A super-sensitivity incoherent optical method with time-delay embedding for pixel-limited dynamic displacement measurements

Document Type

Article

Publication Date

5-1-2025

Abstract

The sensitivity of incoherent optical methods using digital cameras (e.g., photogrammetry with optical flows and digital image correlation) for full-field displacement measurements, defined as the minimum measurable displacement, is inherently limited by the bit depth B of the digital camera due to the quantization error, with a single-pixel theoretical sensitivity limit as δp=1/(2B−1) [pixel]; however, it is not achievable due to the existence of noise in practice. Recent research has leveraged the random noise and the spatial-pixel-averaging with dithering to exceed such a sensitivity limit, achieving super-sensitivity displacement measurements. In particular, the derived mathematical model indicates that the achieved super sensitivity is inversely proportional to the square root of the number of the effective pixel (those record actual intensity changes due to the object displacement), i.e., a larger effective pixel number for averaging is advantageous for super sensitivity. However, it may not be realized practically in common situations where the number of effective spatial pixels available for averaging is limited. To address the limitation of limited pixels available for super-sensitivity measurements, in this study we incorporate the time-delay embedding into the pixel-limited dynamic displacement measurements to achieve super sensitivity. The key innovation is to leverage abundant temporal samplings to compensate for the pixel-limited spatial-dimensional samplings. Specifically, we leverage the time-delay embedding of temporal samplings to construct pseudo-phase-space samplings, allowing an application of an SVD-based adaptive averaging over the pseudo-phase space with natural dithering to achieve super-sensitivity displacement estimations with limited pixels. For validations, the most challenging pixel-limited scenario, i.e., only one-effective-pixel measurement case, is focused on in this study. Numerical simulations, including theoretical and optical flow simulations, and laboratory experiments on the incoherent optical measurement using a representative 8-bit video camera of the vibration of a bench-scale three-story building structure and a cantilever beam structure are conducted, respectively. The results indicate that the developed method is able to achieve super-sensitivity measurement of dynamic displacements with limited effective pixels, especially approaching the coherent laser displacement sensor (LDS) performance with increasing number of pseudo-phase-space dimensions. Furthermore, a mathematical model is derived for the achievable super-sensitivity (minimum measurable displacement) as δp∗∝[σn(1/ND+1/NL)]δp, where σn is the noise level, ND and NL the number of dimensions of the constructed pseudo-phase space and the number of temporal samplings in the pseudo-phase space for averaging, respectively, and δp=1/(2B−1) [pixel] the nominal single-pixel sensitivity limit. We also discuss the current limitations of the developed method and the required further work.

Publication Title

Mechanical Systems and Signal Processing

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