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.
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
Man please don’t ever stop making these videos, they are extremely well done and edited and very entertaining while magnificently informative for such complex topics!!!
This is like a third of a semester of intro to signals processing in computer science curriculum, packed into one half-hour video, and I actually understood more from it now than I did from lectures. Huge thanks for doing this! For those who wonder whether to watch: notable things include good mental models for complex numbers, Fourier transform, convolution and its relationship with vector dot product and functions as infinite-dimensional vectors, with an unexpected cameo from Heisenberg's uncertainty principle. This video is gold.
I can't imagine the amount of work that has gone into this masterpiece of a science yt video❤️🔥 Thank you very much, more content like this is needed😍!
@@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.
This video literally blown my mind about wavelets. There're been several weeks of works studying wavelets (in the discrete domain) for the work of my thesis. So far, more or less I have all the concepts explained in the video clear, but the amazing graphic representation of the signals and wavelets in the video, and also of the entire process of wavelet analysis almost filled all my remaining gaps! This video is incredible to understand wavelets!
@@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.
The subject is highly interesting. On top of that your video is amazing with all details. The music is very quiet but "opens" the space, the subtle effects on "static" graphs that make them dynamic, the not-so-subtle but entertaining and functional use of sound effects and the use of special effects in manim make this very nice to watch. I've played around with manim a bit and can only imagine how much work this must've been, holy heck.
This made perfect sense after knowing only a few things about wavelets, convolution and Fourier transform. I am in disbelief that I sat through the whole video and it was quite captivating with smooth transitions and excellent presentation. I won't even mention the animations, it was like jigsaw fall into place. I wish all engineering concepts was explained this way. Thank you a lot, keep up the good work!
This is absolutely mind blowing - especially when you bring in the complex wavelet. The gradual addition of concepts is extremely well done, and everything is well explained.
Holy moly, with my startup, I am working on an image analysis project collaborating with hospitals in Spain and the next steps on the project are similar to what you just showed to us. You just gave me more ideas to test and your visualizations are the best! (it reminds me of 3b1b videos) I will send you some results as soon as we finish it :)
I've been using and teaching the mathematics of the Fourier Transform, but had never seen a description of wavelets. This is a beautifully done presentation. And, I can tell that once I get to correlation and convolution with my students, they will really enjoy this description of wavelets. To be human and alive these days is just amazing. We are truly in an explosive evolutionary phase of information. And, I am astounded by how many people are so incredibly intelligent, able to describe complex topics so well, and willing to take the time to make these beautiful animated videos.... I wish I could give 1000 thumbs up.
This is literally the best video on yt discussing about wavelets and wavelet transforms along with equally good visualizations and animations. I mean I don't need to write anything more. Just watch it and know for yourself. You wont regret watching it. Read the other comments and it will tell you how good the video actually is. The amount of effort put into the video is commendable. I would like to end this comment saying that I actually learnt more from this half hour video than going through hours of lectures. Huge respect and love from India!
So many details touched upon, such clear imagination of the underlying geometric intuition. So many little programs written to produce those graphs, diagrams, and visualizations. So refreshing to not rely on the Haar wavelet for an introduction to the topic. This video is going to leave many lasting memories in many minds. I am in awe.
I can substitute this video for entertainment during meals, it's that simplified and fun to watch. Thank you for such videos. I hope your channel grows and reaches wider audience!
27' 33" is just gorgeous. It's wonderful to see visualisation tools that were undreamed of when I was studying Maths in the 1970s, and how expertly people like you can use them.
One of the best videos I have ever seen, and the best explanation of wavelet transform on the internet. I can't imagine how many hours of work you invested here , but it tells a lot about your passion on knowledge sharing. Kudos ! 👏
One misconception about the Fourier transform: it is *not* completely blind to time (cf. 7:17). If that were the case, the inverse Fourier transform would be ill-defined. However, *any* signal can be transformed into a Fourier representation and back to the *exact* original signal again, with all time information intact! It doesn't change to a different time series signal with the same frequencies present after this back-and-forth transformation, but it *always* remains the same signal no matter how many times you transform it. This means, that time is somehow present in the Fourier transform. What this video forgets, like many people, is that Fourier is not only a magnitude spectrum (as often visualized, like here). The result obtained from a Fourier transform, analyzing a time series signal, is first and foremost a series of complex numbers (which then usually get transformed into a magnitude spectrum). Within this complex number result, you have both the magnitude (the length of each complex number, so to speak) and the *phase* (the angle of the complex number with respect to the real-value axis). So, time is very much present in Fourier, but it is "hidden" in the phase information. I completely understand that phase is a very abstract and complex way of thinking of time for us humans. Phase by itself doesn't tell us anything. Only in combination with the magnitude spectrum can we understand how each frequency present overlaps with the others, which creates the very specific signal that we analyzed previously. Think of the spectrum of an impulse: it ideally has *all* frequencies present, because they need to overlap so precisely that they all cancel each other out at every point in the time series signal apart from where the actual impulse happens. It is essentially an incredibly short white noise burst. Yet, in the time series signal, all you see is a spike in an otherwise silent signal. No indication of oscillating frequency anywhere to be seen. More importantly, what this video is talking about in regards to the time frequency duality is not the Fourier series, but the problem we face when we want to generate *spectrograms*. Here we take tiny sub-windows of our signal and analyze each using the Fourier transform. As the frequencies in the original time series fade in and out, each window potentially contains different frequencies or a different proportion of the frequencies. The problem is, that the discrete Fourier analysis is only as high in resolution, as the window is big: a bigger window (more time series samples) results in the same (bigger) amount of frequency bins (more frequency values represented individually). But, the bigger the window, the bigger the time frame that is hidden in the phase of this analysis, which we normally just throw away, since we only look at the magnitude spectrum anyway. Again, as long as you keep the phase information for every window, you can reverse an STFT (basically a spectrogram analysis) and receive the original signal again, which shows that time information is maintained (as long as you do not throw information like phase away).
