infer a 2D filter from its frequency spectrum

Dear all,
I am working on a 2D convolution problem. I have estimated the
frequency spectrum |H(u, v)| now I wish to infer the 2D filter f(x,
y).
I just don't know a way to work it out from the spectrum, although I
may have several constrains of f(x, y) in the spatial domain for
assistance.

It will also be helpful if there's a good way to easily compute
derivative d|H(u, v)|/df(x, y), that I can optimize f(x, y) in the
spatial domain.

Any ideas?

Thanks!