Fourier analysis and its applications

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Fourier Analysis and its Applications
D. McLean Snyder III
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What Is Fourier Series?
A method for solving some differential
equations An approximation for a complex
function with an infinite sine and cosine series
A foundation of Fourier Transformation which is
used for various analyses such as sounds and
images From Elementary Differential
Equations and Boundary Value Problems(Ninth
Edition), William E. Bryce and Richard C. Prima,
John Wiley and Sons, Inc. 2009
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The General Formula for a Fourier Series
FromFourier Series, University of
Hawaii, http//www.phys.hawaii.edu/teb/java/ntnuj
ava/sound/Fourier.html
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The full rectifier can be approximated with
Fourier series.
Full rectifier as the series
FromFourier Series, University of
Hawaii, http//www.phys.hawaii.edu/teb/java/ntnuj
ava/sound/Fourier.html
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The Computational Result
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One Dimensional Fourier Transformation

• An example function
• The test function has four different frequencies
  and these generate several periods as a wave
  function.

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The time series of the function
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1
3
2
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This is the Fourier transformed graph. Four peaks
are found in the plot.
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Time series
Fourier Transform
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Fourier Transform using Sine Functions
Fourier Transforms using Cosine Functions
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Graph with six sine functions
Graph with six cosine functions
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2D Fourier Transformation (Image Processing)

• One of the most popular uses of the Fourier
  Transform is in image processing.
• Fourier Transforms represents each image as an
  infinite series of sines and cosines.
• Images consisting of only cosines are the simplest

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Cosine Image and its Transform
The higher frequency colors on each image
generate the patters of dots in their Fourier
Transform.
From Introduction to Fourier Transforms in
Image Processing,The University of Minnesota ,
http//www.cs.unm.edu/brayer/vision/fourier.html
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For all REAL (not imaginary or complex) images,
Fourier Transforms are symmetrical about the
origin.
From Introduction to Fourier Transforms in
Image Processing,The University of Minnesota ,
http//www.cs.unm.edu/brayer/vision/fourier.html
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What happens when you rotate the image? The
Fourier Transform creates a much more complex
image. What causes the shaped vertical and
horizontal components?
From Introduction to Fourier Transforms in
Image Processing,The University of Minnesota ,
http//www.cs.unm.edu/brayer/vision/fourier.html
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Fourier Transforms are INFINITE series of sines
and cosines. The edges of the arrays affect each
other.
From Introduction to Fourier Transforms in
Image Processing,The University of Minnesota ,
http//www.cs.unm.edu/brayer/vision/fourier.html
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Putting a frame around the image creates a more
accurate Fourier Transform
Image with the edges covered by a gray frame
Transform of original image
Transform of gray framed image
Actual transform of original image framed image
From Introduction to Fourier Transforms in
Image Processing,The University of Minnesota ,
http//www.cs.unm.edu/brayer/vision/fourier.html
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Effect of noise on a Image
From Introduction to Fourier Transforms in
Image Processing,The University of Minnesota ,
http//www.cs.unm.edu/brayer/vision/fourier.html
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From Introduction to Fourier Transforms in
Image Processing,The University of Minnesota ,
http//www.cs.unm.edu/brayer/vision/fourier.html
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Fourier Transforms of more general images have
very little structure
The more symmetrical baboon has a more
symmetrical Fourier Transform
From Introduction to Fourier Transforms in
Image Processing,The University of Minnesota ,
http//www.cs.unm.edu/brayer/vision/fourier.html
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Data set for a two dimensional map
0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 100, 100,
100, 100, 0, 0, 0, 0, 0, 0, 100, 100, 100,
100, 0, 0, 0, 0, 0, 0, 100, 100, 100, 100, 0,
0, 0, 0, 0, 0, 100, 100, 100, 100, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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Two Dimensional Fourier Transform of the data
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Data set for two dimensional map with noise'
around the edges
50, 50, 50, 50, 50,
50, 50, 50, 50, 50, 50, 0, 0, 0, 0,
0, 0, 0, 0, 50, 500, 0, 0, 0, 0, 0,
0, 0, 0, 50, 50, 0, 0, 100, 100, 100, 100, 0,
0, 50, 50, 0, 0, 100, 100, 100, 100, 0, 0,
50, 50, 0, 0, 100, 100, 100, 100, 0, 0,
50, 50, 0, 0, 100, 100, 100, 100, 0, 0,
50, 50, 0, 0, 0, 0, 0, 0, 0, 0,
50, 50, 0, 0, 0, 0, 0, 0, 0, 0,
50, 50, 50, 50, 50, 50, 50, 50,
50,50, 50
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Two Dimensional Fourier Transform with noise
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Data set of a Two Dimensional map with random
numbers
49, 29, 13, 69, 39,
62, 03, 97, 0, 44, 18, 4,46,66, 41,
39, 44, 57, 27, 59, 26, 30, 98, 74, 88,
89, 84, 1, 98, 46, 0, 40,35, 100, 100,
100, 100, 76, 4, 48, 98, 15, 46, 100, 100,
100, 100, 34, 55, 86, 73, 29, 40, 100, 100,
100, 100, 35, 34, 9, 7, 61, 99, 100, 100,
100, 100, 40, 67, 61, 25, 77, 53, 84, 72,
63, 18, 13, 69, 31, 81, 52, 20, 91,
76, 63, 6, 8, 23, 73, 21, 59, 76, 68,
79, 44, 20, 48, 53, 19
Values used came from the middle two terms of
phone numbers from a random page in the telephone
directory
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Two Dimensional Fourier Transform with Random
Noise
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Original Fourier Transform versus Transform with
Random Noise
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Summary

• Fourier series and transformation are used for
  various scientific and engineering applications,
  such as heat conduction, wave propagation,
  potential theory, analyzing mechanical or
  electrical systems acted on by periodic external
  forces, and shock wave analysis