Sampling (Section 4.3)

CS474/674 – Prof. Bebis

Sampling

• How many samples should we obtain to minimize information loss during sampling?

• Hint: take enough samples to allow reconstructing the “continuous” image from its samples.

Example

Sampled signal looks like a sinusoidal of a lower frequency !

Definition: “band-limited” functions

• A function whose spectrum is of finite duration

• Are all functions band-limited?

max frequency

NO!!

Properties of band-limited functions

• Band-limited functions have infinite duration in the time domain.

• Functions with finite duration in the time domain have infinite duration in the frequency domain.

Sampling a 1D function

• Multiply f(x) with s(x)

sampled f(x)x

Hint: use convolution theorem!

Question: what is the DFT of f(x) s(x)?

Sampling a 1D function (cont’d)

• Suppose f(x) F(u)

• What is the DFT of s(x)?

Sampling a 1D function (cont’d)

* =

So:

x1

x

1

x1

x

1

Sampling a 2D function (cont’d)

s(x,y)

Δy Δxx y

• 2D train of impulses

Sampling a 2D function (cont’d)

• DFT of 2D discrete function (i.e., image)

f(x,y)s(x,y) F(u,v)*S(u,v)

Reconstructing f(x) from its samples

• Need to isolate a single period:

– Multiply by a window G(u)

x

Reconstructing f(x) from its samples (cont’d)

• Then, take the inverse FT:

What is the effect of Δx?

• Large Δx (i.e., few samples) results to overlapping periods!

Effect of Δx (cont’d)

• But, if the periods overlap, we cannot anymore isolate a single period aliasing!

x

What is the effect of Δx? (cont’d)

• Smaller Δx (i.e., more samples) alleviates aliasing!

What is the effect of Δx? (cont’d)

• 2D case

u u

v v

umaxvmax

Example

• Suppose that we have an imaging system where the number of samples it can take is fixed at 96 x 96 pixels.

• Suppose we use this system to digitize checkerboard patterns.

• Such a system can resolve patterns that are up to 96 x 96 squares (i.e., 1 x 1 pixel squares).

• What happens when squares are less than 1 x 1 pixels?

Examplesquare size: 16 x 16 6 x 6

square size: 160.9174 0.4798

(same as12 x 12 squares)

How to choose Δx?

• The center of the overlapped region is at

How to choose Δx? (cont’d)

•Choose Δx as follows:

where W is the max frequency of f(x)

Practical Issues

• Band-limited functions have infinite duration in the time domain.

• But, we can only sample a function over a finite interval!

•

Practical Issues (cont’d)

• We would need to obtain a finite set of samples

by multiplying with a “box” function:

[s(x)f(x)]h(x)

x =

Practical Issues (cont’d)

• This is equivalent to convolution in the frequency domain! [s(x)f(x)]h(x) [F(u)*S(u)] * H(u)

Practical Issues (cont’d)

•

instead of this!

*

How does this affect things in practice?

• Even if the Nyquist criterion is satisfied, recovering a function that has been sampled in a finite region is in general impossible!

• Special case: periodic functions– If f(x) is periodic, then a single period can be isolated

assuming that the Nyquist theorem is satisfied!

– e.g., sin/cos functions

Anti-aliasing

• In practice, aliasing in almost inevitable!

• The effect of aliasing can be reduced by smoothing the input signal to attenuate its higher frequencies.

• This has to be done before the function is sampled.– Many commercial cameras have true anti-aliasing filtering

built in the lens of the sensor itself.

– Most commercial software have a feature called “anti-aliasing” which is related to blurring the image to reduced aliasing artifacts (i.e., not true anti-aliasing)

Example

50% less samples3 x 3 blurring and50% less samples