Three-dimensional narrow volume reconstruction method with unconditional stability based on a phase-field Lagrange multiplier approach

TL;DR

This paper introduces a stable, efficient 3D reconstruction algorithm using a phase-field model with a Lagrange multiplier, suitable for complex shapes and validated through numerical experiments.

Contribution

The paper develops a novel, unconditionally stable 3D reconstruction method based on an Allen--Cahn-type model with a Lagrange multiplier, enhancing stability and accuracy.

Findings

Demonstrates unconditional stability of the scheme

Successfully reconstructs complex 3D objects

Provides open-source code and data for reproducibility

Abstract

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.