Sampling and Aliasing

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

-

-

= + × = -

+ = - × = -