Fourier Transform. Any signal can be expressed as a linear combination of a bunch of sine gratings of different frequency Amplitude Phase

Transcription

1 Fourier Transform

2 Fourier Transform Any signal can be expressed as a linear combination of a bunch of sine gratings of different frequency Amplitude Phase 2

3

4 1 sin sin 3 1 sin sin sin 5 1 sin sin sin 5 1 sin 7 7 4

5 Fourier Transform Input: infinite periodic signal Output: set of sine and cosine waves which together provide the input signal 5

6 Fourier Transform Digital Signals Hardly periodic Never infinite 6

7 Fourier Transform in 1D 7

8 Representation in Both Domains 2 Frequency Domain Amplitude Frequency Time Domain Phase 0 Frequency 8

9 Discrete Fourier Transform DFT decomposes x into 1cosine and sine waves Each of a different frequency 9

10 DFT - Rectangular Representation Decomposition of the time domain signal x to the frequency domain and 10

11 Polar Notation Sine and cosine waves are phase shifted versions of each other Amplitude Phase 11

12 Polar Representation 12

13 Polar Representation Unwrapping of phase 13

14 Properties Homogeneity Additivity 14

15 Properties Linear phase shift 15

16 Symmetric Signals Symmetric signal always has zero phase 16

17 Symmetric Signals Frequency response and circular movement 17

18 Amplitude Modulation 18

19 Periodicity of Frequency Domain Amplitude Plot 19

20 Periodicity of Frequency Domain Phase Plot 20

21 Aliasing 21

22 Sampling Frequency Domain Convolution -f s 2 f s 2 -f s f s 2 f s 3 f s -f s f s 2 f s 3 f s

23 Reconstruction Frequency Domain -f s f s 2 f s 3 f s -f s 2 f s 2 Multiplication -f s 2 f s 2

24 Reconstruction (Wider Kernel) -f s f s 2 f s 3 f s -f s 2 f s 2 PIXELIZATION: Lower frequency aliased as high frequency -f s 2 f s 2

25 Reconstruction (Narrower Kernel) -f s f s 2 f s 3 f s -f s 2 f s 2 BLURRING: Removal of high frequencies -f s 2 f s 2

26 Aliasing artifacts (Right Width)

27 Wider Spots (Lost high frequencies)

28 Narrow Width (Jaggies, insufficient sampling)

29 DFT extended to 2D : Axes Frequency Only positive Orientation 0 to 180 Repeats in negative frequency Just as in 1D

30 Example

31 How it repeats? Just like in 1D Even function for amplitude Odd function for phase For amplitude Flipped on the bottom

32 Why all the noise? Values much bigger than 255 DC is often 1000 times more than the highest frequencies Difficult to show all in only 255 gray values

33 Mapping Numerical value = i Gray value = g Linear Mapping is g = ki Logarithmic mapping is g = k log (i) Compresses the range Reduces noise May still need thresholding to remove noise

34 Example Original DFT Magnitude In Log scale Post Thresholding

35 Low Pass Filter Example

36 Additivity + Inverse DFT =

37 Nuances

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39

40 Rotation What is this about?

41 X

42 X

43 More examples: Blurring Note energy reduced at higher frequencies What is direction of blur? Horizontal Noise also added DFT more noisy

44 More examples: Edges Two direction edges on left image Energy concentrated in two directions in DFT Multi-direction edges Note how energy concentration synchronizes with edge direction

45 More examples: Letters DFTs quite different Specially at low frequencies Bright lines perpendicular to edges Circular segments have circular shapes in DFT

46 More examples: Collections Concentric circle Due to pallets symmetric shape DFT of one pallet Similar Coffee beans have no symmetry Why the halo? Illumination

47 More examples: Natural Images Natural Images Why the diagonal line in Lena? Strongest edge between hair and hat Why higher energy in higher frequencies in Mandril? Hairs

48 More examples Spatial Repeatation makes perfect periodic signal Therefore perfect result perpendicular to it Frequency

49 More examples Spatial Just a gray telling all frequencies Why the bright white spot in the center? Frequency

50 Amplitude How much details? Sharper details signify higher frequencies Will deal with this mostly 50

51 Cheetah 51

52 Magnitude 52

53 Phase 53

54 Zebra 54

55 Magnitude 55

56 Phase 56

57 Reconstruction Cheetah Magnitude Zebra Phase 57

58 Reconstruction Zebra magnitude Cheetah phase 58

59 Uses Notch Filter 59

60 Uses

61 Smoothing Box Filter 61

62 Smoothing Gaussian Filter 62