# Multi-Reconstruction from Points Cloud by Using a Modified Vector-Valued Allen–Cahn Equation

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## Abstract

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## 1. Introduction

- The first algorithm using a modified vector-valued Allen–Cahn equation for multi-reconstruction from point clouds, which can reconstruct multicomponent surfaces without self-intersections;
- Based on operator splitting techniques, the proposed numerical scheme is simple to implement;
- The algorithm can be straightforwardly applied to a graphics processing unit (GPU), allowing for accelerated implementation that performs many times faster than other central processing units (CPU).

## 2. Related Work

#### 2.1. Implicit Surface Reconstruction

#### 2.2. Multi-Reconstruction

## 3. Preliminary: Poisson Surface Reconstruction

## 4. Methodology

## 5. Numerical Method

## 6. Experimental Results

#### 6.1. Basic Mechanism of the Algorithm

#### 6.2. Performance on the Complex Reconstruction

#### 6.3. Multicomponent Surface Reconstruction

#### 6.4. Multi-Reconstruction with Different Density of Point Data

#### 6.5. Parameter Sensitivity Analysis

#### 6.6. Performance of Our Method

#### 6.7. Comparisons with Related Works and Accuracy Test

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Shammaa, M.H.; Ohtake, Y.; Suzuki, H. Segmentation of multi-material ct data of mechanical parts for extracting boundary surfaces. Comput. Aided Des.
**2010**, 42, 118–128. [Google Scholar] [CrossRef] - Zhang, Y.; Hughes, T.J.; Bajaj, C.L. An automatic 3d mesh generation method for domains with multiple materials. Comput. Methods Appl. Mech. Eng.
**2010**, 199, 405–415. [Google Scholar] [CrossRef] [Green Version] - Li, Y.; Shin, J.; Choi, Y.; Kim, J. Three-dimensional volume reconstruction from slice data using phase-field models. Comput. Vis. Image Underst.
**2015**, 137, 115–124. [Google Scholar] [CrossRef] - Kim, J.; Lee, C.O. Three-dimensional volume reconstruction using two-dimensional parallel slices. SIAM J. Imaging Sci.
**2019**, 12, 1–27. [Google Scholar] [CrossRef] - Zou, Q. A PDE model for smooth surface reconstruction from 2d parallel slices. IEEE Signal Process. Lett.
**2020**, 27, 1015–1019. [Google Scholar] [CrossRef] - Kazhdan, M.; Bolitho, M.; Hoppe, H. Poisson surface reconstruction. In Proceedings of the Fourth Eurographics Symposium on Geometry, Cagliari, Sardinia, 26–28 June 2006; Volume 7, pp. 61–70. [Google Scholar]
- Allen, S.M.; Cahn, J.W. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall.
**1979**, 27, 1085–1095. [Google Scholar] [CrossRef] - Ilmanen, T. Convergence of the Allen-Cahn equation to brakke’s motion by mean curvature. J. Differ. Geom.
**1993**, 38, 417–461. [Google Scholar] [CrossRef] - Li, Y.; Jeong, D.; Kim, H.; Kim, J. Comparison study on the different dynamics between the Allen–Cahn and the Cahn–Hilliard equations. Comput. Math. Appl.
**2019**, 77, 311–322. [Google Scholar] [CrossRef] - Long, J.; Luo, C.; Yu, Q.; Li, Y. An unconditional stable compact fourth-order finite difference scheme for three dimensional Allen-Cahn equation. Comput. Math. Appl.
**2019**, 77, 1042–1054. [Google Scholar] [CrossRef] - Li, Y.; Kim, J. An unconditionally stable hybrid numerical method for solving the Allen-Cahn equation. Comput. Math. Appl.
**2010**, 60, 1591–1606. [Google Scholar] [CrossRef] [Green Version] - Karma, A.; Rappel, W.J. Quantitative phase-field modeling of dendritic growth in two and three dimensions. Phys. Rev. E
**1998**, 57, 4323. [Google Scholar] [CrossRef] [Green Version] - Li, Y.; Lee, H.; Kim, J. A fast, robust, and accurate operator splitting method for phase-field simulations of crystal growth. J. Cryst. Growth
**2011**, 321, 176–182. [Google Scholar] [CrossRef] - Li, Y.; Lee, H.; Kim, J. Phase-field simulations of crystal growth with adaptive mesh refinement. Int. J. Heat Mass Transf.
