Fourier Analysis What is it? Modern Applications 2D Shape Analysis Jessica Curry PH 313 What is it? Modern Applications Fourier transform (FT) is an algorithm or mathematical procedure used to simplify a set of data in order to bring out reoccurring events that otherwise may not be seen. The equation for the FT of a function is: The FT changes your data from a time domain to a frequency domain, essentially adding up each signal per time to convert into amplitude per frequency. The FT is a summation of cosine and sine waves, which is equal to the real and imaginary parts of a complex exponential. 2D Shape Analysis FT analysis has proven to be very helpful in modern technologies. Retina scanning, fingerprint recognition, mp3 compression, sound filtering and DNA structural recognition are all current applications that rely on FT. (This method works for any data, not just something as function of time). The same FT analysis can be used in 2D and 3D for the same purposes. Below you’ll see an original image of evenly spaced horizontal lines. Next to it is a picture of the FT of the original. As you can see in the FT, only one vertical line exists. This represents the one reoccurring event, which is the repeating horizontal line. You’ll also notice, that when doing FT in 2D the symmetries are maintained in both the original and FT data. For example look at the rotational symetry….. Like below, the original can be rotated 180˚ and you still have the same image. The same can be said for its FT. Pictured above is a typical fingerprint, and to the right is the FT of it. As you can see there are very unique patterns in the FT, and this is true for every set of prints. Imaginary Now lets take a look at a more complex image; pictured below. Again in this example, rotational symmetry exists at 180˚ and is retained by both images. You can also see some light yellow diagonal lines which can be explained by the fact that the original image is not precise, and therefore has diagonal lines reoccurring at a different frequency. (mirror symtry) Real On the left is an image of our DNA double helix structure created with crystallography and on the right is the FT. As you can see, DNA has a very complex and unique structure, which scientists were able verify using FT analysis. 1D Application Sound Filtering Above is a retina and its FT, again you can see the distinct patterns that are unique to this retina. Any other retina will have a different FT image. Below are two graphs, the green is a typical recording from that would be seen from a tuning fork. The graph to the right shows the FT of the original tuning fork graph. As you can see, there is a sharp peak at 481 Hz, which is the frequency that this particular tuning fork resonates at. Another feature you can also see is the smaller peaks that are less significant. These can be noise and are background tones picked up during recording. In order to get the purest recording, you can flat line those frequencies whose amplitude is below a threshold and eliminate the noise. Then you can perform a reverse FT to reproduce the “original” recording, except the tone in the new recording will be purer and lacking background noise. Above is a graph of Facebook breaks-ups over the time period of a year. On the surface you can see some large peaks around spring break, middle of summer, and before Christmas. When perform a FT of this data (shown below), you can see there are two distinctive frequency peaks. One is at a low frequency, which corresponds to the three large peaks on the original graph. The second is at a high frequency, which corresponds to the weekly break-ups that are not as clearly seen on the original plot. References http://www.relisoft.com/science/physics/complex.html http://www.youtube.com/watch?v=ObklYbQaX24 http://www.geekosystem.com/facebook-breakup-graph/ http://blinkdagger.com/matlab/a-super-simple-introduction-to-fourier-analysis/ http://www.dspguide.com/ch8.htm vnatsci.ltu.edu/natsci/physics/labs/C2Lab02_Sound.pdf http://www.gadgetrivia.com/photos/o/24228-best_fingerprint_scanner.jpg http://www.art-dept.com/artists/rankin/portfolio/specialprojects/images/Eye%20Scapes%20-%2001.jpg Amplitude Acknowledgements Thank you to Ryan Stanley for your help with the GI software. Dr. KC Walsh and my minion Frequency