**Figure 1.**
Flowchart of the automatic SORs reconstruction from point clouds.

**Figure 1.**
Flowchart of the automatic SORs reconstruction from point clouds.

**Figure 2.**
Two kinds of SORs. (**a**) A tall-thin type SOR and (**b**) a short-wide type SOR.

**Figure 2.**
Two kinds of SORs. (**a**) A tall-thin type SOR and (**b**) a short-wide type SOR.

**Figure 3.**
Rotation axis extraction for a short-wide SOR of a straw hat. (**a**) The point cloud of a straw hat with three axial directions, (**b**) Two parts of the point cloud divided by the plane $Y=0$ according to axial direction 1, (**c**) Two parts of the point cloud divided by the plane $Y=0$ according to axial direction 2, and (**d**) Two parts of the point cloud divided by the plane $Y=0$ according to axial direction 3.

**Figure 3.**
Rotation axis extraction for a short-wide SOR of a straw hat. (**a**) The point cloud of a straw hat with three axial directions, (**b**) Two parts of the point cloud divided by the plane $Y=0$ according to axial direction 1, (**c**) Two parts of the point cloud divided by the plane $Y=0$ according to axial direction 2, and (**d**) Two parts of the point cloud divided by the plane $Y=0$ according to axial direction 3.

**Figure 4.**
Circular contour fitting. (**a**) A point set of the top of a straw-hat, (**b**) Constructed TIN model, (**c**) Fitted circular contour by the RANSAC algorithm, and (**d**) Fitted circular contour only by the RANSAC algorithm.

**Figure 4.**
Circular contour fitting. (**a**) A point set of the top of a straw-hat, (**b**) Constructed TIN model, (**c**) Fitted circular contour by the RANSAC algorithm, and (**d**) Fitted circular contour only by the RANSAC algorithm.

**Figure 5.**
Original point-cloud (gray points) and the extracted projection profile of a straw hat (magenta points).

**Figure 5.**
Original point-cloud (gray points) and the extracted projection profile of a straw hat (magenta points).

**Figure 6.**
Schematic diagram of extracting the boundary X of the projection profile.

**Figure 6.**
Schematic diagram of extracting the boundary X of the projection profile.

**Figure 7.**
Overflow points processing. (**a**) The extracted point set of boundary X containing overflow points and (**b**) the processed point set of boundary X without overflow points.

**Figure 7.**
Overflow points processing. (**a**) The extracted point set of boundary X containing overflow points and (**b**) the processed point set of boundary X without overflow points.

**Figure 8.**
3D spatial data hyperfine modeling system.

**Figure 8.**
3D spatial data hyperfine modeling system.

**Figure 9.**
Reconstructed SOR of a cylinder. (**a**) A photo of the cylinder, (**b**) Original point cloud of the cylinder, (**c**) Reconstructed SOR of the cylinder by the curvature computation method, (**d**) Cross-section of the reconstructed SOR of the cylinder by the curvature computation method, (**e**) Reconstructed SOR of the cylinder by the proposed method, and (**f**) Cross-section of the reconstructed SOR of the cylinder by the proposed method.

**Figure 9.**
Reconstructed SOR of a cylinder. (**a**) A photo of the cylinder, (**b**) Original point cloud of the cylinder, (**c**) Reconstructed SOR of the cylinder by the curvature computation method, (**d**) Cross-section of the reconstructed SOR of the cylinder by the curvature computation method, (**e**) Reconstructed SOR of the cylinder by the proposed method, and (**f**) Cross-section of the reconstructed SOR of the cylinder by the proposed method.

**Figure 10.**
Reconstructed SOR of a frustum of a cone. (**a**) A photo of the frustum of a cone, (**b**) Original point cloud of the frustum of a cone, (**c**) Reconstructed SOR of the frustum of a cone by the curvature computation method, (**d**) Cross-section of the reconstructed SOR of the frustum of a cone by the curvature computation method, (**e**) Reconstructed SOR of the frustum of a cone by the proposed method, and (**f**) Cross-section of the reconstructed SOR of the frustum of a cone by the proposed method.

**Figure 10.**
Reconstructed SOR of a frustum of a cone. (**a**) A photo of the frustum of a cone, (**b**) Original point cloud of the frustum of a cone, (**c**) Reconstructed SOR of the frustum of a cone by the curvature computation method, (**d**) Cross-section of the reconstructed SOR of the frustum of a cone by the curvature computation method, (**e**) Reconstructed SOR of the frustum of a cone by the proposed method, and (**f**) Cross-section of the reconstructed SOR of the frustum of a cone by the proposed method.

**Figure 11.**
Reconstructed SOR of a vase. (**a**) A photo of the frustum of the vase, (**b**) Original point cloud of the vase, (**c**) Reconstructed SOR of the vase by the curvature computation method, (**d**) Cross-section of the reconstructed SOR of the vase by the curvature computation method, (**e**) Reconstructed SOR of the vase by the proposed method, and (**f**) Cross-section of the reconstructed SOR of the vase by the proposed method.

