Hidden Markov Tree Models for Complex Wavelet Transforms
Baraniuk, Richard G.
Kingsbury, Nicholas G.
Multiresolution models such as the hidden Markov tree (HMT) aim to capture the statistical structure of signals and images by leveraging two key wavelet transform properties: wavelet coefficients representing smooth/singular regions in a signal have small/large magnitude, and small/large magnitudes persist through scale. Unfortunately, the HMT based on the conventional (fully decimated) wavelet transform suffers from shift-variance, making it less accurate and realistic. In this paper, we extend the HMT modeling framework to the complex wavelet transform, which features near shift-invariance and improved directionality compared to the standard wavelet transform. The complex HMT model is computationally efficient (with linear-time computation and processing algorithms) and applicable to general Bayesian inference problems as a prior density for images. We demonstrate the effectiveness of the model with two applications. In a simple estimation experiment, the complex wavelet HMT model outperforms a number of high-performance denoising algorithms, including redundant wavelet thresholding (cycle spinning) and the redundant HMT. A multiscale maximum likelihood texture classification algorithm produces fewer errors with the new model than with a standard HMT.