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

2. Contents Fourier Transform of sine and cosine 2D Fourier Transform Properties of the Discrete Fourier Transform 2

3. Euler’s formula 3

4. Cosine 4 Recall

5. Sine 5

6. Contents Fourier Transform of sine and cosine 2D Fourier Transform Properties of the Discrete Fourier Transform 6

7. Discrete Fourier Transform Forward Inverse 7

8. Formulation in 2D spatial coordinates Discrete Fourier Transform (2D) Inverse Discrete Transform (2D) 8 f(x,y) digital image of size M x N

9. Spatial and Frequency intervals Inverse proportionality Suppose function is sampled M times in x, with step, distance is covered, which is related to the lowest frequency that can be measured And similarly for y and frequency v 9

10. Examples 10

11. Examples 11

12. Periodicity 2D Fourier Transform is periodic in both directions 12

13. Periodicity 2D Inverse Fourier Transform is periodic in both directions 13

14. Contents Fourier Transform of sine and cosine 2D Fourier Transform Properties of the Discrete Fourier Transform 14

15. Properties of the 2D DFT 15

16. 16 Real Imaginary Sin (x) Sin (x + π/2) Real

17. Note: translation has no effect on the magnitude of F(u,v) 17

18. Symmetry: even and odd Any real or complex function w(x,y) can be expressed as the sum of an even and an odd part (either real or complex) 18

19. Properties Even function (symmetric) Odd function (antisymmetric) 19

20. Properties - 2 20

21. FT of even and odd functions FT of even function is real FT of odd function is imaginary 21

22. 22 Real Imaginary Cos (x) Even

23. 23 Real Imaginary Sin (x) Odd

24. 24 Real Imaginary F(Cos(x))F(Cos(x+k)) Even

25. 25 Real Odd Sin (x)Sin(y)Sin (x) Imaginary

26. Consequences for the Fourier Transform FT of real function is conjugate symmetric FT of imaginary function is conjugate antisymmetric 26

27. Scaling property Scaling t with a 27

28. a 28 Imaginary parts

29. Differentiation and Fourier Let be a signal with Fourier transform Differentiating both sides of inverse Fourier transform equation gives: 29

30. Examples – horizontal derivative 30

31. Examples – vertical derivative 31

32. Examples – hor and vert derivative 32

33. Thanks and see you next Wednesday!☺ 33