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 $p(\boldsymbol{y} \mid \boldsymbol{x})$, often resulting in artifacts and instability under complex or high-noise conditions. To address these limitations, we propose a novel framework called \emph{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 $\{\boldsymbol{y}_t\}$, evolving in parallel with the data-space diffusion $\{\boldsymbol{x}_t\}$, which enables the derivation of a closed-form posterior $p(\boldsymbol{x}_{t-1} \mid \boldsymbol{x}_t, \boldsymbol{y}_{t-1})$. 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.

翻译：逆问题的目标是从含噪或不完整的测量中恢复未知信号，是医学成像、遥感与计算生物学等应用的核心。扩散模型最近已成为解决此类问题的强大先验。然而，现有方法要么依赖于基于投影的技术，通过启发式更新强制测量一致性；要么近似似然 $p(\boldsymbol{y} \mid \boldsymbol{x})$，这通常在复杂或高噪声条件下导致伪影与不稳定。为应对这些局限，我们提出一种称为**数据与测量空间耦合扩散后验采样**（C-DPS）的新框架，该框架无需约束调优或似然近似。C-DPS在测量空间 $\{\boldsymbol{y}_t\}$ 中引入一个前向随机过程，与数据空间扩散 $\{\boldsymbol{x}_t\}$ 并行演化，从而能够推导出闭式后验 $p(\boldsymbol{x}_{t-1} \mid \boldsymbol{x}_t, \boldsymbol{y}_{t-1})$。这种耦合使得能够基于定义明确的后验分布进行精确且递归的采样。实证结果表明，在多个逆问题基准测试中，C-DPS在定性与定量上均持续优于现有基线方法。