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

Author

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  • Jin Wang

    (School of Humanities and Social Science, Xi’an Jiaotong University, Xi’an 710049, China)

  • Zhengyuan Shi

    (School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China)

Abstract

The Poisson surface reconstruction algorithm has become a very popular tool of reconstruction from point clouds. If we reconstruct each region separately in the process of multi-reconstruction, then the reconstructed objects may overlap with each other. In order to reconstruct multicomponent surfaces without self-intersections, we propose an efficient multi-reconstruction algorithm based on a modified vector-valued Allen–Cahn equation. The proposed algorithm produces smooth surfaces and closely preserves the original data without self-intersect. Based on operator splitting techniques, the numerical scheme is divided into one linear equation and two nonlinear equations. The linear equation is discretized using an implicit method, and the resulting discrete system of equation is solved by a fast Fourier transform. The two nonlinear equations are solved analytically due to the availability of a closed-form solution. The numerical scheme has merit in that it can be straightforwardly applied to a graphics processing unit, allowing for accelerated implementation that performs much faster than central processing unit alternatives. Various experimental, numerical results demonstrate the effectiveness and robustness of the proposed method.

Suggested Citation

  • Jin Wang & Zhengyuan Shi, 2021. "Multi-Reconstruction from Points Cloud by Using a Modified Vector-Valued Allen–Cahn Equation," Mathematics, MDPI, vol. 9(12), pages 1-15, June.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:12:p:1326-:d:571475
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    References listed on IDEAS

    as
    1. Yu, Qian & Wang, Kunyang & Xia, Binhu & Li, Yibao, 2021. "First and second order unconditionally energy stable schemes for topology optimization based on phase field method," Applied Mathematics and Computation, Elsevier, vol. 405(C).
    2. Li, Yibao & Guo, Shimin, 2017. "Triply periodic minimal surface using a modified Allen–Cahn equation," Applied Mathematics and Computation, Elsevier, vol. 295(C), pages 84-94.
    3. Guido Marseglia & Carlo Maria Medaglia & Francisco A. Ortega & Juan A. Mesa, 2019. "Optimal Alignments for Designing Urban Transport Systems: Application to Seville," Sustainability, MDPI, vol. 11(18), pages 1-14, September.
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