1 Spatial Frequency or How I learned to love the Fourier Transform Jean Baptiste Joseph Fourier.

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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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A

g(x) = A cos(x)

2

x

g(x)

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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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A

g(x) = A cos(x 2f)

x

g(x)

(1/f)

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g(x) = A cos(x 2f + )

x

g(x)

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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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g(x) = A cos(x 2f + )

x

g(x)

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gi(x) = Ai cos(x 2fi + i), i = 0,1,2,3,...

x

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gi(x) = Ai cos(x 2i/N + i), i = 0,1,2,3,…,N-1

f=i/N

0 N

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gi(x) = Ai cos(x 2i/N + i), i = 0,1,2,3,…,N-1

f=i/N

0 N

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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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i=N/2-1i=0


Lowest frequency Highest frequency

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i=N/2i=0


Lowest frequency Too high Redundant frequency

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12/

0

12/

0

/2cos)(Ni

iii

Ni

ii NixAxgxg

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+

+=

Low frequency

Medium frequency

High frequency

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+

+=

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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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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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+

+=

Low frequency

Medium frequency

High frequency

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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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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

1 Spatial Frequency or How I learned to love the Fourier Transform Jean Baptiste Joseph Fourier.

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