Measurement Embedded Schr\"odinger Bridge for Inverse Problems

TL;DR

This paper introduces the Measurement Embedded Schr"odinger Bridge (MESB), a novel method leveraging optimal transport theory to improve inverse problem solutions by directly modeling the relationship between corrupted and clean images conditioned on measurements, outperforming existing Schr"odinger Bridge approaches.

Contribution

The paper proposes MESB, a new Schr"odinger Bridge framework that incorporates observed measurements, enhancing inverse problem solving with better performance and efficiency.

Findings

MESB outperforms existing Schr"odinger Bridge methods in visual quality.

MESB achieves superior quantitative metrics on diverse inverse problems.

The approach effectively models the relationship between corrupted and clean images.

Abstract

Score-based diffusion models are frequently employed as structural priors in inverse problems. However, their iterative denoising process, initiated from Gaussian noise, often results in slow inference speeds. The Image-to-Image Schr\"odinger Bridge (I$^2$SB), which begins with the corrupted image, presents a promising alternative as a prior for addressing inverse problems. In this work, we introduce the Measurement Embedded Schr\"odinger Bridge (MESB). MESB establishes Schr\"odinger Bridges between the distribution of corrupted images and the distribution of clean images given observed measurements. Based on optimal transport theory, we derive the forward and backward processes of MESB. Through validation on diverse inverse problems, our proposed approach exhibits superior performance compared to existing Schr\"odinger Bridge-based inverse problems solvers in both visual quality and quantitative metrics.