Aliasing And Anti-Aliasing Sampling and Reconstruction

Transcription

1 Aliasing And Anti-Aliasing Sampling and Reconstruction An Introduction Computer

2 Overview Intro - Aliasing Problem definition, Examples Ad-hoc Solutions Sampling theory Fourier transform Convolution Reconstruction Sampling theorem Reconstruction in theory and practice 2

3 Aliasing - a Common Problem Errors incurred during analog-digital conversions: Geometric (pixel artefacts, jaggies ) Colour quality Errors in animation sequences 3

4 Aliasing: Jaggies 4

5 Aliasing: Sampling Errors 5

6 Aliasing: Sampling Errors 5

7 Aliasing: Sampling Errors 5

8 Aliasing: Sampling Errors 5

9 Aliasing: Colour Ramps 6

10 Animation Aliasing Jumpy images "worming Wheels turning backwards 7

11 Wheels turning backwards 8

12 Wheels turning backwards 8

13 Solution Strategies More effort Ad-hoc solutions Higher resolution Higher colour depth Faster image refresh Use your brain Understanding the problem Efficient counter-measures 9

14 Solution Strategies More effort Ad-hoc solutions Higher resolution Higher colour depth Faster image refresh Oft aufwendig, nicht immer zielführend, manchmal unmöglich Use your brain Understanding the problem Efficient counter-measures 9

15 Solution Strategies More effort Ad-hoc solutions Higher resolution Higher colour depth Faster image refresh Oft aufwendig, nicht immer zielführend, manchmal unmöglich Use your brain Understanding the problem Efficient counter-measures machbar 9

16 Simple Splatting Kernels box filter cone Gaussian filter 10

17 Filter Kernels in Action 11

18 Filter Kernels in Action 11

19 Filter Kernels in Action 11

20 Scanline as Signal 12

21 Scanline as Signal 12

22 Scanline as Signal 12

23 Scanline as Signal 12

24 Storing a Scanline Original Sampling Digitalisiertes Signal Reconstruction Wiedergabe 13

25 Sampling: Basic Problem Without at least some knowledge about the sampled signal, we cannot guarantee that this will work! At all! 14

26 Square Wave Decomposition 15

27 Signal Decomposition 16

28 Fourier Transform Connects time and frequency domain Applicable for arbitrary signals f(x) time domain F(u) frequency domain F(u) = + f (x)[cos 2πux isin 2π ux] dx + f (x) = F(u)[cos 2πux isin 2π ux] du 17

29 Box & Sawtooth f (x) F(u) 18

30 Box & Sawtooth f (x) F(u) 18

31 Box & Sawtooth f (x) F(u) 18

32 Square Wave & Scanline f (x) F(u) 19

33 Square Wave & Scanline f (x) F(u) 19

34 Square Wave & Scanline f (x) F(u) 19

35 Fourier Transform Yields complex-valued frequency space functions The imaginary part contains phase information - this is usually omitted Can be generalised to higher dimensions (2D, 3D) Alternatives, such as the Hartley transform, exist 20

36 Discrete Fourier Transform For discrete signals F(u) = f (x) = N 1 0 N 1 0 f (x)[cos (2π ux /N) isin (2πux /N)],0 u N 1 F(u)[cos (2π ux /N) isin (2πux /N)],0 x N 1 N samples: complexity O(N^2) Fast FT (FFT): O(N log N) 21

37 Sampling Function: Comb A single Dirac pulse corresponds to sampling an image at a single location Regular sampling can be seen as multiplication with n evenly spaced Dirac pulses (comb function) 22

38 FT of a Comb A time domain comb corresponds to a frequency domain comb with inverse pulse distance comb T (x) comb 1/T (ω) 23

39 Convolution Mathematical operator: Two functions as input A new function as output Weighing the first function with the second ( f * g)(t) = f (τ)g(t τ)dτ D 24

40 Convolution Examples 25

41 Convolution Theorem Convolution in the time domain corresponds to point wise multiplication in the frequency domain And vice versa f * g F G f g F *G 26

42 Low Pass Filter Goal: low pass filtering of a scanline Time domain: convolution with a Sinc() Frequency domain: cutting off high frequencies - multiplication with a box function Sinc() in the time domain corresponds to a box in the frequency domain 27

43 Sinc() Unlimited domain Perfect reconstruction filter Fourier transform of a box function Sinc(x) = Sin(x) x falls x 0 0 falls x = 0 28

44 Low Pass (time domain) 29

45 Low Pass (time domain) 30

46 Low Pass (freqency domain) 31

47 Sampling Sampling is a multiplication of the source signal with a comb function: f s (x) = f (x) comb T (x) In the frequency domain, this corresponds to a convolution (!) with an inversely spaced comb: F s (ω) = F(ω) * comb 1/T (ω) 32

48 Spectrum of a Sampled Function 33

49 Perfect Reconstruction The copies of the original frequency spectrum must not overlap in the frequency domain Multiplication of the spectrum with a box function is equivalent to cutting out of the original spectrum This corresponds to a convolution with a Sinc function in the time domain 34

50 Reconstruction Examples Sampling and Reconstruction of a scanline With sufficient bandwidth With insufficient bandwidth With band limiting With sinc() and tent kernels 35

51 Sampling > w 36

52 Sampling > w: Sinc() kernel 37

53 Sampling > w: Tent Kernel 38

54 Sampling < w 39

55 Sampling < w: Sinc() 40

56 Sampling < w: Tent Kernel 41

57 Band Limiting the Signal 42

58 Band Limiting the Signal 43

59 Band Limiting the Signal 44

60 Band Limiting 45

61 Sampling 46

62 Reconstruction 47

63 Reconstruction in Practice An ideal reconstruction filter exists: Sinc(x) Problem: Sinc(x) is unusable in practice, as it has infinite extent And to add insult to injury, truncated Sinc(x) is actually worse than many alternatives In practice, one uses a number of sub-optimal filters, depending on application 48

64 Sinc() & cut-off Sinc() 49

65 Blender 50

66 Blender 50

67 Blender 50

68 Blender 51

69 Grid Example 52

70 Computer Grid Example 52

71 Computer Grid Example 52