But what is a convolution?

Discrete convolutions, from probability to image processing and FFTs.
Video on the continuous case: • Convolutions | Why X+Y...
Help fund future projects: / 3blue1brown
Special thanks to these supporters: 3b1b.co/lessons/convolutions#...
An equally valuable form of support is to simply share the videos.
Other videos I referenced
Live lecture on image convolutions for the MIT Julia lab
• Convolutions in Image ...
Lecture on Discrete Fourier Transforms
• What is a Discrete Fou...
Reducible video on FFTs
• The Fast Fourier Trans...
Veritasium video on FFTs
• The Remarkable Story B...
A small correction for the integer multiplication algorithm mentioned at the end. A “straightforward” application of FFT results in a runtime of O(N * log(n) log(log(n)) ). That log(log(n)) term is tiny, but it is only recently in 2019, Harvey and van der Hoeven found an algorithm that removed that log(log(n)) term.
Another small correction at 17:00. I describe O(N^2) as meaning "the number of operations needed scales with N^2". However, this is technically what Theta(N^2) would mean. O(N^2) would mean that the number of operations needed is at most constant times N^2, in particular, it includes algorithms whose runtimes don't actually have any N^2 term, but which are bounded by it. The distinction doesn't matter in this case, since there is an explicit N^2 term.
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Italian: Emanuele Vezzoli
Vietnamese: lkhphuc
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These animations are largely made using a custom python library, manim. See the FAQ comments here:
www.3blue1brown.com/faq#manim
github.com/3b1b/manim
github.com/ManimCommunity/manim/
You can find code for specific videos and projects here:
github.com/3b1b/videos/
Music by Vincent Rubinetti.
www.vincentrubinetti.com/
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
Timestamps
0:00 - Where do convolutions show up?
2:07 - Add two random variables
6:28 - A simple example
7:25 - Moving averages
8:32 - Image processing
13:42 - Measuring runtime
14:40 - Polynomial multiplication
18:10 - Speeding up with FFTs
21:22 - Concluding thoughts
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with KZbin, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
Various social media stuffs:
Website: www.3blue1brown.com
Twitter: / 3blue1brown
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Instagram: / 3blue1brown
Patreon: / 3blue1brown
Facebook: / 3blue1brown

[@reillymcdowell]

Ok… the fact that a 23 minute video about an advanced math topic is #42 in the youtube trending charts is incredible and represents the sheer explanatory power of 3B1B. Congratulations, Grant!

[@fredinit]

That gives me hope for the masses.

[@XTSonic]

@@fredinit I feel countries should be buying the rights to many of these videos to use in classes. The way 3b1b can help students understand math and make it more interesting is valuable and his view counts show that. Many of his videos definitely have their place in high school or university and it would make more sense to use that than some grimy old book if the goal of a country/university is to educate and engage, with 3b1b well remunerated for the licensing sales to all these potential buyers combined.

[@shaunoconnell9506]

I’d like to see the demographic breakdown on videos like this. It would be interesting who’s interested in learning

[@Cyclone1024]

To be fair a lot of engineers and engineering students are really familiar with convolutions, making this video more approachable for a lot compared to any other “advanced math topic”

[@ryuusensei8677]

how do you know that position (#42)? :0

[@lioreshkhar2752]

Anyone else would love to see a Stats series from 3b1b? I feel like that's a topic that's so often very confusingly and unintuitively explained.

[@djredrover]

No stats. lol I hate stats. Actually, maybe the reason I hate it is because I don't fully understand it. Soo... yeah! A 3b1b stats series would be AWESOME! PS. Are you doing a Duality section on the transforms? Don't get it to this day.

[@SiddharthPrabhu1983]

I'd recommend the channel jbstatistics. Brilliantly explained statistics fundamentals.

[@glokta1]

@@SiddharthPrabhu1983 +1. I used it when I took AP Stats and it was wonderful, especially the hypothesis testing parts.

[@whenurefree]

I would love for him to explain random variables, clt, and maybe some other obscure topics such as assumptions in models. That would be really cool :)

[@3blue1brown]

I've certainly thought about doing a probability/stats. In fact, I'd be very disappointed with myself if it didn't happen next year. What parts do you find most confusing?

[@dude157]

I did my PhD in image processing, detecting the surface interfaces of different structures in volumetric imagery such as MRI data to support tasks such as segmentation. Convolution is such a key process for so many of the different stages. This video is the most elegant and clear explanation of convolution I've ever come across.

[@S1m0nTG]

dude157 I would love to read your PhD.

[@algeriapower7242]

Hi I also did my dissertation in image processing, my topic was about the osmosis variational method for image Fusion, top tip: the new chat gpt ai is insane and could help me code all image processing corses that i learned theoretically and didn't implement. Give it a try. What was your topic btw ?

[@S1m0nTG]

@@algeriapower7242 Thank you, I am working on automatic ways to extract property boundaries from surveying documents and implement the methodology inside a GIS software.

[@algeriapower7242]

@@S1m0nTG i checked some of the articles you were involved in with a google scholars search, you are great i hope you keep up the good work and good luck.

[@stewartdahamman]

Interesting, but what do you get out of life?

