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 Are all functions band-limited? max frequency 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) * = So:

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)

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 vmax 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 6 x 6 (same as 12 x 12 squares)

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 “anti-aliasing” which is related to blurring the image to reduce aliasing artifacts (i.e., not true anti-aliasing)

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