Finally a video that uses manim without being a 3b1b clone. There's clearly a distinct personality here through the sound effects, fonts, and animations. Thinking about the "personality" of your math explainer is important, but unfortunately is neglected often.
Wow! This brought together so many complicated topics so seamlessly and intuitively. Was more engaging than a Netflix episode. So good! Thanks for all your efforts!
I literally just burst out with a loud "whoa" at 21:14, about the insight of similarity as captured by the inner-product interpretation of the integral. This video is too well done!!!!
You can tell how much thoughtfulness went into every visualization here. For example, during the dot product explanation the value of the dot product was mapped onto the distance of the angle marker from the origin, and scaled such that the right angle location made a perfect square. Little touches like that were abound in the video and really help drive home intuition. Every bit of information was there for a reason!
More than 7 years after I first learned about the Fourier transform, this video finally helped me wrap my mind around Wavelet transforms. I GASPED when the time/frequency/amplitude graph at 23:02 came up - everything just clicked. Really great presentation!
Good introduction to wavelets! But you give the Fourier transform too little credit :).. It DOES contain information about the time sequence/"order" of the frequency components, after all it's a "dual" representation of the time-domain signal, right? That temporal order is contained in the spectral phase - and that's what most people miss about the Fourier transform, since they only plot the magnitude (or power) spectrum but forget about the phase and lose half of the contained information (which happens to be about the timing order).
This video is so absolutely incredible, I'm in awe. Your script, your animations, your understanding and explanation of the mathematics... This is a masterclass in education videos.
This has to be the most well edited video I've ever seen. Can't imagine watching this all for free. Thank you so much for your contribution towards Science.
i finaly came back after watching both 3b1b video about Fourier Transforms and Convolutions and now i can appriciate this goldmine of a video :D explained amazingly here
Excellent. You have an intuitive sense of pace and information that keeps the viewer fascinated and intrigued. This video alone should be mandatory viewing in any university's physics or electronics courses, and I hope you follow it up with others in the same vein.
This video is incredible! High production value and amazingly clear explanations! Not enjoying this kind of math is almost impossible after watching your beautiful video!
As someone with a background in signal processing: amazing video, explanation wise as well as animations. I wish that would have been the introduction at university. beautiful work!
This is just amazing. The level of detail in this is just baffling. Keep it coming. Your videos are scintillating. I have read wavelet transforms back when i was in Undergrad but this level of detail, wish I had known these intuitive interpretations behind this. All the best to you. This made my day.
You are my favorite channel I’ve found all year. The production and information value of your videos is absolutely unheard of. Please keep doing this, it’s an incredible contribution to the informational commons!
I don't know where you got this stuff, but it is in an important respect incorrect. The Fourier transform of a function, with certain technical qualifications, has complete information about that function. The Fourier transform contains both amplitude and phase information about the sine-like functions it refers to (alternatively, amplitude information about both the sine and cosine functions at each frequency); this gives it the power to represent the function it is a transform of completely accurately (in the L2 sense), including complete information about when the various parts of the function occur. (The Heisenberg uncertainty relation is related to the inability of a (non-trivial) physical wavefunction to have both position and momentum amplitudes both non-zero at only one value, or even a range of values whose product is less than a certain physical constant, which is important in quantum mechanics.) The rest of the video, about wavelets, may be mostly accurate and useful; I don't have an evaluation of that, since I'm not familiar with that subject and don't want to spend more time on it in the near future.
For my aerospace engineering undergrad I had to take a class on wireless communication systems. It was all about how we encode data in a radio wave and I always had a really hard time getting an intuitive understanding of how the very abstract concept of taking this matrix of data, doing a convolution with an encoding wave form and some carrier frequency and then just feeding the result of that as a voltage to an antenna would result in a radio wave that when received by another antenna we could just reverse the process and get the data back out no problem even in really really high noise environments. It was really hard seeing how the very abstract concepts of linear algebra could correspond so directly to these real tangible systems. This video really helped clear some of that up and gives me a better mental picture of what is happening there so thank you!