[@misterjigolo]

I'm amazed by the fact that Grant is basically providing so much more intuition and understanding in 25 minutes about a complex subject than what I got from 10 hours classes during my AI master degree. And that's free. You really are a legend Grant, thank you.

[@ResilientFighter]

agree^

[@d1str1be]

I think that animated video format is very helpful in studying. You directly see what comes from what. It should be implemented in courses. I hope it will have improvement in 20 years

[@leonardopsantos]

I'll 100 % second that. I was thinking about the "sneaky backdoor" way to calculate a convolution using the FFT/IFFT. Anyone who has studied the Fourier transform knows that the equivalent of a time domain convolution is just multiplication, so using the FFT/IFFT makes perfect sense. But how would I have loved to have access to these videos when I was buried in math trying to figure out that equivalency. Having that intuitive understanding before diving into the math would have been so, so great. The quality of these videos is simply astounding. Thank you very much for creating these videos.

[@hugodaniel8975]

This video should be taught in poor neighborhoods

[@SuperYtc1]

What did you do with your AI master degree?

[@liweicai2796]

<a href="#" class="seekto" data-time="1340">22:20</a> Hidden fun fact here: the O(NlogN) algorithm is only discovered in 2019, and for it to be actually fast the number would be too long to fit in our universe. Naturally utilizing the FFT convolution introduced in this video yields an algorithm of complexity O(NlogNloglogN), known as the Schönhage-Strassen algorithm, discovered in 1971. It is fast for numbers with thousands of digits and is built-in for big integer computation in some programming languages.

[@Filipnalepa]

Is it one of so called "galactic algoruthms"? Algorithm optimal for sufficiently large N, but "sufficently large" means here it's too large to be ever used.

[@2fifty533]

@@multiarray2320 that's one way of saying "i have no idea what this means and i think it's boring"

[@jamaluddin9158]

@@multiarray2320 fun != funny

[@notEphim]

Oh, I didn't know that the algorithm with roots of 1 had additional loglogN. I was told it's NlogN

[@arpyzero]

It's funny, I was looking at this galactic algorithm just yesterday! It uses a 1729-dimensional version of the FFT, which is hilarious... was trying to figure out why the taxicab number showed up, but I got lost in the math. I hope they find better versions of the O(NlogN) algorithm over time, it'd be neat if it ever sees actual use

[@nosy-cat]

That transition animation at <a href="#" class="seekto" data-time="328">5:28</a> is a work of art!

[@3blue1brown]

That one was definitely very fun. Once the data was well organized it was one of those that was more emergent than planned, after setting the transition I remember thinking "Whoa! That turned out nicer than expected"

[@samgolden14]

was about to comment this exact thing; it is awesome

[@victor_creator]

I had to just stop for a moment and appreciate just how beautifully perfect that transition was when I saw it!

[@tedsheridan8725]

I especially loved the dimension transition at 10:19.

[@GabrielPettier]

@@tedsheridan8725 I was so not expecting it, that i got the impression my screen was reclining 😅

[@mandarbamane4268]

Video: But what is a convolution? - LTI systems entered the chat. - Laplace transform entered the chat. - Fourier transform entered the chat. - Z transform entered the chat. - Correlation entered the chat.

[@Sugarman96]

AKA, the good stuff.

[@ME1D]

@@Sugarman96 The most complicated stuff

[@mandarbamane4268]

@@ME1D if that's "most" complicated, then study Electromagnetics, transmission lines, antennas waveguides, wave propagation

[@ME1D]

@@mandarbamane4268 Sadly, Im an electronics and communication engineer 😣

[@mandarbamane4268]

@@ME1D lol me too xD Good luck studying all those xD

[@janiKB]

You honestly bring me near to tears because I always thought I was "bad at math". But binge watching your videos and understanding concepts I never could have grasped have made me realize that I can learn math, with the right teacher. Thank you so much for making these videos!

[@exylic]

<a href="#" class="seekto" data-time="645">10:45</a> One correction: Putting your lens out of focus tends to actually produce, rather than a gaussian blur, a disc blur, which is actually a lot *more* similar to a box blur, just circular in shape rather than square. It's harder to compute than a gaussian or box blur, because with those two, you are able to use the 1D kernel twice rather than use a more expensive 2d kernel. Using a box blur is often undesirable because it produces a "bokeh" square in shape rather than circular, and other directional artifacts. The gaussian blur, on the other hand, is more akin to putting a translucent film in front of the camera, or covering the lens in vaseline.

[@3blue1brown]

Very interesting! Would you have some kind of dissipation in the disc though? Surely for a large enough disc, the influence of different sources in the image don't necessarily have equal influence on a particular pixel after going through the (unfocused) lens. I can imagine exp(-x^2) is too steep a dropoff, but what's the more accurate dropoff?

[@stephaneduhamel7706]

@@3blue1brown As I said in my other comment, the bokeh blur is exactly the same shape as the aperture of the camera. It's often a circle, but can actually be a square in some cameras. This is because the rays of light which reach the camera sensor have to pass through the aperture to be able to go through the lens and then hit the sensor. When the lens is properly focused, all the rays that come from some point from the real object converge to a single point on the sensor, and this point is actually the vertex of a cone whose cross section is the shape of the aperture. But if the focus is wrong, then this cone of light don't hit the sensor at it's vertex, so the sensor is activated on an area corresponding to the cross-section of the cone. So the accurate dropoff is a step function, which is pretty steep.

