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

M Nikolova1, J Idier, A Mohammad-Djafari

  • 1Lab. des Signaux et Syst., CNRS, Gif-sur-Yvette, France. nikolova@lss.supelec.fr

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
|February 16, 2008
PubMed
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We present a new method for reconstructing partially observed signals and images. Our approach uses a piecewise Gaussian (PG) Markov random field (MRF) prior and an extended graduated nonconvexity (GNC) algorithm for optimization, improving reconstruction accuracy.

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).

Related Experiment Videos

  • 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.