Let $X$ and $Y$ be two closed connected Riemannian manifolds of the same dimension and $\phi: S^*X \mapsto S^*Y$ a contact diffeomorphism. We show that the index of an elliptic Fourier operator $\Phi$ associated with $\phi$ is given by $\int_{B^*(X)} {\rm e}^{ \theta_0}\hat{A}(T^*X) - \int_{B^*(Y)} {\rm e}^{ \theta_0}\hat{A}(T^*Y)$ where $\theta_0$ is a certain characteristic class depending on the principal symbol of $\Phi$ and, $B^*(X)$ and $B^*(Y)$ are the unit ball bundles of the manifolds $X$ and $Y$. The proof uses the algebraic index theorem of Nest-Tsygan for symplectic Lie Algebroids and an idea of Paul Bressler to express the index of $\Phi$ as a trace of $1$ in an appropriate deformed algebra.\\ \mbox{\quad} In the special case when $X=Y$ we obtain a different proof of a theorem of Epstein-Melrose conjectured by Atiyah and Weinstein.