[@Tumbolisu]

You could argue that due to weird quantum effects of light, that for instance give it the ability to bend around a sharp corner, it isn't totally a step function. You could also argue that light that hits the sensor at a 0.01° angle will be dimmer compared to light that hits it straight on. Both of these effects are completely insignificant though.

[@aka5]

@@stephaneduhamel7706 great explanation. I imagine there is also a subtle effect from the inverse square law. In the simplest case of a point source with the aperture orthogonal to it, the source is slightly further from the edges than the centre. Therefore the dropoff would be very slight until the step

[@stephaneduhamel7706]

@@Tumbolisu actually, diffraction around the edges of the aperture does produce some kind of blur, which we often refer to as "bloom". Is pretty insignificant when the image is exposed properly, and doesn't depend on focus anyways.

[@MrSpeakerCone]

Great stuff as always! I'm an audio engineer and we generally use FFT convolution algorithms for creating reverb. Long story short, we can record how a physical space reacts to an impulse, then convolve other audio such that it sounds like it was produced in that space. It honestly feels like magic :)

[@djmips]

Jane street, has a good video lecture on this called 'Echoes of Fourier' @3Blue1Brown the topic would make a good 3B1B format video.

[@holomorphmusic]

Ohh so thats how convolution reverb works.

[@thebaker8637]

In particular: this is essentially the same thing that 3B1B did here with the moving averages. If you emit a sound with a step-like pattern and record the reverb in the environment, that gives you an acoustic fingerprint you can apply to any audio source! If you take any source of audio and convolve it with that fingerprint, you basically get how that audio source would sound like in the room. This is because a sound file is just a sequence of steps (samples) and your signature tells you how one such step responds to the room. The size of the step determines the granularity of the fingerprint -- ideally it's just an impulse like an infinitely short clap, then applying the signature to each sample trivially gives the exact behavior of the room.

[@anachromium]

That's just what I wanted to comment. This lecture is great and fantastic as always, but as a musician I just miss the audio part because this is such a beautiful way to make a reverb. (Or other sound effects as it is an extremely powerful tool!) Though I do understand that the visual way is much more in the style of 3b1b. Maybe a topic for a possible future SoMe3? :)

[@MrSpeakerCone]

@@thebaker8637 yup! there's two main ways we do this, 1) a sharp percussive sound which excites the air across the entire frequency domain like a small explosion for example, or if you're oldschool a balloon popping. 2) a long sine-sweep across the audible range from a really well tuned loudspeaker equally loud at all frequencies. We then deconvolve that against the time domain which we represent as a sine wave with a frequency of 1 cycle per second (time moves at one second per second, after all!) so that gives us a super accurate impulse response across the frequency domain

[@konstantine8054]

We also use convolution in audio. It's the equivalent of adding one sound inside the "space" of another sound. It is highly applicable to the reverberation of spaces but not limited to just that. Excellent video. Subscribed for more!

[@Phobos221B]

I've been struggling to visualise convolution for my University's Signal course, I've even watched some animations online and tried to get a hang of the discrete convolution and nothing, NOTHING comes close to this absolutely vivid imagery carefully chaperoned by your voice. You're the Richard Feynman of this century and Thank you for continuing making these videos. :)

[@sharkinahat]

Fun trick for doing blur - it's separable. If you have a 9x9 kernel, you just sample along one axis using a 1x9 kernel and do the same using the results along the other axis (9x1 kernel). You go down from 81 samples per pixel to just 18.

[@AstroTibs]

I'm not an expert on this, but isn't this only true if the values in the kernel aren't correlated? That is-true for Gaussian (or flat), because the kernels are each symmetric about their x and y axes through their centers; but no longer true if you had a more "skewed" distribution within your kernel?

[@ShankarSivarajan]

@@AstroTibs Yes, that's essentially what "separable" is a technical definition for. Not all kernels are separable (obviously?).

[@Asuryante]

Separability property. It works good on Gaussian Filter since it reduces computing time by A LOT. Instead of 2(k×k)^2 operations you do 2×2×(k×k), where k is the kernel size.

[@azmodanpc]

The video I needed back in college when I was struggling to understand these concepts for my umpteenth math exam.

[@Wypipo]

Do not try to ”understand”, student. Just apply ze algorithm!!!

[@N0Xa880iUL]

@@Wypipo Spoken like a true engineer!

[@jimnewton4534]

I think umpteen is an imaginary number.

[@hyronvalkinson1749]

@@jimnewton4534 imaginary numbers are easy. Umpteen is complex Wait a second...

[@Wypipo]

@@hyronvalkinson1749 umpteen is such that when evaluated as an integer it yields at least like a bunch. Fun fact: the square root of umpteen is also umpteen.

[@debaelwyn]

As someone who's spent much of the last ten years doing complex signal processing professionally and using FFTs and convolutions in a bunch of contexts, this is phenomenal and does a great job of capturing the intuitions I built up over the last decade and summarizing them in 23 minutes. Thank you!

