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Fourier Analysis and its Applications. What Is Fourier Series?. A method for solving some differential equations An approximation for a complex function with an infinite sine and cosine series

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Fourier Analysis and its Applications

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Fourier Analysis and its

Applications


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


The General Formula for a Fourier Series

From:”Fourier Series”, University of Hawaii,

http://www.phys.hawaii.edu/~teb/java/ntnujava/sound/Fourier.html


The full rectifier can be approximated with Fourier series.

Full rectifier as the series

From:”Fourier Series”, University of Hawaii,

http://www.phys.hawaii.edu/~teb/java/ntnujava/sound/Fourier.html


The Computational Result


One Dimensional Fourier Transformation

  • An example function:

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


The time series of the function


1

3

2

4

This is the Fourier transformed graph. Four peaks are found in the plot.


Time series

Fourier Transform


Fourier Transform using

Sine Functions

Fourier Transforms using

Cosine Functions


Graph with six sine functions

Graph with six cosine functions


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

2D Fourier Transformation (Image Processing)


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


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


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


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


Putting a frame around the image creates a more accurate Fourier Transform

Transform of original image

Image with the edges covered by a gray frame

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


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


From: “Introduction to Fourier Transforms in Image Processing”,The University of Minnesota , http://www.cs.unm.edu/~brayer/vision/fourier.html


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


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


Two Dimensional Fourier Transform of the data


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


Two Dimensional Fourier Transform with noise


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


Two Dimensional Fourier Transform with Random Noise


Original Fourier Transform versus Transform with Random Noise


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


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