Sampling (Section 4.3)

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

2. 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.

3. Example Sampled signal looks like a sinusoidal of a lower frequency !

4. Definition: “band-limited” functions • A function whose spectrum is of finite duration • Are all functions band-limited? max frequency NO!!

5. 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.

6. 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!

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

8. = * Sampling a 1D function (cont’d) So:

9. Sampling a 2D function (cont’d) • 2D train of impulses s(x,y) x y Δy Δx

10. 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)

11. x Reconstructing f(x) from its samples • Need to isolate a single period: • Multiply by a window G(u)

12. Reconstructing f(x) from its samples (cont’d) • Then, take the inverse FT:

13. What is the effect of Δx? • Large Δx (i.e., few samples) results to overlapping periods!

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

15. What is the effect of Δx? (cont’d) • Smaller Δx (i.e., more samples) alleviates aliasing!

16. What is the effect of Δx? (cont’d) • 2D case u u vmax umax v v

17. 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?

18. Example square size: 16 x 16 6 x 6 (same as 12 x 12 squares) square size: 160.9174 0.4798

19. How to choose Δx? • The center of the overlapped region is at

20. How to choose Δx? (cont’d) • Choose Δx as follows: where W is the max frequency of f(x)

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

22. 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)

23. 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)

24. instead of this! Practical Issues (cont’d) *

25. 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

26. 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)

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