Hierarchical Bayesian sparse image reconstruction with application to MRFM.

Area of Science:

• Image reconstruction
• Bayesian inference
• Signal processing

Background:

• Image reconstruction from linear transformations with additive white Gaussian noise is a challenging problem.
• Existing sparse reconstruction methods often provide only point estimates, lacking comprehensive posterior information.

Purpose of the Study:

• To develop a hierarchical Bayesian model for sparse image reconstruction.
• To incorporate image properties like sparsity and positivity through tailored Bayes priors.
• To provide a fully Bayesian approach offering complete posterior distributions for all parameters.

Main Methods:

• A hierarchical Bayesian model is proposed, utilizing a weighted mixture prior (positive exponential distribution and mass at zero).
• Hyperparameter tuning is achieved automatically via marginalization within the hierarchical model.
• A Gibbs sampling strategy is employed to manage the posterior distribution's complexity.
• Image estimation is performed by maximizing the estimated posterior distribution.

Main Results:

• The proposed model successfully reconstructs sparse images from linear transformations corrupted by Gaussian noise.
• The method effectively accounts for image sparsity and positivity using a novel Bayes prior.
• The Gibbs sampling strategy provides a feasible approach for complex posterior distributions.
• The fully Bayesian approach yields complete posterior distributions, offering richer information than point estimates.

Conclusions:

• The developed hierarchical Bayesian model offers a robust and informative method for sparse image reconstruction.
• The automatic hyperparameter tuning and Gibbs sampling strategy enhance model applicability.
• The method demonstrates superior performance compared to existing techniques, as shown on synthetic and real MRFM data.