sampling and aliasing
DESCRIPTION
Sampling and Aliasing. Gilad Lerman Math 5467 (stealing slides from Gonzalez & Woods, and Efros). The Sampling Theorem. Theorem: If f is in L 1 ( ) & supported on [- B 0 , B 0 ], then Recall Proof: We view as (2 B 0 )-periodic function with coefficients: - PowerPoint PPT PresentationTRANSCRIPT
Sampling and Aliasing
Gilad LermanMath 5467
(stealing slides from Gonzalez & Woods, and Efros)
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
00 0
( ) sinc 22 2k
k kf x f B x
B BÎ
æ öæ ö æ ö÷ç÷ ÷ç ç ÷÷ ÷= × -çç ç ÷÷ ÷çç ç ÷÷ ÷ç ç ÷çè ø è øè øå
¢
0 0[ , ]ˆ ˆ( ) ( ) ( )B Bf fx x c x-= ×
f̂
0 0
1ˆ( )2 2k
kc f f
B B
æ ö÷ç ÷= × ç ÷ç ÷çè ø
f̂
f̂
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
02BW=0
1
2T
B=
[ , ]ˆ ˆ( ) ( ) ( )B Bf fx x c x-= ×
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…
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)
2 2
( ) cos(2 ) .2
iax iaxe ef x ax
p p
p-+
= =
Example: for all 3 functions freq: 0.5, time: 1
The sampled function has different aliases…
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
Example: Increased Frequency
Picket fence recedingInto the distance willproduce aliasing…
Input signal: Related Image:
x = 0:.05:5; imagesc(sin((2.^x).*x))
Matlab output:
One more example at the Fourier domain
Aliasing in Images (Fourier domain)
Good and Bad Sampling
Good sampling:•Sample often or,•Sample wisely
Bad sampling:•see aliasing in action!
Texture makes its worse(high frequencies)
Even worse for synthetic images
Slide by Steve Seitz
Really bad in video
Slide by Paul Heckbert
Wheels of Wagons in Westerns
• 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.
Moiré pattern
Moiré pattern due to undersampling
Original image downsampled image
Antialiasing• What can be done?
Sampling rate ≥ 2 * max frequency in the image
1. 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
Gaussian pre-filtering
G 1/4
G 1/8
Gaussian 1/2
• Solution: filter the image, then subsample– Filter size should double for each ½ size reduction.
Subsampling with Gaussian pre-filtering
G 1/4 G 1/8Gaussian 1/2
Compare with...
1/4 (2x zoom) 1/8 (4x zoom)1/2
Correcting some Moiré patterns
Rethinking of the Cooley-Tukey FFT
• Step 1(top to bottom): Create two subsampled signals (even and odd coordinates)
• Note that the two subsampled signals are associated with half bands in the frequency domain (Shannon)
• Step 2 (bottom-up): Combine the two signals by the formulas:
• Interpretation: combining the two half bands in the right way (in frequency domain) to exactly recover the signal
2even odd 2
2
2even odd 2
2
ˆ ˆ ˆ( ) ( ) ( ) , 0,..., 1,
ˆ ˆ ˆ( ) ( ) ( ) , 0,..., 1.
L L
L L
πin
LL
πin
LL
x n x n x n e n L
x n L x n x n e n L
-
-
= + × = -
+ = - × = -