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We recently introduced a novel approximation of the intractable two-dimensional hidden Markov model (2D HMM), the turbo-HMM (T-HMM), which consists of a set of interconnected horizontal and vertical 1D HMMs. In this paper, we consider the extension of this framework to the continuous state HMM, generally referred to as the state-space model (SSM). We provide efficient approximate answers to the three following problems: (1) how to compute the likelihood of a set of observations; (2) how to find the sequence of states that best "explains" a set of observations; and (3) how to estimate the model parameters given a set of observations. The application of this work to the challenging problem of face recognition, in the presence of large illumination variations, illustrates the potential of our approach.