Separability Membrane: 3D Active Contour for Point Cloud Surface Reconstruction

TL;DR

The paper tackles robust 3D surface reconstruction from unstructured point clouds by introducing Separability Membrane, a 3D active contour that deforms a cubic B-spline surface to maximize the separability between inner and outer regions of the object via Fisher ratio. It defines point separability directly on augmented point clouds, integrates multiple attributes with weighted separability, and uses a dynamic membrane that adaptively adjusts control points and sampling during optimization, all without training data or voxelization. The method is validated on synthetic data and the 3DNet dataset, showing strong robustness to noise and outliers and superior performance relative to traditional and learning-based baselines. This unsupervised framework offers a scalable and flexible alternative for point-cloud surface reconstruction with potential extensions to real-time processing and higher-dimensional data.

Abstract

This paper proposes Separability Membrane, a robust 3D active contour for extracting a surface from 3D point cloud object. Our approach defines the surface of a 3D object as the boundary that maximizes the separability of point features, such as intensity, color, or local density, between its inner and outer regions based on Fisher's ratio. Separability Membrane identifies the exact surface of a 3D object by maximizing class separability while controlling the rigidity of the 3D surface model with an adaptive B-spline surface that adjusts its properties based on the local and global separability. A key advantage of our method is its ability to accurately reconstruct surface boundaries even when they are ambiguous due to noise or outliers, without requiring any training data or conversion to volumetric representation. Evaluations on a synthetic 3D point cloud dataset and the 3DNet dataset demonstrate the membrane's effectiveness and robustness under diverse conditions.

Figures (13)

• Figure 1: Basic idea of finding the surface of a 3D point cloud object based on class separability. Our method identifies the object boundary by gradually deforming the active surface while maximizing the class separability between the inner volume region ($R_1$) and the outer volume region ($R_2$).
• Figure 2: Definition of edge based on separability. Unlike traditional methods that rely on pixel intensity gradients, this approach defines edges by statistically analyzing the separability between adjacent regions.
• Figure 3: Non-data points estimation scenarios. (a) Ideal case: non-data points (gray points) match the local density of actual points (blue points). (b) Undersampling case: insufficient non-data points in dense regions. (c) Oversampling case: excessive non-data points in sparse regions.
• Figure 4: Adaptive non-data points estimation. (a) Adjacent regions containing actual points from point cloud data. (b) Local density estimation based on k-nearest neighbors on the merged region (c) Visualization of how non-data points would uniformly fill the empty space based on its local density. Note that non-data points are not explicitly added in the implementation, but their count is used in calculations.
• Figure 5: Comparison of 3D separability filter approaches. (a) 3D separability filter using inside/outside cube volumesYataka2017FeaturePE. (b) Our proposed approach that maintains the essence of the original 2D separability filter fukui1995edge by comparing adjacent regions, but extends it to 3D space with arbitrary orientation enabled by the point cloud representation.