Injecting Measurement Information Yields a Fast and Noise-Robust Diffusion-Based Inverse Problem Solver

2025-08-04

AI Summary

Existing zero-shot inverse problem solvers based on diffusion models rely on the Tweedie formula for sampling guidance but fail to explicitly model the conditional constraint imposed by the measurement $y$ on latent variables, resulting in delayed integration of measurement information downstream and limiting reconstruction efficiency and noise robustness. This paper proposes MeasDiff, a novel inverse problem solving framework that directly injects measurement information into the diffusion process. Its core innovation is the first introduction of a learnable conditional posterior mean estimator, jointly optimized via a lightweight single-parameter maximum likelihood objective and a noise-aware early-stopping mechanism to enable real-time incorporation of observational data during sampling. Experiments demonstrate that MeasDiff achieves or surpasses state-of-the-art performance across multiple image inverse problems—including super-resolution, denoising, and compressive sensing—while significantly improving convergence speed and robustness to measurement noise.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Web measurements
• Security and Privacy: Large-scale security measurements

Abstract

Diffusion models have been firmly established as principled zero-shot solvers for linear and nonlinear inverse problems, owing to their powerful image prior and iterative sampling algorithm. These approaches often rely on Tweedie's formula, which relates the diffusion variate $mathbf{x}_t$ to the posterior mean $mathbb{E} [mathbf{x}_0 | mathbf{x}_t]$, in order to guide the diffusion trajectory with an estimate of the final denoised sample $mathbf{x}_0$. However, this does not consider information from the measurement $mathbf{y}$, which must then be integrated downstream. In this work, we propose to estimate the conditional posterior mean $mathbb{E} [mathbf{x}_0 | mathbf{x}_t, mathbf{y}]$, which can be formulated as the solution to a lightweight, single-parameter maximum likelihood estimation problem. The resulting prediction can be integrated into any standard sampler, resulting in a fast and memory-efficient inverse solver. Our optimizer is amenable to a noise-aware likelihood-based stopping criteria that is robust to measurement noise in $mathbf{y}$. We demonstrate comparable or improved performance against a wide selection of contemporary inverse solvers across multiple datasets and tasks.

Problem

• Improving diffusion models for inverse problems
• Integrating measurement information into diffusion process
• Enhancing noise robustness in inverse solvers

Innovation

• Estimates conditional posterior mean with measurement info
• Uses lightweight single-parameter MLE formulation
• Noise-aware likelihood-based stopping criteria