Surfaces on the Severi line

Let $S$ be a minimal complex surface of general type and
of maximal Albanese dimension; by the Severi inequality one has $K^2_S\ge 4\chi(\mathcal O_S)$. We prove that the equality $K^2_S=4\chi(\mathcal O_S)$ holds if and only if $q(S):= h^1(\mathcal O_S)=2$ and the canonical model of $S$ is a double cover of the Albanese surface branched on an ample divisor with at most negligible singularities.