spatial frequencies spatial frequencies. why are spatial frequencies important? efficient data...
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Spatial Spatial Frequencies Frequencies
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Why are Spatial Frequencies Why are Spatial Frequencies important?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?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?What is spatial frequency?
A
g(x) = A cos(x)
2
x
g(x)
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What is spatial frequency?What is spatial frequency?
Period (L)Wavelength ()Frequency f=(1/ )
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?What is spatial frequency?
A
g(x) = A cos(x 2f)
x
g(x)
(1/f)(1/f)
period
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But what if cosine is shifted in phase?But what if cosine is shifted in phase?
g(x) = A cos(x 2f + )
x
g(x)
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What is spatial frequency?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...
Let us take arbitrary g(x)
We substitute values of A, f and
We calculate discrete values of g(x) for various values of x
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What is spatial frequency?What is spatial frequency?
g(x) = A cos(x 2f + )
x
g(x)We calculate discrete values of g(x) for various values of x
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What is spatial frequency?What is spatial frequency?
12/
0
12/
0
/2cos)(Ni
iii
Ni
ii NixAxgxg
g(x) = A cos(x 2f + )
gi(x) = Ai cos(x 2i/N + i), i = 0,1,2,3,…,N/2-1
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We try to approximate a periodic We try to approximate a periodic function with standard trivial function with standard trivial (orthogonal, base) functions(orthogonal, base) functions
+
+=
Low frequency
Medium frequency
High frequency
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We add values from component We add values from component functions functions point by pointpoint by point
+
+=
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g(x)
i=1
i=2
i=3
i=4
i=5
i=63
0 127
xExample of periodic function created by summing standard trivial functions
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g(x)
i=1
i=2
i=3
i=4
i=5
i=10
0 127x
Example of periodic function created by summing standard trivial functions
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g(x)
g(x)
64 terms
10 terms
Example of periodic function created by summing standard trivial functions
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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)
Example of periodic function created by summing standard trivial functions
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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)
Example of periodic function created by summing standard trivial functions
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Main concept – summation of base Main concept – summation of base functionsfunctions
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
Observe two numbers for every i
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Information is not lost when we Information is not lost when we change the domainchange the domain
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
SpatialSpatial Domain
Frequency Domain
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What is spatial frequency?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
Information is not lost when we Information is not lost when we change the domainchange the domain
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EXAMPLE
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A cos(x2i/N)frequency (f) = i/Nwavelength (p) = N/I
N=512i f p0 0 infinite1 1/512 51216 1/32 32256 1/2 2
Substitute values
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 NotationFourier 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)
• G[g(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 SpectrumPower Spectrum and Phase Spectrum
• |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-1[Im(G(f))/Re(G(f))] is the phase spectrum of G(f)– 0.0, -27.12, 145.89, …, 0.0, -145.89, 27.12
complex
Complex conjugate
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1-D DFT and IDFT1-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 time intervals
Equal frequency intervals
1N
0k
nkN2
j;e ]k[x]n[X
1N
0n
nkN2
j;e ]n[X
N1
]k[x
n = 0, 1, 2,….., N-1
k = 0, 1, 2,….., N-1