A Hierarchical Bayesian Model for Frame Representation

TL;DR

This paper introduces a hierarchical Bayesian model for frame representation in signal processing, enabling joint estimation of frame coefficients and hyper-parameters using MCMC algorithms, with applications in image denoising.

Contribution

It proposes a novel hierarchical Bayesian framework and sampling algorithms for estimating frame coefficients and hyper-parameters in non-bijective frame representations.

Findings

Accurate estimation of frame coefficients and hyper-parameters.

Improved image denoising results using Bayesian estimation.

Effective MCMC algorithms for posterior sampling.

Abstract

In many signal processing problems, it may be fruitful to represent the signal under study in a frame. If a probabilistic approach is adopted, it becomes then necessary to estimate the hyper-parameters characterizing the probability distribution of the frame coefficients. This problem is difficult since in general the frame synthesis operator is not bijective. Consequently, the frame coefficients are not directly observable. This paper introduces a hierarchical Bayesian model for frame representation. The posterior distribution of the frame coefficients and model hyper-parameters is derived. Hybrid Markov Chain Monte Carlo algorithms are subsequently proposed to sample from this posterior distribution. The generated samples are then exploited to estimate the hyper-parameters and the frame coefficients of the target signal. Validation experiments show that the proposed algorithms provide an accurate estimation of the frame coefficients and hyper-parameters. Application to practical problems of image denoising show the impact of the resulting Bayesian estimation on the recovered signal quality.