Wavelets: a mathematical microscope

Wavelet transform is an invaluable tool in signal processing, which has applications in a variety of fields - from hydrodynamics to neuroscience. This revolutionary method allows us to uncover structures, which are present in the signal but are hidden behind the noise. The key feature of wavelet transform is that it performs function decomposition in both time and frequency domains.

In this video we will see how to build a wavelet toolkit step by step and discuss important implications and prerequisites along the way.

This is my entry for Summer of Math Exposition 2022 ( #SoME2 ).
My name is Artem, I'm a computational neuroscience student and researcher at Moscow State University.
Twitter: @artemkrsv


OUTLINE:
00:00 Introduction
01:55 Time and frequency domains
03:27 Fourier Transform
05:08 Limitations of Fourier
08:45 Wavelets - localized functions
10:34 Mathematical requirements for wavelets
12:17 Real Morlet wavelet
13:02 Wavelet transform overview
14:08 Mother wavelet modifications
15:46 Computing local similarity
18:08 Dot product of functions?
21:07 Convolution
24:55 Complex numbers
27:56 Wavelet scalogram
30:46 Uncertainty & Heisenberg boxes
33:16 Recap and conclusion

Credits:
Vector assets: freepik.com
- Microscope vector created by freepik -https://www.freepik.com/vectors/microscope
- Lab room vector created by upklyak: https://www.freepik.com/vectors/lab-room
- Semaphore vector created by macrovector: https://www.freepik.com/vectors/semaphore

Mathematical animations were done using manim (https://docs.manim.community/en/stable/) and matplotlib python libraries.
3D animations were done in Blender