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Prediction techniques for dynamic imaging with online primal-dual methods

Neil Dizon, Jyrki Jauhiainen, Tuomo Valkonen

TL;DR

The paper addresses online, time-evolving inverse problems by proposing a Predictive Online Primal-Dual Proximal Splitting (POPD2) framework with symmetric dynamic regret guarantees. It introduces a relaxed, versatile predictor design (pseudo-affine, including total-variation preserving, inner-product preserving, and dual scaling variants) and derives a symmetric regret bound based on a temporal sub-infimal convolution, linking performance to predictor quality and comparison-set richness. The approach is validated on challenging dynamic imaging tasks—image stabilisation and dynamic PET—showing that carefully crafted dual predictors substantially improve reconstruction quality (PSNR/SSIM) under motion, suggesting practical viability for real-time motion-robust imaging. Overall, the work provides a simpler, theoretically grounded online optimization tool with concrete predictor strategies for dynamic inverse problems, enabling more reliable real-time imaging in medical and industrial settings.

Abstract

Online optimisation facilitates the solution of dynamic inverse problems, such as image stabilisation, fluid flow monitoring, and dynamic medical imaging. In this paper, we improve upon previous work on predictive online primal-dual methods on two fronts. Firstly, we provide a more concise analysis that symmetrises previously unsymmetric regret bounds, and relaxes previous restrictive conditions on the dual predictor. Secondly, based on the latter, we develop several improved dual predictors. We numerically demonstrate their efficacy in image stabilisation and dynamic positron emission tomography.

Prediction techniques for dynamic imaging with online primal-dual methods

TL;DR

The paper addresses online, time-evolving inverse problems by proposing a Predictive Online Primal-Dual Proximal Splitting (POPD2) framework with symmetric dynamic regret guarantees. It introduces a relaxed, versatile predictor design (pseudo-affine, including total-variation preserving, inner-product preserving, and dual scaling variants) and derives a symmetric regret bound based on a temporal sub-infimal convolution, linking performance to predictor quality and comparison-set richness. The approach is validated on challenging dynamic imaging tasks—image stabilisation and dynamic PET—showing that carefully crafted dual predictors substantially improve reconstruction quality (PSNR/SSIM) under motion, suggesting practical viability for real-time motion-robust imaging. Overall, the work provides a simpler, theoretically grounded online optimization tool with concrete predictor strategies for dynamic inverse problems, enabling more reliable real-time imaging in medical and industrial settings.

Abstract

Online optimisation facilitates the solution of dynamic inverse problems, such as image stabilisation, fluid flow monitoring, and dynamic medical imaging. In this paper, we improve upon previous work on predictive online primal-dual methods on two fronts. Firstly, we provide a more concise analysis that symmetrises previously unsymmetric regret bounds, and relaxes previous restrictive conditions on the dual predictor. Secondly, based on the latter, we develop several improved dual predictors. We numerically demonstrate their efficacy in image stabilisation and dynamic positron emission tomography.
Paper Structure (13 sections, 11 theorems, 109 equations, 9 figures, 1 table, 1 algorithm)

This paper contains 13 sections, 11 theorems, 109 equations, 9 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. An online primal-dual method
  3. Assumptions and definitions
  4. A general regret estimate
  5. Pseudo-affine predictors
  6. Prediction bounds and penalties
  7. Total variation preserving predictors
  8. Inner product preserving predictors
  9. Dual scaling predictors
  10. Numerical experiments
  11. Predictors
  12. Image stabilisation
  13. Dynamic positron emission tomography

Key Result

Lemma 2.3

Suppose $E: X \to \overline \mathbb{R}$ is convex, proper, and lower semicontinuous, and has $L$-Lipschitz gradient. Then If $E$ is, moreover, $\gamma_{E}$-strongly convex, then for any $\beta>0$ and ${\bar{x}}, z, x \in X$, also

Figures (9)

  • Figure 1: Test image, added noise, and stationary reconstruction for comparison.
  • Figure 2: Iteration-wise SSIM and PSNR for the image stabilisation experiment.
  • Figure 3: Image stabilisation results for several predictors when $\alpha = 0.25$. The numbers on the left indicate the iteration/frame.
  • Figure 4: Shepp-Logan phantom, true sinogram, noisy subsampled sinogram, and static reconstruction. The colours represent values in $[0,1]$ as .
  • Figure 5: Brain phantom belzunce2020ultra, true sinogram, noisy subsampled sinogram, and static reconstruction. The colours represent values in $[0,1]$ as .
  • ...and 4 more figures

Theorems & Definitions (32)

  • Example 2.2
  • Lemma 2.3: tuomov-proxtest or clasonvalkonen2020nonsmooth
  • Corollary 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • Remark 2.7: Comparison set solution discrepancy
  • Lemma 3.1
  • ...and 22 more