Nearly Exact Discrepancy Principle for Low-Count Poisson Image Restoration

Area of Science:

• Image restoration
• Computational imaging
• Applied mathematics

Background:

• Variational methods are crucial for image restoration, but their effectiveness hinges on regularization parameter selection.
• The discrepancy principle, a common method for parameter selection, performs poorly with low-count Poisson noise due to theoretical approximations.

Purpose of the Study:

• To address the limitations of the traditional discrepancy principle in low-count Poisson noise image restoration.
• To propose and validate a modified, nearly exact discrepancy principle for improved parameter selection.

Main Methods:

• Theoretical analysis of the limitations of the original discrepancy principle.
• Development of a modified discrepancy principle using Monte Carlo simulation and weighted least-square fitting.
• Numerical experiments to evaluate performance across different Poisson noise levels.

Main Results:

• The proposed modified discrepancy principle significantly outperforms the original method in low-count Poisson noise regimes.
• Similar or slightly improved performance is observed in mid- and high-count Poisson noise regimes compared to the original method.
• Validation of the theoretical improvements through extensive numerical simulations.

Conclusions:

• The modified discrepancy principle offers a robust solution for parameter selection in variational image restoration, particularly for low-count Poisson noise.
• This enhanced approach provides more accurate and reliable image restoration results, expanding the applicability of variational methods.
• The study highlights the importance of refining theoretical approximations for practical performance gains in scientific computing.