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Spatial Spatial Frequencies Frequencies
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
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
What is spatial frequency?What is spatial frequency?
A
g(x) = A cos(x)
2
x
g(x)
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)
What is spatial frequency?What is spatial frequency?
A
g(x) = A cos(x 2f)
x
g(x)
(1/f)(1/f)
period
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)
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
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
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
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
We add values from component We add values from component functions functions point by pointpoint by point
+
+=
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
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
g(x)
g(x)
64 terms
10 terms
Example of periodic function created by summing standard trivial functions
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
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
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
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
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
EXAMPLE
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
More examples to give you some intuition….
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)
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
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
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