tags: math,scientific computing, computer vision
What is Fourier transform
The best way to layout Fourier transform is to start with the formula Let’s look deeper in the formula, first thing we have Euler’s formula here , Euler’s formula represents rotation on a unit circle in the complex plane, so we can deduce that we are rotating a unit vector and we are also scaling it with the magnitude of the function at that point.
This would give us the center of mass of the object - that the vector that is rotating and scaled by the function - formed. They decided to remove the division for the center of mass since we are not really interested in the center of mass itself, rather in a value that represents the value of strength of presence of this frequency
Or it gives the measure of correlation between the signal and the revolution around the unit circle. That’s why during cos and sin signal transform, they are two rotations of the unit circle but with different directions/phases. TODO: elaborate more.
We can move a function to the frequency domain and and then do some computations and revert it back to to the time domain using the inverse formula:
Time-frequency uncertainty principle
Why do functions defined in a small range have a really broad transform transform, well, just imagine you defined a wave that is in a unit range, and that function actually does not go through 1 in number line, instead it goes through 0.9999999. So you can’t really tell its frequency from observing for a short period, you can’t be certain really, maybe after some long time you will realize that there was really small frequency shift.
The same thing happens in Fourier transform, the more you measure the wave, the more you are sure about its frequency components