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.
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).
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.
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
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!!!
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 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.
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
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.
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 ! 👏
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'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.
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 :)
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 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!!!!
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!
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!
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!
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.
Came here with nothing but a rough concept of what a wavelet is, gained the insight about it being a kind of convolution, understood its advantages and shortcomings and now I have an intuitive understanding of the uncertainty principle! Wow. The visualization here really is the key to understanding. Thank you very much!
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.
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.
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.
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.
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!
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!
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!
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!
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.
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 😅
The animation is incredible, the content is rigurous yet approachable, the communication approach is intuitive : 11/10!!! I worked with wavelet transform for paleo-oceanography data, and I now I finally understand it. Thank you so much!!
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
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.
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!
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!
wow, this video is the probably the best video out there that talks about wavelet transform. The visuals, the explanations, all are top tier. It not only explains wavelets but explains the need for wavelet transform. Great Job.
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. ❤❤
Keep up the great work. I've watched a few of your videos so far and all of them are so well made and they convey very complex and difficult topics in such a clear and understandable way. PS - just don't get drafted...
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!
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.
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!
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.
Just WOW. Thank you so much for the great job. This is indeed a work of art. This is indeed a work of art; you convey so much information in such a short time and in the best manner. Please do not stop making such videos.
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 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.
Amazing ! I've never thought about the relation between "similarity / the dot product" and "the formula of Fourier / wavelet transformations", this video really illuminates that idea! The animations also helps a lot to explain !
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).
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.
This is brilliant. Thank you, Sensai. Your ability to explain a complex mathematical principle in a way that makes intuitive sense demonstrates, in addition to intellectual mastery of the material, a rare and humane understanding of the way people embrace and incorporate new information.
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
Such an amazing quality of explanations! The small details and animations to clearly show your arguments make this just so much better and superior to anything else!
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
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!
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
This is the best discussion of wavelets I Have seen. Your graphics are in the best tradition of 3B1B.
More please.
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
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).
I hadn’t thought about this but it makes a lot of sense actually. Been looking at a lot of Bode plots recently…
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.
what even is an pln
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
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!!!
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.
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😍!
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.
Hands down the best video on Wavelets. This video packs so much information but in such a succinct & intuitive way, that makes watching it a delight.
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.
As an engineer, I can only regret I was born a bit too soon... how lucky of those who are learning thest things with amazing videos like this!
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
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!
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.
I can't tell you how much I learnt from this one video. Thanks a lot ! Please keep making these videos.
How this is free but explained so well is honestly so humbling. Thank you.
everyone talks about how amazing are the animations, and forget how amazing is the explanation,
Artem Kirsanov is truly a genius
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.
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
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 ! 👏
Thanks man. This kind of matirials makes me want to be a teacher. Thanks so much for this class
What an insightful masterpiece! Such an elegant balance of simplicity, entertainment, and information
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'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.
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 :)
Wow, that's fascinating! Good luck ;)
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.
Me too. Cookie cutters you can exit with a cube can leave many questions.
Thank you for this quick but thorough refresher!
Awesome content and animation! First time I've ever felt motivated to donate. Way more bang for your buck than my vibes classes were
Never expected I am MOVED by a math video explaining wavelets.
Thank you VERY MUCH! I really enjoyed every second of it, and as a graduate student I can really appreciate how insightful this is.
You just saved me at least a full day of stressfully reading huge swaths of the internet. Thank you.
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!!!!
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!
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!
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!
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.
Came here with nothing but a rough concept of what a wavelet is, gained the insight about it being a kind of convolution, understood its advantages and shortcomings and now I have an intuitive understanding of the uncertainty principle! Wow. The visualization here really is the key to understanding. Thank you very much!
It's out! I can't wait to finish it-- the first few minutes is already fantastic!
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 is a MASTERPIECE, thanks for you for the huge effort to come up with such video.
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.
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.
This video was amazing, thank you! The ideas seem very helpful in quantum mechanics as well
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.
