Basis beeldverwerking (8D040)

dr. Andrea FusterProf.dr. Bart ter Haar Romenydr. Anna VilanovaProf.dr.ir. Marcel Breeuwer

The Fourier Transform II

Contents

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

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Euler’s formula

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Cosine

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Recall

Sine

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Contents

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

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Discrete Fourier Transform

• Forward

• Inverse

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Formulation in 2D spatial coordinates

• Discrete Fourier Transform (2D)

• Inverse Discrete Transform (2D)

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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

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Examples

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Examples

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Periodicity

• 2D Fourier Transform is periodic in both directions

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Periodicity

• 2D Inverse Fourier Transform is periodic in both directions

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Contents

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

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Properties of the 2D DFT

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Real

ImaginarySin (x)Sin (x + π/2)

Real

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

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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)

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Properties

• Even function (symmetric)

• Odd function (antisymmetric)

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Properties - 2

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FT of even and odd functions

• FT of even function is real

• FT of odd function is imaginary

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Real

Imaginary

Cos (x)

Even

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Real

Imaginary

Sin (x)

Odd

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Real

ImaginaryF(Cos(x))F(Cos(x+k))

Even

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RealOdd

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

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Scaling property

• Scaling t with a

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• a

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Imaginary parts

Differentiation and Fourier

• Let be a signal with Fourier transform

• Differentiating both sides of inverse Fourier transform equation gives:

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Examples – horizontal derivative

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Examples – vertical derivative

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Examples – hor and vert derivative

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Thanks and see you next Wednesday!☺

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