Coupled Data and Measurement Space Dynamics for Enhanced Diffusion Posterior Sampling

Inverse problems, where the goal is to recover an unknown signal from noisy or incomplete measurements, are central to applications in medical imaging, remote sensing, and computational biology. Diffusion models have recently emerged as powerful priors for solving such problems. However, existing methods either rely on projection-based techniques that enforce measurement consistency through heuristic updates, or they approximate the likelihood p(boldsymboly mid boldsymbolx), often resulting in artifacts and instability under complex or high-noise conditions. To address these limitations, we propose a novel framework called coupled data and measurement space diffusion posterior sampling (C-DPS), which eliminates the need for constraint tuning or likelihood approximation. C-DPS introduces a forward stochastic process in the measurement space boldsymbolyt, evolving in parallel with the data-space diffusion boldsymbolxt, which enables the derivation of a closed-form posterior p(boldsymbolxt-1 mid boldsymbolxt, boldsymbolyt-1). This coupling allows for accurate and recursive sampling based on a well-defined posterior distribution. Empirical results demonstrate that C-DPS consistently outperforms existing baselines, both qualitatively and quantitatively, across multiple inverse problem benchmarks.