# FiRe: Fixed-points of Restoration Priors for Solving Inverse Problems

arXiv: https://arxiv.org/abs/2411.18970 · HTML: https://mlsanity.com/paper/2411.18970/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 38 references that are in the map.
- First posted on arXiv 28 Nov 2024.

## Builds on (most cited first, 27 of 27)

- [SwinIR: Image Restoration Using Swin Transformer](https://mlsanity.com/paper/2108.10257/) (2021)
- [Restormer: Efficient Transformer for High-Resolution Image Restoration](https://mlsanity.com/paper/2111.09881/) (2021)
- [Resolution-robust Large Mask Inpainting with Fourier Convolutions](https://mlsanity.com/paper/2109.07161/) (2021)
- [The Perception-Distortion Tradeoff](https://mlsanity.com/paper/1711.06077/) (2017)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [Plug-and-Play Image Restoration with Deep Denoiser Prior](https://mlsanity.com/paper/2008.13751/) (2020)
- [Denoising Diffusion Models for Plug-and-Play Image Restoration](https://mlsanity.com/paper/2305.08995/) (2023)
- [Cold Diffusion: Inverting Arbitrary Image Transforms Without Noise](https://mlsanity.com/paper/2208.09392/) (2022)
- [A Survey on Diffusion Models for Inverse Problems](https://mlsanity.com/paper/2410.00083/) (2024)
- [Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restoration](https://mlsanity.com/paper/2303.11435/) (2023)
- [Deep Plug-and-Play Super-Resolution for Arbitrary Blur Kernels](https://mlsanity.com/paper/1903.12529/) (2019)
- [Regularization by Denoising: Clarifications and New Interpretations](https://mlsanity.com/paper/1806.02296/) (2018)
- [Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie](https://mlsanity.com/paper/2103.04715/) (2021)
- [Practical Blind Image Denoising via Swin-Conv-UNet and Data Synthesis](https://mlsanity.com/paper/2203.13278/) (2022)
- [Proximal Denoiser for Convergent Plug-and-Play Optimization with Nonconvex Regularization](https://mlsanity.com/paper/2201.13256/) (2022)
- [An Online Plug-and-Play Algorithm for Regularized Image Reconstruction](https://mlsanity.com/paper/1809.04693/) (2018)
- [Direct Diffusion Bridge using Data Consistency for Inverse Problems](https://mlsanity.com/paper/2305.19809/) (2023)
- [Learning Maximally Monotone Operators for Image Recovery](https://mlsanity.com/paper/2012.13247/) (2020)
- [Regularization by Denoising via Fixed-Point Projection (RED-PRO)](https://mlsanity.com/paper/2008.00226/) (2020)
- [Provable Convergence of Plug-and-Play Priors with MMSE denoisers](https://mlsanity.com/paper/2005.07685/) (2020)
- [Tuning-free Plug-and-Play Proximal Algorithm for Inverse Imaging Problems](https://mlsanity.com/paper/2002.09611/) (2020)
- [Stochastic Deep Restoration Priors for Imaging Inverse Problems](https://mlsanity.com/paper/2410.02057/) (2024)
- [A Restoration Network as an Implicit Prior](https://mlsanity.com/paper/2310.01391/) (2023)
- [A Unified Convergence Theorem for Stochastic Optimization Methods](https://mlsanity.com/paper/2206.03907/) (2022)
- [The Perception-Robustness Tradeoff in Deterministic Image Restoration](https://mlsanity.com/paper/2311.09253/) (2023)
- [Ensemble Super-Resolution with A Reference Dataset](https://mlsanity.com/paper/1905.04696/) (2019)
- [Convergent plug-and-play with proximal denoiser and unconstrained regularization parameter](https://mlsanity.com/paper/2311.01216/) (2023)

## Built on by (most cited first, 6 of 6)

- [MRI Super-Resolution with Deep Learning: A Comprehensive Survey](https://mlsanity.com/paper/2511.16854/) (2025)
- [NPN: Non-Linear Projections of the Null-Space for Imaging Inverse Problems](https://mlsanity.com/paper/2510.01608/) (2025)
- [Reconstruct Anything Model: a lightweight general model for computational imaging](https://mlsanity.com/paper/2503.08915/) (2025)
- [h-control: Training-Free Camera Control via Block-Conditional Gibbs Refinement](https://mlsanity.com/paper/2605.11871/) (2026)
- [SP3: Spherical Priors for Plug-and-Play Restoration](https://mlsanity.com/paper/2606.16396/) (2026)
- [Analysis Plug-and-Play Methods for Imaging Inverse Problems](https://mlsanity.com/paper/2509.15422/) (2025)

## Abstract (arXiv)

Selecting an appropriate prior to compensate for information loss due to the measurement operator is a fundamental challenge in imaging inverse problems. Implicit priors based on denoising neural networks have become central to widely-used frameworks such as Plug-and-Play (PnP) algorithms. In this work, we introduce Fixed-points of Restoration (FiRe) priors as a new framework for expanding the notion of priors in PnP to general restoration models beyond traditional denoising models. The key insight behind FiRe is that smooth images emerge as fixed points of the composition of a degradation operator with the corresponding restoration model. This enables us to derive an explicit formula for our implicit prior by quantifying invariance of images under this composite operation. Adopting this fixed-point perspective, we show how various restoration networks can effectively serve as priors for solving inverse problems. The FiRe framework further enables ensemble-like combinations of multiple restoration models as well as acquisition-informed restoration networks, all within a unified optimization approach. Experimental results validate the effectiveness of FiRe across various inverse problems, establishing a new paradigm for incorporating pretrained restoration models into PnP-like algorithms. Code available at https://github.com/matthieutrs/fire.