The Fourier Transform II
description
Transcript of The Fourier Transform II
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
ImaginaryCos (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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