Quantification of Spatiotemporal Equivalence and Fundamental Sensitivity Limit of SVD-Based Super-Sensitivity Full-Field Incoherent Optical Measurements

The sensitivity limit (minimum measurable displacement) of incoherent optical methods with digital cameras (e.g., photogrammetry with optical flow and digital image correlation) may be exceeded, according to recent studies, through a singular value decomposition (SVD)-based adaptive spatial pixel averaging with natural dithering to overcome the quantization effect, enabling super-sensitivity full-field displacement measurements. Particularly, it was found that the achievable sensitivity limit is constrained by the number of effective spatial pixels used for SVD-based weighted averaging, that is, a large number of spatial pixels is typically required to achieve super-sensitivity. Nevertheless, the spatiotemporal dynamics in this process are not fully understood and quantified. In this study, we theoretically quantify the roles of spatiotemporal samplings and derive the fundamental sensitivity limit of the SVD-based super-sensitivity photogrammetry method using the Cramér-Rao lower bound (CRLB) theory and the uncertainty propagation principle. Specifically, the equivalence between spatial and temporal samplings for the achievable super-sensitivity of the SVD-based method is established; it implies that increasing either the number of spatial pixels or the number of temporal frames for the SVD-based weighted averaging equivalently improves the measurement sensitivity limit. Furthermore, the fundamental sensitivity limit of the SVD-based super-sensitivity photogrammetry method is theoretically derived, as the minimum measurable displacement is found to be δ p∗ propto σn(1N_rfsti+1N t , quantitatively dependent on the number of spatial pixels Ns and the number of temporal frames Nt for weighted averaging, and the noise level σ n in the measurement system. Validations on the principle and derived model are conducted through numerical simulations and laboratory experiments. Overall, this work contributes to a rigorous general formula for the achievable sensitivity limit of the SVD-based super-sensitivity photogrammetry for full-field displacement measurements. In addition, the revealed quantitative spatiotemporal equivalence indicates that spatial and temporal samplings can compensate for each other to achieve super-sensitivity dynamic displacement measurements. Some remaining challenges and required future research are also discussed.