**Figure 11.**
Reconstructed SOR of a vase. (**a**) A photo of the frustum of the vase, (**b**) Original point cloud of the vase, (**c**) Reconstructed SOR of the vase by the curvature computation method, (**d**) Cross-section of the reconstructed SOR of the vase by the curvature computation method, (**e**) Reconstructed SOR of the vase by the proposed method, and (**f**) Cross-section of the reconstructed SOR of the vase by the proposed method.

**Figure 12.**
Reconstructed SOR of a pillar of an ancient building. (**a**) An image of the pillar of an ancient building, (**b**) Original point cloud of the pillar of an ancient building, (**c**) Reconstructed SOR of the pillar of an ancient building by the curvature computation method, (**d**) Cross-section of the reconstructed SOR of the pillar of an ancient building by the curvature computation method, (**e**) Reconstructed SOR of the pillar of an ancient building by the proposed method, and (**f**) Cross-section of the reconstructed SOR of the pillar of an ancient building by the proposed method.

**Figure 12.**
Reconstructed SOR of a pillar of an ancient building. (**a**) An image of the pillar of an ancient building, (**b**) Original point cloud of the pillar of an ancient building, (**c**) Reconstructed SOR of the pillar of an ancient building by the curvature computation method, (**d**) Cross-section of the reconstructed SOR of the pillar of an ancient building by the curvature computation method, (**e**) Reconstructed SOR of the pillar of an ancient building by the proposed method, and (**f**) Cross-section of the reconstructed SOR of the pillar of an ancient building by the proposed method.

**Figure 13.**
Reconstructed SOR of a pot. (**a**) A photo of the frustum of the pot, (**b**) Original point cloud of the pot, (**c**) Reconstructed SOR of the pot by the curvature computation method, (**d**) Cross-section of the reconstructed SOR of the pot by the curvature computation method, (**e**) Reconstructed SOR of the pot by the proposed method, and (**f**) Cross-section of the reconstructed SOR of the pot by the proposed method.

**Figure 13.**
Reconstructed SOR of a pot. (**a**) A photo of the frustum of the pot, (**b**) Original point cloud of the pot, (**c**) Reconstructed SOR of the pot by the curvature computation method, (**d**) Cross-section of the reconstructed SOR of the pot by the curvature computation method, (**e**) Reconstructed SOR of the pot by the proposed method, and (**f**) Cross-section of the reconstructed SOR of the pot by the proposed method.

**Figure 14.**
Reconstructed SOR of a ceramic. (**a**) A photo of the frustum of the ceramic, (**b**) Original point cloud of the ceramic, (**c**) Reconstructed SOR of the ceramic by the curvature computation method, (**d**) Cross-section of the reconstructed SOR of the ceramic by the curvature computation method, (**e**) Reconstructed SOR of the ceramic by the proposed method, and (**f**) Cross-section of the reconstructed SOR of the ceramic by the proposed method.

**Figure 14.**
Reconstructed SOR of a ceramic. (**a**) A photo of the frustum of the ceramic, (**b**) Original point cloud of the ceramic, (**c**) Reconstructed SOR of the ceramic by the curvature computation method, (**d**) Cross-section of the reconstructed SOR of the ceramic by the curvature computation method, (**e**) Reconstructed SOR of the ceramic by the proposed method, and (**f**) Cross-section of the reconstructed SOR of the ceramic by the proposed method.

**Figure 15.**
Reconstructed SOR of a pillar of an ancient building. (**a**) Reconstructed SOR by the Delaunay-based SOR reconstruction method, (**b**) Reconstructed SOR by the Poisson SOR reconstruction method, (**c**) Reconstructed SOR by the RBF SOR reconstruction method, and (**d**) Reconstructed SOR by the proposed method.

**Figure 15.**
Reconstructed SOR of a pillar of an ancient building. (**a**) Reconstructed SOR by the Delaunay-based SOR reconstruction method, (**b**) Reconstructed SOR by the Poisson SOR reconstruction method, (**c**) Reconstructed SOR by the RBF SOR reconstruction method, and (**d**) Reconstructed SOR by the proposed method.

**Figure 16.**
Reconstructed SOR of a simple SOR with different sampling rate. (**a**) Reconstructed SOR with a sampling rate of 100%, (**b**) Reconstructed SOR with a sampling rate of 75%, (**c**) Reconstructed SOR with a sampling rate of 50%, and (**d**) Reconstructed SOR with a sampling rate of 50%.

**Figure 16.**
Reconstructed SOR of a simple SOR with different sampling rate. (**a**) Reconstructed SOR with a sampling rate of 100%, (**b**) Reconstructed SOR with a sampling rate of 75%, (**c**) Reconstructed SOR with a sampling rate of 50%, and (**d**) Reconstructed SOR with a sampling rate of 50%.

**Figure 17.**
Reconstructed SOR of a tall-thin SOR with different sampling rate. (**a**) Reconstructed SOR with a sampling rate of 100%, (**b**) Reconstructed SOR with a sampling rate of 75%, (**c**) Reconstructed SOR with a sampling rate of 50%, and (**d**) Reconstructed SOR with a sampling rate of 50%.

