Lecture 17/18 Sampling Theory/2D Curves

Question 1

What is sampling?

Answer 1

Sampling: capturing the value of a function at specific points

Question 2

What are examples of regular sampling?

Answer 2

1D: digitised music
2D: digitized pictures
3D: voxel volumes
4D: video (3 space + 1 time)

Question 3

What are examples of Irregular sampling

Answer 3

1D: Gieger counters, buses
2D: cities on a map
3D: bees in a swarm
ND: lots of data!

Question 4

How can the original continuous function be RECONSTRUCTED from samples?

Answer 4

Want to reconstruct the original continuous signal from samples.
This is done using convolution, here a linear kernel is used.

Question 5

What are the issues that might occur when reconstructing a continuous function from samples?

Answer 5

Question 6

Aliasing in terms of frequency is caused as follows:

Answer 6

Aliasing is caused by using low sampling frequency to sample a high-frequency function.

Aliasing can effect all signals

Question 7

Visualise spatial and temporal aliasing

Answer 7

Question 8

What is the The Nyquist Limit?

Answer 8

Question 9

What is the Nyquist–Shannon sampling theorem?

Answer 9

Question 10

What function kernel is needed for the sampling theorem to be obeyed?

Answer 10

If the sampling theorem is obeyed,it is possible to reconstruct the continuous signal exactly from its discrete samples.

A sinc function kernel is needed sinc(x) = sin(px)/ (px)

Question 11

How can aliasing be avoided?

Answer 11

Two things:

Low pass filter
+ This removes troublesome high frequency components
- But blurs the signal

Add noise !!
+ Used in ray-tracing and other areas of graphics
- Possibly counter intuitive
~ Basic idea is to “scramble” the picture just a little, so it looks less regular

Question 12

How can aliasing be removed by sample rate alteration?

Answer 12

Naïve downsampling causes overlapping (aliased) spectra

Higher sample rates move overlapping spectra further apart in frequency domain, thus reducing aliasing

…eventually overlap doesn’t matter

Question 13

How to remove aliasing if the sample size cannot be altered?

Answer 13

remove high frequencies in spectrum
i.e. low-pass filter signal
e.g. Gaussian blur

Question 14

How does a low pass filter remove aliasing?

Answer 14

overlapping spectra without filtering:
- low-pass filter can narrow the spectrum enough to eliminate overlap
- produces a well-sampled representation of the filtered signal
Avoids aliasing artefacts, but loses high frequencies