Reminder

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Reminder
t?0,1
Fourier Basis
n?Z
Fourier Series
Fourier Coefficient
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Example - Sinc
rect(t)
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Sinc - Pictures
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Discrete Fourier Transform
Fourier Transform ( notations f(x) s(x/N),
F(u) ? auNk )
Inverse Fourier Transform
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Fourier of Delta
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2D Discrete Fourier
Fourier Transform
Inverse Fourier Transform
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Display Fourier Spectrum as Picture
1. Compute
2. Scale to full range
3. Move (0,0) to center of image (Shift by N/2)
Example for range 0..10
Original f 0 1 2 4 100 Scaled to
10 0 0 0 0 10 Log (1f) 0 0.69 1.01 1.61 4.62 Sca
led to 10 0 1 2 4 10
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Fourier Displays
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Decomposition
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Decomposition (II)

• 1-D Fourier is sufficient to do 2-D Fourier
• Do 1-D Fourier on each column. On result
• Do 1-D Fourier on each row
• (Multiply by N?)
• 1-D Fourier Transform is enough to do Fourier for
  ANY dimension

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Decomposition Example
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Translation
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Periodicity Symmetry
(Only for real images)
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Rotation
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Linearity
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Derivatives I
Inverse Fourier Transform
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Derivatives II

• To compute the x derivative of f (up to a
  constant)
• Computer the Fourier Transform F
• Multiply each Fourier coefficient F(u,v) by u
• Compute the Inverse Fourier Transform
• To compute the y derivative of f (up to a
  constant)
• Computer the Fourier Transform F
• Multiply each Fourier coefficient F(u,v) by v
• Compute the Inverse Fourier Transform

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Convolution Theorem
Convolution by Fourier
Complexity of Convolution O(N logN)
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Filtering in the Frequency Domain
Filtered Picture
Filtered Fourier
Fourier
Filter
Picture
Low-Pass Filtering
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Low Pass Frequency Image

• (0 0 1 1 0 0) ? Sinc
• (0 0 1 1 0 0) (0 0 1 1 0 0 ) (0 1 2 1 0 0) ?
  Sinc2
• (0 1 4 6 4 1 0) (0 0 1 1 0 0 ) 4 ? Sinc4
• Fourier (Gaussian) ? Gaussian

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Continuous Sampling