Generalized Barycentric Coordinates

重心坐标已成为计算机图形学中的一种标准插补技术。这些坐标解决了边值内插问题，可用于对不规则镶嵌上的离散标量场、矢量场甚至多维场进行内插。重心坐标在1975年由Wachspress首次推广用于有限元分析，而图形界在纹理映射、多边形光栅化、光线相交计算和样条线曲面等应用程序中大量使用了这些坐标。最近，重心坐标的新研究在曲面参数化、实体纹理和曲面变形方面有了更多的应用。尽管取得了这些进展，但目前存在的重心坐标的概括很少，而且大多数都包含限制。例如，许多重心坐标结构要求输入形状必须是凸的，即使这样，坐标可能并不是到处都定义得很好。目前，只有一种类型的重心坐标，称为平均值坐标，用于定义任意的闭合形状。然而，这些坐标仍然有很多不足之处，因为它们可能是负的(对于插补应用来说是有问题的)，并且缺乏局部控制，从而产生伪影。理想情况下，重心坐标应具有恒定和线性精度(如变形等应用所需)、产生平滑函数、具有局部影响并且仅包含正值。目前没有一种解析公式包含所有这些性质。这项研究通过研究一种构建重心坐标的新方法来推广重心坐标构造，该方法直接求解具有这些期望属性的坐标。一旦理论完成，新的重心坐标将被应用于几个应用领域，包括：边值插值、曲面变形和基于实例的变形合成。

Barycentric coordinates have become a standard interpolation technique in Computer Graphics. These coordinates solve a boundary value interpolation problem and can be used to interpolate discrete scalar fields, vector fields or even multidimensional fields over irregular tessellations. While barycentric coordinates were first generalized in 1975 by Wachspress for Finite Element Analysis, the Graphics community has made heavy use of these coordinates for applications such as texturing mapping, polygonal rasterization, ray-intersection calculations and spline surfaces. More recently, new research in barycentric coordinates has led to additional applications in surface parameterization, solid texturing and surface deformation. Despite these advances, very few generalizations of barycentric coordinates exist today and most contain restrictions. For instance, many barycentric coordinate constructions require that the input shapes must be convex and, even then, the coordinates may not be well-defined everywhere. Currently, there is only one type of barycentric coordinates, called Mean Value Coordinates, defined for arbitrary, closed shapes. However, these coordinates still leave much to be desired as they can be negative (problematic for interpolation applications) and lack local control, thus producing artifacts. Ideally, barycentric coordinates should have constant and linear precision (needed for applications such as deformation), produce smooth functions, have local influence and contain only positive values. No current analytic formulation contains all of these properties. This research is generalizing barycentric coordinate construction by investigating a novel approach to building barycentric coordinates that directly solves for coordinates that have these desirable properties. Once the theory is completed, the new barycentric coordinates will be applied to several application domains including: Boundary Value Interpolation, Surface Deformation, and Example-based Deformation Synthesis.