[@douglascheng9794]

I just graduated from school where I used convolution for image processing, signals, etc. and you just explained it more intuitively than anything I got out of school.

[@liammccreary2941]

I love Grant’s respect for his students’ variable exposure to math concepts and problem solving. Instead of saying “you should know by now that…” he says “just know that there are certain paths you could have walked in math that make this more of an expected step.” This really helps me feel more validated in my own experience with math.

[@mikkolukas]

A lot of teachers could learn from this!

[@CarbonRollerCaco]

Actually sounds MORE guilting to me: "you not only remembered poorly, but you chose poorly as well". IOW, "you calculated poorly". :

[@simonmasters3295]

So I walked a SQL programming path select a, b, a*b from A, B -- full outer join group by a*b ...does something similar?

[@nodvick]

"you should know by now" from a teacher, as opposed to someone giving a presentation on something interesting online without the context of a proper course-structure, is a prompt that you missed something and to seek to fill the gap, it is necessary, and the "invalidated"(?) feeling you get is the intent of the assertion. Instead of getting mad at the sound, answer the door when opportunity knocks, find out what you missed and catch up to where you should be by that point in your course, the best person to ask is usually the teacher themselves.

[@jeremydeal230]

If you SELECT a*b, SUM(a*b), you might be on to something 😀

[@N0Xa880iUL]

It's the most simple yet convoluted concept in engineering!

[@michaelleue7594]

convolved

[@sacharjawellmer5530]

pun intended? I like it hahahaha

[@N0Xa880iUL]

@@sacharjawellmer5530 Yes! Thanks.

[@jasonreed7522]

Convolution is really powerful, yet its pain to do it normally so we actuvely avoid them by first converting to frequency domain where it becomes basic multiplication and then convert back. At some point in your education as an electrical engineer you realize that complex numbers are pure magic in just how much effort they save. (And I'm sure other engineering fields make heavy use of them too)

[@joyboricua3721]

I C wat U did

[@wyattr7982]

My final project for my MS in Computer Engineering was implementing an FPGA based guitar speaker simulator. The convolution of an audio signal with an impulse response of a particular speaker will impart the characteristics of that speaker onto the signal (though the actual implementation was FFT based so it would work im real time). Thanks for this explanation so others in my life can understand what i was losing my mind over for months

[@joelaul]

Made the same connection while watching this - in terms of the image processing example, you're BLURRING the amp signal with the impulse response. I made a similar albeit much simpler web app that simulates a tube screamer and I'd love to see how much deeper your project goes, if you'd share?

[@muhilan8540]

Yes convolution was a big part of my signals and systems course in computer engineering

[@jeffreygordonmusic]

Hey - I know this comment is a year old, but this sounds like a really cool project! I'm currently considering Computer/Electrical Engineering MS programs specifically because I'm interested in audio signal processing - would you mind sharing the name of the institution you went to?

[@jacemc9852]

This is so great. I would have loved if he included a bit on audio too. The natural phenomena of sound reverberating in a large hall or cave is a convolution as detected by the ear. The hall's impulse response can be captured (e.g record a gun-shot/ ballon-pop, in a large hall) and used as the filter kernel to convolve a dry audio signal. The result is to imprint the room's response on the input signal. Many famous cathedrals, theatres and basilicas have been modelled this way, so that singers and musicians can benefit in the studio with modern convolution dsp techniques. It also provides methods for echo removal in noisy recordings.

[@birdwalkin]

creepy, like simulating the ghost of a location using only its captured shadow 🤯

[@physics_hacker]

I also love the sound design applications of non-reverberation IRs, they always give such interesting sounds

[@jacemc9852]

For sure. You can model a rack of chained outboard effects, going beyond echo and reverb to things like EQ, delay and chorus etc., as long as they are linear processes. You can't model non-linear processes such as dynamic compression though.@@physics_hacker

[@Holko96]

Seeing so many different lectures on youtube on different topics is already an amazement in itself, but watching the highqualitity output, which has been kickstarteted by 3b1b with manim, transition to ofther channels, just increased youtubes potential as a learning platform. It is really amazing to see others using the library and creating their own content and I, as a humble viewer, am thankful for this addition.

[@kurtu5]

This was once the promise of TV. But with the small finite number of channels, TV degenerated into, well what we see today. KZbin with its larger finitity of channels allows both great learning content and also well what we see today.

[@jursamaj]

@@kurtu5 Ufortunately, YT *also* has flat-earthers, creationists, "Trump won" nuts, etc., many of whom are get *way* more views than Grant. The crap on TV *and* YT is there because there's an audience for it that is very profitable, much more profitable than educational info.

[@kurtu5]

@@jursamaj Thats fine. Let them have their ideas out in the open where you can clearly see them and be able to present counter ideas right back. People are always going to believe different things. My point is there is enough room for all the things. It hasn't devolved into a place where Grant is not able to reach millions and unlike TV, it never likely will.

[@jursamaj]

@@kurtu5 The problem with that attitude is that a large fraction of the population prefers a nice, simple, wrong explanation that they can understand, versus a a complicated but correct explanation that hey'd have to think hard about and probably still not understand.

[@MrMarc1991]

It's awesome that he programmed it and open sourced it. It's such a cool library.

[@sentinelaenow4576]

Imagine how many awesome technologies will derive from a few minutes of extremely sophisticated math visualizations. It's like a hyper efficient way to advance human race through talent, knowledge and a warm heart. Thank you very much Sir Sanderson. Humanity owes you a lot. Please continue for good.

[@NigelThorne]

These videos are gold. Thank you for putting in all the time to think up ways to explain complex concepts so clearly. Also thanks for all the patreons that support you to allow you to make this stuff publicly available. We are so lucky to be alive at such a time.

[@Nossairito]

I cannot put into words just how outstanding your videos are. My absolute favorite thing about them is how they bring together notions that would always pop up in my daily life like gaussian blur and make them suddenly emerge from something I just learned, as if to reassure me that the world is always within understanding's reach if you don't let it intimidate you. I love it so much.

[@arjunraja8143]

When I was doing my bachelor's and studying the Taylor series you dropped your Taylor series video and saved me. Now I'm doing my master's and studying image processing and you've dropped this to save me again ❤

[@Nishantpatale9423]

Msc maths? Or stat?

[@Stewi1014]

Just wanted to say - I think the real power of what you're teaching is knowing that *certain mathematical tool set* exists, and understanding enough about it to go "Aha! I know what tools I need to deep dive into to solve this problem" when we encounter different problems.

[@vigilantcosmicpenguin8721]

That's what it's all really about.

[@richardpike8748]

Great way to put it. Extra tool in your toolboxes

[@LeoPlaw]

This was excellent! It was the missing piece in a jigsaw of what I previously thought of as disparate applications, (trading indicators, image processing...) but are all tied together with a few core mathematical concepts. I'm going to watch this again a few more times to fully absorb it.

[@jjohansen86]

The conclusion about the FFT also suggests one more important thing: How to do a deconvolution. It's hard to see how to undo that integral of a product... thing... that is the definition of a convolution. But when you understand that the Fourier transform of a convolution of two functions is equal to the product of the Fourier transforms of the functions, then you don't have to undo that integral, you just take some Fourier transforms, divide, and do an inverse Fourier transform. So in addition to giving you a speedup, the FFT method suggests something that you just can't do otherwise. Why do we care? Maybe we have a signal where what we measure is a convolution of what we want to know about with the response of some detection method. That's because there's one other answer to the question "What is a convolution?" that comes to mind for me more than anything: Say you have a thing that has a known response to an input. Now replace that thing with a bunch of things with some distribution of important properties. The convolution tells you how the whole distribution responds to the input. For a concrete example, I'm an atomic physicist, and sometimes I want to know about the distribution of atoms in a gas, things like their velocities. I shine a laser through the cloud of atoms, and the velocity information is there because of each atom's Doppler shift, but what I measure directly is the convolution of that distribution of Doppler shifts of all the atoms in the gas with the Lorentzian response of an atom to a laser. The deconvolution can let me take out the Lorentzian response of an individual atom and leave me with the actual distribution of Doppler shifts, so now I actually know what the cloud of atoms is doing. Or maybe you are trying to detect some radio signal with an antenna, but what you read just isn't as clean as you'd like it to be because the response of your antenna is actually being convolved with the original signal; a deconvolution can help you remove the way your antenna is influencing the signal from what you measure. There's a ton of different signal and response problems like this, and an FFT deconvolution can let you get a better signal in each of those cases.

[@notEphim]

I once multiplied two 3-digit numbers by hand using fft just for fun. It took the whole blackboard

[@vigilantcosmicpenguin8721]

Mathematicians have a very interesting idea of fun.

[@Wypipo]

I’m a software engineer. You described this all AMAZINGLY.

[@Snowflake_tv]

He's genius at explaining Math lol

[@spacejunk2186]

I'm not an engineer and I agree.

[@leparraindufromage366]

I really love the connection between the multiplication of polynomials and convolution of their coefficients, it arises a lot in the theory of (old school algebraic) error correcting codes as well and it's just so satisfying how probability, algebra and geometry are intertwined that way

[@LegendaryKenneth]

<a href="#" class="seekto" data-time="102">1:42</a> That little snippet of animation in the bottom left was enough to give me an "ah-ha!" moment, and I finally understand how convolution reverb works in audio. Amazing!

[@huhneat1076]

<a href="#" class="seekto" data-time="1130">18:50</a> PLEASE keep giving the 3 blues different personalities like this in the future. It's so cute.

[@Xoque551]

I've literally waited years to for 3b1b to make this exact video because convolution has never made sense to me! I'm so glad it's finally here and it's as illuminating as I'd hoped!! Can't get enough of this channel, a gem of the internet age :D

[@DrsharpRothstein]

I have come across convolution functions and have never understood them beyond a 'round robin' way of multiplication. Then I discovered its use for image manipulation and was confused by kernels. Thanks, you have guided my further interest in the subject.

[@donaldmcronald2331]

As a university student who currently learns the maths behind signals, this is a very interesting perspective. Visualizing the convolution as two functions or two sets of numbers which are sliding through one and another really helped me to wrap my head around it. It's always helpful to see a different perspective, many thanks for your effort!

