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This dissertation introduces the negative binomial (NB) model in handling overdispersed count data, particularly in low-photon imaging and related applications. This dissertation explores both mathematical and deep learning approaches to improve image reconstruction and parameter estimation under NB noise conditions.
First, we develop multiple image reconstruction frameworks based on gradient-based optimization and the alternating direction method of multipliers (ADMM). By incorporating the NB likelihood function into the reconstruction objective function, we demonstrate improved robustness and accuracy compared to traditional Poisson-based methods. Our theoretical analysis guarantees the existence of a global minimizer, and experiments validate the superiority of our approach in reconstructing noisy and blurred images. Additionally, we discuss the convergence properties of the ADMM-NB algorithm.
Next, we integrate deep learning with statistical modeling by designing a hybrid NB deep learning framework. This includes (1) enhancing image reconstruction by embedding NB priors into loss functions and (2) estimating the NB dispersion parameter using a novel two-head U-Net architecture. The latter model predicts the dispersion parameter and the corresponding image probability matrix, providing accurate parameter estimation crucial for downstream reconstruction algorithms. Our experimental results show that incorporating prior statistical knowledge into deep learning training significantly enhances performance, particularly in low-photon imaging scenarios.
Furthermore, we explore applications of our framework in practical settings, such as data completion and medical image completion, where NB noise naturally arises. Additionally, we apply deep learning reconstruction to X-ray luminescence computed tomography (XLCT) under NB noise conditions, demonstrating the generalization of our approach in different imaging modalities.
This dissertation contributes to the development of NB models and advances the intersection of deep learning and statistical modeling. The findings highlight the advantages of (1) using a purely statistical NB model for fast image reconstruction and (2) integrating statistical knowledge into deep learning frameworks, paving the way for more accurate and interpretable solutions in image reconstruction and parameter estimation under NB noise conditions.