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

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.

• The paper introduces an unconditionally energy stable algorithm for 3D narrow volume reconstruction using a phase‐field Allen–Cahn model enhanced with a Lagrange multiplier.
• It employs a stabilized Crank–Nicolson scheme, achieving second-order temporal accuracy and preserving the discrete energy dissipation law.
• Extensive numerical experiments demonstrate the method's robustness to noise and undersampling, outperforming previous SAV and BDF2-based approaches in efficiency and stability.

Three-dimensional Narrow Volume Reconstruction with Unconditional Stability via Phase-Field Lagrange Multiplier Approach

Introduction

This paper presents a robust and efficient algorithm for three-dimensional (3D) volume reconstruction from unorganized point clouds, leveraging a phase-field Allen–Cahn (AC) model reformulated with a Lagrange multiplier to guarantee unconditional energy stability. The method is motivated by the need for accurate, stable, and parameter-insensitive reconstruction in applications such as medical imaging, additive manufacturing, and computer graphics, where point cloud data is often noisy and lacks explicit connectivity.

Mathematical Formulation

The reconstruction is formulated as a variational problem using the AC phase-field model, where the interface between object and background is represented by the zero level set of a scalar field $\phi(\mathbf{x}, t)$. The free energy functional is

$$E(\phi) = \int_\Omega \left[ \frac{F(\phi)}{\epsilon^2} + \frac{1}{2}|\nabla\phi|^2 \right] d\mathbf{x},$$

with $F(\phi) = 0.25(\phi^2-1)^2$ and $\epsilon$ controlling the interface thickness. The evolution follows the $L^2$-gradient flow:

$$\frac{\partial\phi}{\partial t} = -g(\mathbf{x})\mu, \quad \mu = \frac{F'(\phi)}{\epsilon^2} - \Delta\phi,$$

where $g(\mathbf{x})$ is an edge detection function derived from the unsigned distance to the point cloud, localizing the evolution to a narrow band around the initial interface.

To preserve the original energy dissipation law in the discrete setting, a time-dependent scalar Lagrange multiplier $Q$ is introduced, yielding the modified system:

$$\frac{\partial \phi}{\partial t} = -g(\mathbf{x})\mu, \quad \mu = -Q \frac{F'(\phi)}{\epsilon^2} + \Delta\phi,$$

with $Q$ determined by enforcing the discrete chain rule for the nonlinear energy. This approach maintains the variational structure and avoids the energy modification inherent in scalar auxiliary variable (SAV) methods.

Numerical Scheme

Spatial discretization is performed on a uniform Cartesian grid using standard finite differences, which ensures exact summation-by-parts and underpins the discrete energy-dissipation law. Temporal integration employs a stabilized Crank–Nicolson (CN) scheme, achieving second-order accuracy and unconditional stability. The stabilization constant $S$ is set empirically, with $S=2/\epsilon^2$ providing a robust balance between stability and accuracy.

The CN update is decoupled via an affine-in-$Q$ splitting:

• Solve two linear subproblems for $Q$-independent and $Q$-dependent components of $\phi$ and $\mu$.
• Determine $Q$ by solving a scalar nonlinear equation via Newton iteration, enforcing the discrete energy constraint.

This structure enables large time steps, efficient linear solves, and avoids the need for auxiliary variables or ad hoc reinitialization of signed distance fields.

Theoretical Properties

The scheme is analytically proven to be unconditionally energy stable, both in the continuous and fully discrete settings:

$\frac{d}{dt}E(\phi) = -\|\sqrt{g}\mu\|^2 \le 0,$

and

$$\tilde{E}(\phi^{n+1},\phi^n) \le \tilde{E}(\phi^n,\phi^{n-1}),$$

where $\tilde{E}$ is the discrete energy functional. The method achieves second-order accuracy in time, as verified by $L^2$-error convergence rates in numerical experiments.

Numerical Experiments

Extensive simulations demonstrate the method's accuracy, stability, and versatility:

• Reconstruction of complex geometries (teapot, Costa–Hoffman–Meeks surface, horse, happy Buddha, Armadillo, Stanford dragon, C-3PO, Darth Vader helmet) from point clouds.
• Energy dissipation is monotonic for all tested time steps and geometries.
• Second-order temporal accuracy is confirmed by $L^2$-error analysis.
• The method is robust to point cloud density and noise, with the interface thickness parameter $\epsilon$ providing a tunable trade-off between detail preservation and smoothing.
• Comparative studies with SAV and BDF2-based schemes show that the Lagrange multiplier approach preserves the original energy landscape and achieves comparable or superior accuracy and stability, with reduced parameter sensitivity.

Implementation and Performance

The algorithm is implemented with Gauss–Seidel iteration for the linear subproblems and Newton iteration for the scalar constraint. Convergence is rapid except for initial transients, and the per-step computational cost scales linearly with grid resolution, not with the number of input points. CPU timings on commodity hardware confirm practical efficiency for moderate-to-large reconstructions.

Practical Implications and Guidelines

• The method is suitable for unorganized, noisy point clouds, requiring no training data or explicit connectivity.
• The interface thickness $\epsilon$ should be chosen based on the desired balance between detail and smoothness; larger $\epsilon$ improves robustness to undersampling and noise but may oversmooth fine features.
• The stabilization constant $S$ enables large time steps; $S=2/\epsilon^2$ is recommended for most cases.
• Sufficient point cloud density is necessary for high-fidelity reconstruction; undersampling leads to rough or incomplete surfaces.
• The method is ready for deployment in medical imaging, 3D printing, and computer graphics, with open-source code provided.

Future Directions

The authors propose integrating adaptive mesh refinement (AMR) to enable scalable, high-resolution reconstructions with localized refinement near complex geometries. This will further improve computational efficiency and memory usage, facilitating large-scale engineering applications.

Conclusion

This work introduces a structure-preserving, unconditionally stable, and efficient algorithm for 3D volume reconstruction from point clouds, based on a phase-field AC model with a Lagrange multiplier. The method achieves second-order accuracy, robust energy dissipation, and practical efficiency, outperforming or matching prior SAV and BDF2-based approaches. The framework is broadly applicable and provides practical guidelines for parameter selection and implementation, with future work aimed at scalable AMR-based extensions.