Gaussian surrogates do well on Poisson inverse problems

2026-02-19

AI Summary

This study addresses the inconsistency between objective functions and performance metrics in inverse imaging problems under Poisson noise, where methods typically optimize Poisson likelihood yet are evaluated using mean squared error (MSE). The authors systematically investigate the MSE performance of Gaussian surrogate objectives—both homoscedastic and heteroscedastic quadratic forms—in low-dose settings, comparing them against Poisson maximum likelihood estimation (MLE) and maximum a posteriori (MAP) approaches. By leveraging theoretical analysis within a diagonalized model, they derive closed-form MSE expressions for four representative methods and validate their findings through CT reconstruction experiments. The results demonstrate that, even at low doses, simple Gaussian surrogates can achieve MSE performance comparable to regularized Poisson MAP while offering significantly higher computational efficiency, thereby challenging the prevailing paradigm that Poisson likelihood is indispensable for accurate low-dose reconstruction.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In imaging inverse problems with Poisson-distributed measurements, it is common to use objectives derived from the Poisson likelihood. But performance is often evaluated by mean squared error (MSE), which raises a practical question: how much does a Poisson objective matter for MSE, even at low dose? We analyze the MSE of Poisson and Gaussian surrogate reconstruction objectives under Poisson noise. In a stylized diagonal model, we show that the unregularized Poisson maximum-likelihood estimator can incur large MSE at low dose, while Poisson MAP mitigates this instability through regularization. We then study two Gaussian surrogate objectives: a heteroscedastic quadratic objective motivated by the normal approximation of Poisson data, and a homoscedastic quadratic objective that yields a simple linear estimator. We show that both surrogates can achieve MSE comparable to Poisson MAP in the low-dose regime, despite departing from the Poisson likelihood. Numerical computed tomography experiments indicate that these conclusions extend beyond the stylized setting of our theoretical analysis.

Problem

• Poisson inverse problems
• mean squared error
• Gaussian surrogates
• low-dose imaging
• reconstruction objectives

Innovation

• Poisson inverse problems
• Gaussian surrogate
• mean squared error
• low-dose imaging
• maximum a posteriori estimation