What's going on in a recursive picture?

1. Start with an image showing the frame you want to use for your Escher "Print Gallery" photograph. The highest the resolution, the better.




2. Rotate and crop the source image so that the center frame is dead in the center, and the aspect ratio of the whole image is the same as the center frame. Also, now may be a good time to correct any problem in the source image, check levels and sharpen a bit, especially the center (where the "blowing out" will be highest).




3. This step is where the mathmap magic starts to come in. By using a complex logarithm on the pixels, the image is "unrolled" around the center.

4. The previous image is cut and pasted several times, so that the whole canvas is covered.

5. This step undoes what was done in Step 3. This is done by "rolling up" the image in Step 4 by using the inverse of the complex logarithm, i.e. the complex exponential.

You might have said, why bother? I could have easily gimped the source image to get this. And you would be right. But there is more to it...

6. If you rotate the image produced by Step 4 a bit, and then roll it up as in Step 5, you get this image. Voilà!

Why you should correct for barrel distortion

Why you should correct for barrel distortion
Originally uploaded by Seb Przd.

All zoom lenses are prone to a type of distortion known as barrel distortion, where straight lines appear curved as if the center of the image had been inflated. This is especially the case with the less expensive zoom lenses; usually more expensive lenses, lenses with less zoom range and prime (fixed) lenses are corrected and are closer to a perfcet rectilinear lens. My new lens, the Nikkor 18-200mm, shows a lot of barrel distortion at 18mm.

Fortunately, quite a few tools are available to correct for this predictable type of distortion. I use clens, a command-line utility distributed with panotools, with the lens profiles that come with PTlens, a windows alternative to clens (you can find the database in panotools or here).

The top image above is a straight grid, as shot. You can see the curves that should be straight. The bottom image shows the automatic correction with clens, with the lens profile from PTlens.

clens is run like this:

clens -p profile.txt -s nona -l "Nikkor 18-200mm f/3.5-5.6 G ED-IF AF-S VR DX" image.jpg

If it run like this:

clens -p profile.txt -s nona --Lenses image.jpg

it outputs a list of possible lenses, according to the make of the camera and the focal length.
The correction is not perfect, since the Nikkor 18-200mm shows a more complicated (higher order) type of distortion that clens cannot correct for.

Turkish Delight

Turkish Delight
Originally uploaded by Seb Przd.

Having fun with conformal mappings.

Honey I Escherized the kids!

Honey I Escherized the kids!
Originally uploaded by Seb Przd.

Conformal mappings are great: it's a very powerful branch of mathematics, which basically says you can deform almost any 2D object and transform it into something else while keeping the shape of objects the same (on a small scale). Hence this picture: you see no holes, no obvious distortion, but this image cannot be real. This trick is inspired by a painting by M.C. Escher: some Dutch mathematicians have found what the transformation is and explained in (almost) layman terms.
This image was done in The Gimp and transformed with MathMap.

Shaft the tilt-shift lenses

Why do you still need tilt-shit lenses with digital cameras ?


The first image was shot "as is" of the Saint Ambroise church in Paris with a wide-angle lens. The second one is the same one, but corrected for perspective distortion with hugin.

Equirectangular obsession

Bois de Vincennes (3)
Originally uploaded by Seb Przd.

I've been going through a full immersive panorama phase recently that has me going shooting many pictures from one spot in all directions. I've even bought a new ultra-wide-angle lens (Canon EF-S 10-22mm) so that I have to take fewer pictures. I'm waiting for my spherical tripod head to arrive (Nodal Ninja). You can see more of my pictures here.