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Sparse Signal Reconstruction for Overdispersed Low-photon Count Biomedical Imaging Using $\ell_p$ Total Variation

Yu Lu, Roummel F. Marcia

TL;DR

This work extends sparse signal reconstruction for photon-limited biomedical imaging by adopting a negative binomial likelihood to model overdispersed low-photon counts and promoting sparsity with an $\ell_p$ total variation quasi-seminorm. It presents a gradient-based optimization framework using a Barzilai-Borwein Hessian approximation and a reweighted $\ell_p$ TV scheme to convert to a weighted $\ell_1$ problem, solved via Fast Gradient Projection. Across experiments, the negative binomial model with $\ell_p$ TV quasi-seminorm consistently outperforms the Poisson model and standard regularizers, showing robustness to the choice of $p$ and convergence toward Poisson performance as dispersion grows. The approach provides improved reconstruction quality for sparse, piecewise-constant images in low-photon settings, with practical implications for biomedical imaging under photon-limited conditions.

Abstract

The negative binomial model, which generalizes the Poisson distribution model, can be found in applications involving low-photon signal recovery, including medical imaging. Recent studies have explored several regularization terms for the negative binomial model, such as the $\ell_p$ quasi-norm with $0 < p < 1$, $\ell_1$ norm, and the total variation (TV) quasi-seminorm for promoting sparsity in signal recovery. These penalty terms have been shown to improve image reconstruction outcomes. In this paper, we investigate the $\ell_p$ quasi-seminorm, both isotropic and anisotropic $\ell_p$ TV quasi-seminorms, within the framework of the negative binomial statistical model. This problem can be formulated as an optimization problem, which we solve using a gradient-based approach. We present comparisons between the negative binomial and Poisson statistical models using the $\ell_p$ TV quasi-seminorm as well as common penalty terms. Our experimental results highlight the efficacy of the proposed method.

Sparse Signal Reconstruction for Overdispersed Low-photon Count Biomedical Imaging Using $\ell_p$ Total Variation

TL;DR

This work extends sparse signal reconstruction for photon-limited biomedical imaging by adopting a negative binomial likelihood to model overdispersed low-photon counts and promoting sparsity with an total variation quasi-seminorm. It presents a gradient-based optimization framework using a Barzilai-Borwein Hessian approximation and a reweighted TV scheme to convert to a weighted problem, solved via Fast Gradient Projection. Across experiments, the negative binomial model with TV quasi-seminorm consistently outperforms the Poisson model and standard regularizers, showing robustness to the choice of and convergence toward Poisson performance as dispersion grows. The approach provides improved reconstruction quality for sparse, piecewise-constant images in low-photon settings, with practical implications for biomedical imaging under photon-limited conditions.

Abstract

The negative binomial model, which generalizes the Poisson distribution model, can be found in applications involving low-photon signal recovery, including medical imaging. Recent studies have explored several regularization terms for the negative binomial model, such as the quasi-norm with , norm, and the total variation (TV) quasi-seminorm for promoting sparsity in signal recovery. These penalty terms have been shown to improve image reconstruction outcomes. In this paper, we investigate the quasi-seminorm, both isotropic and anisotropic TV quasi-seminorms, within the framework of the negative binomial statistical model. This problem can be formulated as an optimization problem, which we solve using a gradient-based approach. We present comparisons between the negative binomial and Poisson statistical models using the TV quasi-seminorm as well as common penalty terms. Our experimental results highlight the efficacy of the proposed method.
Paper Structure (7 sections, 15 equations, 3 figures, 1 table)

This paper contains 7 sections, 15 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Algorithm
  4. Experiments
  5. Experiment I: $p$-value exploration
  6. Experiment II: Comparison between Poisson and negative binomial models with mutiple regularizations
  7. Conclusion

Figures (3)

  • Figure 1: Example of observation model. (a) The true image $f^*$. (b) The expected observation at the detector stage. Here, the measurement operator $A$ is a Gaussian blur. (c) Observed measurement $y_{r=1}$ drawn from a negative binomial (NB) distribution with dispersion parameter $r = 1$. (d) Observed measurement $y_{r=10}$ drawn from a NB distribution with $r = 10$. (e) Observed measurement $y_{1000}$ drawn from a NB distribution with $r = 1000$.
  • Figure 2: Experiment I: RMSE analysis for 2D data reconstruction employing the negative binomial model in conjunction with the $\ell_p$ TV quasi-seminorm. The evaluation spans across multiple noise levels, i.e., $r=1, 10, 25$ and $1,000$ for different $p$ values. Observe that the RMSE values do not change significantly as the value of $p$ changes.
  • Figure 3: Experiment II: 2D data from a negative binomial distribution ($r=10$). The negative binomial model yields a lower RMSE than the Poisson model with $\ell_p$ TV quasi-seminorm. The difference between isotropic and anisotropic versions is small.