Spatial%20Frequencies

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Spatial Frequencies
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Why are Spatial Frequencies important?

• 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?
g(x) A cos(x)
g(x)
2?
A
x
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What is spatial frequency?
A cos(x ? 2?/L) g(x) A cos(x ? 2?/?)
A cos(x ? 2?f)
g(x)
Period (L) Wavelength (?) Frequency f(1/ ?)
Amplitude (A) Magnitude (A)
x
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What is spatial frequency?
g(x) A cos(x ? 2?f)
g(x)
A
x
(1/f)
period
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But what if cosine is shifted in phase?
g(x) A cos(x ? 2?f ?)
g(x)
x
?
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What is spatial frequency?
Let us take arbitrary g(x)
x g(x) 0.00 2 cos(0.25?)
0.707106... 0.25 2 cos(0.50?) 0.0 0.50 2
cos(0.75?) -0.707106... 0.75 2 cos(1.00?)
-1.0 1.00 2 cos(1.25?) -0.707106 1.25 2
cos(1.50?) 0 1.50 2 cos(1.75?)
0.707106... 1.75 2 cos(2.00?) 1.0 2.00 2
cos(2.25?) 0.707106...
g(x) A cos(x ? 2?f ?) A2 m f 0.5 m-1
0.25? 45? g(x) 2 cos(x ? 2?(0.5) 0.25?)
2 cos(x ? ? 0.25?)
We calculate discrete values of g(x) for various
values of x
We substitute values of A, f and ?
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What is spatial frequency?
g(x) A cos(x ? 2?f ?)
g(x)
We calculate discrete values of g(x) for various
values of x
x
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What is spatial frequency?
g(x) A cos(x ? 2?f ?)
gi(x) Ai cos(x ? 2?i/N ?i), i
0,1,2,3,,N/2-1
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We try to approximate a periodic function with
standard trivial (orthogonal, base) functions
Low frequency

Medium frequency


High frequency
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We add values from component functions point by
point



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g(x)
i1
i2
i3
i4
i5
i63
x
0
127
Example of periodic function created by summing
standard trivial functions
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g(x)
i1
i2
i3
i4
i5
i10
x
0
127
Example of periodic function created by summing
standard trivial functions
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64 terms
g(x)
10 terms
g(x)
Example of periodic function created by summing
standard trivial functions
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Fourier Decomposition of a step function (64
terms)
g(x)
i1
i2
i3
i4
i5
Example of periodic function created by summing
standard trivial functions
x
i63
0
127
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Fourier Decomposition of a step function (11
terms)
g(x)
i1
i2
i3
Example of periodic function created by summing
standard trivial functions
i4
i5
i10
x
0
63
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Main concept summation of base functions
Any function of x (any shape) that can be
represented by g(x) can also be represented by
the summation of cosine functions
Observe two numbers for every i
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Information is not lost when we change the domain
Spatial Domain
gi(x) 1.3, 2.1, 1.4, 5.7, ., i0,1,2N-1
N pieces of information
Frequency Domain
N pieces of information N/2 amplitudes (Ai,
i0,1,,N/2-1) and N/2 phases (?i, i0,1,,N/2-1)
and
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What is spatial frequency?
Information is not lost when we change the domain
gi(x)
and
Are equivalent They contain the same amount of
information
The sequence of amplitudes squared is the SPECTRUM
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EXAMPLE
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Substitute values
A cos(x?2?i/N) frequency (f) i/N wavelength (p)
N/I N512 i f p 0 0
infinite 1 1/512 512 16 1/32
32 256 1/2 2
Assuming N we get this table which relates
frequency and wavelength of component functions
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More examples to give you some intuition.
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Fourier Transform Notation

• g(x) denotes an spatial domain function of real
  numbers
• (1.2, 0.0), (2.1, 0.0), (3.1,0.0),
• G() denotes the Fourier transform
• G() is a symmetric complex function
• (-3.1,0.0), (4.1, -2.1), (-3.1, 2.1), (1.2,0.0)
  , (-3.1,-2.1), (4.1, 2.1), (-3.1,0.0)
• Gg(x) G(f) is the Fourier transform of g(x)
• G-1() denotes the inverse Fourier transform
• G-1(G(f)) g(x)

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Power Spectrum and Phase Spectrum
complex
Complex conjugate

• G(f)2 G(f)?G(f) is the power spectrum of
  G(f)
• (-3.1,0.0), (4.1, -2.1), (-3.1, 2.1),
  (1.2,0.0),, (-3.1,-2.1), (4.1, 2.1)
• 9.61, 21.22, 14.02, , 1.44,, 14.02, 21.22
• tan-1Im(G(f))/Re(G(f)) is the phase spectrum of
  G(f)
• 0.0, -27.12, 145.89, , 0.0, -145.89, 27.12

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1-D DFT and IDFT

• Discrete Domains
• Discrete Time k 0, 1, 2, 3, , N-1
• Discrete Frequency n 0, 1, 2, 3, , N-1
• Discrete Fourier Transform
• Inverse DFT

Equal frequency intervals
n 0, 1, 2,.., N-1
k 0, 1, 2,.., N-1