Who Expected That? Extreme close-ups create a Klein Bottle.

http://www.thebigquestions.com/2012/02/23/who-expected-that/

by ColinWright • 14 years ago

110 32 14 years ago

fridek2• 14 years ago

Have you considered that this pattern may be a result of camera internal structure rather than structure of objects?

jamii4• 14 years ago

The linked paper shows how the klein bottle maps onto the space. Basically, grids which have an orientation (horizontal or vertical stripes) are more common and join together to form a klein bottle. That tendency towards orientation might be an artefact of the camera or might be because cameras are usually held horizontal or vertical relative to the sky.

Drbble2• 14 years ago

The paper notes that the vertical lines are gravity aligned, not camera aligned.

The paper desperately needs a accompanying translation from extreme mathematician to mortal PhD level.

teaspoon• 14 years ago

Are you talking about things like sensor noise and chromatic aberration? It would be interesting to see if downsampling the image beforehand affects the result.

However, it's hard to separate image patterns from camera structure insofar as linear projection is a result of camera structure.

fridek2• 14 years ago

I was thinking about CFA mosaic and JPG compression, I think these may introduce some axis aligned artifacts. But maybe they took it into account (using raw format?) or effect is not relevant in this case.

obtu2• 14 years ago

It's mentioned in the abstract of [1], but not really expanded upon.

[1] http://www.dam.brown.edu/people/mumford/Papers/DigitizedVisi...

jamii4• 14 years ago

The klein bottle is relatively uninteresting. The technique which identified it is the interesting part here. The linked paper explains a method to take a high-dimensional point cloud and compute a 'bar code' which encodes fundamental geometric features of the solid of which the cloud is an approximation. Its a way of visualizing high-dimensional data without using dimensional reduction.

Drbble2• 14 years ago

Does the point cloud approximate the shape, or does the shape approximate the point cloud? That distinction informed a recent discussion on the origin of the word "regression", which on the surface seems a weird term for curve-fitting, at least from my point of view.

jamii4• 14 years ago

I suppose the idea is that there is an underlying shape which the point cloud approximates. The fitted shape is then a statistical guess at the underlying shape. In which case it would be nice to see a proof that as more points are added the fitted shape eventually converges to the underlying shape.

yread• 14 years ago

from the article

"The full data set consists of roughly 8,000,000 points in E9. By normalizing with respect to mean intensity and restricting attention to high-contrast images (those away from the origin), the data set is projected to a set of points M a topological seven-sphere S7 ⊂ E8"

http://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf

TheBoff• 14 years ago

Does this allow for better compression, for example?

glogla• 14 years ago

I don't know about compression (though there's allegedly some research using fractals in video compression, but I have no idea if and how that works), but there is image pattern recognition method that is very similar. It's called Local Binary Patterns and it cuts image into square blocks and uses a clever way to turn that blocks into binary string, for example 3x3 pixels block into 8 bits. You run this process across a texture, and get a histogram of features the texture has, like corners, black spots, white spots, gradients and similar. LBP is also inherently invariant to lightning (it uses "lighter than/darker than" instead of absolute values) and there are modifications that make it rotation invariant too (rotating the strings to biggest sequence of ones, for example).

It's pretty smart, really.

grannyg00se3• 14 years ago

"a Klein bottle, which is a two-dimensional surface that can’t be squeezed into three dimensions, but fits perfectly well in nine (or for that matter in four). "

I'm having trouble squeezing this concept into my mind.

wisty2• 14 years ago

Well, a Möbius strip is a 2D surface that can't fit in 2 dimensions. Knots are 1D, and won't fit in 2 dimensions.

grannyg00se3• 14 years ago

Is it common to refer to the Möbius strip as a 2D surface? You can't uniquely identify each point on its surface using only two dimensions, so I wouldn't call it a 2D surface at all. It lives in 3D, therefore it is a 3D surface, no?

ebrink• 14 years ago

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obtu2• 14 years ago

Shouldn't zoomed photographs of smooth objects have mostly identical pixels, with the 9-dimension coordinates clustering around the all-pixels-at-the-same-intensity axis?

Edit: Oh, the papers mention this is about feature extraction and they filter for high-contrast patches.

nooneelse• 14 years ago

Yeah, not mentioning the filtering for high contrast makes this more confusing than it need be. Actually, the main link is worse than just not mentioning it... it says "randomly choose 3-by-3 pixel patches" and leaves it at that. Saying "random" rather than "random then filtered" is worse then just saying nothing and letting the reader guess that maybe there is something going on in the choosing.

teamonkey• 14 years ago

This strongly reminds me of the Bible Code[1], or rather of the Moby Dick Code[2]

Take any large body of work, manipulate it in some way, then patterns will emerge that make it resemble something else.

I mean seriously: they take a photo and reduce it down to a 3x3 square, arbitrarily convert the numerical values of those pixels to numbers, then take those nine numbers as 9-dimensional coordinates. The result is a surface that looks a little bit like a Klien bottle, but isn't really, since the surface is 4-dimensional instead of 2.

[1] http://en.wikipedia.org/wiki/Bible_code

[2] http://cs.anu.edu.au/people/bdm/dilugim/moby.html

aGHz• 14 years ago

This is a terribly incorrect trivialization of the work they did. The idea is not to observe some meaningless pattern emerge. Instead, they're trying to describe the statistics of random 3x3 pixels taken from photographs of nature (in fact, only of the luminosity of each pixel, hence the black-and-white photos). At a first thought, you'd expect this to be completely random, but it looks like there is a certain statistical structure to it. You only find this structure far-fetched because you're reading a blogger's vulgarization of the scientific paper whose concepts you're not familiar with.

Unlike the unreasonable bible and Moby Dick codes, this has practical applications in image manipulation and compression.

Who Expected That? Extreme close-ups create a Klein Bottle.

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