Sampling

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

2. Image Digitization • How many samples should we obtain to minimize information loss? • Hint: take enough samples to allow reconstructing the “continuous” image from its samples.

3. Example Very few samples: 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 spatial domain. • Functions with finite duration in the spatial domain have infinite duration in the frequency domain.

6. Sampling a 1D function • Multiply f(x) with s(x) (i.e., train of impulses) 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. x Reconstructing f(x) from its samples • Need to isolate a single period: • Multiply by a window G(u)

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

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

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

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

14. How should we choose Δx? • The center of the overlapped region is at

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

16. Example

17. Example (cont’d)

18. Sampling a 2D function • 2D train of impulses s(x,y) x y Δy Δx f(x,y)s(x,y) F(u,v)*S(u,v)

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

20. Example • Suppose that we need to digitize checkerboard patterns.

21. Example (cont’d) • Suppose that the number of samples is fixed at 96 x 96 pixels. • We can resolve patterns that are up to 96 x 96 pixel squares (i.e., each square is 1 x 1). • What happens when squares are less than 1 x 1 pixels?

22. Example (cont’d) square size: 16 x 16 6 x 6 (same as 12 x 12 squares) square size: 0.9174 x 0.9174 0.4798 x 0.4798

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

24. 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)

25. 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)

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

27. 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

28. 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 trueanti-aliasing)

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