[Paper Review] Automatic parameter selection for the TGV regularizer in image restoration under Poisson noise

Daniela di Serafino, Monica Pragliola|arXiv (Cornell University)|May 26, 2022

Sparse and Compressive Sensing Techniques4 citations

TL;DR

This paper proposes a hierarchical Bayesian framework with Maximum A Posteriori estimation to automatically select parameters in the TGV² regularization for image restoration under Poisson noise. By coupling a discrepancy principle with an ADMM-based alternating minimization scheme, the method jointly estimates the image and regularization parameters (α₀, α₁, λ), achieving high-quality restorations comparable to manual tuning, especially in moderate to high counting regimes.

ABSTRACT

We address the image restoration problem under Poisson noise corruption. The Kullback-Leibler divergence, which is typically adopted in the variational framework as data fidelity term in this case, is coupled with the second-order Total Generalized Variation (TGV$^2$). The TGV$^2$ regularizer is known to be capable of preserving both smooth and piece-wise constant features in the image, however its behavior is subject to a suitable setting of the parameters arising in its expression. We propose a hierarchical Bayesian formulation of the original problem coupled with a Maximum A Posteriori estimation approach, according to which the unknown image and parameters can be jointly and automatically estimated by minimizing a given cost functional. The minimization problem is tackled via a scheme based on the Alternating Direction Method of Multipliers, which also incorporates a procedure for the automatic selection of the regularization parameter by means of a popular discrepancy principle. Computational results show the effectiveness of our proposal.

Motivation & Objective

• Address the challenge of automatic parameter selection in TGV² regularization for Poisson-noise image restoration, where manual tuning is impractical and error-prone.
• Overcome limitations of Total Variation (TV) regularization, such as the staircasing effect, by employing higher-order TGV² to preserve both edges and smooth regions.
• Develop a robust, fully automatic method that jointly estimates the image and regularization parameters without requiring user intervention.
• Ensure the regularization parameter λ is adaptively selected via a discrepancy principle to maintain data fidelity while avoiding over- or under-smoothing.

Proposed method

• Formulate the image restoration problem as a hierarchical Bayesian model, treating the unknown image and TGV² parameters as random variables to be estimated.
• Use Maximum A Posteriori (MAP) estimation to derive a joint minimization problem for the image and parameters, incorporating the Kullback-Leibler divergence as the data fidelity term.
• Implement an alternating minimization scheme based on the Alternating Direction Method of Multipliers (ADMM) to solve the non-smooth, non-convex optimization problem efficiently.
• Incorporate a discrepancy principle into the cost functional to automatically select the regularization parameter λ during the optimization process.
• Introduce a two-stage initialization: first solve the TV-KL model to obtain a good starting point, then use its solution to initialize the TGV²-regularized problem.
• Monitor convergence via ISNR and SSIM values, parameter evolution, and relative change in iterates to validate stability and effectiveness.

Experimental results

Research questions

• RQ1 Can a fully automatic parameter selection strategy be developed for TGV² regularization in Poisson-noise image restoration without manual tuning?
• RQ2 How does the proposed hierarchical Bayesian approach with MAP estimation compare to manual parameter selection in terms of image quality and robustness?
• RQ3 To what extent does the discrepancy principle improve the selection of the regularization parameter λ in the presence of varying noise levels?
• RQ4 How does the method perform in low-counting regimes or when the observed data contain a high number of zero entries?
• RQ5 Does the alternating ADMM scheme converge reliably, and can it avoid the oversmoothing typical of suboptimal parameter choices?

Key findings

• The proposed method achieves image restorations with ISNR and SSIM values very close to those obtained via manual tuning of TGV² parameters, demonstrating high effectiveness.
• For the penguin image, the method performs well even at low counting levels (κ=30), with ISNR values above 15 dB and SSIM above 0.85, indicating robustness to low signal levels.
• In the brain image test case, which contains many zero entries, the method produces smoother outputs than the target, suggesting a tendency toward oversmoothing due to suboptimal λ selection.
• The estimated parameters α₀ and α₁ stabilize quickly during iterations, and their evolution is well-captured by the algorithm, indicating reliable convergence.
• Convergence of the alternating scheme is observed within 10–15 iterations, as evidenced by stabilization of ISNR, SSIM, and parameter values in the penguin test case with κ=50.
• The discrepancy principle effectively guides λ selection, enhancing robustness, though performance degrades in very low-counting regimes or with high zero counts, indicating a need for further refinement.