Sampling (Section 4.3)

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Sampling (Section 4.3)

• CS474/674 Prof. Bebis

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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.

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Example
Sampled signal looks like a sinusoidal of a lower
frequency !
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Definition band-limited functions

• A function whose spectrum is of finite duration
• Are all functions band-limited?

NO!!
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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.

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Sampling a 1D function

• Multiply f(x) with s(x)

sampled f(x)
x
Question what is the FT of f(x) x s(x)?
Hint use convolution theorem!
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Sampling a 1D function (contd)

• Suppose f(x) ?? F(u)
• What is the FT of s(x)?

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Sampling a 1D function (contd)
So
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Sampling a 2D function (contd)

• 2D train of impulses

s(x,y)
x
y
?y
?x
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Sampling a 2D function (contd)

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

f(x,y)s(x,y) ?? F(u,v)S(u,v)
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Reconstructing f(x) from its samples

• Need to isolate a single period
• Multiply by a window G(u)

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Reconstructing f(x) from its samples (contd)

• Then, take the inverse FT

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What is the effect of ?x?

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

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Effect of ?x (contd)

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

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What is the effect of ?x? (contd)

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

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What is the effect of ?x? (contd)

• 2D case

u
u
vmax
umax
v
v
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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?

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Example
square size 16 x 16
6 x 6
(same as 12 x 12 squares)
square size 160.9174
0.4798
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How to choose ?x?

• The center of the overlapped region is at

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How to choose ?x? (contd)

• Choose ?x as follows

where W is the max frequency of f(x)
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Practical Issues

• Band-limited functions have infinite duration in
  the time domain.
• But, we can only sample a function over a finite
  interval!

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Practical Issues (contd)

• We would need to obtain a finite set of samples
• by multiplying with a box function
• s(x)f(x)h(x)

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Practical Issues (contd)

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

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Practical Issues (contd)

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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

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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
  true anti-aliasing)

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Example
3 x 3 blurring and 50 less samples
50 less samples