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

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。

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 r,~constructed by the partial differential surface modeling method. The result shows that the method is easy to implement and th2 reconstructed surface is smooth naturally, as well as can interpolate the given scattered data points.