How and why to take a logarithm of an image [video]

https://www.youtube.com/watch?v=ldxFjLJ3rVY

by jgwil2 • 4 months ago

177 69 4 months ago

youtube.com

boriskourt2• 4 months ago

This video is an absolute tour de force of communicating a complex concept.

dmbche• 4 months ago

All of 3Blue1Brown is - hoghly highly recommend

boriskourt2• 4 months ago

I've seen most! Highlighting this one out of them all. Exemplary! : D

JKCalhoun• 4 months ago

Seems like you could apply the clever transforms to generate a displacement map (that then allows you to move it across any source image and quickly get the Droste effect).

(I still have not made it all the way to the end of the video though, perhaps that is where they end up.)

Dwedit• 4 months ago

"Complex concept"

I see what you did there.

coldpie2• 4 months ago

Clickbait title could use another pass. What is this about?

nticompass• 4 months ago

I clicked on the link and the video title is "Decoding Escher's most mind-bending piece", which is a lot better. I also had no idea what "3B1B video" meant, apparently it's a channel called "3Blue1Brown".

hnuser123456• 4 months ago

It's about examining the mathematical methods MC Escher used in one of his recursive drawings.

coldpie2• 4 months ago

Examining the mathematical methods MC Escher used in one of his recursive drawings

This would be an excellent title :)

rcxdude3• 4 months ago

Probably he didn't use these techniques explicity: the video mentions but doesn't emphasise that he probably sketched out the map by feel instead of analytically, which is probably one reason why he didn't fill in the center.

wodenokoto• 4 months ago

It makes more sense when seen on YouTube where you get the thumbnail of one of M. C. Eschers famous drawings is shown.

It’s a drawing of a guy looking at a picture of a town with himself standing in the town, but it’s all twirled and twisted so it’s self repetition isn’t obvious.

jgwil2• 4 months ago

This was the title used when I came across the video. Apparently YouTube uses many different titles for A/B testing but this is the one I got. Can't edit it now, unfortunately.

EDIT: seems like dang or team took care of it, thanks!

m-hodges2• 4 months ago

The title I get when I click on this is, "How (and why) to take a logarithm of an image"

peesem• 4 months ago

YouTube has A/B testing features that allow videos to have multiple titles and/or thumbnails.

m-hodges2• 4 months ago

Right. So I thought it would be helpful to share the more-descriptive title that I got.

dandanua2• 4 months ago

I'm sorry, what? Can people now see different titles? Insanity, if true.

sva_• 4 months ago

For me it is "Decoding Escher's most mind-bending piece"

emmelaich• 4 months ago

Same for me and this is why I had not clicked on it before.

close04• 4 months ago

How (and why) to take a logarithm of an image

I watched it a few days ago and this descriptive title was part of the reason I clicked. I generally trust 3B1B anyway but normally a title like "This picture broke my brain" would put me off.

3b1b2• 4 months ago

In case you're curious, when I ran that title/thumbnail AB test, the option "This picture broke my brain" did end up winning. I was a bit disappointed, because I didn't really _want_ it to win, but I did include it out of curiosity. Ultimately, I changed it to the other title, mostly because I like it better, and the margin was small.

I was genuinely torn about how to title this, because one of my aims is that it stands to be enjoyed by people outside the usual online-math-viewing circles, especially the first 12 minutes, and leaning into the idea of a complex log risks alienating some of those.

john_strinlai4• 4 months ago

i see "Decoding Escher's most mind-bending piece".

fascinating, and absurdly confusing, that there are multiple titles.

eblume• 4 months ago

It's a pretty common feature of youtube creator studio. https://www.theverge.com/news/840789/youtube-video-title-a-b...

Jeff_Brown• 4 months ago

I love 3B1B but generally don't have time to watch long videos. Can anyone sum up the punchline?

ahns2• 4 months ago

One of Dutch artist M.C. Escher's works is a man is admiring a piece of art that itself depicts the building the (very same) man is in [0]. Escher left out the middle bit of the painting, probably since it's fairly complicated, putting his signature there instead. The video itself is about the complex analysis used to fill in that missing middle, based on a paper ~20 years ago.

[0] https://en.wikipedia.org/wiki/Print_Gallery_(M._C._Escher)

rcxdude3• 4 months ago

I think the gap also has a compositional purpose: the viewer's eye is meant to travel around the image in a circle, and the gap helps anchor that in a way that the filled-in version might not.

ahns2• 4 months ago

Hm, since I can't edit my comment: link to paper [0]

[0] https://pub.math.leidenuniv.nl/~smitbde/papers/2003-de_smit-...

oulipo2• 4 months ago

The whole point is the explanation... it's a bit like someone telling you to take a 2 week holidays somewhere and you'd just say: it's too long, can't someone just get me a plane ticket there and back the same day so I can compress the stay?

yread• 4 months ago

The print gallery is just Aw^c in the complex plane

griffzhowl2• 4 months ago

Answers that are only comprehensible to those who already know the answer:

pdpi• 4 months ago

The punchline is that you can fill in the centre of Escher's piece by using complex analysis, and it produces a very satisfying, "obviously correct", solution.

