Presentation is loading. Please wait.

Presentation is loading. Please wait.

Basis beeldverwerking (8D040) dr. Andrea Fuster Prof.dr. Bart ter Haar Romeny dr. Anna Vilanova Prof.dr.ir. Marcel Breeuwer The Fourier Transform I.

Similar presentations


Presentation on theme: "Basis beeldverwerking (8D040) dr. Andrea Fuster Prof.dr. Bart ter Haar Romeny dr. Anna Vilanova Prof.dr.ir. Marcel Breeuwer The Fourier Transform I."— Presentation transcript:

1 Basis beeldverwerking (8D040) dr. Andrea Fuster Prof.dr. Bart ter Haar Romeny dr. Anna Vilanova Prof.dr.ir. Marcel Breeuwer The Fourier Transform I

2 Contents Complex numbers etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform 2

3 Introduction Jean Baptiste Joseph Fourier (* ) French Mathematician La Théorie Analitique de la Chaleur (1822) 3

4 Fourier Series Any periodic function can be expressed as a sum of sines and/or cosines Fourier Series 4 (see figure 4.1 book)

5 Fourier Transform Even functions that are not periodic and have a finite area under curve can be expressed as an integral of sines and cosines multiplied by a weighing function Both the Fourier Series and the Fourier Transform have an inverse operation: Original Domain Fourier Domain 5

6 Contents Complex numbers etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform 6

7 Complex numbers Complex number Its complex conjugate 7

8 Complex numbers polar Complex number in polar coordinates 8

9 Eulers formula 9 Sin (θ) Cos (θ) ? ?

10 10 Re Im

11 Complex math Complex (vector) addition Multiplication with i is rotation by 90 degrees in the complex plane 11

12 Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform 12

13 Unit impulse (Dirac delta function) Definition Constraint Sifting property Specifically for t=0 13

14 Discrete unit impulse Definition Constraint Sifting property Specifically for x=0 14

15 What does this look like? Impulse train 15 ΔT = 1 Note: impulses can be continuous or discrete!

16 Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform 16

17 Fourier Series with 17 Series of sines and cosines, see Eulers formula Periodic with period T

18 Fourier transform – 1D cont. case 18 Symmetry: The only difference between the Fourier transform and its inverse is the sign of the exponential.

19 Fourier Euler Fourier and Euler 19

20 If f(t) is real, then F(μ) is complex F(μ) is expansion of f(t) multiplied by sinusoidal terms t is integrated over, disappears F(μ) is a function of only μ, which determines the frequency of sinusoidals Fourier transform frequency domain 20

21 Examples – Block W/2W/2 A

22 Examples – Block 2 22

23 Examples – Block 3 23 ?

24 Examples – Impulse 24 constant

25 Examples – Shifted impulse 25 Euler

26 Examples – Shifted impulse 2 26 Real partImaginary part impulseconstant

27 Also: using the following symmetry 27

28 Examples - Impulse train 28 Periodic with period ΔT Encompasses only one impulse, so

29 Examples - Impulse train 2 29

30 Intermezzo: Symmetry in the FT 30

31 31

32 So: the Fourier transform of an impulse train with period is also an impulse train with period 32

33 Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform 33

34 Fourier + Convolution What is the Fourier domain equivalent of convolution? 34

35 What is 35

36 Intermezzo 1 What is ? Let, so 36

37 Intermezzo 2 Property of Fourier Transform 37

38 Fourier + Convolution contd 38

39 Convolution theorem Convolution in one domain is multiplication in the other domain: And also: 39

40 And: Shift in one domain is multiplication with complex exponential (modulation) in the other domain And: 40

41 Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform 41

42 Sampling Idea: convert a continuous function into a sequence of discrete values. 42 (see figure 4.5 book)

43 Sampling Sampled function can be written as Obtain value of arbitrary sample k as 43

44 Sampling

45 Sampling

46 Sampling

47 FT of sampled functions Fourier transform of sampled function Convolution theorem From FT of impulse train 47 (who?)

48 FT of sampled functions 48

49 Sifting property of is a periodic infinite sequence of copies of, with period 49

50 Sampling Note that sampled function is discrete but its Fourier transform is continuous! 50

51 Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples 51

52 Sampling theorem Band-limited function Sampled function lower value of 1/ΔT would cause triangles to merge 52

53 Sampling theorem 2 Sampling theorem: If copy of can be isolated from the periodic sequence of copies contained in, can be completely recovered from the sampled version. Since is a continuous, periodic function with period 1/ΔT, one complete period from is enough to reconstruct the signal. This can be done via the Inverse FT. 53

54 Extracting a single period from that is equal to is possible if Sampling theorem 3 54 Nyquist frequency

55 Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples 55

56 Aliasing If, aliasing can occur 56

57 Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform 57

58 Discrete Fourier Transform Continuous transform of sampled function 58

59 is continuous and infinitely periodic with period 1/ΔT 59

60 We need only one period to characterize If we want to take M equally spaced samples from in the period μ = 0 to μ = 1/Δ, this can be done thus 60

61 Substituting Into yields 61 Note: separation between samples in F. domain is

62 By now we probably need some … 62


Download ppt "Basis beeldverwerking (8D040) dr. Andrea Fuster Prof.dr. Bart ter Haar Romeny dr. Anna Vilanova Prof.dr.ir. Marcel Breeuwer The Fourier Transform I."

Similar presentations


Ads by Google