**2012**, 55, 7926–7932. [Google Scholar] [CrossRef] - Beneš, M.; Chalupeckỳ, V.; Mikula, K. Geometrical image segmentation by the Allen–Cahn equation. Appl. Numer. Math.
**2004**, 51, 187–205. [Google Scholar] [CrossRef] - Esedog, S.; Tsai, Y.H.R. Threshold dynamics for the piecewise constant Mumford—Shah functional. J. Comput. Phys.
**2006**, 211, 367–384. [Google Scholar] [CrossRef] [Green Version] - Kay, D.A.; Tomasi, A. Color image segmentation by the vector-valued Allen–Cahn phase-field model: A multigrid solution. IEEE Trans. Image Process.
**2009**, 18, 2330–2339. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Li, Y.; Kim, J. Multiphase image segmentation using a phase-field model. Comput. Math. Appl.
**2011**, 62, 737–745. [Google Scholar] [CrossRef] [Green Version] - Li, Y.; Guo, S. Triply periodic minimal surface using a modified Allen—Cahn equation. Appl. Math. Comput.
**2017**, 295, 84–94. [Google Scholar] [CrossRef] - Li, Y.; Xia, Q.; Yoon, S.; Lee, C.; Lu, B.; Kim, J. A simple and efficient volume merging method for triply periodic minimal structure. Comput. Phys. Commun.
**2021**, 264, 107956. [Google Scholar] [CrossRef] - Li, Y.; Lee, D.; Lee, C.; Lee, J.; Lee, S.; Kim, J.; Ahn, S.; Kim, J. Surface embedding narrow volume reconstruction from unorganized points. Comput. Vis. Image Underst.
**2014**, 121, 100–107. [Google Scholar] [CrossRef] - Li, Y.; Kim, J. Fast and efficient narrow volume reconstruction from scattered data. Pattern Recognit.
**2015**, 48, 4057–4069. [Google Scholar] [CrossRef] - Li, Y.; Lan, S.; Liu, X.; Lu, B.; Wang, L. An efficient volume repairing method by using a modified Allen-Cahn equation. Pattern Recognit.
**2020**, 107, 107478. [Google Scholar] [CrossRef] - Yu, Q.; Wang, K.; Xia, B.; Li, Y. First and second order unconditionally energy stable schemes for topology optimization based on phase field method. Appl. Math. Comput.
**2021**, 405, 126267. [Google Scholar] - Tang, Y.; Feng, J. Multi-scale surface reconstruction based on a curvature-adaptive signed distance field. Comput. Graph.
**2018**, 70, 28–38. [Google Scholar] [CrossRef] - Morel, J.; Bac, A.; Véga, C. Surface reconstruction of incomplete datasets: A novel poisson surface approach based on csrbf. Comput. Graph.
**2018**, 74, 44–55. [Google Scholar] [CrossRef] - Hoppe, H.; DeRose, T.; Duchamp, T.; McDonald, J.; Stuetzle, W. Surface reconstruction from unorganized points. In Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques, New York, NY, USA, 1 July 1992; pp. 71–78. [Google Scholar]
- Carr, J.C.; Beatson, R.K.; Cherrie, J.B.; Mitchell, T.J.; Fright, W.R.; McCallum, B.C. Reconstruction and representation of 3d objects with radial basis functions. In Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, Los Angeles, CA, USA, 12–17 August 2001; pp. 67–76. [Google Scholar]
- Ohtake, Y.; Belyaev, A.; Alexa, M.; Turk, G.; Seidel, H.P. Multi-level partition of unity implicits. In Proceedings of the ACM SIGGRAPH 2003 Papers, Los Angeles, CA, USA, 1 July 2003; p. 173-es. [Google Scholar]
- Manson, J.; Petrova, G.; Schaefer, S. Streaming surface reconstruction using wavelets. In Computer Graphics Forum; Blackwell Publishing Ltd.: Oxford, UK, 2008; Volume 27, pp. 1411–1420. [Google Scholar]
- Kazhdan, M.; Hoppe, H. Screened poisson surface reconstruction. ACM Trans. Graph.
**2013**, 32, 1–13. [Google Scholar] [CrossRef] [Green Version] - Eliasof, M.; Sharf, A.; Treister, E. Multimodal 3D shape reconstruction under calibration uncertainty using parametric level set methods. SIAM J. Imaging Sci.
**2020**, 13, 265–290. [Google Scholar] [CrossRef] - Hirt, C.W.; Nichols, B.D. Volume of fluid (vof) method for the dynamics of free boundaries. J. Comput. Phys.
**1981**, 39, 201–225. [Google Scholar] [CrossRef] - Dyadechko, V.; Shashkov, M. Moment-of-Fluid Interface Reconstruction. Technical Report; Los Alamos National Laboratory (LA-UR-05-7571). 2005. Available online: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.77.7998&rep=rep1&type=pdf (accessed on 29 April 2021).
- Dyadechko, V.; Shashkov, M. Reconstruction of multi-material interfaces from moment data. J. Comput. Phys.
