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Measurement Embedded Schrödinger Bridge for Inverse Problems

Yuang Wang, Pengfei Jin, Siyeop Yoon, Matthew Tivnan, Quanzheng Li, Li Zhang, Dufan Wu

TL;DR

Measurement Embedded Schrödinger Bridge (MESB) introduces a data-consistent Schrödinger-Bridge formulation for inverse problems by embedding measurements in the margin conditions, linking corrupted-image distributions to clean-image distributions conditioned on observed data. It derives forward and backward stochastic differential equations under optimal-transport theory and demonstrates that MESB yields improved visual fidelity and LPIPS/SSIM metrics across natural and medical imaging tasks, with favorable efficiency versus prior Schrödinger-Bridge methods. The approach unifies and extends existing methods such as I^2SB and CDDB, showing analytical tractability (f=0) and practical performance gains, while highlighting flexibility via the transformation matrix T and data-consistency terms. The results suggest MESB as a robust, scalable priors-based framework for solving ill-posed inverse problems in imaging, with potential extensions to unpaired data and broader modalities.

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 Schrödinger 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 Schrödinger Bridge (MESB). MESB establishes Schrödinger 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 Schrödinger Bridge-based inverse problems solvers in both visual quality and quantitative metrics.

Measurement Embedded Schrödinger Bridge for Inverse Problems

TL;DR

Measurement Embedded Schrödinger Bridge (MESB) introduces a data-consistent Schrödinger-Bridge formulation for inverse problems by embedding measurements in the margin conditions, linking corrupted-image distributions to clean-image distributions conditioned on observed data. It derives forward and backward stochastic differential equations under optimal-transport theory and demonstrates that MESB yields improved visual fidelity and LPIPS/SSIM metrics across natural and medical imaging tasks, with favorable efficiency versus prior Schrödinger-Bridge methods. The approach unifies and extends existing methods such as I^2SB and CDDB, showing analytical tractability (f=0) and practical performance gains, while highlighting flexibility via the transformation matrix T and data-consistency terms. The results suggest MESB as a robust, scalable priors-based framework for solving ill-posed inverse problems in imaging, with potential extensions to unpaired data and broader modalities.

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 Schrödinger Bridge (ISB), 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 Schrödinger Bridge (MESB). MESB establishes Schrödinger 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 Schrödinger Bridge-based inverse problems solvers in both visual quality and quantitative metrics.
Paper Structure (24 sections, 3 theorems, 50 equations, 2 figures, 5 tables, 1 algorithm)

This paper contains 24 sections, 3 theorems, 50 equations, 2 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Schrödinger Bridge
  4. Score-based Diffusion Models
  5. Image-to-Image Schrödinger Bridge
  6. Method
  7. Measurement Embedded Schrödinger Bridge for Inverse Problems
  8. Understanding of Measurement Embedded Schrödinger Bridge
  9. Simplification
  10. Relationship with CDDB and CDDB deep
  11. Implementation Details
  12. Results
  13. Conclusion and Discussion
  14. Appendix
  15. Proof
  16. ...and 9 more sections

Key Result

Theorem 1

If $p_A$ and $p_B$ are defined according to equations (eq:pA) and (eq:pB), and $f$ is set to $0$, then the PDEs (eq:PDE) and (eq:PDE_margin) have the following analytical solutions: where and

Figures (2)

  • Figure 1: Visualization results for the deblur-gauss (no noise) task and the sr4x-bicubic (1% noise) task. The details within the blue and yellow boxes are zoomed in for enhanced visual clarity.
  • Figure 2: Visualization results for the CT sparse view reconstruction (0.1% noise) task. The details within the blue and yellow boxes are zoomed in for enhanced visual clarity. The display window for the entire images is set to [-1000HU, 200HU], for the zoomed regions outside the lungs is set to [-160HU, 200HU], and for the zoomed regions inside the lungs is set to [-1000HU, -550HU].

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3