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: - Microscope vector created by freepik - - Lab room vector created by upklyak: - Semaphore vector created by macrovector: Mathematical animations were done using manim () and matplotlib python libraries. 3D animations were done in Blender