Sampling (Section 4.3)

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Sampling (Section 4.3). CS474/674 – Prof. Bebis. Sampling. How many samples should we get so that no information is lost during the sampling process? Hint: take enough samples so that the “continuous” image can be reconstructed from its samples. Example.

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### Sampling (Section 4.3)

CS474/674 – Prof. Bebis

Sampling
• How many samples should we get so that no information is lost during the sampling process?
• Hint: take enough samples so that the “continuous” image can be reconstructed 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

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

Hint: use convolution theorem!

Sampling a 1D function (cont’d)
• Suppose f(x) F(u)
• What is the DFT of s(x)?
Sampling a 2D function (cont’d)
• 2D train of impulses

s(x,y)

x

y

Δy

Δx

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)

x

Reconstructing f(x) from its samples
• Need to isolate a single period:
• Multiply by a window G(u)
What is the effect of Δx?
• Large Δx (i.e., few samples) results to overlapping periods!

x

Effect of Δx (cont’d)
• But, if the periods overlap, we cannot anymore isolate
• a single period  aliasing!
What is the effect of Δx? (cont’d)
• Smaller Δx (i.e., more samples) alleviates aliasing!
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?
Example

square size: 16 x 16 6 x 6

(same as

12 x 12

squares)

square size: 160.9174 0.4798

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!

x

=

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)
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)
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) isperiodic, 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 surface 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 trueanti-aliasing)
Example

3 x 3 blurring and

50% less samples

50% less samples