**Figure 17.**
Reconstructed SOR of a tall-thin SOR with different sampling rate. (**a**) Reconstructed SOR with a sampling rate of 100%, (**b**) Reconstructed SOR with a sampling rate of 75%, (**c**) Reconstructed SOR with a sampling rate of 50%, and (**d**) Reconstructed SOR with a sampling rate of 50%.

**Figure 18.**
Reconstructed SOR of a short-wide SOR with different sampling rate. (**a**) Reconstructed SOR with a sampling rate of 100%, (**b**) Reconstructed SOR with a sampling rate of 75%, (**c**) Reconstructed SOR with a sampling rate of 50%, and (**d**) Reconstructed SOR with a sampling rate of 50%.

**Figure 18.**
Reconstructed SOR of a short-wide SOR with different sampling rate. (**a**) Reconstructed SOR with a sampling rate of 100%, (**b**) Reconstructed SOR with a sampling rate of 75%, (**c**) Reconstructed SOR with a sampling rate of 50%, and (**d**) Reconstructed SOR with a sampling rate of 50%.

**Table 1.**
Relative deviations of the three rotation axes from axial directions 1, 2, and 3 after quaternion rotation.

**Table 1.**
Relative deviations of the three rotation axes from axial directions 1, 2, and 3 after quaternion rotation.

Axial Direction | Number of Points M | Number of Points N | Number of Points M-N | Relative Deviation |
---|

1 | 21,668 | 15,620 | 6048 | 0.613 |

2 | 21,668 | 10,966 | 10,702 | 0.02 |

3 | 21,668 | 6043 | 15,625 | 1.586 |

**Table 2.**
Parameters comparison for the two reconstructed simple SORs between the curvature computation method and the proposed method.

**Table 2.**
Parameters comparison for the two reconstructed simple SORs between the curvature computation method and the proposed method.

Objects | Parameters | Curvature Computation Method | Proposed Method | Percentage Improvement |
---|

Cylinder | RMS (mm) | 0.42 | 0.29 | 30.1% |

Time (ms) | 2151 | 1039 | 51.7% |

Frustum of a cone | RMS (mm) | 0.56 | 0.29 | 41.1% |

Time (ms) | 1928 | 1001 | 48.1% |

**Table 3.**
Parameters comparison for the two reconstructed tall-thin SORs between the curvature computation method and the proposed method.

**Table 3.**
Parameters comparison for the two reconstructed tall-thin SORs between the curvature computation method and the proposed method.

Objects | Parameters | Curvature Computation Method | Proposed Method | Percentage Improvement |
---|

Vase | RMS (mm) | 0.35 | 0.24 | 31.4% |

Time (ms) | 2450 | 1835 | 25.1% |

Pillar | RMS (mm) | 0.43 | 0.30 | 30.2% |

Time (ms) | 1836 | 1349 | 26.5% |

**Table 4.**
Parameters comparison for the two reconstructed short-wide SORs between the curvature computation method and the proposed method.

**Table 4.**
Parameters comparison for the two reconstructed short-wide SORs between the curvature computation method and the proposed method.

Objects | Parameters | Curvature Computation Method | Proposed Method | Percentage Improvement |
---|

Pot | RMS (mm) | 0.51 | 0.21 | 58.8% |

Time (ms) | 4020 | 3012 | 25.1% |

Ceramic | RMS (mm) | 0.33 | 0.23 | 30.3% |

Time (ms) | 1548 | 1113 | 28.1% |

**Table 5.**
Parameters comparison for the reconstructed SOR of a pillar of an ancient building by the Delaunay-based method, Poisson method, RBF method, and the proposed method.

**Table 5.**
Parameters comparison for the reconstructed SOR of a pillar of an ancient building by the Delaunay-based method, Poisson method, RBF method, and the proposed method.

Parameters | Delaunay | Poisson | RBF | Proposed Method |
---|

RMS (mm) | 0.06 | 0.58 | 0.45 | 0.30 |

Time (ms) | 1936 | 1489 | 1523 | 1349 |

**Table 6.**
Accuracy comparison of the reconstructed SORs with different sampling rates for a simple SOR, a tall-thin SOR, and a short-wide SOR.

**Table 6.**
Accuracy comparison of the reconstructed SORs with different sampling rates for a simple SOR, a tall-thin SOR, and a short-wide SOR.

| Sampling Rate | 100% | 75% | 50% | 25% |
---|

Simple SOR | Number of points | 148,400 | 111,300 | 74,200 | 37,100 |

RMS (mm) | 0.28 | 0.88 | 3.9 | 4.5 |

Tall-thin SOR | Number of points | 36,114 | 27,085 | 18,057 | 9028 |

RMS (mm) | 0.32 | 0.85 | 3.77 | 4.81 |

Short-wide SOR | Number of points | 10,131 | 7598 | 5065 | 2532 |

RMS (mm) | 0.26 | 0.81 | 4.47 | 5.54 |