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Sampling Issues
Aliasing
- discretization of signals
- high frequency information appears as low frequency information
- jaggies
Sampling Theorem
A continuous-time signal can be completely recovered
from its samples iff the sampling rate is greater than
twice the maximum frequency present in the signal.
This is also known as Nyquist's or Shannon's Theorem. The minimal sampling rate to recover a signal is called the Nyquist rate.
Example 1
Example 2
The Sampling and Reconstruction Process
Strategies for reducing aliasing
Prefiltering
Unweighted area sampling
- use average intensity of square pixel area
- interpolate between object and background at edges
- better, but still has problems
- `box filter'
sampling frequency: f
signal frequency: 5f/6
output frequency: f/6
Improvements:
- filter needs wider support
- center of pixel should receive more weight
Filter kernels in the spatial and frequency domains:
Comments on Prefiltering
- commonly used for textures
- assumes continuous representation of image exists
Postfiltering
- multiple samples per pixel
- high frequency noise preferable
