Question

After exploring the very excellent answer from Heike to my previous question about anamorphic transformations, I eventually wanted to see an image turned inside out completely.

The idea is that, instead of just stretching the image out with an anamorphic transform, like you’re pulling the edges of the paper around, you can actually turn the paper ‘inside out’. The inside ‘pixels’ will be pulled out to the edges (greatly distorted/stretched), while the outside pixels will be squashed inwards towards the centre (greatly shrunk).

I can’t illustrate it, but another way of trying to describe it is in this picture:

so, the more red the pixels are, the more they are transformed to the edges (and vice versa).

I tried FindGeometricTransform, but it didn’t seem to lead anywhere.

It’s not been easy to google for this, and I’ve yet to find any clues in Mathematica that such a destructive transformation is possible. It’s kind of a 2.5D re-projection.

What do you think? Is it possible?

Edit

So thanks to your great answers I can now illustrate my question properly:

Here’s Leonardo’s famous Anom Asil, the result of subjecting poor Lisa to the inside-out transform ():

and here’s the Prague Orloj:

Practical uses for this will be forthcoming, er, soon…

Thanks!

Answer 1

Perhaps something like this:

f[x_, y_] := {x, y} (1/Norm@{x, y} - 1);
GraphicsGrid[{{ 
       p = Rasterize[Graphics[ {Black, Disk[{0, 0}, 5],
                                Red,   Disk[{0, 0}, 3],
                                Blue,  Disk[{0, 0}, 2]}]],
       ImageTransformation[p, f[#[[1]], #[[2]]] &, 
                           DataRange -> {{-1, 1}, {-1, 1}}]}}, 
       Frame -> All]

Edit

Using Heike’s fthe function is bijective, and it’s own inverse:

f[x_, y_] := {x, y} (1/Norm[{x, y}, Infinity] - 1);
g[x_]:=ImageTransformation[x, f[#[[1]], #[[2]]] &,DataRange ->{{-1, 1}, {-1, 1}}]

GraphicsGrid[{{i, g@i, g@g@i}}, Frame -> All]

Edit

Setting:

f[x_, y_, t_] := {x, y} ((1/Norm[{x, y}, Infinity] - 1 ) t + (1 - t));