Phase-field computation for 3D shell reconstruction with an energy-stable and uniquely solvable BDF2 method

Abstract

Three-dimensional (3D) reconstruction from points cloud is an important technique in computer vision and manufacturing industry. The 3D volume consists of a set of voxels which preserves the characteristics of scattered points. In this paper, a 3D shell (narrow volume) reconstruction algorithm based on the Allen–Cahn (AC) phase field model is proposed, aiming to efficiently and accurately generate 3D reconstruction models from point cloud data. The algorithm uses a linearized backward differentiation formula (BDF2) for time advancement and adopts the finite difference method to perform spatial discretization, unconditional energy stability and second-order time accuracy can be achieved. The present method is not only suitable for 3D reconstruction of unordered data but also has the effect of adaptive denoising and surface smoothing. In addition, theoretical derivation proves the fully discrete energy stability. In numerical experiments, the complex geometric models, such as Asian dragon, owl, and turtle, will be reconstructed to validate the energy stability. The temporal accuracy is validated by the numerical reconstructions of a Costa surface and an Amremo statue. Later, we reconstruct the Stanford dragon, teapot, and Thai statue to further investigate the capability of the proposed method. Finally, we implement a comparison study using a 3D happy Buddha. The numerical results show that the algorithm still has good numerical stability and reconstruction accuracy at large time steps, and can significantly preserve the detailed structure of the model. This research provides an innovative solution and theoretical support for scientific computing and engineering applications in the field of 3D reconstruction.