Source/Filter Factorial Hidden Markov Model, with Application to Pitch and Formant Tracking

Tracking vocal tract formant frequencies ($f_p$) and estimating the fundamental frequency ($f_0$) are two tracking problems that have been tackled in many speech processing works, often independently, with applications to articulatory parameters estimations, speech analysis/synthesis or linguistics. Many works assume an auto-regressive (AR) model to fit the spectral envelope, hence indirectly estimating the formant tracks from the AR parameters. However, directly estimating the formant frequencies, or equivalently the poles of the AR filter, allows to further model the smoothness of the desired tracks. In this paper, we propose a Factorial Hidden Markov Model combined with a vocal source/filter model, with parameters naturally encoding the $f_0$ and $f_p$ tracks. Two algorithms are proposed, with two different strategies: first, a simplification of the underlying model, with a parameter estimation based on variational methods, and second, a sparse decomposition of the signal, based on Non-negative Matrix Factorization methodology. The results are comparable to state-of-the-art formant tracking algorithms. With the use of a complete production model, the proposed systems provide robust formant tracks which can be used in various applications. The algorithms could also be extended to deal with multiple-speaker signals.