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Let $S$ be a minimal surface of general type with $p_{g}(S)=0$ and$K^{2}_{S}=4$. Assume the bicanonical map $\varphi$ of $S$ is a morphism ofdegree $4$ such that the image of $\varphi$ is smooth. Then we prove that thesurface $S$ is a Burniat surface with $K^{2}=4$ and of non nodal type.

A characterization of Burniat surfaces with $K^{2}=4$ and of non nodal type