Derive minimum phase from magnitude
Check out Julius O. Smith III's write up.

There is a Hilbert transform relationship between the magnitude response, $G(\omega)$, and the phase response, $\theta(\omega),$ of the associated minimum phase filter.

If  

$$
H(\omega) = G(\omega) \exp(\jmath \theta(\omega))
$$
then
$$
\ln( H(\omega) ) = \ln(G(\omega)) + j\theta(\omega)
$$ 
and
$$
\theta(\omega) = - {\cal H}\left[ \ln(G(\omega)) \right]
$$
where ${\cal H}$ is the Hilbert transform.

Also check out Wikipedia's write up.