A different digital swirling, or "and you thought little planets were bad"

[inkista]

Hopefully, this is enough in the spirit of the forum, that the mods won't remove it for breaking the letter of the law. My "Before" image is not unaltered straight from camera. It's an equirectangular (360°x180° or spherical pano) that was made by stitching together four hand-held fisheye shots. For the interactive view (needs Flash), go here.

Before:


After:


This is a math-geek remapping thing, like little planets. There are numerous ways to represent a sphere on a flat 2-D plane. The most common way of representing the earth is the equirectangular one: your longitude and latitude become your x and y values on a cartesian plane. But among the more exotic remappings there's one called Peirce Quincuncial, which remaps the sphere to a square, where one pole is smack in the middle of the square, and the other is split among the four corners.

The thing is, if you get a Peice Quincuncial remapping of a sphere, it's tileable, seamlessly. That's half of what's going on here.

The other half is what's called Droste. Classical Droste effects are things like the Land o'Lakes butter logo: where someone's holding a picture of themselves holding a picture of themselves... etc. etc. Back in 2002, Hendrik Lenstra figured out the math for this kind of effect, similar to how Escher would tile things in his paintings. Once you have the math, you code it.

So, Ben Reichardt aka breic on Flickr used Mathmap, a Gimp plugin. The power of Mathmap is that you can write a script for it in a C-like language. Any kind of mathematical transform can be encoded for Mathmap to perform on a graphic file. In this specific case, we're really lucky. Breic, who wrote the code for "Droste" is an open-source guy, and he freed the code. He's also someone who shoots and likes to remap equirectangulars, so he came up with code that not only does the Peirce Quincuncial remapping, it will also "drostify" it. Doing both, results in an image like the one I came up with. And it, too, is freely available on the Flickr Mathmap group.

I am on a Mac, btw, so I didn't use Gimp or the Mathmap plugin directly, but I used an OSX standalone version called Mathmap Cocoa (it's available at the Mathmap website I linked to above). The basic steps in Mathmap Cocoa are:

1. Open up breic's Peirce Quincuncial script in the application.
2. Click on the User Values tab.
   
3. Drag and drop your equirectangular on to the "in image" box (shaded in red).
4. Set ImageH to the height of the equirectangular you're feeding it.
5. Pull sliders about. :^)
6. Check the "drostify" checkbox.
7. Pull some more sliders about until you like the image.
8. Export the image to a .TIFF file.

If you don't have any equirectangulars to play with, you can skip the Peirce's Quincuncial, and just use the Droste filter (it comes with Mathmap these days), and just do a straight-up Droste-swirly instead:



The steps are pretty much the same, only you won't have to check a "drostify" checkbox. The Flickr Escher Droste group has more information on how you can swirl around shapes other than circular by using masks, as well as how to use PS CS4 and Pixel Blender to do Droste.

[Sterling]

Those are really cool effects. But all that math talk is making me a little woozy.

[OsmosisStudios]

Dont get me wrong, Inkista: you're the goddess of all that is awesome and I would be amazed to meet you... but all I got from this was "ooh pretty picture... mathmathmath... ooh pretty swirly picture!"

[Indigo November]

Originally Posted by OsmosisStudios

Dont get me wrong, Inkista: you're the goddess of all that is awesome and I would be amazed to meet you... but all I got from this was "ooh pretty picture... mathmathmath... ooh pretty swirly picture!"

LOL, yeah me too!

[inkista]

And I didn't even post any equations!

Sorry, guys, I knew I was pressurizing it at firehose levels, but if I broke it down into bite-sized chunks, it would've been PAGES and PAGES. And DPS limits you to only four graphics per post. Enjoy the pretty swirlies; don't worry about the math. Just look at the numbered steps.

[Nicole]

You come up with the neatest looking things

[Aegea]

My brain hurts ... I'm gonna reread this after coffee ...

[Jill H]

I have no idea what you are talking about - and I am too sick with a cold to even try to comprehend it - but the effect is cool.

[Aegea]

OK, I think I get part of it. The pano is actually an equirectangular mapping from a sphere, and the Pierce quincuncial is a different mapping to the plane. Then you do the Droste magic on the plane. I like it! Very, very cool!

[inkista]

Originally Posted by Aegea

OK, I think I get part of it. The pano is actually an equirectangular mapping from a sphere,

Yup. The sphere's been mapped to a 2x1 rectangle like this:



Essentially, your latitude is mapped to the y coordinate, and your longitude is mapped to the x coordinate. The top and bottom of the rectangle are the top and bottom of the sphere, and the right and left sides wrap around to seam together. Most of the distortion happens at the poles.

and the Pierce quincuncial is a different mapping to the plane.

Yup. For the earth, it looks like this:



If you do one from the top of the sphere, and one from the bottom, they join up seamlessly, and if you flip that 1x2 rectangle, it also tiles seamlessly, like this:

Then you do the Droste magic on the plane.

Yay!! You got it!!

I like it! Very, very cool!

Thanks! I like it, too.