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, we’ll assume that the samples are evenly spaced
In the “Stochastic Sampling” section, we’ll 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

28. Supersampling: Limitations
Q: What practical consideration hampers super-sampling?
A: Storage goes up quadratically
Q: What theoretical problem does supersampling suffer?
A: Doesn’t eliminate aliasing! Supersampling simply shifts the Nyquist limit higher

29. Supersampling: Worst Case
Q: Give a simple scene containing infinite frequencies
A: A checkered ground plane receding into infinity
See next slide…

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

32. Supersampling The process:
Create virtual image at higher resolution than the final image
Apply a low-pass filter
Resample filtered image

33. Supersampling: Step 1
Create virtual image at higher resolution than the final image
This is easy

34. 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
Isn’t this a lot of work?

35. 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

36. Supersampling: Digital Convolution
Q: What does convolving a filter with an image entail at each sample point?
A: Multiplying and summing values
Example: 1 2 4  = …

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

38. Supersampling Common filters: Truncated sinc Box Triangle Guassian
sinc(x) = sin(x)/x
Box
Triangle
Guassian

39. 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

40. 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

41. 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

42. 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 function’s FT in space domain
Which equates to a weighted average of nearby samples at each pixel

43. Summary So Far Supersampling cons Supersampling pros
Doesn’t eliminate aliasing, just shifts the Nyquist limit higher
Can’t 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

45. 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

46. Antialiasing in the Continuous Domain
Pixel
Grid
Polygons
Filter kernel

47. 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

48. 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. Catmull’s Algorithm A2 A1 AB A3 Find fragment areas
Multiply by fragment colors
Sum for final pixel color A1 AB A3

51. Catmull’s Algorithm First real attempt to filter in continuous domain
Very expensive
Clipping polygons to fragments
Sorting polygon fragments by depth (What’s 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

55. 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

56. 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
Doesn’t 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
Doesn’t 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

60. 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

61. 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

62. Stochastic Sampling
Problem: given a pixel, how to distribute points (samples) within it?
Grid Random Poisson Disc Jitter

63. Stochastic Sampling Poisson distribution: Completely random
Add points at random until area is full.
Uniform distribution: some neighboring samples close together, some distant

64. 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

65. Stochastic Sampling Jittered distribution
Start with regular grid of samples
Perturb each sample slightly in a random direction
More “clumpy” or granular in appearance

66. 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.

67. Stochastic Sampling Watt & Watt, p. 134 See Foley & van Dam, p 644-645
Should have images next time