Recovering a signal from its degraded measurements is a long standing challenge in science and engineering. Recently, zero-shot diffusion based methods have been proposed for such inverse problems, offering a posterior sampling based solution that leverages prior knowledge. Such algorithms incorporate the observations through inference, often leaning on manual tuning and heuristics. In this work we propose a rigorous analysis of such approximate posterior-samplers, relying on a Gaussianity assumption of the prior. Under this regime, we show that both the ideal posterior sampler and diffusion-based reconstruction algorithms can be expressed in closed-form, enabling their thorough analysis and comparisons in the spectral domain. Building on these representations, we also introduce a principled framework for parameter design, replacing heuristic selection strategies used to date. The proposed approach is method-agnostic and yields tailored parameter choices for each algorithm, jointly accounting for the characteristics of the prior, the degraded signal, and the diffusion dynamics. We show that our spectral recommendations differ structurally from standard heuristics and vary with the diffusion step size, resulting in a consistent balance between perceptual quality and signal fidelity.

翻译：从退化测量中恢复信号是科学与工程领域长期存在的挑战。近期，针对此类逆问题提出了基于零样本扩散的方法，提供了一种利用先验知识的后验采样解决方案。这类算法通过推理过程整合观测信息，通常依赖于手动调参和启发式策略。本研究提出对此类近似后验采样器的严格分析，其基于先验分布的高斯性假设。在此框架下，我们证明了理想后验采样器与基于扩散的重建算法均可通过闭式表达，从而能够在谱域对其进行深入分析和比较。基于这些表示形式，我们还提出了参数设计的原理性框架，以替代当前使用的启发式选择策略。所提出的方法具有算法无关性，可为每种算法生成定制化的参数选择，同时综合考虑先验特征、退化信号特性及扩散动力学特性。研究表明，我们的谱域建议在结构上不同于标准启发式方法，且随扩散步长变化而变化，从而在感知质量与信号保真度之间实现稳定平衡。