But what is a convolution?

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!

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.

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.

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.

22:20 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.

That transition animation at 5:28 is a work of art!

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.

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!

10:45 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.

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 :)

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!

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. :)

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.

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

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!

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.

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.

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

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

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.