Generalized Barycentric Coordinates

重心坐标已成为计算机图形学中的标准插值技术。 这些坐标解决了边界值插值问题，并可用于插值离散标量场，矢量场，甚至多维场不规则镶嵌。虽然重心坐标在1975年由Wachspress首次推广用于有限元分析，但图形社区已大量使用这些坐标用于纹理映射，多边形光栅化，光线相交计算和样条曲面等应用程序。最近，新的研究重心坐标导致额外的应用在表面参数化，固体纹理和表面变形。 尽管有这些进步，今天很少有重心坐标的推广存在，大多数包含限制。例如，许多重心坐标构造要求输入形状必须是凸的，即使这样，坐标也可能不是处处都定义良好。目前，只有一种类型的重心坐标，称为均值坐标，定义为任意的，封闭的形状。然而，这些坐标仍然有很多需要改进的地方，因为它们可能是负的（对于插值应用来说是有问题的），并且缺乏局部控制，从而产生伪像。理想情况下，重心坐标应该具有恒定和线性精度（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。目前还没有一个解析公式包含所有这些性质。本研究通过研究一种新的方法来建立重心坐标，直接求解具有这些理想属性的坐标，从而推广重心坐标的构建。 一旦理论完成，新的重心坐标将被应用到几个应用领域，包括：边界值插值，表面变形，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.