But, as with all jokes, the punchline isn't funny at all without the setup.

ant6n• 4 months ago

The joke is that if you fill in the center, it shows the Droste effect of the image and kind of diminishes the magic of it.

rcxdude3• 4 months ago

The image is essentially a self-similar 'droste-effect' image in disguise. The warping of that image shifts that self- similarity into a visual loop, but the warped image still has a droste-style self-similarity in the center as well.

Lerc• 4 months ago

Awᶜ

This kind of risks obscuring what's actually going on.

db48x2• 4 months ago

Obscurity indeed. This morning shortly after I woke up I read your message and could only think that you were referring to Awk somehow. That didn’t make much sense to me. Now, hours later and after finally eating something, it does make sense! You really do have to have your faculties about you.

amelius• 4 months ago

Clickbait title broke my brain.

OscarCunningham• 4 months ago

I've been wondering if you could do a similar thing for a Droste effect image containing two copies of itself. Packs of Laughing Cow cheese show a cow with two earrings, each of which is a pack of the cheese.

gowld2• 4 months ago

What "similar thing" are you asking for? The Laughing Cow image exists. The Print Gallery is an object itself existing at 2 zoom levels in the same place, but the cheese exists in different places. You can't have two copies of the same image in the same place - that's not a copy; it's just itself.

manudaro2• 4 months ago

I've been loking into how 3B1B builds their rendering pipeline, and it's honestly mind blowing. They use Python along with custom OpenGl shaders to handle most of geometric transformations, shich seems to be what creates those "brain breaking" visual effects.It's fascinating how our visual cortex tries to interpret overlapping geometric patterns and ends up producing such counterintuitive perceptions. Shat I still can't quite wrap my hand around is... to what extent are these effects caused by the rendering itself, and how much of it is just how our brain interprets the visual information?

john_strinlai4• 4 months ago

there is at least one video, if not a series of videos, where he explains his process in detail, showing the coding process and everything. it is a collaboration with another person where, if i recall, he is teaching them.

i will see if i can find it.

edit: "How I animate 3Blue1Brown | A Manim demo with Ben Sparks" https://www.youtube.com/watch?v=rbu7Zu5X1zI

more on workflow: https://github.com/3b1b/videos?tab=readme-ov-file#workflow

code for videos: https://github.com/3b1b/videos

manim: https://github.com/3b1b/manim

manim community edition: https://github.com/ManimCommunity/manim/

manudaro2• 4 months ago

This is great, thanks for digging these up. The workflow doc on Github is exactly what I wat looking for - didn't realize they had that level of documentation publicly available.

pierrec2• 4 months ago

This kind of technique can be used in 3D space as well! The analysis here represents Escher's techniques as conformal maps in the complex plane. Conformal maps are also possible, though more limited, in R^3. This is something that I explored some years ago and wrote an article about it, though it focuses more on graphics than math: https://www.osar.fr/notes/logspherical/

toxik4• 4 months ago

So to do this same Droste effect in 3D you would need a self-similar volume? Though since we can't really see 3D, we could never have that "one circle zooms in" effect.

Or could you walk around in such a world? That would be a very cool concept for a game.

pierrec2• 4 months ago

Though since we can't really see 3D, we could never have that "one circle zooms in" effect.

Well, the 3D structure just needs to be sufficiently "holey" for the effect to become apparent. For example a cage-like structure, or a house with no roof (when seen from above).

qoez• 4 months ago

Makes me wonder how this would look/feel interactively if a game world was rendered like this

marginalia_nu2• 4 months ago

Manifold Garden isn't far off.

aprentic• 4 months ago

Those videos are awesome! 3B1Bs visualizations finally made e^(pi*i) make sense.

His videos on Euler's formula inspired me to make a silly toy so I could play with it myself.

https://gitlab.com/aprentic/complex-viz/

rappatic• 4 months ago

Similarly, it's possible to take the derivative of a song. You can use a Fourier transform to express the song's waveform as a series of sin and cosine functions, then take the derivative.

Imagine, for the sake of simplicity, you could express the song's waveform with the function 13 * sin(41x).

The derivative of this function is 533 * cos(41x).

Cosine, of course, is just a phase shifted sine, and the constant coefficient inside the function stays the same. So you're not changing anything about the shape of the wave, just stretching it vertically.

This has the effect of mimicking a "high pass filter," amplifying the volume of the highs.

toxik4• 4 months ago

Well, you get the frequency domain derivative. This is the same as scaling the time domain by a linear ramp. Not exactly hugely useful, unless you happen to be in radar.

You can take the finite difference with eg np.diff(waveform) though.

toxik4• 4 months ago

So in other words you can take any Droste image and make an Escher zooming spiral effect. Neat.

Also curious what happens if you take Escher's painting and undo the effect. Probably not great since it wasn't in the video.

What a cool video.

db48x2• 4 months ago

It was in the video, actually.

toxik4• 4 months ago

I'm somewhat sure that was the recreated version.

How and why to take a logarithm of an image [video]

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