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

• Published in: IEEE transactions on signal processing Vol. 52; no. 9; pp. 2551 - 2560
• Main Authors: Durand, J.-B., Goncalves, P., Guedon, Y.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.09.2004; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Computer Science; Context modeling; Equivalence; Exact sciences and technology; Hidden Markov models; Image restoration; Image segmentation; Information, signal and communications theory; Markov processes; Mathematical methods; Mathematical models; Modeling and Simulation; Regularity; Signal processing algorithms; Signal restoration; Smoothing methods; Telecommunications and information theory; Trees; Viterbi algorithm; Wavelet; Wavelet coefficients; Wavelet transforms; Change detection; upward-downward algorithm; wavelet decomposition; Scaling law; hidden Markov tree model; Viterbi detection; Wavelet transformation; hidden state tree restoration; Hidden Markov models; scaling laws; Diagnosis; EM algorithm; Computing method; Signal detection
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2004.832006

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.