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

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

Euler’s formula 3

Cosine 4 Recall

Sine 5

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

Discrete Fourier Transform Forward Inverse 7

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

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

Examples 10

Examples 11

Periodicity 2D Fourier Transform is periodic in both directions 12

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

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

Properties of the 2D DFT 15

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

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

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

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

Properties

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

22 Real Imaginary Cos (x) Even

23 Real Imaginary Sin (x) Odd

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

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

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

Scaling property Scaling t with a 27

a 28 Imaginary parts

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

Examples – horizontal derivative 30

Examples – vertical derivative 31

Examples – hor and vert derivative 32

Thanks and see you next Wednesday!☺ 33