Download presentation

Presentation is loading. Please wait.

Published byZane Ambrose Modified about 1 year ago

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

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

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google