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

TL;DR

The paper addresses robust 3D reconstruction from unorganized point clouds by formulating an edge-weighted Allen–Cahn phase-field model augmented with a time-dependent Lagrange multiplier to preserve the original energy and achieve unconditional stability. A Crank–Nicolson time discretization yields a decoupled, linear-solve scheme comprising two subproblems plus a scalar constraint for the multiplier, guaranteeing energy dissipation and efficient computation. Through extensive 3D experiments (including complex shapes like C-3PO and Darth Vader), the method demonstrates accurate surface recovery, insensitivity to point-density, and clear guidance on parameter choices such as the interface thickness $\epsilon$ and stabilization $S$. The approach offers practical advantages over SAV and BDF2 schemes, including a structure-preserving energy law, straightforward implementation, and favorable computational performance on standard hardware. Overall, the framework provides a principled, stable, and scalable route to high-fidelity 3D reconstructions from scattered data.

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.

Figures (19)

• Figure 1: 3D Reconstruction. The unorganized points, 3D object and narrow band are displayed.
• Figure 2: Schematic illustrations of signed distance function, phase-field function, and edge detection function. From the left to right in top row, the 2D illustrations of $d({\bf x})$, $\phi({\bf x},0)$, and $g({\bf x})$ are plotted. The bottom row shows the 1D cross sections. The figures were adapted from JJYangJCP with the permission of Elsevier.
• Figure 3: 3D narrow volume reconstructions of three different objects.
• Figure 4: Energy stability for three different objects with different time steps.
• Figure 5: Accuracy experiment of our method. In each figure, the errors versus time steps are plotted.