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 $ECODF$? $\text{(A)}\ \frac {8\sqrt {2}}{3} \qquad$ $ \text{(B)}\ 8\sqrt {2} - 4 - \pi \qquad$ $ \text{(C)}\ 4\sqrt {2} \qquad$ $ \text{(D)}\ 4\sqrt {2} + \frac {\pi}{8} \qquad$ $ \text{(E)}\ 8\sqrt {2} - 2 - \frac {\pi}{2}$