On calculating the mean values of quantum observables in the optical   tomography representation
  Given a density operator $\hat \rho$ the optical tomography map defines a one-parameter set of probability distributions $w_{\hat \rho}(X,\phi),\ \phi \in [0,2\pi),$ on the real line allowing to reconstruct $\hat \rho $. We introduce a dual map from the special class $\mathcal A$ of quantum observables $\hat a$ to a special class of generalized functions $a(X,\phi)$ such that the mean value $<\hat a>_{\hat \rho} =Tr(\hat \rho\hat a)$ is given by the formula $<\hat a>_{\hat \rho}= \int \limits_{0}^{2\pi}\int \limits_{-\infty}^{+\infty}w_{\hat \rho}(X,\phi)a(X,\phi)dXd\phi$. The class $\mathcal A$ includes all the symmetrized polynomials of canonical variables $\hat q$ and $\hat p$. 
Grigori G. Amosov, Yakov A. Korennoy, Vladimir I. Man'ko
quant-ph math-ph math.MP