We propose a new approach to study the stability of the optimal filter w.r.t. its initial condition, by introducing a "robust" filter, which is exponentially stable and which approximates the optimal filter uniformly in time. The "robust" filter is obtained here by truncation of the likelihood function, and the robustification result is proved under the assumption that the Markov transition kernel satisfies a pseudo-mixing condition (weaker than the usual mixing condition), and that the observations are "sufficiently good". This robustification approach allows us to prove also the uniform convergence of several particle approximations to the optimal filter, in some cases of nonergodic signals.