Improving the Shape of Surfaces by Perturbation

许多制造或可视化中的几何建模应用都需要切平面连续曲面来插值数据。虽然存在许多保证局部解析平滑的算法，但与（非光滑）分段线性表面相比，得到的表面通常具有不希望的无关特征。这些特征是弯曲，曲率的急剧变化，数据点之间的膨胀等。这些“形状缺陷”对建模对象的功能和美学影响可能是严重的。该项目将查明特定表面结构特征的来源和性质，并设计算法来系统地修改数据和/或底层表面表示，以减少或删除特征。特别是，该研究将建立在许多算法的经验基础上，并将调查和量化以下影响：跨越斑块边界的连续性约束的变化，特别是再参数化的选择；2)数据点和法线位置的扰动；3)斑块表示的变化；特别是，从多项式到有理表示的变化；4)对数据之间的邻域关系进行扰动，例如通过移除或插入边界来重新划分网格。

Many applications of geometric modeling in manufacturing or for visualization require tangent plane continuous surfaces to interpolate data. While a number of algorithms exist that guarantee local, analytic smoothness, the resulting surfaces often have undesirable extraneous features when compared to a (nonsmooth) piecewise linear surface. Such features are inflection, sharp changes of curvature, bulging between the data sites, etc. The functional and aesthetic implications of these 'shape defects' on the modeled object can be severe. This project will pinpoint sources and nature of the features for specific surface constructions and devise algorithms to systematically modify data and/or the underlying surface representation in order to reduce or remove the features. In particular, the research will build on the experience with a number of algorithms and will investigate and quantify the effects of: 1. a change of continuity constraints across patch boundaries, in particular, the choice of reparametrization; 2) a perturbation of the location of data sites and normals; 3) a change of the representation of the patches; in particular, changing from a polynomial to a rational representation; 4) a perturbation of the neighborhood relations between the data, such as remeshing by removing or inserting boundaries.