Physics-informed and vision-guided deep learning for robust 3D shape reconstruction of multi-core FBG sensors

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This paper proposes PINN+RMF, a physics-informed and vision-guided framework for robust 3D shape reconstruction using multi-core fiber Bragg grating (MCFBG) sensors. It maps node-wise spectral responses to interpretable local states—including curvature, bending direction, and temperature variation—using calibration parameters as physical priors. A differentiable rotation-minimizing-frame (RMF) layer integrates these states into the global centerline while suppressing torsion-induced orientation drift. Synchronized stereo vision provides coordinate-level supervision, and coarse-to-fine optimization first enforces local physical consistency and then refines global geometry. Experiments on a 1000-mm flexible rod yield a mean point error of 0.93 mm and a mean tip error of 1.62 mm (P95 tip error 5.18 mm, i.e., 0.52% of the rod length). RMF integration substantially reduces frame drift under 3D twisting: in a spatial-twist case, the maximum point-wise error drops from 9.40 mm with Frenet integration to 1.44 mm with RMF. Under a 3-mm tip-error tolerance, PINN+RMF consistently achieves higher success rates than the Frenet-based and end-to-end baselines across the test set. Simulations under calibration perturbations and multi-source uncertainties further demonstrate progressive rather than abrupt degradation, with the mean tip error increasing gradually from 1.86 mm (clean) to 4.16, 8.39, and 22.47 mm under 1%, 2%, and 5% wavelength noise, respectively. These results establish PINN+RMF as a robust and interpretable approach for precision-critical MCFBG shape sensing.