Ian Thomas

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???????????????????????? Ian Thomas
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The Fourier transform
www.univ-tln.fr
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The Fourier transform has many applications.
http//www.intmath.com/Fourier/6_linesp.php
http//www.arahne.si/tutorial22.html
http//www3.ocn.ne.jp/hanbei/eng-f80spec.html
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http//btc.montana.edu/messenger/instruments/mascs
.htm
And more applications.
http//mediabyran.kib.ki.se/projects/cns/mri/mri-p
ictures/complete.jpg
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The meaning of the Fourier transform
A
f0
t
f
3f0
f0
-f0
3f0
-3f0
-5f0
5f0
5f0
A function is split up into its frequency
components by the Fourier transform.
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Waveforms consist of harmonics of the fundamental
added together.
Waveadd
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The Fourier transform is reversible.
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An example of a function and its Fourier
transform.
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???????????????????????? Ian Thomas
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F(u)
u
Re(F(u)) Im(F(u))
u
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A function in two dimensions
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???????????????????????? Ian Thomas
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Fourier Transform in 2 Dimensions
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???????????????????????? Ian Thomas
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Some examples of Fourier transforms in 2
dimensions
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Two-dimensional Fourier transform pairs
Image
Power spectrum
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Two-dimensional Fourier transform pairs
Image
Power spectrum
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???????????????????????? Ian Thomas
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Properties of the Fourier Transform for Image
Processing
http//www.ph.tn.tudelft.nl/Courses/FIP/noframes/f
ip-Properti-2.html
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???????????????????????? Ian Thomas
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The Fourier transform can be written in terms of
its magnitude and phase.
A(u,v)
a(x,y)
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???????????????????????? Ian Thomas
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You cannot reconstruct an image from the
magnitude of its FT, but you can from the phase
of its FT
Reconstructed from phase
Reconstructed from magnitude
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The FT and the Inverse FT are linear operations.

Image

g(x,y)
f(x,y)
f(x,y) g(x,y)
Power spectrum


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If you expand the image, its Fourier Transform
gets smaller.
A(u,v)
a(x,y)
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???????????????????????? Ian Thomas
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If you rotate and image, its FT rotates too.
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???????????????????????? Ian Thomas
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The mean of an image is the value a(0,0).
Mean(a(x,y))
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???????????????????????? Ian Thomas
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You can measure the energy of an image in either
normal space or spatial frequency space
(Parsevals Theorem).
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???????????????????????? Ian Thomas
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You can differentiate using the FT.
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???????????????????????? Ian Thomas
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You can shift the FT by multiplying the image by
a phase term.
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???????????????????????? Ian Thomas
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Convolution in one domain is equivalent to
multiplication in the other.
Spatial domain
Spatial frequency domain