Bayesian Wavelet Domain Image Modeling using Hidden Markov Trees
Justin Romberg (jrom@rice.edu)
Electrical and Computer Engineering, Rice University
Hyeokho Choi (choi@ece.rice.edu)
Electrical and Computer Engineering, Rice University
Richard Baraniuk (richb@rice.edu)
Electrical and Computer Engineering, Rice University
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 statistics 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 EM algorithm, for example). In this paper, we propose two
reduced-parameter HMT models that capture the general structure of a
broad class of grayscale images. The {\em image HMT (iHMT)} model
leverages 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 {\em 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 two new models retain
nearly all of the key structures modeled by the full HMT.
Based on these new models, we develop a shift-invariant wavelet
denoising scheme that outperforms all algorithms in the
current literature.