[Paper Review] Locally controlled globally smooth ground surface reconstruction from terrestrial point clouds

I. Rychkov|arXiv (Cornell University)|Jun 4, 2012

Advanced Numerical Analysis Techniques2 references3 citations

TL;DR

This paper presents a novel method for reconstructing globally smooth ground surfaces from terrestrial LiDAR point clouds using locally controlled, curvature-continuous surfaces. By combining local least squares planes with Hermite Radial Basis Functions and Partition of Unity methods using B-splines or compactly supported exponential functions, the approach achieves both local adaptability and global smoothness, enabling fast evaluation and high-fidelity modeling of complex terrain.

ABSTRACT

Approaches to ground surface reconstruction from massive terrestrial point clouds are presented. Using a set of local least squares (LSQR) planes, the "holes" are filled either from the ground model of the next coarser level or by Hermite Radial Basis Functions (HRBF). Global curvature continuous as well as infinitely smooth ground surface models are obtained with Partition of Unity (PU) using either tensor product B-Splines or compactly supported exponential function. The resulting surface function has local control enabling fast evaluation.

Motivation & Objective

• To address the challenge of creating accurate, globally smooth ground surface models from massive terrestrial point clouds.
• To enable local control over surface reconstruction while maintaining global curvature continuity or infinite smoothness.
• To improve computational efficiency and accuracy in terrain modeling for applications such as topographic mapping and environmental monitoring.
• To integrate local geometric fitting with global approximation techniques for robust hole-filling and surface smoothing.
• To develop a method that supports fast evaluation and high-fidelity representation of complex ground surfaces.

Proposed method

• The method uses local least squares (LSQR) planes to model small patches of the ground surface, ensuring local adaptability and accuracy.
• Missing regions ('holes') are filled using either a coarser-level ground model or Hermite Radial Basis Functions (HRBF) for smooth interpolation.
• Partition of Unity (PU) method combines local basis functions into a global surface representation, ensuring smooth transitions across patches.
• Global smoothness is achieved using tensor product B-splines or compactly supported exponential functions as the PU weighting functions.
• The resulting surface function supports local control, enabling efficient and fast evaluation across large datasets.
• The approach ensures curvature continuity or infinite smoothness depending on the choice of basis functions.

Experimental results

Research questions

• RQ1 How can local geometric fitting be combined with global approximation to achieve smooth, accurate ground surface reconstruction?
• RQ2 What basis functions enable both local control and global smoothness in surface modeling from point clouds?
• RQ3 Can hole-filling in ground models be effectively achieved using hierarchical models or radial basis functions?
• RQ4 How does the use of Partition of Unity improve surface continuity and computational efficiency?
• RQ5 What trade-offs exist between smoothness, accuracy, and computational cost in terrain reconstruction from terrestrial point clouds?

Key findings

• The method successfully generates globally smooth ground surfaces with curvature continuity or infinite smoothness using local basis functions.
• Hermite Radial Basis Functions (HRBF) effectively fill gaps in the ground model, preserving surface continuity and accuracy.
• The use of Partition of Unity with B-splines or compactly supported exponential functions ensures smooth blending across local patches.
• The resulting surface function supports local control, enabling fast and efficient evaluation on large-scale point clouds.
• The approach achieves high-fidelity reconstruction of complex terrain features while maintaining computational efficiency.
• The method demonstrates robustness in handling noisy or sparse point cloud data through adaptive local fitting and smooth global blending.