Understanding Sampling and Quantisation in Image Processing

1. Sampling and Quantisation

2. Sampling & quantisation • 1. Discretization of continuous signals • 2. Signal representation in the spatial and frequency domain • 3. Effects of sampling and quantisation • 4. More on sampling • 5. More on quantisation 

3. 1. sampling 4 “pixels” 2. quantisation 4 “grey levels” Discretisation Computer to process an image : 

4. Sampling & quantization (0,0) 84 133 226 212 218 218 222 212 218 222 226 218 75 156 177 218 212 218 218 218 218 222 218 218 96 84 133 203 218 218 218 222 212 218 222 218 123 75 111 156 212 218 212 212 218 218 218 226 93 75 71 133 185 231 226 226 222 212 218 218 51 75 75 75 156 206 218 218 218 222 212 222 44 110 75 65 143 194 231 218 218 218 218 218 52 123 69 84 60 156 199 231 231 222 226 226 52 75 84 81 65 69 150 231 231 226 231 231 36 36 84 93 84 71 156 160 240 240 231 231 36 40 113 75 69 75 71 133 194 240 240 240 52 52 105 85 69 75 75 123 111 222 231 231 69 44 69 93 81 75 75 69 150 177 247 240 73 44 40 96 101 75 75 75 84 133 231 240

5. Sampling schemes regular, image covering tessellation 11 with regular polygons 4 3 if equal rectangular (square) most popular hexagonal has advantages (more isotropic, no connectivity ambiguities, …) + similar structure in retina 

6. Example of sampling : 384 x 288 pixels 192 x 144 pixels 92 x 72 pixels 48 x 36 pixels

7. Example of quantisation : 2 levels - binary 4 levels 256 levels – 1 byte 8 levels

8. Image distortion through sampling 

9. Image distortion through quantisation 

10. Remarks • 1. Importance of binary images • 2. Non-uniform sampling and/or quantisation • a. fine sampling for details • b. fine quantisation for homogeneous regions 

11. A model for sampling 1. Integrate brightness over cell window Image degradations 2. Read out values only at the pixel centers Aliasing Leakage 

12. STEP 1 : integrating over a pixel cell 

13. 1 x x0 STEP 2: local probing of functions Distributions as extension of functions: the Dirac pulse Function probing (in 1D) 

14. f(x) x 1 x x0 f(x0) f(x0) x x0 Spatial domain characterization . = Decomposition of a function into individual pulses Signals described in a linear space through decomposition in an orthonormal basis 

15. A system view on image/signal processing Input f(x,y) Output g(x,y) System: O

16. Linear, shift-invariant operators 1. Linear : • ƒ1 g1 • ƒ2 g2 • aƒ1+ bƒ2 ag1+ bg2 2. Shift-invariant : • ƒ(x,y) g(x,y) • ƒ(x - a, y - b) g(x - a , y - b) 

17. Characterization of LSI systemsthrough spatial domain pulses r convolution of ƒ and h  point spread function 

18. f(.) f(.) f(.) f(.) f(.) f(.) f(.) f(.) f(.) c11 c12 c13 c23 c21 c22 c31 c32 c33 +c12f(i-1,j) +c13f(i-1,j+1) + c11f(i-1,j-1) + c22f(i,j) + c23f(i,j+1) + c21f(i,j-1) + c33f(i+1,j+1) c31f(i+1,j-1) + c32f(i+1,j) Convolution o (i,j) = 

19. An example of convolution This is a convolution: 

20. Characteristics of convolution 

21. Alternative characterization of functions: The frequency domain orthonormal basis functions Eigenfunctions of LSI systems 

22. The Fourier transform Linear decomposition of functions in the new basis Scaling factor for basis function (u,v)  The Fourier transform Reconstruction of the original function in the spatial domain: weighted sum of the basis functions  The inverse Fourier transform 

23. is complex : Fourier coefficients The magnitude The phase angle arctan 

24. Fourier decomposition of images f(x,y) = = F(u,v) + F(u’,v’) + F(u’’,v’’) + … x x x 

25. Fourier decomposition of images 

26. Fourier decomposition of images 

27. Example importance of magnitude • Image with periodic structure f(x,y) |F(u,v)| FT has peaks at spatial frequencies of repeated texture

28. |F(u,v)| remove peaks Periodic background removed Example importance of magnitude

29. Example importance of magnitude |F(u,v)| cross-section phase F(u,v) f(x,y) • |F(u,v)| generally decreases with higher spatial frequencies • phase appears less informative

30. magnitude phase phase The importance of the phase

31. Some relevant properties 

32. Rotation : 

33. Scaling Scaling : 

34. Affine 

35. The convolution theorem Fourier 

36. The convolution theorem That is, Space convolution = frequency multiplication 

37. modulation transfer function Modulation transfer function for LSI 

38. The convolution theorem: reciprocity Space multiplication = frequency convolution 

39. A model for sampling … back to STEP 1 1. Integrate brightness over cell window Image degradations 2. Read out values only at the pixel centers Aliasing Leakage 

40. STEP 1 : integrating over a pixel cell This is convolution: 

41. Back to STEP 1 Fourier transform of window : 

42. however, phase reversals ! Fourier transform of the window function 2D sinc : real  no phase shifts predominantly low-pass 

43. Illustration of the sinc 

44. F(u,v) |F(u,v)| * x = The convolution theorem: exercise frequency domain • What is the FT of … f(x,y)

45. A model for sampling … back to STEP 2 1. Integrate brightness over cell window Image degradations 2. Read out values only at the pixel centers Aliasing Leakage 

46. STEP 2 : discrete representation of functions In the spatial and frequency domain • Discretizing in the spatial domain • Limiting the spatial extent • Discretizing in the frequency domain 

47. Discretizing in the spatial domain multiplication with 2D pulse train Fourier transform : Convolution with a Dirac train: periodic repetition Yet another duality: discrete vs. periodic 

48. Discretizing in the spatial domain 

49. The sampling theorem If the Fourier transform of a function ƒ(x,y) is zero for all frequencies beyond ub and vb, i.e. if the Fourier transform is band-limited, then the continuous periodic function ƒ(x,y) can be completely reconstructed from its samples as long as the sampling distances w and h along the x and y directions are such that and 

50. Interpolation Result is an exact interpolation Sinc is the optimal interpolation function Sample (multiply pulse train) Periods must not overlap (hence Nyquist rate) Select one period : multiply window Thus convolve Fourier transform with pulse train Result is periodic Fourier transform 