  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.