[@appelnonsurtaxe]

What I find so satisfying with all your videos is that I don't just learn things; I really - more than ever - feel that I've managed to connect concepts in my head. - Convolutions ("learnt" as a math concept in annoying signal processing class) - Gaussian blur (I have average photo editing skills) - CNNs (of course it has something to do with convolutions but I like knowing the details: the kernel is what we're trying to "find" during training) - Sum of 2 dice (glanced over in stats class) - Polynomials (algebra class) - etc. I often feel powerless knowing about how much knowledge and skills I lack (even though I have a very decent education), and it' sooo incredibly satisfying to see that everything is actually linked together. Idk if that's actually the case but I also have this feeling that by connecting concepts together I'm less likely to forget them, as if my brain was a search engine with a page ranking algorithm.

[@lizzzylavender]

I’m a sophomore in a CS degree and it’s actually really exciting to understand a lot more of what goes on in these videos from all my college level stem classes. Even just being familiar with some of the concepts makes the video tie together so much nicer in my head.

[@mattp1337]

Man, two years of post-secondary math and this topic never came up once. Thanks for the brief and elegant introduction. I had a month's worth of ecstatic "aha" moments in less than half an hour.

[@PtakubJ]

Important in audio processing too! Analogous to blur in graphics, we use convolution in audio for reverb. You may record your electric guitar with some interface. Then record an impulse response of some cathedral or something. Impulse response is the reverberation that the room gives you in response to an impulse - ideally that impulse should be an infinitely high pressure for infinitesimaly small time, but you may use a gun shot, balloon pop or a specifically constructed noise (sounding like white noise) to calculate it. And then you convolute the guitar track with the cathedral response and here you go - you have a track of your guitar playing in the said cathedral. And then of course it's done not with the pure convolution, but rather multiplying the FFTs - which are in and of themselves important for audio as the sound spectra.

[@andrespulido8]

I believe this is my first time paying back to a channel for all the value you provide me. Thank you Grant!

[@adityamrai3892]

All these years of doing math and I never saw it in this manner. Probability or even simple multiplication of numbers. Always grateful towards 3blue1brown for bringing such awesome intuition out! 🙇‍♂️

[@ahmedhani9854]

I have to say I’m thankful that I’ve watched veritasium’s video on FT and FFT before hand. The same concept feels so much more familiar and easier to digest when viewed from two different but very similar takes that covers ever so slightly different aspects of the same topic. I’d highly recommend anyone to go watch the video on Derek’s channel if you haven’t already. The two videos just meld together so well it’s unbelievable!

[@aryanverma6939]

DFT and FFT are the concepts i studied in my University curriculum but visualization of these makes it so much better to understand and fun to watch. This is one of the best channel that i came across recently, absolutely love it!

[@cuadernoazul5958]

I loved the way you explain this topic using arrays and polimiomies. I remember had had a hard times trying to figure out convolution for signal analisys. Thank you very much!

[@sacharjawellmer5530]

I think it is noteworthy here that the convolution operation in German is called a "Faltung" or translated a "Fold" as you fold one function/list into the other, a very fitting name I think. Great video by the way, keep up the good work

[@3blue1brown]

Every time I learn what the german word is for a math concept, it feels so wonderfully precise in contrast to English.

[@vigilantcosmicpenguin8721]

That makes it sound a lot simpler than the English term does.

[@Jack-tk3ub]

@@3blue1brown Or alternatively that the English version is more convoluted (pun intended)

[@TheF3AR98]

This month is awesome, thank you 3b1b for the inspiration to study.

[@antoineprosper8645]

Amn, I wish those videos had been around during my undergrad, you make some brain twisting concepts appear in such a clear and motivated light, I have been following your channel for a few years now and I truly love what you are doing. I sincerely thank you for this, and for all the content you make

[@glitchy_weasel]

My professor just introduced convolutions in my differential equations class, so this is a perfect coincidence. Hoping to see the continuous case. Fantastic as always.

[@JoeShmowYo]

Somehow I did grant’s homework assignment before this video came out. I do signal processing related optimization at work and just last week I was showing my coworker why I made the connection between squaring a string of 1s and the binomial coefficients / Pascal’s triangle, and how that relates to an iterative algorithm we were trying to vectorize (basically do it in steps of 4 at a time using special processor instructions).

[@SoniasWay]

As someone who studied engineering I’m really glad to see this

[@nicolasoche7993]

A Laplace transform video would be great. Congratulations for your excellent content.

[@kirankumar31]

Goodness me. This kind of explanation is truly a bliss. I have fallen in love with your creation of the content. Its clarity, conciseness just hooked me on to it. Immensely grateful.

[@rz7464]

I've followed you for years and you keep making amazingly useful and clean videos. Hard work, impressive. Greetings from France !

[@mjrmls]

I love the recent wave of "DSP explained" videos in the YT education space, hope to see more!

[@samcarswell9890]

Ooooh if you have any recommendations for any other videos, I'd love to know 🙂 Really enjoying diving further into DSP at the moment!

[@ti84satact12]

What is DSP?

[@samcarswell9890]

@@ti84satact12 Digital Signal Processing :)

[@lawrencedoliveiro9104]

Next: how to explain to all those computer graphics noddies that “pixels are not little squares” ...

