Hidden Markov Tree Models for Complex Wavelet Transforms
Romberg, Justin; Choi, Hyeokho; 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.