Title: Measurement Geometry and Design for Trustworthy Generative Inverse Problems

Original: arXiv:2606.02309v1 Announce Type: new

Abstract: Generative models are increasingly used as priors for inverse problems, but their ability to produce realistic images creates a basic trust problem: a plausible reconstruction may be supported by the measurements, or it may be filled in by the prior along unobserved directions. This distinction is especially important in medical imaging, where acquisition operators are designed under scan-time, dose, and calibration constraints. We study generative inverse problems from a measurement-geometry perspective. The central question is whether a fixed measurement operator can distinguish nearby images that are plausible under the generative prior, and whether this relationship can guide better measurements. We introduce a local measurement-manifold compatibility measure that quantifies how well the operator observes prior-relevant tangent directions. Under local regularity assumptions, we prove that this quantity controls the stable part of the reconstruction error, while the generative prior controls off-manifold drift. This worst-direction certificate motivates practical fixed and sequential acquisition rules based on overall local volume preservation, including a posterior-cloud design that adapts measurements at test time without training a sampling policy. Across row-sampling, tomographic, and MR acquisition settings, the proposed scores predict failure modes, explain measurement-induced hallucinations, and guide better sampling. In fastMRI Cartesian sampling, posterior-cloud measurement design improves over strong non-learned ACS-preserving baselines, including variable-density and Poisson-like masks.

Rewritten:

Title: Geometric Measurement Strategies and Design for Reliable Generative Inverse Problems

Abstract: The growing adoption of generative models as priors in inverse problems introduces a fundamental challenge to reliability: while these models excel at generating realistic imagery, they raise concerns about whether a reconstructed image is genuinely supported by the acquired data or if the prior has simply hallucinated details in unobserved regions. This issue is particularly critical in medical imaging, where the design of acquisition operators must balance strict limitations on scan duration, radiation dose, and calibration. This paper examines generative inverse problems through the lens of measurement geometry. The primary inquiry focuses on whether a static measurement operator can differentiate between nearby images that are both plausible under the generative prior, and whether such insights can inform improved measurement strategies. To address this, we propose a metric for local measurement-manifold compatibility, which evaluates the extent to which an operator captures tangent directions relevant to the prior. Assuming local regularity, we demonstrate that this metric governs the stable component of reconstruction error, whereas the generative prior dictates drift off the manifold. This "worst-direction" certificate informs the development of practical fixed and sequential acquisition protocols centered on local volume preservation. Notably, this includes a posterior-cloud design approach that adjusts measurements during test time without requiring the training of a sampling policy. Validated across row-sampling, tomographic, and magnetic resonance (MR) contexts, our proposed scores successfully predict failure modes, elucidate measurement-induced hallucinations, and facilitate superior sampling strategies. In the context of fastMRI Cartesian sampling, the posterior-cloud measurement design outperforms robust non-learned baselines that preserve ACS, such as variable-density and Poisson-like masks.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC