Computational methods for hidden Markov tree models - An application to wavelet trees.

Abstract : The hidden Markov tree models were introduced by Crouse et al. in 1998 for modeling nonindependent, non-Gaussian wavelet transform coefficients. In their paper, they developed the equivalent of the forward-backward algorithm for hidden Markov tree models and called it the 'upward-downward algorithm'. This algorithm is subject to the same numerical limitations as the forward-backward algorithm for hidden Markov chains (HMCs). In this paper, adapting the ideas of Devijver from 1985, we propose a new 'upward-downward' algorithm, which is a true smoothing algorithm and is immune to numerical underflow. Furthermore, we propose a Viterbi-like algorithm for global restoration of the hidden state tree. The contribution of those algorithms as diagnosis tools is illustrated through the modeling of statistical dependencies between wavelet coefficients with a special emphasis on local regularity changes.
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IEEE Transactions on Signal Processing, Institute of Electrical and Electronics Engineers, 2004, 52 (9), pp.2551-2560. 〈10.1109/TSP.2004.832006〉
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Jean-Baptiste Durand, Paulo Goncalvès, Yann Guédon. Computational methods for hidden Markov tree models - An application to wavelet trees.. IEEE Transactions on Signal Processing, Institute of Electrical and Electronics Engineers, 2004, 52 (9), pp.2551-2560. 〈10.1109/TSP.2004.832006〉. 〈hal-00830078〉

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