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.
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 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
Effect of Δx (cont’d)
• But, if the periods overlap, we cannot anymore isolate a single period aliasing!
x
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?
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)
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)