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.

翻译：从点云重建物体在假肢学、医学成像、计算机视觉等领域至关重要。我们提出了一种基于Allen-Cahn型重建模型的有效算法，采用拉格朗日乘子法。利用物体散乱数据点，通过求解由无符号距离函数导出的边缘检测函数增强的控制方程，重建窄壳层。专门设计的边缘检测函数确保了能量稳定性。通过拉格朗日乘子技术重构控制方程并采用Crank-Nicolson时间离散化，我们能够以稳定且解耦的方式更新解。空间运算采用有限差分法近似，并解析证明了全离散格式的无条件稳定性。综合数值实验（包括《星球大战》字符等复杂三维体积的重建）验证了算法的准确性、稳定性和有效性。此外，我们分析了特定参数选择如何影响重建体积的细节水平和精细度。为便于读者理解算法，我们在https://github.com/cfdyang521/C-3PO/tree/main分享了计算代码与数据。