Puzzle: Sudoku Stories

Today I discovered a little book called *Sudoku Stories*
(Oscar Seurat, 2014). Each page presents a little encyclopedia entry on some
random topic accompanied by a sudoku puzzle whose pre-filled cells trace out
a shape corresponding to the entry. For example:

This got me thinking: There are 2^{81} different “images” possible
on a 9x9 grid. For some of these images (e.g. the all-filled-in image, the
moose image), it’s clearly possible to create a sudoku from them. For others
(e.g. any image containing fewer than 17 pixels),
it’s clearly impossible.

Roughly how many “sudoku-able” 9x9 images exist? Is it on the order of 2^{81}?
On the order of 2^{30}?

We can imagine a “meta-sudoku” version of the moose puzzle. Given *only*
the moose image,

I ask you to assign numbers to the black squares so that the resulting grid is a valid sudoku puzzle (with only one possible completion, of course).

One solution to the moose meta-puzzle is Seurat’s original moose puzzle.

Does the moose meta-puzzle have a *unique* solution? (Trivially, no, because you
can always e.g. substitute `1`

for `9`

and `9`

for `1`

, to produce another
valid sudoku. But does it have a unique solution *up to* swaps of that kind?)

Off the top of my head, I suspect that the moose meta-puzzle does *not* have
a unique solution. Can you come up with any meta-puzzle that *does* have a unique
solution (up to swaps)?

How many of the 2^{81} metapuzzles have unique solutions?