Dear Artem, your didactics is a kind of world reference of Distance Education. In times of Engineering graduation I asked for a hardcopy of some papers from Bethesda Naval Warfare Center in 1992. I could understand every Transfom expressed there, including FFT, DCT, Eigenvectors (Principal Components) , Hadamard, FIR, IIR, MPG Layer 3, JPG, work signed by Sanjit K Mitra and Charles D Creusere, a great work I think, but just today I could understand the Wavelets. Thank you very much for your precious work! Amazing!
Outstanding explanation ever!!!! I have never come across something this clear. Please don't ever stop making such videos please you are helping mankind to grow at multiple dimensions. I support your work from my heart. ❤❤
You are unparalleled. I have never seen such a master piece on youtube. Please continue the noble efforts. Hope you will make more videos sooner than later . stay no blessed
Wow man what a video! Can't imagine how much work must have gone into producing such a great explanation of such an interesting and useful technique, really really good job. I'm currently doing a PhD in systems neuroscience and your videos like this really make me feel like I need to up my game when it comes to learning complex topics like this. Convinced I'll find the technique or insight that makes my work next level from this channel, I can't wait to go look into how this has been used. Is this all self researched or do you have a seriously top notch neuroscience professor somewhere?
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 😅
I cant believe how clearly and well you covered some of the most complex topics of math such as Orthogonal Functions and the fourier transform. I feel like you explained it in such an intuitive fasion that anyone, even with a very crude math backround, could comprehend it.
The bit at the end where you talk about the wavelet transform's adaptive uncertainty is neat, and explains something I was wondering about the entire time -- how is the wavelet transform different from a time-windowed Fourier transform? This seems to be the answer! Because the support of a wavelet varies over frequency, unlike the static window size of a windowed FFT, you can get more information where it matters.
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.
As a french junior engineer working in vibration analysis, I want to thank you for this wonderful explanation of the wavelet transform, the mathematical animations were so good and beautiful. This Wavelet transform can be such a powerful tool for vibration analysis and signal processing in general.
This video is an absolute masterpiece! Not only do you clearly have a gift when it comes to explaining things, but you clearly have an amazing work ethic as well. I can't even imagine how much effort must've gone into making all these gorgeous animations! Definitely gained a subscriber!
Great video, really appreciated the explanations and cool animations! I've been wanting to understand this topic for a while but couldn't quite get my hands on as I'd like, so this served as a great push. I'm getting close to using this technique in my work (not neuroscience though), so this was a nice way of getting a bit of contact with the topic before having to go deeper in the subject. It's funny that I found you a while ago by your videos about Obsidian and Zotero and didn't know you did videos like this one, now I'm definitely subscribed. Keep up the great videos!
When you show the individual convolution around 21:00 the top graph is a little bit out of phase with the product graph. Just thought I would mention it... Anyway this is an awesome work, best I've seen to date and I too like a few others have been struggling to really understand this. I totally got Fourier when I started this study of wavelets months ago now. Never, without these pictures would I see how the two signals resonate It's just such a great contribution for the rest of us who truly want to understand. thank you, thank you, and thank you again!
Amazing video! Just a remark about 5:25 : it's not that we lose sense of time, rather that the decomposition gives us pure sine waves, meaning they stretch from -inf to inf.
This is indescribably well explained, I can't thank you enough for this feat! I have been looking into this subject for some time every once in a while, but could never accomplish something that could be honestly called a grasp on this matter. My work is related to non-destructive testing and the analysis of acquired signals, so, Wavelet Transform can obviously very much enhance the way in which we handle the signals, store them and derive useful information from them. I know that medical ultrasonics is relying heavily on such signal processing, like IQ-demodulation for the sake of Doppler-measurements of blood stream velocity differences. Applied to non-biological targets, we are dealing with different challenges, but Wavelet Transform is bound to improve the way we handle ultrasonic echoes, once we get to harness initial successes on this path.
This video is the best summary of the Fourier Transform I've ever seen, it's given me greater insight into what it even means, and what it's transform cousins are really about.
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
lol this is such a hilariously ambitious video, summarizing so many semesters of grad school stuff into one video. good job man, I hope you get whatever you're aiming for
8:00 I don't think that uncertainty in the time domain would mean that you're not sure what a value is at a given moment. Rather I see it as when you take a fourier transform of a signal that is defined over a long period of time, it will have a more specific fourier transform. Think of a cosine in the time domain which translates to a delta function in the frequency domain. This function is defined at exactly one value for the frequency. Therefore we observe that the longer and less determined a signal is in the time domain (cosine's domain extends from minus infinity to infinity) , the more determined it gets in the frequency domain and visa versa. The problem therefore is that when you take a fourier transform of a too short signal, that the frequent domain will start to show less specifically which frequencies are contained. That's the trade off you need to make.
@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.
Thanks for nice content dude.I would like to know how to learn this complex topics in neuroscience, math, programming and have one of the best video compositions (about the visual effects and aesthetics as whole)I ever seen on youtube
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
I have been wondering about using wavelets for my analysis and this video really helped me to concretely understand what I i could do with it. Спасибо Артём! Я даже не могу передать насколько это мне помогло❤
