Hierarchical GraphCut Phase Unwrapping based on Invariance of Diffeomorphisms Framework

Recent years have witnessed rapid advancements in 3D scanning technologies, with applications spanning VR/AR, digital human creation, and medical imaging. Structured-light scanning with phase-shifting techniques is preferred for its use of low-intensity visible light and high accuracy, making it well suited for capturing 4D facial dynamics. A key step is phase unwrapping, which recovers continuous phase values from measurements wrapped modulo 2pi. The goal is to estimate the unwrapped phase count k in the equation Phi = phi + 2pi k, where phi is the wrapped phase and Phi is the true phase. Noise, occlusions, and complex 3D geometry make recovering the true phase challenging because phase unwrapping is ill-posed: measurements only provide modulo 2pi values, and estimating k requires assumptions about surface continuity. Existing methods trade speed for accuracy: fast approaches lack precision, while accurate algorithms are too slow for real-time use. To overcome these limitations, this work proposes a phase unwrapping framework that reformulates GraphCut-based unwrapping as a pixel-labeling problem. This framework improves the estimation of the unwrapped phase count k through the invariance property of diffeomorphisms applied in image space via conformal and optimal transport (OT) maps. An odd number of diffeomorphisms are precomputed from the input phase data, and a hierarchical GraphCut algorithm is applied in each domain. The resulting label maps are fused via majority voting to robustly estimate k at each pixel. Experimental results demonstrate a 45.5x speedup and lower L2 error in real experiments and simulations, showing potential for real-time applications.