[@mjrmls]

@@samcarswell9890 check out Veritasium and Reducible's recent videos on FFTs, + stand-up math just posted a video about using convolution to identify whales

[@sinan4495]

I continue to be in awe of how your videos enrich my understanding of so many topics. Thank you

[@larrysal8866]

It's quite funny. I study an engineering field, and we have a module called system and signal theory. I watched this video, and after maybe half a week, we had the topic "convolutions". I scratched my head, knowing i had heard it before, then remembered ir was the video. In the lecture it was really confusing, as we did the convolution as a new function on which the parameter was an offset of one of the functions compared to the other. Now i come back to this video, and it just suddenly makes sense. Truly amazing.

[@reilandeubank]

As a CS major taking a break from studying for my test by watching some youtube, I really didn't expect to be hearing about big O time complexity right now!

[@alid9519]

Bro I JUST had this lecture today in Numerical Mathematics

[@Karreth]

Fascinating! This would have been extremely useful when I was trying to learn this stuff in uni. I've always felt like this kind of math is a problem you just bang your head against until your brain changes to a new shape capable of understanding the problem, but these clear explanations with excellent visual helpers definitively helps. Great work!

[@ts4gv]

This is the third time in recent memory you’ve uploaded a video covering a topic I recently became interested in thanks

[@user-se9kr9pw5q]

I watched this video yesterday, and today the topic of convolution came up in my Applied Mathematics course. You're always one step ahead!

[@QuinnDuPont]

I love how Grant speaks about different journeys in math, and doesn't presuppose a traditional schooling. He's the best.😃

[@joyrc01]

I am really grateful to this channel 💖💝💝 I am studying engineering in electronics and communication systems So I deal with these theories everyday. And being able to visualise these makes things great to study.... Love from India.

[@nirmaladhikaree9609]

Damn !! In the midway of the video currently, and I have to say, your teaching makes concept assimilation smooth as butter. Cant thank you enough for this. Keep up the good work !

[@SuperLinnea96]

This is the best video on convolution i have seen! thank you so much! i've "always" understood convolution in therms of 2D signals like your examples with images, but not with 1D signals. With your polynomial examples it makes much more sense and especially in relation of the effeciency of the fft! thank you SO much! so many years and videos and now i finally get it!

[@ashwinnicholas497]

3B1B never ceases to amaze. Truly one of the best channels on youtube.

[@tlniec]

One thing I really like about how you sequence and pace your videos is that you give the viewer a chance to make connections in their head before revealing them. For example, I realized "oh, that's how a gaussian blur works" just before you stated it in the video. Those a-ha moments are a great way to keep viewers/students engaged!

[@MsDontBlink]

I started Googling convolutions a few days ago for some image processing in my postdoc research, and was so glad to see this one just come out. What a lovely coincidence. Thank you so much, Grant.

[@SF-fb6lv]

Only 59 seconds into my Saturday morning, I've already gotten a whole new insight into convolution! Thanks Grant; you never cease to amaze!

[@whitenat]

As a uni student currently stuggling to understand what a convolution is, it's perfect timing ! thank you !

[@threemr01]

Grant, this time I literally screamed at the screen “nooo, not yet” when you got to “the clicky stuff”! This was the intuition I was missing back in signal processing class, and loved that explanation of the FFT trick. Awesome video (as always). Please don’t keep us waiting two years for the next one 😉… still waiting for the next video after the one for the binomial distribution.

[@noraxi]

Amazing, always nice to see math/science videos not being over-simplified

[@EricZuckerman1]

Really enjoyed the ground up approach to the topic of convolution. My first exposure came in a graduate course that discussed signal analysis (it was in analytical chemistry). We were shown the reverse combing, but it really came alive with the dice example...so simple yet deep. For me, the end result was to always think of the convolution as a way to apply a filter to some data, which you showed with the blurring examples. Great stuff, and I look forward to the next installment!!

[@KelfranGt]

Thanks 3b1b, I've been doing a project on Convolutional Neural Network for structural response and this video really helped me gain a deeper understanding to convolutions and the math behind my code. Your videos have helped me a lot throughout the years and I hope to see more amazing videos from you in the future!

[@Snowflake_tv]

Wow your reply would be honor for him

[@iamsushi1056]

Oh, boy, I can’t wait to fftconvolve all my matrix operations

[@kaijiechow275]

What a video! Just love the whole concept of this topic and how Grant approached it starting from the dice example. Marvelous!

[@DistortedV12]

You’ve outdone yourself. This is my favorite video you’ve done. Your video flow has gotten a lot better .

[@genessab]

I’m in the middle of a class on quantum field theory and I was literally asking just yesterday “well..what exactly *is* a convolution?” As always, the best math KZbinr out there has got my back :)

[@gurpremsingh]

I just finished my project on Sobel Edge Detection, wish this video existed when I started. Very well explained!

[@rohitchan007]

Wow, this changed my perspective about how I think about convolutions now. Thank you so much, Grant. Please continue making more videos on statistics.

[@yashkokane4841]

I cannot thank you enough for this video. 3Blue1Brown, you're an absolute genius for this.

[@gokularamm]

As always, a great video! Whatever follows after <a href="#" class="seekto" data-time="880">14:40</a> is a nice visualization to appreciate the convolution theorem of Z transform (by letting x = z^-1) and discrete time Fourier transform!

