Wavelets: a mathematical microscope
=====
My name is Artem, I'm a neuroscience PhD student at Harvard University.
🌎 Website and Social links: https://kirsanov.ai/
📥 "Receptive Field" neuro-newsletter: https://artemkirsanov.substack.com/
✨ Support me on Patreon to get access to Discord community: / artemkirsanov
=====
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 ).
🕒 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
=====
Special thanks to Crimson Ghoul for providing English subtitles!
Vector assets: freepik.com
Microscope vector created by freepik - https://www.freepik.com/vectors/micro...
Lab room vector created by upklyak: https://www.freepik.com/vectors/lab-room
Semaphore vector created by macrovector: https://www.freepik.com/vectors/semap...
Mathematical animations were done using manim (https://docs.manim.community/en/stable/) and matplotlib python libraries.
3D animations were done in Blender
=====
Disclaimer: This channel is my personal project. The views and content expressed here are my own and are separate from my research role at Harvard University.
#Wavelets #SignalProcessing #SoME2
Description remastered: February 2026. Links & Bio updated; original context preserved.