Inversion of large-support ill-posed linear operators using a piecewise Gaussian MRF.

Area of Science:

• Image reconstruction
• Signal processing
• Computational imaging

Background:

• Ill-posed inverse problems arise from partial observations via large linear operators.
• Reconstruction requires incorporating prior information about object properties.
• Piecewise Gaussian (PG) Markov random fields (MRFs) model smooth regions with sharp transitions.

Purpose of the Study:

• To develop a robust method for reconstructing signals and images from partial, large-support linear observations.
• To address the challenge of minimizing multimodal posterior energy functions in ill-posed inverse problems.
• To extend the graduated nonconvexity (GNC) algorithm for effective global optimization in this context.

Main Methods:

• Modeling object priors using piecewise Gaussian (PG) Markov random fields (MRFs).
• Defining reconstruction via maximum a posteriori (MAP) estimation.
• Extending the graduated nonconvexity (GNC) algorithm to handle ill-posed linear inverse problems and their associated optimization challenges.

Main Results:

• The proposed method effectively reconstructs signals and images from partial observations.
• The extended GNC algorithm provides a practical and efficient solution for minimizing complex posterior energies.
• Theoretical analysis offers new insights into the GNC algorithm's application to ill-posed problems.

Conclusions:

• The extended GNC algorithm is a powerful tool for ill-posed inverse problems with PG MRF priors.
• This approach offers significant improvements in reconstruction quality for applications like diffraction tomography.
• The method demonstrates practical efficiency and theoretical grounding for complex signal and image reconstruction tasks.