A Hierarchical Bayesian Model for Frame Representation

In many signal processing problems, it is fruitful to represent the signal under study in a frame. If a probabilistic approach is adopted, it becomes then necessary to estimate the hyperparameters 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 hyperparameters 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 hyperparameters and the frame coefficients of the target signal. Validation experiments show that the proposed algorithms provide an accurate estimation of the frame coefficients and hyperparameters. Application to practical problems of image denoising in the presence of uniform noise illustrates the impact of the resulting Bayesian estimation on the recovered signal quality.