[@hobbified]

When I was a kid in the 90s I had a computer and a copy of Paint Shop Pro (long before Corel bought and ruined it). It had a bunch of built-in filters, but it also had a "Filter Matrix" option that would let you put in an arbitrary kernel, up to 5x5. The manual explained what it did mathematically, but not really anything on how to use it - just sort of "you'll know if you want this" - but the first things I figured out were that something like [[1, 1, 1], [1, 1, 1], [1, 1, 1]] would blur an image (there was an option to automatically divide the kernel by the sum of its elements, which made things easy when you didn't know what you were doing), [[-1, -1, -1], [-1, 8, -1], [-1, -1, -1]] would do an omnidirectional edge detect, and things like [[0, 0, 0], [0, 1, -1], [0, 0, 0]] would do an edge detect in just one direction.

[@MathiasVerhasselt]

ohh I remember this! as a kid I didnt really understand what was going on but it was fun to play with!

[@fredrik-ekelund]

Thanks! You have an amazing way of explaining and conveying the logic without simplifying too much, as far as I can understand. Again, thanks! Love your stuff!

[@simonbeaumont1911]

Amazingly clear. I was struggling to understand this concept in signal processing and now I've got it in full generality 10 minutes in. Thanks Grant!

[@jb76489]

Im a chem PhD candidate and for a lab I taught a student had some data from a potentiometric titration that was unusable. The overall sigmoid was there but it was very noisy which due to the way the software they used worked meant it was useless. However, thanks to the math in curve smoothing bits of the Julia course, I was able to fix that students data, thanks Grant!

[@justafish5559]

This is actually, hands down, the most interesting video you have uploaded recently. The recent content has just been superb so this is amazing! My class is about probability right now too so this was very cool to see.

[@hmm.3367]

In just 3 minutes you completed 23 minute video !! 😆

[@Embassy_of_Jupiter]

@@hmm.3367 He is the product of a science experiment that genetically engineered a fish to think at 10x the speed of a normal human. Our team actually had to build a special KZbin app for him just so he could watch videos at 10x speed or else he's get bored.

[@kasiphia]

You've figured out before watching, must have just been a great thumbnail.

[@frankjohnson123]

gotta get that early comment for the thumbs

[@justafish5559]

to all the comments talking about how i was able to watch the video so fast: i don’t know what’s so weird about watching videos so fast.. is that not normal? if not, i apologize for spoiling the video for the late comers.. sorry.

[@rttuo1259]

I've been studying EE and Physics for a while now, and I had my mind completely blown away after less than four minutes, when that dice analogy clicked. Absolutely amazing, thank you!

[@RHSGH]

Genius, Excellent, Calculating the convolution and understanding its identity is really simple using this method. You are the best.

[@guilhermemotapereira5917]

Wow, what a coincidence! I'm doing a discipline over Signal Processing and Wavelets this semester in my CS bachelors. So, this came in great hand! Thank you 3b1b!!! :)

[@samyount2546]

I just finished the series on neural networks and was getting into CNNs with PyTorch when this video drops. 3B1B you are an absolute gem!

[@dcterr1]

I really enjoy all your videos and this is one of the best I've seen! Can't wait for your follow up involving how to compute 2D convolutions efficiently!

[@MrSafeTCam]

I had an idea for a video game I wanted to make about 5 years ago that I never completed because I was trying to make an algorithm to do a particular thing, and it would never work properly. At <a href="#" class="seekto" data-time="557">9:17</a> of this video I paused it and said "CONVOLUTION!" because I realised, THIS is the algorithm I needed. One of my two dream video games finally has a way to exist!

[@AJMansfield1]

Very insightful! There's also another _massive_ "aha" moment in the N log N multiplication where you switch from using the complex numbers of the Fourier transform to the ring integers of the number theoretic transform. Because, once you make _that_ jump, the fifth root of unity isn't 0.3090 + 0.95106 i anymore -- it's just 2 (mod 5).

[@kyay10]

This video feels a bit different, idk how to explain it, but it's a nice, refreshing change. Keep it up!

[@stevenclarke2655]

I'm glad I'm not the only one who feels this way. I am a huge fan of all Grant's videos, but there is something about this one that is just making it "pop" a bit more for me. I'm still trying to figure out why that is.

[@nicolasdezadiaz1658]

This is by far the best KZbin account, congrats again for another great video!

[@mohammadjoudivand7523]

You are a valuable asset to the mathematics community. Your ability to present complex concepts with a smooth narrative flow is truly amazing. Not to mention, your graphics are awesome. Thank you man

[@peterboneg]

Something I find amazing is how image convolutions / correlations can be done entirely optically because of how an image encoded in a laser beam can be Fourier transformed when passed through a lens. It doesn't seem possible that all that maths gets reduced to lenses. A follow up video to explain that would be great!

[@kidgroovie]

This used to be a nightmare to wrap my head around during college, thank you for creating this

[@deborazum3468]

You're one of the best KZbin channel that has ever existed! Thanks for all, that's amazing

[@mrtoast244]

This guy is crazy, he covers pretty technical topics but I'm literally disappointed whenever one of his videos end because of how simple and interesting he makes it.