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Alternating Direction Method of Multipliers for Negative Binomial Model with The Weighted Difference of Anisotropic and Isotropic Total Variation

Yu Lu, Kevin Bui, Roummel F. Marcia

TL;DR

This work tackles image reconstruction under overdispersed Poisson noise by adopting a negative binomial model with parameters $r$ and $p$, coupled with a weighted difference of anisotropic and isotropic TV (AITV) regularization. An ADMM framework is developed where each subproblem admits a closed-form solution, including an FFT-based update for the image, a per-pixel cubic solve for the auxiliary variable, and a proximal step for the AITV term. The method, NBirthed by combining NB data fidelity $F(f)$ with $\|f\|_{\mathrm{TV}^{(\mathrm{AI})}}$, demonstrates improved PSNR and SSIM over Poisson-based baselines in very photon-limited settings, with performance converging to Poisson as the dispersion parameter grows. The approach offers a practical and effective tool for denoising and reconstructing NB-corrupted images in applications such as medical imaging, where photon counts are limited.

Abstract

In many applications such as medical imaging, the measurement data represent counts of photons hitting a detector. Such counts in low-photon settings are often modeled using a Poisson distribution. However, this model assumes that the mean and variance of the signal's noise distribution are equal. For overdispersed data where the variance is greater than the mean, the negative binomial distribution is a more appropriate statistical model. In this paper, we propose an optimization approach for recovering images corrupted by overdispersed Poisson noise. In particular, we incorporate a weighted anisotropic-isotropic total variation regularizer, which avoids staircasing artifacts that are introduced by a regular total variation penalty. We use an alternating direction method of multipliers, where each subproblem has a closed-form solution. Numerical experiments demonstrate the effectiveness of our proposed approach, especially in very photon-limited settings.

Alternating Direction Method of Multipliers for Negative Binomial Model with The Weighted Difference of Anisotropic and Isotropic Total Variation

TL;DR

This work tackles image reconstruction under overdispersed Poisson noise by adopting a negative binomial model with parameters and , coupled with a weighted difference of anisotropic and isotropic TV (AITV) regularization. An ADMM framework is developed where each subproblem admits a closed-form solution, including an FFT-based update for the image, a per-pixel cubic solve for the auxiliary variable, and a proximal step for the AITV term. The method, NBirthed by combining NB data fidelity with , demonstrates improved PSNR and SSIM over Poisson-based baselines in very photon-limited settings, with performance converging to Poisson as the dispersion parameter grows. The approach offers a practical and effective tool for denoising and reconstructing NB-corrupted images in applications such as medical imaging, where photon counts are limited.

Abstract

In many applications such as medical imaging, the measurement data represent counts of photons hitting a detector. Such counts in low-photon settings are often modeled using a Poisson distribution. However, this model assumes that the mean and variance of the signal's noise distribution are equal. For overdispersed data where the variance is greater than the mean, the negative binomial distribution is a more appropriate statistical model. In this paper, we propose an optimization approach for recovering images corrupted by overdispersed Poisson noise. In particular, we incorporate a weighted anisotropic-isotropic total variation regularizer, which avoids staircasing artifacts that are introduced by a regular total variation penalty. We use an alternating direction method of multipliers, where each subproblem has a closed-form solution. Numerical experiments demonstrate the effectiveness of our proposed approach, especially in very photon-limited settings.
Paper Structure (8 sections, 23 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 8 sections, 23 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Negative binomial model
  4. Regularization in image processing
  5. Proposed Model and Algorithm
  6. Experiments
  7. Results
  8. Conclusion

Figures (4)

  • Figure 1: True images used for testing. All five images are of pixel size $321 \times 481$. Results for (a) Airplane will be visually shown in full, while partial results for (b) Train and (c) Elephant will also be displayed. For (d) Elk and (e) Skier, only quantitative results are presented.
  • Figure 2: Full results for the Airplane image. First row: noisy images. Second row: results from the NB model. Third row: results from the Poisson model. Each column corresponds to a different noise level.
  • Figure 3: Partial results of the Train image. First row: noisy images. Second row: results from the NB model. Third row: results from the Poisson model. The first column correspond to $r = 1$ and the second column correspond to $r = 100$.
  • Figure 4: Partial results of the Elephant image. First row: noisy images. Second row: results from the NB model. Third row: results from the Poisson model. The first column correspond to $r = 1$ and the second column correspond to $r = 100$.