1. Sampling, Aliasing

2. Examples of Aliasing

3. What is a Pixel? A pixel is not:
a box
a disk
a tiny little light
A pixel “looks different” on different display devices
A pixel is a sample
it has no dimension
it occupies no area
it cannot be seen
it has a coordinate
it has a value

4. Philosophical perspective
The physical world is continuous, inside the computer things need to be discrete
Lots of computer graphics is about translating continuous problems into discrete solutions
e.g. ODEs for physically-based animation, global illumination, meshes to represent smooth surfaces, antialiasing
Careful mathematical understanding helps do the right thing

5. More on Samples
The process of mapping a continuous function 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
Focus on the effects of sampling and how to overcome them
discrete value
discrete position

6. Sampling & reconstruction
The visual array of light is a continuous function
1/ we sample it
with a digital camera, or with a ray tracer
This gives us a finite set of numbers, not really something we can see
We are now inside the discrete computer world
2/ we need to get this back to the physical world: we reconstruct a continuous function
for example, the point spread of a pixel on a CRT or LCD
Both steps can create problems
But we’ll focus on the first one (you are not display manufacturers)

7. Examples of Aliasing

8. Examples of Aliasing

9. Examples of Aliasing
Texture Errors
point sampling

10. Sampling Density If we’re lucky, sampling density is enough Input
Reconstructed

11. Sampling Density
If we insufficiently sample the signal, it may be mistaken for something simpler during reconstruction (that's aliasing!)

12. Aliasing insufficient sampling (sampling rate is too small)
Aliasing: a high-frequency signal masquerading as a low frequency
Actual (high-frequency) signal
Sampling
Interval
Sampled (aliased) signal

13. Discussion Two types of aliasing Edges Repetitive textures More tricky
Harder to solve

14. Solution?
How do we avoid the high-frequency patterns distort our image?
We blur!
For ray tracing: compute at higher resolution, blur, resample at lower resolution
For textures, we can also blur the texture image before doing the lookup

15. Aliasing and Line Drawing
We draw lines by sampling at intervals of one pixel and drawing the closest pixels
Results in stair-stepping
Sampling
Interval
Sampling
Interval

16. Anti-aliasing Lines Idea: Make line thicker
Fade line out (removes high frequencies)
Now sample the line

17. Anti-aliasing Lines Solution – Unweighted Area Sampling:
Treat line as a single-pixel wide rectangle
Colour pixels according to the percentage of each pixel covered by the rectangle.

18. Solution : Unweighted Area Sampling
Pixel area is unit square
Constant weighting function
Pixel colour is determined by computing the amount of the pixel covered by the line, then shading accordingly
Easy to compute, gives reasonable results
One Pixel
Line

19. Solution : Weighted Area Sampling
Treat pixel area as a circle with a radius of one pixel
Use a radially symmetric weighting function (e.g., cone):
Areas closer to the pixel centre are weighted more heavily
Better results than unweighted, slightly higher cost
One Pixel
Line

20. Re-hashing
Your intuitive solution is to compute multiple colour values per pixel and average them
A better interpretation of the same idea is that
You first create a higher resolution image
You blur it (average)
You resample it at a lower resolution

21. 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)

22. Filters (convolution kernel)
Weighting function
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

23. 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

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

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

26. The Ideal Filter Unfortunately it has infinite spatial extent
Every sample contributes to every interpolated point
Expensive/impossible to compute
spatial
frequency

27. 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

28. Gaussian Filter
This is what a CRT does for free!
spatial
frequency

29. Box Filter / Nearest Neighbour
Pretending pixels are little squares.
spatial
frequency

30. Tent Filter / Bi-Linear Interpolation
Simple to implement
Reasonably smooth
spatial
frequency

31. Bi-Cubic Interpolation
Begins to approximate the ideal spatial filter, the sinc function
spatial
frequency

32. 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!

33. 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

34. Supersampling in graphics
Pre-filtering is hard
Requires analytical visibility
Then difficult to integrate analytically with filter
Possible for lines, or if visibility is ignored
usually, fall back to supersampling

35. Solution: Super-sampling
Divide pixel up into “sub-pixels”: 22, 33, 44, etc.
Sub-pixel is coloured if inside line
Pixel colour is the average of its sub-pixel colours
Easy to implement (in software and hardware)
No anti-aliasing
Anti-aliasing (22 super-sampling)

36. Many Types of Supersampling
Grid
Random
Jittered
Poisson Disc

37. Polygon Anti-aliasing
To anti-alias a line, we treat it as a rectangle
Anti-aliasing a polygon is similar.
Some concerns:
Micro-polygons: smaller than a
pixel
Super-sampling: There may still
be polygons that “slip between
the cracks”

38. Uniform supersampling
Compute image at resolution k*width, k*height
Downsample using low-pass filter (e.g. Gaussian, sinc, bicubic)

39. Uniform supersampling
Advantage:
The first (super)sampling captures more high frequencies that are not aliased
Issues
Frequencies above the (super)sampling limit are still aliased
Works well for edges
Not as well for repetitive textures

40. 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

41. Jittered supersampling
Regular, Jittered Supersampling

42. 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!

43. Reconsider Uniform supersampling
Problem: supersampling only pushes the problem further: The signal is still not bandlimited
Aliasing still happens
We need a better solution for high frequency sections of the image

44. Adaptive sampling
Adjust your sampling rate and/or bucket size on-the-fly, depending on results of previous samples
attempt to concentrate more samples in high frequency areas (e.g., edges of objects) and fewer samples in low frequency areas (constant regions)
get an approximate view of the function by first sampling with a low sampling rate and/or large bucket size over the entire domain
then use that approximation to estimate where more samples should be taken by focusing on regions of high variance
generally a recursive process
Much better than naïve supersampling

45. Adaptive supersampling
Use more sub-pixel samples around edges

46. Adaptive supersampling
Use more sub-pixel samples around edges
Compute colour at small number of sample
If variance with neighbour pixels is high
Compute larger number of samples

47. Adaptive supersampling
Use more sub-pixel samples around edges
Compute colour at small number of sample
If variance with neighbour pixels is high
Compute larger number of samples