Ken, thanks for moving it here... I was going to do the same, but you was first :)
Now, the answer:
Actually "nothing is impossible" as some recent TV advertisment is trying to say..
(yeah well, I know.. they are using reversed version)
Anyway...
It's called "resynthesis" or "synthesis by example".
There are some guys working on it.
I don't know that much about it, because haven't tried it yet myself,
but after reading few papers about this topic (search for siggraph papers for example and other computer graphics conferences),
I can only say, that:
1. It is very hard to produce exact results and usually it is not what we want, because "synthesis by example" is more about producing variations (and gaining some edit control over resynthesised result) or repairing the damaged part of the picture (for a example filling up a hole in a bricky wall) or increase dimension (2d->3d) and not just for compression.
(look for "synthesis by example" or "texture resynthesis" in google)
2. It is hard to generalize the solution for aribtrary shapes or pictures. The problem is: procedural methods (like fractal shapes) are usually limited to just single class of shapes/pictures, which usually means that not every possible mathematical function/surface can be evaluated out of them.
Ken,Tom is there any chance that some kind of such feature/possibility could appear
in the later versions of VOXLAP library and/or CAVEGEN/CAVETEX applications?
The main reason for procedural textures in Voxlap/CaveTex is: they can be defined in 3d easily (taking 3d coordinates as input).
Although it could be possible to give 2d tile and generate 3d texture from it, for example by matching some correlations in it (by translation/angle) and generating new slices according to them for 3d tile (I suspect this is what
http://www.cc.gatech.edu/~turk/ was trying to do in one of his paper)... but I should try it, do some experiments, before saying anything about usefulness of this stuff.