Measurement Geometry and Design for Trustworthy Generative Inverse Problems

2026-06-01

AI Summary

This work addresses the issue of “hallucinated” content in reconstructions from generative models used as priors for inverse problems—particularly problematic in medical imaging due to its impact on diagnostic credibility—arising when measurements are insufficient. From a geometric perspective, the paper introduces a novel local measurement–manifold compatibility metric that, for the first time, links the stable component of reconstruction error to the measurement operator’s ability to observe tangent directions of the generative manifold. Leveraging differential geometry and inverse problem theory, and incorporating assumptions of local regularity and volume preservation, the authors develop a training-free posterior-aware measurement design strategy. The proposed metric accurately predicts failure modes in reconstruction and elucidates the origins of hallucinations, significantly outperforming multiple non-learning-based sampling baselines in tasks such as fastMRI.

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.

Problem

• generative inverse problems
• measurement geometry
• trustworthy reconstruction
• medical imaging
• hallucination

Innovation

• measurement geometry
• generative inverse problems
• posterior-cloud design
• manifold compatibility
• adaptive acquisition