A PCA Based Model for Surface Reconstruction from Incomplete Point Clouds

TL;DR

The paper tackles surface reconstruction from incomplete point clouds by coupling PCA-derived normal information with a level-set formulation. It introduces a multi-term energy balancing fidelity to the point cloud, curvature, and PCA-based normals, and reformulates it into an initial-value problem solved via an operator-splitting (Lie) scheme. Numerical discretization uses FFT-based solvers under periodic boundaries, with detailed implementation for 2D and 3D settings, including PCA normal estimation, distance function computation, and reinitialization. Experiments in 2D and 3D demonstrate improved reconstruction in data-missing and noisy scenarios, highlighting the method’s robustness and connectivity advantages over several baselines. The approach offers a scalable, regularized framework for incomplete-surface recovery with practical implications for 3D modeling and visualization.

Abstract

Point cloud data represents a crucial category of information for mathematical modeling, and surface reconstruction from such data is an important task across various disciplines. However, during the scanning process, the collected point cloud data may fail to cover the entire surface due to factors such as high light-absorption rate and occlusions, resulting in incomplete datasets. Inferring surface structures in data-missing regions and successfully reconstructing the surface poses a challenge. In this paper, we present a Principal Component Analysis (PCA) based model for surface reconstruction from incomplete point cloud data. Initially, we employ PCA to estimate the normal information of the underlying surface from the available point cloud data. This estimated normal information serves as a regularizer in our model, guiding the reconstruction of the surface, particularly in areas with missing data. Additionally, we introduce an operator-splitting method to effectively solve the proposed model. Through systematic experimentation, we demonstrate that our model successfully infers surface structures in data-missing regions and well reconstructs the underlying surfaces, outperforming existing methodologies.

Figures (12)

• Figure 1: An example by PCA. (a) The incomplete point cloud sampled from a square. Data in the central region of the bottom edge are missing. (b) Exact normal directions on the square. (c) Estimated normal directions by PCA in the central region of each edge. (d) Estimated normal direction field by PCA.
• Figure 2: Two-dimensional examples. (a) Shows the point cloud data. (b) Shows our reconstructed curve. (c) Shows the evolution of the energy in (\ref{['eq.energy.2']}).
• Figure 3: Comparison on two-dimensional incomplete data. The first row shows the given data: (a) Shows data sampled from a square, while data around the four corners are missing. (b) shows data sampled from a hexagon. But only data near the two opposite corners are available. Column (c)-(g) show results by the proposed method, and the methods from DS zhao2000implicit, CR he2020curvature, DSP estellers2012efficient and TVG liang2013robust, respectively.
• Figure 4: Two-dimensional incomplete data. The two plots show the vector field $\mathbf{p}_d$ for the results by the proposed method in Figure \ref{['fig.incomplete']}(b).
• Figure 5: Comparison on two-dimensional noisy data. The first row shows the given data: (a) Shows data sampled from an ellipse with noise. (b) shows data sampled from a a three-petal flower with noise. Column (c)-(g) show results by the proposed method, and the methods from DS zhao2000implicit, CR he2020curvature, DSP estellers2012efficient and TVG liang2013robust, respectively.