1. Gonzalez & Woods, and Efros)
Sampling and Aliasing
Gilad Lerman
Math 5467
(stealing slides from
Gonzalez & Woods, and Efros)

2. The Sampling Theorem
Theorem: If f is in L1() & supported on [-B0, B0], then
Recall Proof:
We view as (2B0)-periodic function with coefficients:
At last, find f using IFT and using FS of
Shannon sampling theorem, and is also known as Nyquist–Shannon–Kotelnikov, Whittaker–Shannon–Kotelnikov, Whittaker–Nyquist–Kotelnikov–Shannon, WKS, etc.

3. More on the Sampling Theorem
Frequency band: Time:
Note: Theorem holds for B>B0.
Indeed, then
If B<B0, the above equation is not true for all 

4. Sampling Theorem (meaning)
Interpretation: If a function f(t) contains no frequencies higher than W cps, it is completely determined by giving its ordinates at a series of points spaced 1/(2W) seconds apart
Remark: For L1 function a freq. = W is fine
but for more general functions we need > W…
"cps" for "cycles per second"

5. Simple Example (not L1) Assume a cosine
(it is not L1() but will be instrumental)
Freq: a (“& -a”), Freq Band: =[-a,a], Time: 1/(2a)
Here one needs B>B0 (B=B0 doesn’t work)
Example: for all 3 functions
freq: 0.5, time: 1
The sampled function has different aliases…

6. Aliasing
If the sampling condition is not satisfied, frequencies will overlap (high freq → low freq)
The reconstructed signal is said to be an alias of the original signal

7. Example: Increased Frequency
Input signal:
Related Image:
x = 0:.05:5; imagesc(sin((2.^x).*x))
Matlab output:
Picket fence receding
Into the distance will
produce aliasing…

8. One more example at the Fourier domain

12. Aliasing in Images (Fourier domain)

13. Good and Bad Sampling Good sampling: Bad sampling: Sample often or,
Sample wisely
Bad sampling:
see aliasing in action!

15. Texture makes its worse (high frequencies)

16. Even worse for synthetic images
Slide by Steve Seitz

18. Really bad in video
Slide by Paul Heckbert

19. Wheels of Wagons in Westerns
Can watch
Ballad of the wagon war (1967), start watching around 1:40sec.

20. Moiré pattern
Definition: Interference pattern created, e.g., when two grids are overlaid at an angle, or when they have slightly different mesh sizes.
In images produced e.g., when scanning a halftone picture or due to undersampling a fine regular pattern.
See e.g.
Though if I put it on a slide I could not view aliasing

23. Moiré pattern due to undersampling
See e.g.
Though if I put it on a slide I could not view aliasing
Original image
downsampled image

24. Antialiasing What can be done? Raise sampling rate by oversampling
Sampling rate ≥ 2 * max frequency in the image
Raise sampling rate by oversampling
Sample at k times the resolution
continuous signal: easy
discrete signal: need to interpolate
2. Lower the max frequency by prefiltering
Smooth the signal enough
Works on discrete signals
3. Improve sampling quality with better sampling
Nyquist is best case!
Stratified sampling
Importance sampling
Relies on domain knowledge

25. Gaussian pre-filtering
Solution: filter the image, then subsample
Filter size should double for each ½ size reduction.

26. Subsampling with Gaussian pre-filtering

27. Compare with...
1/2
1/4 (2x zoom)
1/8 (4x zoom)

28. Correcting some Moiré patterns