Fourier Analysis and its Applications

1. Fourier Analysis and its Applications

2. 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

3. The General Formula for a Fourier Series From:”Fourier Series”, University of Hawaii, http://www.phys.hawaii.edu/~teb/java/ntnujava/sound/Fourier.html

4. 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

5. The Computational Result

6. One Dimensional Fourier Transformation • An example function: • The test function has four different frequencies and these generate several periods as a wave function.

7. The time series of the function

8. 1 3 2 4 This is the Fourier transformed graph. Four peaks are found in the plot.

9. Time series Fourier Transform

10. Fourier Transform using Sine Functions Fourier Transforms using Cosine Functions

11. Graph with six sine functions Graph with six cosine functions

12. 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)

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

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

20. 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

21. 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

22. Two Dimensional Fourier Transform of the data

23. 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

24. Two Dimensional Fourier Transform with noise

25. 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

26. Two Dimensional Fourier Transform with Random Noise

27. Original Fourier Transform versus Transform with Random Noise

28. 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