Circles centered at $A$ and $B$ each have radius $2$ , as shown. Point $O$ is the midpoint of $\overline{AB}$ , and $OA = 2\sqrt {2}$ . Segments $OC$ and $OD$ are tangent to the circles centered at $A$ and $B$ , respectively, and $EF$ is a common tangent. What is the area of the shaded region $ECODF$ ?