The Fourier Transform

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The Fourier Transform
Jean Baptiste Joseph Fourier
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A sum of sines and cosines

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Higher frequencies dueto sharp image variations
(e.g., edges, noise, etc.)
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The Continuous Fourier Transform
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Complex Numbers
Imaginary
Z(a,b)
b
Z
?
Real
a
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The 1D Basis Functions
1
x
1/u

• The wavelength is 1/u .
• The frequency is u .

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The Continuous Fourier Transform
1D Continuous Fourier Transform
The Inverse Transform
The Transform
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The 2D Basis Functions
V
u0, v2
u1, v2
u2, v2
u-2, v2
u-1, v2
u0, v1
u1, v1
u2, v1
u-2, v1
u-1, v1
U
u0, v0
u-2, v0
u-1, v0
u1, v0
u2, v0
u-2, v-1
u-1, v-1
u0, v-1
u1, v-1
u2, v-1
u-2, v-2
u-1, v-2
u0, v-2
u1, v-2
u2, v-2
The wavelength is . The
direction is u/v .
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Discrete Functions
f(x)
f(n) f(x0 nDx)
f(x02Dx)
f(x03Dx)
f(x0Dx)
f(x0)
0 1 2 3 ...
N-1
x0
x0Dx
x02Dx
x03Dx
The discrete function f f(0), f(1), f(2),
, f(N-1)
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The Discrete Fourier Transform
1D Discrete Fourier Transform
(u 0,..., N-1)
(x 0,..., N-1)
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The Fourier Image
Fourier spectrum log(1 F(u,v))
Image f
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Frequency Bands
Image
Fourier Spectrum
Percentage of image power enclosed in circles
(small to large) 90, 95, 98, 99,
99.5, 99.9
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Low pass Filtering
90
95
98
99
99.5
99.9
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Noise Removal
Noisy image
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Noise Removal
Noisy image
Fourier Spectrum
Noise-cleaned image
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High Pass Filtering
Original
High Pass Filtered
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High Frequency Emphasis

Original
High Pass Filtered
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High Frequency Emphasis
Original
High Frequency Emphasis
High Frequency Emphasis
Original
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High Frequency Emphasis
Original
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Properties of the Fourier Transform
Developed on the board(e.g., separability of
the 2D transform, linearity, scaling/shrinking,
derivative, rotation, shift ? phase-change,
periodicity of the discrete transform, etc.)We
also developed the Fourier Transform of various
commonly used functions, and discussed
applications which are not contained in the
slides (motion, etc.)
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2D Image
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Fourier Transform -- Examples
Image Domain
Frequency Domain
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Fourier Transform -- Examples
Image Domain
Frequency Domain
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Fourier Transform -- Examples
Image Domain
Frequency Domain
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Fourier Transform -- Examples
Image Domain
Frequency Domain
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Fourier Transform -- Examples
Image
Fourier spectrum
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Fourier Transform -- Examples
Image
Fourier spectrum
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Fourier Transform -- Examples
Image
Fourier spectrum
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Fourier Transform -- Examples
Image
Fourier spectrum
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Fourier Transform -- Examples
Image
Fourier spectrum