# Orebody Modeling from Non-Parallel Cross Sections with Geometry Constraints

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

#### 1.1. Related Works

#### 1.1.1. Implicit Function

#### 1.1.2. Surface Reconstruction from Contours

## 2. Overview of the Method

^{3}, based on the idea of distance field, we can obtain N scattered data points ${\left\{{\mathit{x}}_{i},f\left({\mathit{x}}_{i}\right)\right\}}_{i=1}^{\mu}$ with function values of distance to surface by sampling the unknown geological domain.

## 3. Contours Interpolation

#### 3.1. Implicit Function

**x**) can be defined as

#### 3.2. Adaptive Sampling

#### 3.3. Additional Constraints

#### 3.3.1. Soft Constraints

#### 3.3.2. Hard Constraints

#### 3.4. Distance Field Correction

#### 3.4.1. Distance Estimation

#### 3.4.2. ICPC

## 4. Fast Reconstruction

## 5. Results

#### 5.1. Examples

#### 5.2. Performance

## 6. Discussion and Conclusions

#### 6.1. Limitations

#### 6.2. Extensions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Adaptive resampling of a closed contour, and (

**b**) the reconstruction result. The green boxes are external cells of quadtree and the pink boxes are internal cells of quadtree. The red points are sampled from the cross sections and the blue points are sampled from the internal cells of quadtree.

**Figure 3.**Schematic diagram of distance field scan filling algorithm. The scan line is parallel to the y axis (

**a**) and the x axis (

**b**).

**Figure 4.**Three types of hard constraints: (

**a**) constraint point, (

**b**) constraint line and (

**c**) trend surface; (

**d**) is a sparse example with two parallel sections, and (

**e**) fixes the undesired extrapolation and topological error by adding constraint points.

**Figure 5.**Schematic diagram of the distance field correction principle. The positive values represent the exterior of the geological domain and the negative values represent the interior of the geological domain. The signs of A and B are the same (

**a**) and opposite (

**b**).

**Figure 6.**Reconstructions from input cross sections (

**a**) of a geological example using the method of Liu et al. [18] (

**c**) and the improved method without a constraint line (

**d**) and with a constraint line (

**e**). The geological domain (cutaway view in (

**f**)) is sampled (

**b**) by different function values to form a signed distance field.

**Figure 7.**Reconstruction results for multiple types of data sets. These data sets are parallel (

**a**,

**b**) and non-parallel (

**c**–

**f**) cross sections sampled from ground truth objects.

**Figure 8.**Comparison of reconstructions from cross sections (

**a**) using the method of explicit modeling (

**b**), the method of Zou et al. [22] (

**c**), the unconstrained method with (

**d**) and without (

**e**) distance correction), and the improved method (

**f**) with additional constraints. Arrows point to unexpected errors, for which the results do not recover the original shape. Three constraint lines (red) (

**a**) were added to fix the unexpected errors.

**Figure 9.**Comparison of running time on several examples. The scale of the time axis is a nonlinear scale (logarithmic).

**Figure 10.**The inputs (

**a**) and reconstructions with some (

**b**) and more (

**c**) constraints. Two black boxes point to undesired holes (

**b**), constraint lines (blue) (

**c**) were added to fix the topological issues.

Name of RBFs | Definition |
---|---|

Biharmonic | $\phi \left(r\right)=r$ |

Triharmonic | $\phi \left(r\right)={r}^{3}$ |

Multivariate spline | $\phi \left(r\right)={r}^{2m+1}$ |

Gaussian | $\phi \left(r\right)=\mathrm{exp}\left(-c{r}^{2}\right)$ |

Multiquadric | $\phi \left(r\right)=\sqrt{{r}^{2}+{c}^{2}}$ |

Inverse multiquadric | $\phi \left(r\right)=1/\sqrt{{r}^{2}+{c}^{2}}$ |

Thin-plate spline | $\phi \left(r\right)={r}^{2}\mathrm{log}\left(r\right)$ |

Multivariate spline | $\phi \left(r\right)={r}^{2m}\mathrm{ln}\left(r\right)$ |

**Table 2.**Running time of the solution and reconstruction of our algorithm on several examples. Also showing the number of constraints (N), sampling interval in cross sections (${d}_{sam}$), boundary subdivision level (${l}_{b}$), internal subdivision level (${l}_{i}$), size of resolution (${d}_{res}$) and relative geometric error (Err.).

Models | N | ${\mathit{d}}_{\mathit{s}\mathit{a}\mathit{m}}$ | ${\mathit{l}}_{\mathit{i}}$ | ${\mathit{l}}_{\mathit{b}}$ | ${\mathit{d}}_{\mathit{r}\mathit{e}\mathit{s}}$ | Err. | Time (s) | |||
---|---|---|---|---|---|---|---|---|---|---|

LU | PMC | FMM | SF | |||||||

Figure 7a | 1082 | 2.9 | 2 | 5 | 0.50 | 0.44% | 3.05 | 163.29 | 0.37 | 0.68 |

Figure 7b | 2183 | 9.6 | 2 | 5 | 14.71 | 0.21% | 21.71 | 25.04 | 0.97 | 0.32 |

Figure 7c | 949 | 3.1 | 2 | 5 | 0.50 | 0.71% | 2.12 | 47.18 | 0.25 | 0.62 |

Figure 7d | 2288 | 3.0 | 2 | 5 | 1.33 | 0.28% | 25.35 | 15.31 | 0.75 | 0.23 |

Figure 7e | 4125 | 2.0 | 2 | 6 | 4.51 | 0.47% | 121.81 | 27.82 | 2.27 | 0.41 |

Figure 7f | 3484 | 0.5 | 2 | 6 | 0.80 | 0.89% | 69.65 | 61.24 | 0.99 | 0.39 |

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**MDPI and ACS Style**

Zhong, D.-Y.; Wang, L.-G.; Jia, M.-T.; Bi, L.; Zhang, J. Orebody Modeling from Non-Parallel Cross Sections with Geometry Constraints. *Minerals* **2019**, *9*, 229.
https://doi.org/10.3390/min9040229

**AMA Style**

Zhong D-Y, Wang L-G, Jia M-T, Bi L, Zhang J. Orebody Modeling from Non-Parallel Cross Sections with Geometry Constraints. *Minerals*. 2019; 9(4):229.
https://doi.org/10.3390/min9040229

**Chicago/Turabian Style**

Zhong, De-Yun, Li-Guan Wang, Ming-Tao Jia, Lin Bi, and Ju Zhang. 2019. "Orebody Modeling from Non-Parallel Cross Sections with Geometry Constraints" *Minerals* 9, no. 4: 229.
https://doi.org/10.3390/min9040229