Orienting unorganized points for surface reconstruction

doi:10.1016/j.cag.2010.03.003

Computers & Graphics

Volume 34, Issue 3, June 2010, Pages 209-218

https://doi.org/10.1016/j.cag.2010.03.003 Get rights and content

Abstract

We address the problem of assigning consistently oriented normal vectors to unorganized point cloud with noises, non-uniformities, and thin-sharp features as a pre-processing step to surface reconstruction. The conventional orienting scheme using minimal spanning tree fails on points with the above defects. Different from the recently developed consolidation technique, our approach does not modify (i.e., down-sampling) the given point cloud so that we can reconstruct more surface details in the regions with very few points. The method consists of three major steps. We first propose a modified scheme of generating adaptive spherical cover for unorganized points by adding a sphere splitting step based on eigenvalue analysis. This modification can better preserve the connectivity of surface generated from the spheres in the highly sparse region. After generating the triangular mesh surface and cleaning its topology, a local search based algorithm is conducted to find the closest triangle to every input points and then specify their orientations. Lastly, an orientation-aware principle component analysis step gives correct and consistently oriented normal vectors to the unorganized input points. Conventional implicit surface fitting based approach can successfully reconstruct high quality surfaces from the unorganized point cloud with the help of consistently oriented normal vectors generated by our method.

Introduction

The reverse engineering problem for reconstructing three dimensional models in a computer system from unorganized points that are generated by 3D surface scanning devices has been a subject of intensive research for many years. The scanned 3D surface represented by an unorganized point cloud is typically noisy, contains holes, and has high variations in point density. Oriented normals at the points play a critical role in surface reconstruction. It is because that the oriented normals define the reconstructed surface to the first order and identify the inside/outside information. As will be shown in our tests below, the oriented normals become extremely important at the regions with very sparse points. Also, to generate correctly oriented normal vectors on the points in such regions is a very tough job. The conventional methods using minimal spanning tree (MST) (e.g., [1]) or Voronoi diagram (e.g., [2], [3], [4]) fail. Some recent researches consider estimating normals from captured images using photometric stereo [5], [6], which however suffers from the unideal acquisition conditions like specular reflections, material artifacts, and shadowing. The most recent work presented in [7] does not assign normal vectors directly to the given points. It first adopts a weighted locally optimal projection operator to produce a set of denoised and evenly distributed particles over the original point cloud, and then conducts a priority-driven normal propagation scheme to assign normal vectors to the particles. This down-sampling strategy actually further removes limited information of underlying surfaces from those highly sparse regions, therefore the reconstructed surface in such regions will not be as good as ours (see Fig. 1). Our approach proposed in this paper can assign consistently oriented normal vectors to the scattered points so that the downstream reconstruction algorithm can successfully generate surface in the regions with highly sparse points.

To orient unorganized points effectively and efficiently, we develop two techniques by extending the integrating approach for meshing scattered point data [8]. First, a modified scheme is proposed to generate adaptive spherical cover (ASC) for unorganized points by adding an eigenvalue analysis based sphere splitting step. With this step, our approach can better preserve the surface's connectivity in the regions with highly sparse points. After getting the spherical cover for scattered points, the triangulation and topology cleaning procedure [8] can generate a triangular mesh surface M roughly presenting the underlying surface S. Although this mesh M is not a good approximation of S, it gives a very robust evidence for assigning the orientation of input points. A straightforward way is to find the closest point c_p on M for each input point p, then the normal vector $n_{c_{p}}$ of c_p on M is assigned as the normal vector of p. Nevertheless, as M is an inaccurate approximation of S, such normal vectors give inaccurate surface information to the downstream mesh reconstruction algorithm (e.g., [9]). Therefore, instead of assigning $n_{c_{p}}$ to p, we only let p hold the orientation of n_{c_p}—thus, we name our method as orienting approach (ORT). An orientation-aware principle component analysis (PCA) step is adopted to assign correct and consistently oriented normal vectors to the unorganized points. Moreover, the ASC constructed in the first step will be employed to speed up the closest point search on M. The experimental results demonstrate that our approach can successfully orient the unorganized point clouds for various models—so that conventional schemes like [9] can reconstruct a proper surface for the input data. Fig. 1 shows a comparison of the results between other approaches and ours on a Venus head model with non-uniform point density and noises. Our approach (ORT+RBF) gives the best reconstruction result. The good performance of our approach is benefited by (1) the proposed framework of using adaptive ASC to give the consistent orientation of points and (2) the newly developed sphere splitting step based on eigenvalue analysis.

Section snippets

Related work

The existing work in the literature can be classified into two major groups: (1) computational geometry approaches and (2) volumetric reconstruction techniques, which will be reviewed below.

The computational geometry approaches are usually based on the Voronoi diagram of a given point cloud and reconstruct a mesh surface by directly linking the input samples. Normal information is not required. Amenta et al. [2] gave a provable guarantee of reconstructing a correct model given a minimum

Modified adaptive spherical cover

The adaptive spherical cover (ASC) generated in [8] works well on a noisy point cloud S={p₁, …, p_n} with n scattered points, and outputs a set of covering spheres which will be employed to construct triangular meshes by linking the auxiliary points in the spheres. To compensate the variation of point density on S, every point is assigned with a weight $w_{i} = \frac{1}{k} \sum_{j = 1}^{k} ∥ p_{i} - p_{j} ∥^{2}$ where ${p_{j}}_{j = 1}^{k} \subset S$ are the k-nearest neighbors of p_i. We select k=10 in all our experimental tests, which well balances the speed

Orienting unorganized points

After triangulating the auxiliary points in the modified adaptive spherical cover into a triangular mesh and cleaning its topology, we obtain a rough mesh surface M for approximating the underlying surface, which is represented by the input scattered data point S. Although M does not accurately approximate the shape of the underlying surface H, it gives a very good estimation of H's topology. Therefore, very good estimation of the orientation on H for the points in S can be found from M.

Results

The proposed approach has been implemented in Visual C++. Our implementation has been tested on a variety of models. The statistics in this paper are all tested on a standard PC with Intel Core 2 CPU 6600 at 2.4 GHz plus 2.0 GB RAM.

The first example tested is the Venus head model with noises and non-uniform sparseness, which is shown in Fig. 1. Note that making the density of points much farther sparse may lead our method also failed, so as others. Our method gives better result than other

Conclusion and discussion

In this paper, we have presented a robust and efficient method to assign consistently oriented normal vectors to unorganized points with noises, non-uniformities, and thin sharp features as a pre-processing step to surface reconstruction. The conventional method for this normal assignment step is through the minimal spanning tree based normal propagation, which however is not robust on the unorganized points with noises, non-uniformities, and thin sharp features. The newly developed point

Acknowledgements

The authors would like to thank the authors of [7] for sharing the executable program of their approach. This research is supported by the HKSAR Research Grants Council GRF Grant (Ref.: CUHK/417109), and the Shun Hing Institute of Advanced Engineering (SHIAE) Research Grant (Ref.: CUHK/8115022). The first author is also partially supported by the Open Project Program of the State Key Lab of CAD&CG (Grant no. A0805), Zhejiang University.

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Technical SectionOrienting unorganized points for surface reconstruction