When a continuous-time diffusion is observed only at discrete times with measurement noise, in most cases the transition density is not known and the likelihood is in the form of a high-dimensional integral that does not have a closed-form solution and is difficult to compute accurately. Using Hermite expansions and deconvolution strategy, we provide a general explicit sequence of closed-form contrast for noisy and discretely observed diffusion processes. This work allows the estimation of many diffusion processes. We show that the approximation is very accurate and prove that minimizing the sequence results in a consistent and asymptotically normal estimator. Monte Carlo evidence for the Ornstein-Uhlenbeck process reveals that this method works well and outperforms the Euler expansion of the transition density in situations relevant for financial models.