
Who Expected That? Extreme close-ups create a Klein Bottle. - ColinWright
http://www.thebigquestions.com/2012/02/23/who-expected-that/
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jamii
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.

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Drbble
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.

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jamii
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.

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grannyg00se
"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.

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wisty
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.

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grannyg00se
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?

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dfranke
_You can't uniquely identify each point on its surface using only two
dimensions_

Yes you can. Run a sharpie along the bottom edge of the paper so that it
bleeds onto both sides of the paper. Now tape the ends of the strip together
with a half twist to make a Möbius strip. Place a point anywhere on the strip.
Draw a line through that point such that the line meets both edges of the
strip at a right angle. Measure the distance along that line between your
point and the darkened edge; call this _x_. Now, hold the Möbius strip in your
left hand so that you're pinching it by the tape. With your right hand, run
your finger along the strip, starting from the tape and moving to the right.
Measure how far you have to move your finger in order to reach the line you
drew; this could require up to two loops around the strip. Call this distance
_y_. The tuple (x,y) uniquely identifies your point.

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grannyg00se
Wow. It took me a while to figure out what you were saying, but it was well
said and I eventually got it. Thanks.

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kd0amg
For the record, we use a similar trick on sphere-like surfaces, as latitude
and longitude.

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

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jamii
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.

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Drbble
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.

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jamii
The second (expository) paper is pretty readable. I have only the barest grasp
of algebraic topology but I believe I have a rough high-level understanding of
what it is doing. It would be interesting to implement their barcode program
independently to test whether or not I really understand.

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Tloewald
The comment is interesting. He? argues that any 2 manifold with no edges would
be likely to end up as a Klein bottle. The question is then why a 2 manifold.
Well, the thing we're asking about is a 2d picture, right? Maybe that has
something to do with it. Let's suppose we tried the same trick using 3x3x3
pixel representations of STL models — might that turn out to be form a simple
edge less 3 manifold embedded in 27 dimensional space? (perhaps for simplicity
we could use 2x2x2 cubes and look at 8 dimensional space... In essence we're
talking about a population of 2d data arbitrarily but consistently mapped into
higher dimensional space and we discover it maps to a 2 manifold.

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obtu
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.

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nooneelse
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.

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TheBoff
Does this allow for better compression, for example?

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glogla
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.

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teamonkey
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>

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aGHz
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.

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ebrink
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