I am of science background, but I can only bow. Pls, accept this token. Keep it up for future generation of scientists. L
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!
Thanks. Your video was very valuable.
You’ve truly done the World a great service by putting this together in such beautiful fashion.
I paused the video to say thank you. You unfolded everything, turned it into a comprehensible lesson.
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!
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!
Excellent video! Explained in a really clear and logical way, with impeccable sound design and animations.
It takes a visionary genius to be able to transform complex and abstract mathematical concepts into stunningly beautiful animated works of art - Wow!
You've done an amazing job. By far the best short exposition on wavelets on TH-cam. Please keep sharing your work with us!
I can not explain much and show my gratitude towards your effort explaining such a complex topic that easily. You are genius.
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!
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.
I have been trying to get an intuitive understanding of wavelets for a lot time. This video explained it perfectly!
Fantastic video
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 😅
@@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!
The animation is incredible, the content is rigurous yet approachable, the communication approach is intuitive : 11/10!!!
I worked with wavelet transform for paleo-oceanography data, and I now I finally understand it. Thank you so much!!
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
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.
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!
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!
This is probably one of the best educational videos on youtube. Absolutely superb!
wow, this video is the probably the best video out there that talks about wavelet transform. The visuals, the explanations, all are top tier. It not only explains wavelets but explains the need for wavelet transform. Great Job.
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. ❤❤
Keep up the great work. I've watched a few of your videos so far and all of them are so well made and they convey very complex and difficult topics in such a clear and understandable way. PS - just don't get drafted...
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!
None of my professors back in university has explained things so clearly. Thanks a lot!
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 is a full lecture with a very effective set of graphics. Well done! I think I understood most of it and was never bored or overwhelmed.
Masterfully done. Mindblowing animation, interesting and engaging topic, clear and well-structured script: you've got it all!
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!
The legend is back
Such a brilliant concept. Animation is absolutely phenomenal and makes the content easy to understand.
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.
The relative timing of the different sine waves is represented in the phase of the Fourier transform.
Just WOW. Thank you so much for the great job.
This is indeed a work of art. This is indeed a work of art; you convey so much information in such a short time and in the best manner.
Please do not stop making such videos.
This is incredible! Both, the wavelet transform itself and this amazing video explaining it! :D
the production quality of this video is mind blowing
Просто блестящая работа! Спасибо!
The dot product intutition was mindblowing. It really enhanced my comprehenssion on integral transforms like Fourier, Laplace, etc.
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.
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!
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 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.
I’m blown away by how clear and informative this video was. Nicely done - it’s an inspiration to communicate this clearly.
Masterpiece, as usual. Спасибо!
Amazing ! I've never thought about the relation between "similarity / the dot product" and "the formula of Fourier / wavelet transformations", this video really illuminates that idea! The animations also helps a lot to explain !
Отличная работа! Все понятно и довольно просто, как для введения в вейвлеты. Спасибо за работу!
Valeu!
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).
Thanks!👍
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.
This is brilliant. Thank you, Sensai. Your ability to explain a complex mathematical principle in a way that makes intuitive sense demonstrates, in addition to intellectual mastery of the material, a rare and humane understanding of the way people embrace and incorporate new information.
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
This is the best video I have ever seen not just about signal processing, but about math as a whole. Please, keep on with the amazing work!!
Oh nice topic!
Oh, best topic!
Such an amazing quality of explanations! The small details and animations to clearly show your arguments make this just so much better and superior to anything else!
Oh. My. God. If this video doesn't win SoME2, I'll lose my mind.
What did you use to make the video?
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
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!
No phase in frequency space?!
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!
As an engineer…its embarrassing how videos like this are more clear than hours and hours of class and study. Thank you for this beauty
How about wavelet isit orthogonal matric, like DCT
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
Tbh I would put your channel in 3 Blue One Brown and Veritasium tier. You are truly adding value to TH-cam.