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Spatial Frequencyor
How I learned to love the Fourier Transform
Jean Baptiste Joseph Fourier
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Spatial Frequency
• Efficient data representation
• Provides a means for modeling and removing noise
• Physical processes are often best described in “frequency domain”
• Provides a powerful means of image analysis
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What is spatial frequency?
• Instead of describing a function (i.e., a shape) by a series of positions
• It is described by a series of cosines
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What is spatial frequency?
A
g(x) = A cos(x)
2
x
g(x)
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What is spatial frequency?
Period (L)Wavelength ()Frequency (1/f)
Amplitude (A)Magnitude (A)
A cos(x 2/L)g(x) = A cos(x 2/) A cos(x 2f)
x
g(x)
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What is spatial frequency?
A
g(x) = A cos(x 2f)
x
g(x)
(1/f)
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What is spatial frequency?
g(x) = A cos(x 2f + )
x
g(x)
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What is spatial frequency?
g(x) = A cos(x 2f + )A=2 mf = 0.5 m-1
= 0.25 = 45g(x) = 2 cos(x 2(0.5) + 0.25) 2 cos(x + 0.25)
x g(x)0.00 2 cos(0.25) = 0.707106...0.25 2 cos(0.50) = 0.00.50 2 cos(0.75) = -0.707106...0.75 2 cos(1.00) = -1.01.00 2 cos(1.25) = -0.707106…1.25 2 cos(1.50) = 01.50 2 cos(1.75) = 0.707106...1.75 2 cos(2.00) = 1.02.00 2 cos(2.25) = 0.707106...
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What is spatial frequency?
g(x) = A cos(x 2f + )
x
g(x)
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What is spatial frequency?
gi(x) = Ai cos(x 2fi + i), i = 0,1,2,3,...
x
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What is spatial frequency?
gi(x) = Ai cos(x 2i/N + i), i = 0,1,2,3,…,N-1
f=i/N
0 N
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What is spatial frequency?
gi(x) = Ai cos(x 2i/N + i), i = 0,1,2,3,…,N-1
f=i/N
0 N
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What is spatial frequency?
gi(x) = Ai cos(x 2i/N + i), i = 0,1,2,3,…,N-1
i=0 i=1 i=2
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What is spatial frequency?gi(x) = Ai cos(x 2i/N + i), i = 0,1,2,3,…,N/2-1
i=N/2-1i=0
If N equals the number of pixel in a line, then...
Lowest frequency Highest frequency
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What is spatial frequency?gi(x) = Ai cos(x 2i/N + i), i = 0,1,2,3,…,N/2-1
i=N/2-1i=0
If N equals the number of pixel in a line, then...
Lowest frequency Highest frequency
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What is spatial frequency?gi(x) = Ai cos(x 2i/N + i), i = 0,1,2,3,…,N/2-1
i=N/2i=0
If N equals the number of pixel in a line, then...
Lowest frequency Too high Redundant frequency
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What is spatial frequency?
12/
0
12/
0
/2cos)(Ni
iii
Ni
ii NixAxgxg
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19
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What is spatial frequency?
+
+=
Low frequency
Medium frequency
High frequency
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What is spatial frequency?
+
+=
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g(x)
i=1
i=2
i=3
i=4
i=5
i=63
0 127
x
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g(x)
i=1
i=2
i=3
i=4
i=5
i=10
0 127x
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g(x)
g(x)
64 terms
10 terms
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g(x)
i=1
i=2
i=3
i=4
i=5
i=630 127
x
Fourier Decomposition of a step function (64 terms)
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g(x)
i=1
i=2
i=3
i=4
i=5
i=100 63
x
Fourier Decomposition of a step function (11 terms)
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What is spatial frequency?
12/
0
/2cos)(Ni
iii NixAxg
Any function of x (any shape) that can be represented by g(x) can also be represented by the summation of cosine functions
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What is spatial frequency?
+
+=
Low frequency
Medium frequency
High frequency
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What is spatial frequency?
gi(x) = 1.3, 2.1, 1.4, 5.7, …., i=0,1,2…N-1
N pieces of information
12/
0
/2cos)(Ni
iiii NixAxg
N pieces of informationN/2 amplitudes (Ai, i=0,1,…,N/2-1) andN/2 phases (i, i=0,1,…,N/2-1) and
Spatial Domain
Frequency Domain
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What is spatial frequency?
gi(x)
Are equivalentThey contain the same amount of information
12/
0
/2cosNi
iii NixA and
The sequence of amplitudes squared is the SPECTRUM
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Spatial Frequencyor
How I learned to love the Fourier Transform
Jean Baptiste Joseph Fourier
