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

3. 𝑆𝑖𝑛(𝑥)
1 3 𝑆𝑖𝑛(3𝑥)

4. 𝑆𝑖𝑛 𝑥 sin⁡(3𝑥)
𝑆𝑖𝑛 𝑥 sin 3𝑥 sin⁡(5𝑥)
𝑆𝑖𝑛 𝑥 sin 3𝑥 sin 5𝑥 sin⁡(7𝑥)
𝑆𝑖𝑛 𝑥 sin 3𝑥 sin 5𝑥 sin 7𝑥 +…

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

6. Fourier Transform
Digital Signals
Hardly periodic
Never infinite

7. Fourier Transform in 1D

8. Representation in Both Domains
Frequency Domain 2
Amplitude 1
Frequency
180
Time Domain
Phase
Frequency

9. Discrete Fourier Transform
DFT decomposes x into 𝑁 2 +1 cosine and sine waves
Each of a different frequency

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

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

12. Polar Representation

13. Polar Representation
Unwrapping of phase

14. Properties
Homogeneity
Additivity

15. Properties
Linear phase shift

16. Symmetric Signals
Symmetric signal always has zero phase

17. Symmetric Signals
Frequency response and circular movement

18. Amplitude Modulation

19. Periodicity of Frequency Domain
Amplitude Plot
𝑀 𝑓 =𝑀[−𝑓]

20. Periodicity of Frequency Domain
Phase Plot
𝜃 𝑓 =−𝜃 −𝑓

21. Aliasing

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 Orientation
Only positive
Orientation
0 to 180
Repeats in negative frequency
Just as in 1D

30. Example

31. How it repeats? Just like in 1D For amplitude
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

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
Why the diagonal line in Lena?
Strongest edge between hair and hat
Why higher energy in higher frequencies in Mandril?
Hairs

48. More examples Repeatation makes perfect periodic signal
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

51. Cheetah

52. Magnitude

53. Phase

54. Zebra

55. Magnitude

56. Phase

57. Reconstruction
Cheetah Magnitude
Zebra Phase

58. Reconstruction
Zebra magnitude
Cheetah phase

59. Uses – Notch Filter

60. Uses

61. Smoothing Box Filter

62. Smoothing Gaussian Filter