基于四阶偏微分方程的光滑曲面重构方法

doi:10.11772/j.issn.1001-9081.2015.02.0486

计算机应用 ›› 2015, Vol. 35 ›› Issue (2): 486-489.DOI: 10.11772/j.issn.1001-9081.2015.02.0486

基于四阶偏微分方程的光滑曲面重构方法

邓世武¹, 贾雨¹, 姚兴苗²

1. 成都理工大学核技术与自动化工程学院, 成都 610059;
2. 电子科技大学通信与信息工程学院, 成都 611731

收稿日期:2014-09-02 修回日期:2014-11-25 发布日期:2015-02-12 出版日期:2015-02-10
通讯作者: 邓世武
作者简介:邓世武(1989-),男,甘肃武威人,硕士研究生,主要研究方向:计算机图形图像处理、信号采集与数字处理; 贾雨(1966-),女,四川成都人,副教授,主要研究方向:工程图学、计算机图形图像处理; 姚兴苗(1976-),男,四川南充人,副教授,博士,主要研究方向:复杂多维信息处理与显示、计算机图形学。
基金资助:
国家自然科学基金资助项目(41104067)。

Smooth surface reconstruction based on fourth-order partial differential equation

DENG Shiwu¹, JIA Yu¹, YAO Xingmiao²

1. College of Nuclear Technology and Automation Engineering, Chengdu University of Technology, Chengdu Sichuan 610059, China;
2. School of Communication and Information Engineering, University of Electronic Science and Technology of China, Chengdu Sichuan 611731, China

Received:2014-09-02 Revised:2014-11-25 Online:2015-02-12 Published:2015-02-10

摘要/Abstract

摘要：

常用的基于散点的曲面重构方法如克里金插值法、样条曲面拟合法等存在计算量大、重构曲面不光滑或无法插值已知散点等问题。为此,提出一种基于四阶偏微分方程的曲面重构方法。该方法首先选择一个四阶偏微分方程,并对其构建差分格式,进而分析该差分格式的稳定性和收敛性。在稳定性和收敛性条件下,采用演化的思想,通过有限差分法迭代求解偏微分方程的数值解,并将其稳态解作为原始曲面的逼近。以地质勘探中实际测井数据为例,采用偏微分方程曲面造型方法重构地质曲面,结果表明,该方法计算简便,构造的曲面具有自然光顺性且可以插值于已知散点。

关键词: 偏微分方程, 曲面重构, 插值, 散点, 有限差分

Abstract:

The common surface reconstruction methods based on scattered points, including Kriging interpolation and spline surface fitting, have some problems such as large amount of calculation, unsmooth reconstructed surface and being unable to interpolate the given points. Aiming at this issue, a new surface reconstruction method based on a fourth-order partial differential equation was proposed. In this method, a fourth-order partial differential equation was selected and its difference scheme was built, and then the stability and convergence of the difference scheme was analyzed. On this basis, with the idea of evolution, the finite difference method was used to get the numerical solution of the partial differential equation, and the steady-state solution was treated as an approximation of the original surface. As an example, with the logging data in geological exploration, a geological curved surface was reconstructed by the partial differential surface modeling method. The result shows that the method is easy to implement and the reconstructed surface is smooth naturally, as well as can interpolate the given scattered data points.

Key words: Partial Differential Equation (PDE), surface reconstruction, interpolation, scattered data point, finite difference

中图分类号:

TP391.41

邓世武, 贾雨, 姚兴苗. 基于四阶偏微分方程的光滑曲面重构方法[J]. 计算机应用, 2015, 35(2): 486-489.

DENG Shiwu, JIA Yu, YAO Xingmiao. Smooth surface reconstruction based on fourth-order partial differential equation[J]. Journal of Computer Applications, 2015, 35(2): 486-489.

参考文献

[1] HAN S. Study on surface interpolation algorithms for scattered data [D]. Xi'an: Xi'an University of Technology, 2009. (韩舒然.散乱数据曲面插值算法研究[D].西安:西安理工大学, 2009.)
[2] BLOOR M I G, WILSON M J. Generating blend surfaces using partial differential equations [J]. Computer Aided Design, 1989, 21(3): 165-171.
[3] ZENG J. Some questions of Partial Differential Equation (PDE) method in surface modeling [D]. Dalian: Dalian University of Technology, 2005. (曾杰.偏微分方程曲面造型方法的若干问题[D].大连:大连理工大学, 2005.)
[4] ZHENG C. Some research on the generating and manipulating of PDE transition surface [D]. Dalian: Dalian University of Technology, 2005. (郑彩玲,偏微分方程过渡面构造与曲面控制问题的研究[D].大连:大连理工大学, 2005.)
[5] DU H, QIN H. Free-form geometric modeling by integrating parametric and implicit PDEs [J]. IEEE Transactions on Visualization and Computer Graphics, 2007, 13(3): 549-561.
[6] MA L, ZHU X. Application of PDE method to free form surface design [J]. Chinese Journal of Computers, 1998, 21(S1): 357-362. (马岭,朱心雄. PDE造型方法在复杂曲面设计中的应用[J].计算机学报, 1998, 21(S1): 357-362.)
[7] XU G, WANG G, Bézier approximation of PDE surface [J]. Journal of Software, 2007, 18(11):2914-2920. (徐岗,汪国昭. PDE曲面的Bézier逼近[J].软件学报,2007,18(11): 2914-2920.)
[8] KUBEISA S, UGAIL H, WILSON M. Interactive design using higher order PDE's [J]. The Visual Computer, 2004, 20(10): 682-693.
[9] XU G. Geometric partial differential equation methods in computational geometry [M]. Beijing: Science Press, 2008: 257-258. (徐国良.计算几何中的几何偏微分方程方法[M].北京:科学出版社, 2008: 257-258.)
[10] LIU D, XU G, ZHANG Q. A discrete scheme of Laplace-Beltrami operator and its convergence over quadrilateral meshes [J]. Computers and Mathematics with Applications, 2008, 55(6): 1081-1093.
[11] DAI J, QIU J. Numerical solution of differential equations [M]. 2nd ed. Nanjing: Southeast University Press, 2012: 80-83. (戴嘉尊,邱建贤.微分方程数值解法[M].2版.南京:东南大学出版社, 2012: 80-83.)
[12] YANG J. Application of Parseval equality to the summation of infinite series [J]. College Physics, 2014, 33(5): 42-45. (杨杰.Parseval等式在一类无穷级数求和中的应用[J].大学物理,2014, 33(5):42-45.)
[13] ZONG D. Voronoi Diagrams and its applications [D]. Nanjing: Nanjing University of aeronautics and astronautics, 2006. (宗大伟. Voronoi图及其应用研究[D].南京:南京航空航天大学, 2006.)

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基于四阶偏微分方程的光滑曲面重构方法

Smooth surface reconstruction based on fourth-order partial differential equation

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