Gaussian Surrogates for Poisson Imaging: Some Theoretical and Empirical Results

TL;DR

This paper compares Poisson and Gaussian surrogate objectives for low-dose Poisson imaging, showing Gaussian surrogates can achieve similar MSE to Poisson MAP despite using different likelihood models.

Contribution

It provides theoretical and empirical evidence that Gaussian surrogates can perform comparably to Poisson-based methods in low-dose imaging scenarios.

Findings

Gaussian surrogates achieve MSE comparable to Poisson MAP at low dose.

Poisson maximum-likelihood estimator can have high MSE without regularization.

Numerical CT experiments support theoretical results.

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.