**2008**, 227, 5361–5384. [Google Scholar] [CrossRef] - Ahn, H.T.; Shashkov, M. Multi-material interface reconstruction on generalized polyhedral meshes. J. Comput. Phys.
**2007**, 226, 2096–2132. [Google Scholar] [CrossRef] - Lemoine, A.; Glockner, S.; Breil, J. Moment-of-fluid analytic reconstruction on 2D cartesian grids. J. Comput. Phys.
**2017**, 328, 131–139. [Google Scholar] [CrossRef] - Kikinzon, E.; Shashkov, M.; Garimella, R. Establishing mesh topology in multi-material cells: Enabling technology for robust and accurate multi-material simulations. Comput. Fluids
**2018**, 172, 251–263. [Google Scholar] [CrossRef] - Yuan, Z.; Yu, Y.; Wang, W. Object-space multiphase implicit functions. ACM Trans. Graph.
**2012**, 31, 1–10. [Google Scholar] [CrossRef] - Zhang, Y.; Qian, J. Resolving topology ambiguity for multiple-material domains. Comput. Methods Appl. Mech. Eng.
**2012**, 247, 166–178. [Google Scholar] [CrossRef] - Da, F.; Batty, C.; Grinspun, E. Multimaterial mesh-based surface tracking. ACM Trans. Graph.
**2014**, 33, 112-1–112-11. [Google Scholar] [CrossRef] [Green Version] - Liu, L.; Bajaj, C.; Deasy, J.O.; Low, D.A.; Ju, T. Surface reconstruction from non-parallel curve networks. In Computer Graphics Forum; Blackwell Publishing Ltd.: Oxford, UK, 2008; Volume 27, pp. 155–163. [Google Scholar]
- Bermano, A.; Vaxman, A.; Gotsman, C. Online reconstruction of 3d objects from arbitrary cross-sections. ACM Trans. Graph.
**2011**, 30, 1–11. [Google Scholar] [CrossRef] [Green Version] - Huang, Z.; Zou, M.; Carr, N.; Ju, T. Topology-controlled reconstruction of multi-labelled domains from cross-sections. ACM Trans. Graph.
**2017**, 36, 1–12. [Google Scholar] [CrossRef] [Green Version] - Li, Y.; Wang, J.; Lu, B.; Jeong, D.; Kim, J. Multicomponent volume reconstruction from slice data using a modified multicomponent Cahn–Hilliard system. Pattern Recognit.
**2019**, 93, 124–133. [Google Scholar] [CrossRef] - Li, Y.; Jung, E.; Lee, W.; Lee, H.-G.; Kim, J. Volume preserving immersed boundary methods for two-phase fluid flows. Int. J. Numer. Methods Fluids
**2012**, 69, 842–858. [Google Scholar] [CrossRef] - Li, Y.; Yun, A.; Lee, D.; Shin, J.; Jeong, D.; Kim, J. Three-dimensional volume-conserving immersed boundary model for two-phase fluid flows. Comput. Methods Appl. Mech. Eng.
**2013**, 257, 36–46. [Google Scholar] [CrossRef] - Olshanskii, M.; Xu, X.; Yushutin, V. A finite element method for Allen–Cahn equation on deforming surface. Comput. Math. Appl.
**2021**, 90, 148–158. [Google Scholar] [CrossRef] - Marseglia, G.; Medaglia, C.M.; Ortega, F.A.; Mesa, J.A. Optimal alignments for designing urban transport systems: Application to Seville. Sustainability
**2019**, 11, 5058. [Google Scholar] [CrossRef] [Green Version] - Carrese, S.; Cuneo, V.; Nigro, M.; Pizzuti, R.; Ardito, C.F.; Marseglia, G. Optimization of downstream fuel logistics based on road infrastructure conditions and exposure to accident events. Transp. Policy
**2019**. [Google Scholar] [CrossRef] - The Stanford Volume Data Archive, Copyright©2000 MarcLevoy. 2001. Available online: http://graphics.stanford.edu/data/3Dscanrep (accessed on 29 April 2021).
- Cahn, J.W.; Hilliard, J.E. Free energy of a nonuniform system I. Interfacial free energy. J. Chem. Phys.
**2004**, 28, 258–267. [Google Scholar] [CrossRef] - Li, Y.; Choi, J.I.; Kim, J. Multi-component Cahn–Hilliard system with different boundary conditions in complex domains. J. Comput. Phys.
**2016**, 323, 1–16. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) The point cloud. (

