Introduction to Aliasing and Anti-Aliasing Techniques
Spring 1999
Why does Aliasing Occur?
Where does Aliasing Occur?
An Illustrated Example: A
Spinning Cylinders
Spinning cylinders starting position.
Spinning cylinders after 1/24 second.
What is going on?
Rotation Rate | 0/24 second | 1/24 second | 2/24 second |
---|---|---|---|
3 rps | 0 deg. | 45 deg. | 90 deg. |
5 rps | 0 deg. | 75 deg. | 150 deg. |
11 rps | 0 deg. | 165 deg. | 330 deg. |
12 rps | 0 deg. | 180 deg. | 360 deg. |
Sampling:
How frequently should we sample?
Any signal can be represented as a series of sine waves of different frequencies.
Fourier transformation pair:
F(u) is the sum of sine waves because
u = frequency of sine wave -- frequency domain
x -- spatial domain
if I(x) -> F(u) and g(x) -> G(u)
I(x) * g(x) -> F(u)G(u) (convolve in spatial, multiply in frequency
domain)
I(x)g(x) -> F(u) * G(u) (multiply in spatial, convolve in frequency domain)
So, to create frequency representation of the function, multiply by reconstruction filter ==> original frequency function of the signal.
No information lost iff
or
where x is sampling rate and is the Nyquist Limit.
Super-sampling: sample x times / pixel
Example: Solid Texturing
If(x % .5) < .25 color = black else color = white
Reconstruction/Filtering:
Antialiased texture mapping