A typical image restoration model posits that, starting from a true image $f = f(x,y)$, we observe $$ \tilde f = f \star h + n $$ where $\star$ is convolution, $h$ is the point spread function (caused, e.g., by diffraction in the optics, but hopefully close to a $\delta$ function), and $n$ is noise (typically assumed to be independent between pixels, and possibly i.i.d.).
Question. Given $h$ and a noise model, are there any usefully-computable lower bounds on reconstruction quality?
That is, if $\hat f$ is a reconstruction of the given image $f$, I'm looking for lower bounds on something like $$ \sup_f \mathbb{E}\|\hat f - f\|_2, $$ where the supremum is universal over the class of reconstruction algorithms. I'd expect the lower bound to mention things like $\|f\|_2$, $\|n\|_2$, $\|h\|_2$ (or possibly other moments, or even the exact shape of $h$). If limiting the class of $f$'s is helpful, I'm happy to do that.
The closest inequality I know is the Cramér-Rao lower bound, but that's for unbiased estimates; I don't necessarily care about whether the estimator is biased, although I'm potentially willing to constrain the estimator to e.g. be linear or smooth or local or some such.
