Watertight surface reconstruction method for CAD models based on optimal transport

Fig. 1 Overview: Given the input point cloud (a), construct a convex hull of the point cloud as the initial mesh (b), deform the initial mesh according to the optimal transport plan to obtain the initial approximate mesh (c), and finally perform vertex relocation, edge collapse, and edge flipping on the initial approximate mesh to obtain the reconstructed result (d).

Fig. 2 Computation of the new mesh vertex position. Each face adjacent to vertex $\mathit{a}$ generates a new location (left), and they are averaged to determine the new positions of vertex $\mathit{a}$ (right).

Fig. 3 Result of optimal transport with different cost functions on a 2D shape. The black points are the source, the red points are the target, and the blue lines represent a transport relationship. Left: optimal transport plan using the squared Euclidean distance. Right: result after minimization using the local variance of the neighborhood.

Fig. 4 Effect of vertex relocation. Left: input point cloud. Middle: mesh before vertex relocation. Right: result after vertex relocation.

Fig. 5 Simulation of edge collapse. Collapse the edge (red edge) to its midpoint (blue point), and then the optimal position (yellow point) is obtained by vertex relocation.

Fig. 6 Edge flip in planar areas. Left: before edge flip. Right: after edge flip. The condition for edge flip is ${\alpha }_1+{\alpha }_2>{\beta }_1+{\beta }_2$ .

Fig. 7 Edge flip in non-planar areas. Blue points are the input point cloud. Left: mesh before edge flip. Right: mesh after edge flip.

Fig. 8 Reconstruction result from synthetic data. Left: input point clouds, sampled from a ground truth surface. Middle: reconstruction result of our method. Right: ground truth surface.

Fig. 9 Reconstruction result (middle) on a real scanned point cloud (left) which is obtained by scanning the object (right) with EinScan-SP 3D scanner.

Fig. 10 Comparing the ability of reconstruction methods to preserve sharp features. (a) Input point clouds; (b) screened Poisson; (c) RIMLS; (d) RIMLS over EAR; (e) CVT-based; (f) Point2Mesh; (g) PTN; (h) our method.

Fig. 11 Comparison of reconstruction errors for different surface reconstruction algorithms. The warmer the color, the larger the deviation from the ground truth. (a) Screened Poisson; (b) RIMLS; (c) RIMLS over EAR; (d) CVT-based; (e) Point2Mesh; (f) PTN; (g) our method.

Fig. 12 Preserving features with a small number of faces. Left: input point cloud (7000 points). Middle: result of the method proposed by Digne et al. [ 21 ]. Right: result of our method. The number of vertices of both meshes is 35. Their method takes more than 2 h to reconstruct the surface, whereas our method takes only 0.5 h.

Fig. 13 Reconstruction results with different levels of Gaussian noise. Top: input point clouds (12,000 points) with Gaussian noise. The variances of Gaussian noise are $0.5\mathrm{\%}d$ (left), $1.0\mathrm{\%}d$ (middle), $1.5\mathrm{\%}d$ (right). Bottom: output of our method. The number of vertices of output meshes is 60.

Fig. 14 Reconstruction results for data with missing parts. The blue points are the input. Left: missing data at the sharp edge. Right: missing data at the corner.

Fig. 15 Process of reconstructing a smooth surface. (a) Input point cloud; (b) initial mesh; (c) intermediate mesh between initial mesh and initial approximate mesh; (d) initial approximate mesh; (e) reconstructed result.

Fig. 16 Tests on smooth surface. Left: input point clouds. Right: reconstruction results by our method.

Fig. 17 Peak memory (in MB) and runtime (in second) of our method. We use the fandisk model for testing, where the number of input points ranges from 8000 to 20,000. The initial mesh has 1255 vertices and 2506 faces, and the number of target vertices in Algorithm 2 is 1155.

Fig. 18 A failure case of our method. Left: input points (13,000). Right: reconstruction result.