[Paper Review] Reverse Engineering Point Clouds to Fit Tensor Product B-Spline Surfaces by Blending Local Fits

Lavanya Sita Tekumalla, Elaine Cohen|arXiv (Cornell University)|Nov 21, 2014

Advanced Numerical Analysis Techniques11 references3 citations

TL;DR

This paper presents a novel, efficient method for reverse engineering noisy, incomplete point clouds into tensor product B-spline surfaces by blending local weighted least squares fits across segmented data patches. By using adaptive knot placement and a hierarchical domain decomposition strategy, the approach achieves significant computational speedup—reducing fitting complexity from O(ℓm²) to O(ℓ)—while robustly handling noise, holes, and irregular geometry through an end-to-end pipeline of smoothing, hole filling, parameterization, and surface fitting with a 13×13 control mesh yielding a 0.098 error per point.

ABSTRACT

Being able to reverse engineer from point cloud data to obtain 3D models is important in modeling. As our main contribution, we present a new method to obtain a tensor product B-spline representation from point cloud data by fitting surfaces to appropriately segmented data. By blending multiple local fits our method is more efficient than existing techniques, with the ability to deal with more detail by efficiently introducing a high number of knots. Further point cloud data obtained by digitizing 3D data, typically presents many associated complications like noise and missing data. As our second contribution, we propose an end-to-end framework for smoothing, hole filling, parameterization, knot selection and B-spline fitting that addresses these issues, works robustly with large irregularly shaped data containing holes and is straightforward to implement.

Motivation & Objective

• To address the challenge of fitting high-quality tensor product B-spline surfaces to real-world 3D point clouds that are noisy, incomplete, and contain holes.
• To develop a computationally efficient alternative to global least squares fitting by leveraging local, blended fits with fixed-size control patches.
• To create a complete, end-to-end pipeline that integrates smoothing, hole filling, parameterization, knot selection, and surface fitting for robust reverse engineering.
• To enable high-detail modeling by adaptively placing knots using hierarchical domain decomposition, improving surface fidelity without excessive computational cost.

Proposed method

• The method segments the input point cloud into local patches homomorphic to a disc, each fitted using local weighted least squares with a tensor-product windowing function to emphasize nearby points.
• A blending strategy combines local control points into a global B-spline surface, avoiding the need to solve a large global linear system.
• The knot vector is determined via adaptive domain decomposition, enabling fine resolution in complex regions and coarser resolution elsewhere.
• A two-stage parameterization process is used: initial mapping via mesh-based triangulation, followed by iterative correction to minimize distortion.
• Hole filling is performed using a local surface reconstruction technique that preserves geometric continuity and supports irregularly shaped holes.
• Smoothing is applied to noisy boundary regions using a regularized energy minimization approach to preserve feature integrity.

Experimental results

Research questions

• RQ1 Can local, blended least squares fitting achieve competitive surface quality with significantly reduced computational cost compared to global least squares fitting?
• RQ2 How can an end-to-end framework robustly handle noisy, incomplete point clouds with arbitrary holes and irregular geometry?
• RQ3 To what extent does adaptive knot placement via hierarchical domain decomposition improve surface detail and fitting accuracy?
• RQ4 Can a blending approach maintain geometric fidelity while avoiding rank deficiencies and ill-conditioning in the presence of data gaps?

Key findings

• The blending local fits method reduces the computational complexity of B-spline surface fitting from O(ℓm²) to O(ℓ), achieving substantial speedup even for large control meshes.
• With a 13×13 control grid, the method achieves a fitting error of 0.098 per point, which is acceptable relative to the feature size despite being higher than the global least squares method’s 0.047 error.
• The framework successfully processes noisy, incomplete data with holes, producing a smooth, parameterized, and well-fitted B-spline surface as shown in the final results.
• The adaptive knot placement strategy enables high-resolution detail in complex regions while maintaining efficiency, as demonstrated by the resulting 13×13 control mesh.
• The end-to-end pipeline—including smoothing, hole filling, parameterization, and fitting—proves robust and straightforward to implement on real-world scanned data.