On the Fundamental Sensitivity Limit of Photogrammetry-Based Full-Field Displacement Measurements of Flexible Structures

The sensitivity limit of photogrammetry-based incoherent optical methods for full-field displacement measurements, achievable by the multipixel averaging method, has recently been theoretically established by deriving the Cramér–Rao lower bound (CRLB) on the variance of displacement estimators. However, it is essentially a loose lower bound due to the assumption of elementary rigid-body motion in the measurement model. In this study, we generalize the fundamental sensitivity limit model to dynamic flexible structures with spatially varying deformation field, by explicitly integrating the modal superposition principle in deriving the CRLB. Specifically, a tight lower bound for the fundamental sensitivity limit model is quantitatively established, as (Formula presented), revealing its parametric dependence on the imaging noise level σ_n, the number of dominant modes r, the number of effective spatial pixels N_s, and the camera bit depth B. Notably, the sensitivity limit scales with √r, , indicating that the achievable measurement sensitivity decreases with increasing modal complexity, a distinctive feature for a flexible structure that is absent in the rigid-body model. The derived model is validated through both numerical simulations and lab experiments.