Three-dimensional narrow volume reconstruction method with unconditional stability based on a phase-field Lagrange multiplier approach
This work addresses reconstruction challenges in fields like prosthetics and medical imaging, but it appears incremental as it builds on existing Allen-Cahn models with stability enhancements.
The authors tackled the problem of reconstructing 3D objects from point clouds by developing an algorithm based on an Allen-Cahn model with a Lagrange multiplier approach, achieving unconditional stability and validating it through numerical experiments on complex volumes like Star Wars characters.
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.