Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A convergent Plug-and-Play Majorization-Minimization algorithm for Poisson inverse problems

Thibaut Modrzyk, Ane Etxebeste, Élie Bretin, Voichita Maxim

Abstract

In this paper, we present a novel variational plug-and-play algorithm for Poisson inverse problems. Our approach minimizes an explicit functional which is the sum of a Kullback-Leibler data fidelity term and a regularization term based on a pre-trained neural network. By combining classical likelihood maximization methods with recent advances in gradient-based denoisers, we allow the use of pre-trained Gaussian denoisers without sacrificing convergence guarantees. The algorithm is formulated in the majorization-minimization framework, which guarantees convergence to a stationary point. Numerical experiments confirm state-of-the-art performance in deconvolution and tomography under moderate noise, and demonstrate clear superiority in high-noise conditions, making this method particularly valuable for nuclear medicine applications.

A convergent Plug-and-Play Majorization-Minimization algorithm for Poisson inverse problems

Abstract

In this paper, we present a novel variational plug-and-play algorithm for Poisson inverse problems. Our approach minimizes an explicit functional which is the sum of a Kullback-Leibler data fidelity term and a regularization term based on a pre-trained neural network. By combining classical likelihood maximization methods with recent advances in gradient-based denoisers, we allow the use of pre-trained Gaussian denoisers without sacrificing convergence guarantees. The algorithm is formulated in the majorization-minimization framework, which guarantees convergence to a stationary point. Numerical experiments confirm state-of-the-art performance in deconvolution and tomography under moderate noise, and demonstrate clear superiority in high-noise conditions, making this method particularly valuable for nuclear medicine applications.
Paper Structure (35 sections, 2 theorems, 36 equations, 7 figures, 7 tables)

This paper contains 35 sections, 2 theorems, 36 equations, 7 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Related works
  3. Majorization-Minimization algorithms
  4. Plug-and-Play framework
  5. Plug-and-Play Majorization Minimization
  6. Experiments
  7. Deblurring
  8. Datasets
  9. Experimental setting
  10. Methods
  11. Metrics
  12. Hyper-parameter tuning
  13. Qualitative and quantitative results
  14. Positron Emission Tomography
  15. Dataset and experimental setting
  16. ...and 20 more sections

Key Result

Theorem 1

Let $f : \mathbb{R}^d \rightarrow \mathbb{R} \cup \{+\infty\}$ and $g : \mathbb{R}^d \rightarrow \mathbb{R}$ be proper lower semicontinuous functions, with $f$ convex and $g$ differentiable with an $L$-Lipschitz gradient, and $\lambda > 0$ with $\tau < 1 / \lambda L$. Assume that $h= f+\lambda g$ is

Figures (7)

  • Figure 1: Left: Deconvolution on the Leaves image from the Set3c dataset with the Gaussian kernel and noise level $\zeta=60$. Right: Same image and kernel, with noise level $\zeta=5$. All images were produced from the same measurement data. Evolution of the PSNR and the gap between iterates have been plotted for theoretically convergent methods.
  • Figure 2: Left: Deconvolution on the Butterfly image from the Set3c dataset with the first Levin real world motion blur kernel and noise level $\zeta=60$. Right: Same image and kernel, with noise level $\zeta=5$.
  • Figure 3: Top row: PET reconstructions of a representative mid-axial slice from the BrainWeb test dataset, including two hot lesions. Bottom row: MR slice and the corresponding relative error maps for each reconstruction algorithm with respect to the emission map.
  • Figure 4: NRMSE values for each method and for different region of interest on the Brainweb test dataset
  • Figure 5: Contrast-to-Noise values for each method on the Brainweb test dataset
  • ...and 2 more figures

Theorems & Definitions (4)

  • Definition 1
  • Theorem 1
  • Lemma 1
  • proof