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Provably Accelerated Imaging with Restarted Inertia and Score-based Image Priors

Marien Renaud, Julien Hermant, Deliang Wei, Yu Sun

TL;DR

This work tackles ill-posed imaging inverse problems by enhancing REGULARIZATION BY DENOISING (RED) with Restarted Inertia and Score-based Priors (RISP). The core idea is to fuse inertial acceleration with a restarting mechanism and score-based priors, yielding provable speedups in convergence while preserving high-quality reconstructions. Theoretical results show accelerated stationary-point convergence at $\\mathcal{O}(n^{-4/7})$ for both RISP-GM and RISP-Prox, along with a continuous-time interpretation via a restarted heavy-ball ODE that matches the discrete rate. Empirically, RISP delivers substantial runtime reductions (up to $\\24\\times$) across linear, nonlinear, and large-scale imaging problems, illustrating practical impact for fast, high-fidelity image recovery.

Abstract

Fast convergence and high-quality image recovery are two essential features of algorithms for solving ill-posed imaging inverse problems. Existing methods, such as regularization by denoising (RED), often focus on designing sophisticated image priors to improve reconstruction quality, while leaving convergence acceleration to heuristics. To bridge the gap, we propose Restarted Inertia with Score-based Priors (RISP) as a principled extension of RED. RISP incorporates a restarting inertia for fast convergence, while still allowing score-based image priors for high-quality reconstruction. We prove that RISP attains a faster stationary-point convergence rate than RED, without requiring the convexity of the image prior. We further derive and analyze the associated continuous-time dynamical system, offering insight into the connection between RISP and the heavy-ball ordinary differential equation (ODE). Experiments across a range of imaging inverse problems demonstrate that RISP enables fast convergence while achieving high-quality reconstructions.

Provably Accelerated Imaging with Restarted Inertia and Score-based Image Priors

TL;DR

This work tackles ill-posed imaging inverse problems by enhancing REGULARIZATION BY DENOISING (RED) with Restarted Inertia and Score-based Priors (RISP). The core idea is to fuse inertial acceleration with a restarting mechanism and score-based priors, yielding provable speedups in convergence while preserving high-quality reconstructions. Theoretical results show accelerated stationary-point convergence at for both RISP-GM and RISP-Prox, along with a continuous-time interpretation via a restarted heavy-ball ODE that matches the discrete rate. Empirically, RISP delivers substantial runtime reductions (up to ) across linear, nonlinear, and large-scale imaging problems, illustrating practical impact for fast, high-fidelity image recovery.

Abstract

Fast convergence and high-quality image recovery are two essential features of algorithms for solving ill-posed imaging inverse problems. Existing methods, such as regularization by denoising (RED), often focus on designing sophisticated image priors to improve reconstruction quality, while leaving convergence acceleration to heuristics. To bridge the gap, we propose Restarted Inertia with Score-based Priors (RISP) as a principled extension of RED. RISP incorporates a restarting inertia for fast convergence, while still allowing score-based image priors for high-quality reconstruction. We prove that RISP attains a faster stationary-point convergence rate than RED, without requiring the convexity of the image prior. We further derive and analyze the associated continuous-time dynamical system, offering insight into the connection between RISP and the heavy-ball ordinary differential equation (ODE). Experiments across a range of imaging inverse problems demonstrate that RISP enables fast convergence while achieving high-quality reconstructions.

Paper Structure

This paper contains 37 sections, 27 theorems, 210 equations, 20 figures, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Restarted Inertia with Score-based Priors
  4. Theoretical Analysis
  5. Convergence Rates of RISP-GM & RISP-Prox
  6. Continuous-Time Analysis
  7. Experiments
  8. Linear Inverse Problems
  9. Nonlinear Inverse Problem
  10. Time-Efficiency for Large-Scale Imaging
  11. Conclusion
  12. Extended Related works
  13. Preliminaries
  14. Convergence Analysis of RED
  15. Convergence Analysis of RISP-GM
  16. ...and 22 more sections

Key Result

Proposition 1

Let Assumptions ass:nn_structure-ass:smoothness hold and $\eta = 1/L$. Then, with at most $n$ iterations, RED-GM outputs a point $\hat{\bm{x}}$ such that where $A_0 = \sqrt{2L\Delta_F}$, and $\Delta_F := F(\bm{x}^0) - \mathop{\mathsf{min}} F$. We recall that $\bm x^0$ is the initialization.

Figures (20)

  • Figure 1: Conceptual illustration of how restarting stabilizes inertial algorithms. Without restarting, accumulated inertia can cause overshooting and escape from stationary points (top). Restarting clears inertia and enforces local gradient updates to prevent overshooting (bottom).
  • Figure 2: Visualizations of the gradient‑norm (top row) and PSNR (bottom row) curves for RISP and baseline methods on four linear inverse problems. All curves are averaged over the test dataset, and the $x$‑axis shows the iteration number. By employing the restarted inertia, RISP achieves faster convergence without compromising the performance.
  • Figure 3: Left: Visualization of PSNR curves for RISP and baseline methods on the Rician denoising task with noise level $25.5/255$. The x‑axis shows runtime in milliseconds. RISP‑Prox reaches 31.55 dB in 160 ms, whereas RED‑GM requires roughly ten times longer to achieve comparable performance. Right: Visual comparison of reconstructions produced by each algorithm after 160 milliseconds. Zoomed‑in regions highlight residual noise and artifacts. Note how RISP‑Prox yields a substantially cleaner, nearly noise‑free reconstruction within this time budget.
  • Figure 4: Visual comparison of the reconstructions by RISP and baselines for the inverse scattering task, where the underlying image has a size of $1024\times1024$ pixels. All algorithms are ran until convergence or after reaching the maximum runtime (480 minutes). With only 20 minutes, RISP algorithms can restore clear structures and fine details; on the other hand, RED algorithms still cannot provide a comparable result after 480 minutes. Note the substantial difference in the PSNR values and visual differences highlighted in the zoomed-in regions.
  • Figure 5: Visualization of the PSNR and relative error curves achieved by RISP and baselines. The relative error is computed by $\|\bm{x}^{k+1}-\bm{x}^k\|/\|\bm{x}^0\|$ and is plotted in the log-scale. The $x$-axis shows runtime in minutes. Note the fast convergence of RISP algorithms to high-quality reconstructions.
  • ...and 15 more figures

Theorems & Definitions (52)

  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition
  • proof
  • ...and 42 more