Transcript Sampling (Section 4.3) CS474/674 – Prof. Bebis

Sampling (Section 4.3)
CS474/674 – Prof. Bebis
Sampling
• How many samples should we obtain to minimize
information loss?
• Hint: take enough samples to allow reconstructing the
“continuous” image from its samples.
Example
Very few samples; sampled signal
looks like a sinusoidal of a lower
frequency !
Definition: “band-limited” functions
• A function whose spectrum is of finite duration
max
frequency
• Are all functions band-limited? No!
Properties of band-limited functions
• Band-limited functions have infinite duration in the
spatial domain.
• Functions with finite duration in the spatial domain
have infinite duration in the frequency domain.
Sampling a 1D function
• Multiply f(x) with s(x) (i.e., train of impulses)
x
sampled f(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 1D function (cont’d)
=
*

1
x
1
x
So:

1
x
1
x
Sampling a 2D function (cont’d)
• 2D train of impulses
s(x,y)
x
Δy
Δx
y
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
vmax
u
umax
v
v
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 (cont’d)
Example (cont’d)
square size:
16 x 16
6x6
(same as
12 x 12
squares)
square size:
0.9174 x 0.9174
0.4798 x 0.4798
How should we 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 “antialiasing” which is related to blurring the image to reduce
aliasing artifacts (i.e., not true anti-aliasing)
Example
50% less samples
3 x 3 blurring and
50% less samples