Wigner function as the expectation value of a parity operator

It is pointed out that the Wigner function $f(r, p)$ is $\frac{2}{h}$ times the expectation value of the parity operator that performs reflections about the phase-space point $r$, $p$. Thus $f(r, p)$ is proportional to the overlap of the wave function $\psi$ with its mirror image about $r$, $p$; this is clearly a measure of how much $\psi$ is centered about $r$, $p$, and the Wigner distribution function now appears physically more meaningful and natural than it did previously.