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

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

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Euler’s formula 3

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Cosine 4 Recall

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Sine 5

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

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Discrete Fourier Transform Forward Inverse 7

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

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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

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Examples 10

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Examples 11

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Periodicity 2D Fourier Transform is periodic in both directions 12

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

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

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Properties of the 2D DFT 15

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

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

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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

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

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Properties

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

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22 Real Imaginary Cos (x) Even

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23 Real Imaginary Sin (x) Odd

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

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

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

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Scaling property Scaling t with a 27

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a 28 Imaginary parts

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

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Examples – horizontal derivative 30

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Examples – vertical derivative 31

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Examples – hor and vert derivative 32

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Thanks and see you next Wednesday!☺ 33

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