It's expressed somewhat awkwardly, but what's going on is that R(x) is zero if x is in range of g, and infinite otherwise. Choice of g is such that natural images are in range of g, and non-natural images aren't.
Sorry I still don't understand. They require g to be surjective. Edit: in their paper they call it f_theta and it's explicitly not surjective. Dunno why their writeup is so confused.
