Hierarchical GraphCut Phase Unwrapping based on Invariance of Diffeomorphisms Framework

TL;DR

Phase unwrapping for structured-light 3D scanning is ill-posed due to modulo $2\pi$ ambiguity. The paper introduces an invariance under diffeomorphisms (ID) framework that leverages conformal maps and optimal transport to generate multiple image deformations, which are then reconciled using an ID Hierarchical GraphCut to estimate the unwrapped phase count $k$. The approach achieves a 45.5× speedup and lower $L^2$ error compared with state-of-the-art methods, demonstrated on real and simulated data and enabling robust 3D reconstruction from high-resolution fringe data. This method has strong potential for real-time 4D facial dynamics and other high-detail 3D scanning applications by combining geometric invariances with efficient optimization.

Abstract

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.

Figures (5)

• Figure 1: Fringe patterns ($I_{0,1,2}$) are deformed via Möbius Transformations in Eq. \ref{['eqn:Mobius_transformation']} to produce conformal maps ($\overset{\sim}{I}_{0,1,2}$). These maps are then used to extract the unwrapped phases ($\varPhi_{0,1,2}$) via ID Hierarchical GraphCut Algorithm.
• Figure 2: Fringe patterns ($I_0, I_1, I_2$) are deformed using diffeomorphic Optimal Transport maps computed from $N = 3$ ROIs, which correspond to the circled regions in red. The resulting maps ($\overset{\sim}{I}_0, \overset{\sim}{I}_1, \overset{\sim}{I}_2$) yield wrapped phases ($\varphi_0, \varphi_1, \varphi_2$), which are unwrapped using the ID Hierarchical GraphCut to obtain $\Phi_0, \Phi_1, \Phi_2$.
• Figure 4: Phase Shifting Pipeline: From left to right, we show the captured fringe images (packed into RGB channels), followed by the computed ambient term$A(p)$, the modulation term$R(p)$, and the resulting wrapped phase$\varphi(p)$.
• Figure 7: Qualitative comparison between GraphCut graphcut2007 and our proposed ID Hierarchical GraphCut. The circled regions show sudden surface changes caused by unwrapping errors, which are corrected in our method.
• Figure 8: Illustration of Wrapped and Noise-Corrupted Double Gaussian, along with unwrapped results using GraphCut graphcut2007 and our method. Our method achieves a lower $L^2$ error.

Theorems & Definitions (11)

• Definition 1: Invariance under Diffeomorphism
• Corollary 1: ID property of Phase Unwrapping
• proof
• Definition 2: Diffeomorphism
• Theorem 2: Brenier
• Theorem 3: Caffarelli
• Definition 3: Conformal Map
• Definition 4: Harmonic Energy
• Theorem 4: Main
• Definition 5: Möbius Transformation