**b**) The surfaces that are obtained by signal Poisson surface reconstruction [6]. (

**c**) The result in the cross-view of (

**b**).

**Figure 2.**Reconstructed surface in the cut view. The red and blue regions represent the different materials. The yellow region represents the overlapping region.

**Figure 3.**(

**a**) The point cloud that is downsampled for display. (

**b**,

**c**) The surfaces that are obtained by Poisson surface reconstruction and our method, respectively.

**Figure 4.**The evolution of our method in the cut view. (

**a**–

**c**) The contours from t = 0, 1, 5.5 in the cross-view, respectively.

**Figure 5.**(

**a**,

**c**) The results of Poisson reconstruction and our proposed method in the whole view, respectively. (

**b**,

**d**) The results in the cross-view of (

**a**,

**c**), respectively.

**Figure 6.**Multicomponent reconstruction of our proposed method. (

**a**) Two-component reconstruction. (

**b**) Three-component reconstruction. (

**c**) Four-component reconstruction.

**Figure 7.**Four-component surface reconstruction of real point clouds. (

**a**) Input data set. (

**b**,

**c**) Final reconstructions with different views.

**Figure 8.**Multi-reconstruction with different densities of point data. (

**a**) Initial input data. (

**b**) Reconstructed surfaces. (

**c**,

**d**) The results in the different cross-view of (

**b**).

**Figure 9.**Illustration for the parameter $\lambda $ sensitivity analysis. (

**a**) $\lambda =0.01$. (

**b**) $\lambda =0.1$. (

**c**) $\lambda =10$.

**Figure 10.**Illustration for the parameter $\u03f5$ sensitivity analysis. (

**a**) $\u03f5=0.25$. (

**b**) $\u03f5=0.75$. (

**c**) $\u03f5=2.5$.

**Figure 11.**Comparisons with the related work [45]. (

**a**) Given set of slice data. (

**b**) Transformed point clouds. (

**c**) Reconstructed Schwarz diamond minimal surface. (

**d**) Reconstructed Schoen’s F-RD minimal surface. (

**e**–

**h**) Two-dimensional results at the slice data $z=20,\phantom{\rule{3.33333pt}{0ex}}50,\phantom{\rule{3.33333pt}{0ex}}80$, and 110, respectively.

**Table 1.**List of grid size, iterations, CPU times (second), GPU times (second), and errors. CPU(pro) and GPU(pro) are the times taken to process the multi-reconstruction by using the computer’s central processing unit and graphics processing unit, respectively.

Case | Grid Size | Iteration | CPU (pro) | GPU (pro) |
---|---|---|---|---|

Figure 4 | $(256\times 256\times 256)$ | 24 | 36.2 | 9.8 |

Figure 5 | $(298\times 222\times 156)$ | 12 | 9.2 | 2.5 |

Figure 6a | $(298\times 156\times 212)$ | 14 | 9.6 | 3.0 |

Figure 6b | $(298\times 156\times 212)$ | 18 | 20.6 | 5.6 |

Figure 6c | $(298\times 156\times 212)$ | 30 | 40.2 | 11.3 |

Figure 7 | $(642\times 476\times 484)$ | 20 | 515.9 | 133.4 |

Figure 8 | $(288\times 642\times 288)$ | 13 | 87.7 | 23.2 |

Figure 9b | $(298\times 294\times 240)$ | 20 | 34.3 | 8.8 |

Figure 10b | $(256\times 298\times 236)$ | 15 | 22.4 | 5.4 |

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**MDPI and ACS Style**

Wang, J.; Shi, Z.
Multi-Reconstruction from Points Cloud by Using a Modified Vector-Valued Allen–Cahn Equation. *Mathematics* **2021**, *9*, 1326.
https://doi.org/10.3390/math9121326

**AMA Style**

Wang J, Shi Z.
Multi-Reconstruction from Points Cloud by Using a Modified Vector-Valued Allen–Cahn Equation. *Mathematics*. 2021; 9(12):1326.
https://doi.org/10.3390/math9121326

**Chicago/Turabian Style**

Wang, Jin, and Zhengyuan Shi.
2021. "Multi-Reconstruction from Points Cloud by Using a Modified Vector-Valued Allen–Cahn Equation" *Mathematics* 9, no. 12: 1326.
https://doi.org/10.3390/math9121326