ITR: A Unified Representation for Non-homogeneous Models Manifesting Surface, Volume, and Vector Attributes

该研究项目将开发改进的三维物体的数学表示，用于各种应用。计算机图形学、计算机辅助几何设计、工程分析和医学可视化中的底层模型表示历来受到不同需求的驱动。因此，不同的算法和计算基础产生了两种完全不同的公式。模型表示的差异阻碍了有价值技术的交叉传播，阻碍了模型共享，并限制了跨区域应用。为了解决这个历史上难以解决的问题，将基于扩展的、多维的、参数张量积b样条超体积，开发一种可以在广泛应用中工作的统一数学表示。高维nurb（非均匀有理b样条）由于其在实践中的普遍性、建模的通用性、良好的整体计算和交互行为以及广泛的理解而被用作基础。与直接的泛化不同，扩展表示将支持适合于模型的非超矩形域（对于三元变量来说是非六面体），并包括在公共多维参数域上定义的补充属性。这些属性将对功能特征和表示为标量或向量的数据进行建模。几个关键技术的最新进展正在创造对高维表示和相应的可视化算法的需求，以设计和分析真正的固体几何，即在具有多维属性的非均匀材料的设计空间中工作。这要求我们在计算和可视化环境方面取得根本性的进步。其中一个实际应用是设计光学透镜，通过平滑地改变折射率和保持简单的几何形状来获得折射能力。与传统的多种透镜系统相比，简单的形状与折射率没有不连续的组合提供了实质性的光学和机械优势。该项目还将具有教育效益，在实验室培养学生，帮助他们成为这个日益重要领域的领导者。

This research project will develop improved mathematical representations of three-dimensional objects, for use across a variety of applications. Underlying model representations in computer graphics, computer-aided geometric design, engineering analysis, and medical visualization have historically been driven by different requirements. Thus, different algorithmic and computational foundations have evolved yielding two largely disparate formulations. Differences in model representations impede cross-pollination of valuable techniques, hinder model sharing, and limit cross-area applications. To attack this historically intractable problem, a unified mathematical representation that can work across a broad spectrum of applications will be developed, based on an extended, multidimensional, parametric tensor product B-spline hyper-volume. High dimensional NURBs (Non-Uniform Rational B-Splines) are used as a basis because of their ubiquity in practice, universality in modeling, good overall computational and interactive behavior, and extensive understanding. Distinguishing itself from the straightforward generalization, the extended representation will support a non-hyperrectangular domain (non-hexahedral for the trivariate) appropriate to the model, and include supplementary attributes, defined over a common multidimensional parametric domain. The attributes will model both functional characteristics and data represented as scalars or vectors.Recent advances in several key technologies are creating demand for higher dimensional representations and corresponding visualization algorithms to design and analyze truly solid geometry, i.e., working in a design space of non-homogeneous materials manifesting multidimensional attributes. This calls for fundamental advances in our computing and visualization environments. One of the practical applications is designing optical lenses that achieve refracting power by smoothly varying the index of refraction and keeping a simple geometric shape. The combination of simple shape with no discontinuities in the index of refraction offers substantial optical and mechanical advantages over traditional multifarious lens systems. The project will also have educational benefits, training students in the laboratory to help them become leaders in this increasingly important field.