Off-the-grid regularisation for Poisson inverse problems.

Area of Science:

• Inverse problems
• Image reconstruction
• Computational imaging

Background:

• Off-the-grid regularization is effective for ill-posed inverse problems in continuous settings.
• Traditional methods often use Total Variation regularization with L2 data terms for Gaussian noise.
• Signal-dependent noise, like Poisson noise, requires different modeling approaches.

Purpose of the Study:

• To develop and analyze an off-the-grid regularization framework for sparse reconstruction under Poisson noise.
• To investigate a variational model combining Total Variation regularization with a Kullback-Leibler data term.
• To evaluate the proposed method on simulated and real microscopy data.

Main Methods:

• Formulation of a variational model with Total Variation regularization and a Kullback-Leibler data term.
• Analytical study of optimality conditions and dual problem for the composite functional.
• Development of an homotopy strategy for optimal regularization parameter selection.
• Application of a Sliding Frank-Wolfe algorithm for efficient optimization.

Main Results:

• The proposed model effectively handles signal-dependent Poisson noise in sparse reconstruction.
• Analytical insights into the model's properties and optimization were obtained.
• The homotopy strategy and Sliding Frank-Wolfe algorithm provided efficient parameter selection and reconstruction.
• Successful application to 1D, 2D, and 3D simulated data, as well as real 3D fluorescence microscopy data.

Conclusions:

• The developed off-the-grid regularization framework is suitable for sparse reconstruction with Poisson noise.
• The combination of Total Variation and Kullback-Leibler divergence offers a robust approach for Poisson noise modeling.
• The method demonstrates strong performance on complex imaging data, particularly in microscopy.