Wavelets: a mathematical microscope 

I agree. My masters-level classes covering Fourier and wavelet transforms were some of the only classes I ever really struggled with and resorted to rote in order to pass. I wish I had these videos to watch in concurrence with those classes. I remember almost nothing from them because I had no intuition about the subjects I was learning. This explanation is so simple and intuitive I actually want to revisit the subject and see what I missed by using a purely mathematical approach without a deeper understanding.

@@Fred-mv8fx 2

so true!

Now imagine him and 3b1b and vsauce work together on a topic

I feel the same

i feel like I was born too late, thats so much to learn even if i learn so much id still be behind 😭

I regret I started to appreciate maths too late

😂 even worse for that Fourier guy

bullshit. as an engineer you have had a lot of money to spare in order to buy cheap bitcoin. meanwhile those of those of us born later had shit and were not able to profit

god, someone watches videos from terror*ssians in late 2022 and likes it

@@romanscerbak5167 what are you even trying to say?

@@romanscerbak5167 I don't see any terrorist here, only a scientist. Just go cry anywhere else.

Thank you! ❤️

@@ArtemKirsanov Thanks for sharing Artem. I really hope you can respond to my other comment when you can. Thanks very much.

@@ArtemKirsanov Hey Artem I hope you can respond when when question about the frequency values when you get a chance. I would appreciate it.. Thanks very much.

What is your major?

@@DannyOvox3 I got master's degree in Computer Science at Roma Tre University; we're using wevelets to analyse BGP anomalous traffic

@@samuelequinzi3153 Oh wow, I am going for a CS degree. I know is a masters level where you are at but these topics seem alien to me, I thought this was related more to electrical engineering.

@@DannyOvox3 this goes far deeper. As you drill down through the sciences in search of core truths, You will find that it all leads You to this, the Key to understanding this existence.

@@DannyOvox3 You'll be surprised how much math is there in CS. CS is not the exact same as software engineering.

Wow, thank you so much!! I really appreciate it

I have to say, the subtle effects were a major negative for me - good video though!

@@laurenpinschannels I definitely love those aesthetic subtle effects

Wow, that's fascinating! Good luck ;)

Me too. Cookie cutters you can exit with a cube can leave many questions.

Thank you! I really appreciate it! Well, I’m doing research in the Laboratory of Extrasynaptic signaling, led by Dr. Alexey Semyanov in Moscow, so I’d say I have really great supervisors ;) I’m using Wavelet transform in my work to write code for extraction and analysis of theta rhythms, recorded from hippocampus in freely moving mice. (We are currently preparing a publication on this topic, and I really hope it will be out in a few months) But surely writing a video script requires a lot of additional research. I feel like only after making the animations and going through the process myself, I can finally understand wavelet transform much better, even though I’ve been routinely using it for almost 2 years now 😅

@@ArtemKirsanov Best of luck to you, looking forward to seeing what comes next

@@ArtemKirsanov same for me in my thesis using wavelets. Your animations are amazing!

