基于拉格朗日乘子法的三维窄壳重建，实现无条件稳定
Three-dimensional narrow volume reconstruction method with unconditional stability based on a phase-field Lagrange multiplier approach
- 用拉格朗日乘子重构相场模型，结合无符号距离函数边缘检测
- 数值实验显示对《星战》角色等复杂3D体重建准确且稳定
- 代码开源，适合做医学影像或逆向工程的研究者参考
从点云重建物体在假肢、医学成像、计算机视觉等领域至关重要。本文提出一种针对Allen-Cahn型重建模型的有效算法，采用拉格朗日乘子方法。利用物体散乱点数据，通过引入基于无符号距离函数的边缘检测函数，求解改进的控制方程，重建出狭窄壳体。所设计的边缘检测函数保障了能量稳定性。通过拉格朗日乘子技术重构控制方程，并结合Crank-Nicolson时间离散，实现解的稳定且解耦更新。空间项采用有限差分法逼近，理论上证明了全离散格式的无条件稳定性。大量数值实验，包括《星战》角色等复杂三维体的重建，验证了该算法的精度、稳定性和有效性。此外，分析了特定参数选择对重建细节和精细度的影响。为帮助读者理解，相关计算代码与数据已公开于https://github.com/cfdyang521/C-3PO/tree/main。
原文摘要 · Abstract (English)
Reconstruction of an object from points cloud is essential in prosthetics, medical imaging, computer vision, etc. We present an effective algorithm for an Allen--Cahn-type model of reconstruction, employing the Lagrange multiplier approach. Utilizing scattered data points from an object, we reconstruct a narrow shell by solving the governing equation enhanced with an edge detection function derived from the unsigned distance function. The specifically designed edge detection function ensures the energy stability. By reformulating the governing equation through the Lagrange multiplier technique and implementing a Crank--Nicolson time discretization, we can update the solutions in a stable and decoupled manner. The spatial operations are approximated using the finite difference method, and we analytically demonstrate the unconditional stability of the fully discrete scheme. Comprehensive numerical experiments, including reconstructions of complex 3D volumes such as characters from \textit{Star Wars}, validate the algorithm's accuracy, stability, and effectiveness. Additionally, we analyze how specific parameter selections influence the level of detail and refinement in the reconstructed volumes. To facilitate the interested readers to understand our algorithm, we share the computational codes and data in https://github.com/cfdyang521/C-3PO/tree/main.
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