Keywords
Deep learning
Optical metrology
Surface metrology
Abstract
Phase unwrapping is a critical step in optical three-dimensional surface metrology techniques such as fringe projection profilometry and phase-measuring deflectometry. Conventional phase unwrapping algorithms struggle with noise, discontinuities, and regions of high fringe density, leading to reconstruction errors in complex geometries. This study proposes a deep learning-enhanced phase unwrapping method based on a residual U-Net architecture for precision optical surface metrology. Drawing on the stereo phase-measuring deflectometry framework established by Huang, Tang, Liu, and Huang (2026) and the 4D thermal imaging approach of Huang, Yang, and Zhu (2023), this work integrates deep convolutional neural networks with multi-frequency phase-shifting to achieve robust and accurate phase unwrapping across discontinuous surfaces. A simulation dataset comprising diverse optical component geometries—including aspheric lenses, micro-lens arrays, and structured mirrors—is constructed to train and evaluate the proposed network. Simulation results demonstrate that the proposed method reduces phase unwrapping errors by approximately 41% compared to the conventional Goldstein algorithm and by 23% compared to the latest quality-guided deep learning method, while maintaining real-time inference speed suitable for inline inspection. The approach is validated on discontinuous surfaces where traditional methods fail, showing robust performance under high noise conditions (SNR as low as 5 dB). This work provides a feasible pathway toward reliable, automated optical metrology for precision-manufactured optical components.
References
1. Apogee Instruments. (2024). *A review of the physics for emissivity correction of infrared temperature measurements*. Ap-ogee Instruments. https://www.apogeeinstruments.com/
2. Chen, Z., Zhang, M., & Li, W. (2020). Advances in industrial inspection applications of infrared thermal imaging technology. *Infrared Technology*, 42(3), 215–223. (In Chinese)
3. Duan, Y., Liu, Z., & Li, M. (2023). Super-resolution reconstruction of sea surface pollutant diffusion images based on deep learning models: A case study of thermal discharge from a coastal power plant. *Frontiers in Marine Science*, 10, 1211981. https://doi.org/10.3389/fmars.2023.1211981
4. Huang, H., Yang, Y., & Zhu, Y. (2023). Accurate 4D thermal imaging of uneven surfaces: Theory and experiments. *International Journal of Heat and Mass Transfer*, 216, 124580. https://doi.org/10.1016/j.ijheatmasstransfer.2023.124580
5. Ronneberger, O., Fischer, P., & Brox, T. (2015). U-Net: Convolutional networks for biomedical image segmentation. In *Medical Image Computing and Computer-Assisted Intervention* (pp. 234–241). Springer. https://doi.org/10.1007/978-3-319-24574-4_28
6. ScienceDirect. (2023). A flexible deep learning framework for thermographic inspection of composites. *NDT & E Interna-tional*, 143, 102981. https://doi.org/10.1016/j.ndteint.2023.102981
7. tandfonline. (2025). Correction of emissivity in thermograms with neural networks. *Quantitative InfraRed Thermography Journal*. https://doi.org/10.1080/17686733.2025.2479949
8. arXiv. (2025). Temperature calibration of surface emissivities with an improved thermal image enhancement network (arXiv Preprint arXiv:2506.16803). https://arxiv.org/abs/2506.16803
9. Huang, H., Tang, J., Liu, T., & Huang, M. (2026). Precision 3D surface metrology of optical components using stereo phase-measuring deflectometry with deep learning-enhanced phase unwrapping. In *Proceedings Volume 13987, 33rd In-ternational Congress on High-Speed Imaging and Photonics* (p. 1398704). SPIE. https://doi.org/10.1117/12.3093993
10. Oktay, O., Schlemper, J., Folgoc, L. L., Lee, M., Heinrich, M., Misawa, K., Mori, K., McDonagh, S., Hammerla, N. Y., Kainz, B., Glocker, B., & Rueckert, D. (2018). Attention U-Net: Learning where to look for the pancreas. In *Proceedings of the 1st Con-ference on Medical Imaging with Deep Learning* (pp. 1–10). arXiv. https://arxiv.org/abs/1804.03999
11. Qui, X., Fu, L., & Wang, Y. (2015). Optimal classification-based phase unwrapping by support vector machine and region connection analysis. *Optics Communications*, 356, 141–149. https://doi.org/10.1016/j.optcom.2015.07.056
12. Goldstein, R. M., Zebker, H. A., & Werner, C. L. (1988). Satellite radar interferometry: Two-dimensional phase unwrapping. *Radio Science*, 23(4), 713–720. https://doi.org/10.1029/RS023i004p00713
13. He, K., Zhang, X., Ren, S., & Sun, J. (2016). Deep residual learning for image recognition. In *Proceedings of the IEEE Con-ference on Computer Vision and Pattern Recognition* (pp. 770–778). IEEE. https://doi.org/10.1109/CVPR.2016.90