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Gaussian surrogates do well on Poisson inverse problems

Alexandra Spitzer, Lorenzo Baldassari, Valentin Derbanot, Ivan Dokmanić

TL;DR

This work analyzes MSE performance in Poisson inverse problems for imaging, asking how much Poisson-specific likelihood objectives matter when data are Poisson-distributed. Through a stylized diagonal forward model, it shows that unregularized Poisson MLE can incur large MSE at low dose, while Poisson MAP regularization mitigates this. It introduces two Gaussian surrogates—a heteroscedastic HG and a homoscedastic OG—showing they can achieve MSE comparable to Poisson MAP in the low-dose regime, and provides explicit per-mode MSE bounds as functions of regularization parameters. CT experiments on parallel-beam data and LoDoPaB-CT slices demonstrate that simple quadratic objectives with regularization can match or exceed Poisson MAP in MSE across dose levels, suggesting practical benefits for low-dose tomography where computational simplicity is valuable.

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.

Gaussian surrogates do well on Poisson inverse problems

TL;DR

This work analyzes MSE performance in Poisson inverse problems for imaging, asking how much Poisson-specific likelihood objectives matter when data are Poisson-distributed. Through a stylized diagonal forward model, it shows that unregularized Poisson MLE can incur large MSE at low dose, while Poisson MAP regularization mitigates this. It introduces two Gaussian surrogates—a heteroscedastic HG and a homoscedastic OG—showing they can achieve MSE comparable to Poisson MAP in the low-dose regime, and provides explicit per-mode MSE bounds as functions of regularization parameters. CT experiments on parallel-beam data and LoDoPaB-CT slices demonstrate that simple quadratic objectives with regularization can match or exceed Poisson MAP in MSE across dose levels, suggesting practical benefits for low-dose tomography where computational simplicity is valuable.

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.
Paper Structure (27 sections, 3 theorems, 140 equations, 5 figures)

This paper contains 27 sections, 3 theorems, 140 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related works
  3. A stylized theoretical analysis
  4. Poisson MLE
  5. Surrogate models
  6. Poisson MAP
  7. Homoscedastic Gaussian MAP
  8. Experiments
  9. Experimental setting
  10. LoDoPaB-CT slices
  11. Conclusion
  12. Acknowledgments
  13. Proofs of Section \ref{['SEC:METHODS']}
  14. Proof of Proposition \ref{['prop:per-mode-mse-pmle']}
  15. Per-mode Poisson MAP expression
  16. ...and 12 more sections

Key Result

Proposition 2.1

Define the effective regularization level In the low-dose limit, as $\mu_j=s a_j x_j^\star\to 0$, the per-mode MSE ratio relative to the Poisson MLE satisfies The notation $f(\mu_j)= O(\mu_j)$ means that there exist constants $C,\mu_0>0$ such that $|f(\mu_j)|\leq C\mu_j$ for all $0<\mu_j < \mu_0$.

Figures (5)

  • Figure 1: Qualitative reconstructions for a representative CT slice from the LoDoPaB dataset at two count levels. From left to right: ground truth and reconstructions obtained with regularized HG MAP, homoscedastic LS, PWLS (oracle weights), and Poisson MAP.
  • Figure 2: MSE as a function of the average expected number of counts per detector bin. For HG and all PWLS variants, the stabilization floor $\varepsilon$ was selected from $\{0.1,\,0.5,\,1.0\}$ by minimizing the tuning-set MSE at the lowest count level; the chosen $\varepsilon$ was then fixed and used for all average expected number of counts.
  • Figure 3: MSE against the average expected number of counts per detector bin. For HG and all PWLS variants, the stabilization floor $\varepsilon$ was selected from $\{0.1,\,0.5,\,1.0\}$ by minimizing the tuning-set MSE at the lowest count level; the chosen $\varepsilon$ was then fixed and used for all average expected number of counts.
  • Figure 4: LoDoPaB-CT: sensitivity to the stabilization floor $\varepsilon$. Each panel shows test MSE versus average expected counts $c$ for a fixed method, comparing $\varepsilon\in\{0.1,0.5,1.0\}$ to the Poisson MAP baseline. The results from the Shepp--Logan phantom experiments exhibits the same qualitative behavior.
  • Figure 5: LoDoPaB-CT: MSE as a function of the Tikhonov regularization parameter $\tau$ for three representative average expected counts $c$. We observe qualitatively similar regularization curves on the Shepp--Logan phantom.

Theorems & Definitions (6)

  • Proposition 2.1
  • Proposition 2.2
  • Remark A.1
  • Remark A.2
  • Proposition A.1
  • proof