I was wondering the same thing 🙂

There's two major differences between wavelet transforms (WT) and windowed FTs (say STFT/DFT) that I would highlight, along with their practical implications. 1) First is the multiresolution, stemming from the non-static frequency-time windows (as you've mentioned). Of course, the obvious benefit is that we can collect more time information at frequencies too high to care about discerning accurately instead of simply dropping all that info, which is great for something like any audio processing with a human auditory factor in it, or anything produced by an animal. But the biggest application is that do all sorts of multiresolution analysis like analysing rapidly changing frequencies without having to run FFT several times per frequency or narrowing your frequency as to lose time information. As it turns out, there's a huge amount of nonstationary signals out there in the real world that this perfectly solves. For example, you need to detect gravitational wave which produce a characteristic chirp. Windowed FTs really struggle with these since the output spectrogram ranges from "some ringing artefacting" to "it's literally smaller than my window size". Maybe it shows up somewhat alright, but you may lose some complex features along the way. But if you look at your WT's scalogram, you get a really nice curve, a distinct and empirically detectable feature. This actually works really well for all sorts of transients like discontinuities which may go undetected with windowed FTs. This is great for fault detectors. And detecting and characterising heart irregularities or complex brain wave features. (Technically, there are multiresolution windowed FTs. One of these was a STFT variant called the Constrant-Q transform, developed before wavelet transforms kicked off in full power around the 2000s. In actuality, this is really close to a modern WT, but had certain downsides that come with a less developed understanding of wavelets, like the difficulty in inverting your signal back and some of the jankery that comes with STFTs.) 2) The second is the ability to use different wavelets. This is a much more powerful tool than you would expect. Certain mother wavelets are well suited for certain applications, such as Ricker wavelets for superior seismic processing, or Daubechies for closely spaced features and DWT. A lot of work has been done here, so you have a pretty big toolbox for hotswapping wavelets for your needs. The coolest thing is that you can design your own wavelet tailored for pattern matching your known signal or picking out the set of features you want. Side note (DWT): It's worth noting that there are currently two major categories of WTs, being continuous wavelet transforms (CWT) and discrete wavelet transforms (DWT). Most discussions are implied to be around CWT, since it simply works for both continuous and discrete signals, but DWT offers a whole set of other applications. As you can guess, convolution can be an expensive operation. You are comparing every point of some decently long wavelet to an equal number of points, which is done across every point of the input signal. Sure, you can do some optimisations using FFT itself or adjust your wavelet parameters, but CWT is still generally slow enough that you just can't do certain things with it. Not to mention that its extra redundancy (which windowed FTs also have to some extent) leaves some to be desired for speed and memory performance. The DWT family of algorithms uses a different approach from raw convolution, instead using a fixed set of child wavelets like a filterbank. It loses its redundancy, limits it to certain mother wavelets, and locks it to frequency-time windows to powers of 2. In exchange, it gains better speed and memory performance in a purely discrete environment, allowing it reach its full practical potential. It turns out that this is often enough (or even ideal) for many digital computing applications. The perfect reconstruction with no redundant information makes it an excellent choice for audio/image compression or performant denoising of images. You'll also find it used in real-time applications where CWT just isn't built for, but require multiresolution that FFT can't provide. Damn, that was a long comment.

@@MrSonny6155 This was very informative, thank you!

The relative timing of the different sine waves is represented in the phase of the Fourier transform.

Thank you!! The basis for animations was done in manim and matplotlib python libraries and Blender for 3D surfaces. Then everything was synced and composed in Adobe After Effects

Oh, best topic!

@pyropulse It have to do with uncertainty if we are taking about uncertainty as given in Information theory tho. Its not exactly the same thing as we consider uncertainty in real life, but what it really says is about information entropy.

It's very similar to the waterfall spectrograms in audio software (spectrum analyzers). It just shows you the Fourier transform of the X recent milliseconds of the audio signal, so the frequency definition of a pure sine wave will be limited to the inverse of the chunk length in time. Wavelet transform does basically the same in a mathematically smarter way (convolution instead of Fourier transform, though they are very related) using the optimum window shape, which allows for the "dynamic" trade-off between time and frequency resolution. In a simple waterfall-plot spectrum analyzer this trade-off is fixed and defined by the chunk length.

You're absolutely right - there is definitely a very important notion of phase both in the case Fourier transform and Wavelet transform (computed as the angle of the resultant complex number). I didn't really have the time to mention this in the video, not to make it too overwhelming. But the Morlet wavelet, being a complex function, has amazing capabilities of dealing with phase of the oscillations. One example of such is the Cross Wavelet Analysis, which allows us to compare two signals and study the relative phase shifts. Thank you for pointing this out!

If you are using discrete wavelet transform (DWT), then the wavelets of different scales indeed form an orthogonal basis. The key difference of DWT, compared to the continuous wavelet transform (which I showed in the video), is that the scale parameter (a) can be varied only discretely, to make sure that wavelets of different scales are orthogonal. It depends on the particular application and what type of wavelet you are using. For example, the Morlet is a continuous one, while many other wavelets (such as Haar, Daubechies) are used only in the discrete case