Bayesian tree-structured image modeling using wavelet-domain hidden Markov models

Wavelet-domain hidden Markov models have proven to be useful tools for statistical signal and image processing. The hidden Markov tree (HMT) model captures the key features of the joint density of the wavelet coefficients of real-world data. One potential drawback to the HMT framework is the need for computationally expensive iterative training (using the Expectation-Maximization algorithm, for example). In this paper, we propose two reduced-parameter HMT models that capture the general structure of a broad class of real-world images. In the image HMT (iHMT) model we use the fact that for a large class of images the structure of the HMT is self-similar across scale. This allows us to reduce the complexity of the iHMT to just nine easily trained parameters (independent of the size of the image and the number of wavelet scales). In the universal HMT (uHMT) we take a Bayesian approach and fix these nine parameters. The uHMT requires no training of any kind. While simple, we show using a series of image estimation/denoising experiments that these new models retain nearly all of the key structure modeled by the full HMT. Finally, we propose a fast shift-invariant HMT estimation algorithm that outperforms all other wavelet-based estimators in the current literature, both in mean-square error and visual metrics.