ITR: Procedures for the Rigorous Comparison of Vector and Tensor Fields

在过去的十年中，斯坦福大学科学可视化小组的研究人员一直在开发基于严格数学方法的矢量和张量场的通用数据分析技术。这项工作系统地探讨了直接可视化和特征提取的矢量和张量场在二维和三维使用计算机。最近，我们一直在开发我们可视化工作的下一个阶段：自动atad比较。传统的矢量场比较技术分为三个基本类别：图像，数据和基于特征的比较。在大多数情况下，比较是直观的，而不是自动的。此外，这些现有的比较技术存在根本的局限性。基于图像的比较难以表示二维矢量场以外的图像，基于数据的比较存在网格重叠问题，而基于特征的比较虽然提供了特定特征的良好位置，但可能无法显示该领域的所有全局信息。有一个重要的条件，即我们的特征试图表示场的拓扑结构。这确保了我们不会忽视该领域的任何重要结构。我们的范例是分析研究向量场和张量场，以提取拓扑关键信息，将这些知识转化为有效的计算机程序，然后使用我们的分析结果来可视化这些场。我们已经成功地实现了我们的二维向量场和三维向量场的想法。我们现在打算研究与脑白质功能有关的张量场。我们比较矢量和张量数据的基本技术几乎适用于科学和工程的每一个领域，从太阳的磁场到机翼上的气流。此外，时变场可以通过比较某一固定时刻的场和它以后的状态来研究。我们的方法在科学可视化领域是独一无二的，因为它是基于严格的数学分析，并且它是唯一可用于定量研究NMR张量脑数据的方法。我们在张量和矢量数据集的分析方法，可以更好地设计飞行器，更好地理解电磁问题，并提供大量的时间节省时，处理大型实验数据集。对白色物质脑功能的定量理解有可能开辟一个全新的医学研究领域。

During the last decade, researchers in our Scientific Visualization Group at Stanfor Universityhave been developing general data analysis techniques for vector and tensor fieldsbased on rigorous mathematical approaches. The work has systematically explored directvisualizations and feature extractions of both vector and tensor fields in two and threedimensions using computers. Recently, we have been developing the next stage of ourvisualization efforts: automated atad comparisons.Traditional techniques for vector field comparison fall into three basic categories: image,data, and feature based comparison. In most instances, comparisons are made visually,not automatically. In addition, there are fundamental limitations with these existing comparisontechniques. Image base comparisons suffer from difficulty in representing datasetsbeyond two-dimensional vector fields, data based comparisons suffer from grid alignmentproblems, and feature base comparisons, while providing excellent location of specific features,may not show all the global information in the field.Our new approach to this problem is essentially a feature-based comparison technique,with the important stipulation that our features attempt to represent the topological structureof the field. This ensures that we do not overlook any important structures in thefield. Our paradigm is to analytically study vector and tensor fields to extract topologicallycritical information, transfer this knowledge into effective computer programs, and then tovisualize the fields using the results of our analysis.We have successfully implemented our ideas for two-dimensional vector fields and forthree-dimensional vector fields. We now intend to study tensor fields associated with whitematter brain functions.Our fundamental technique for the comparison of vector and tensord ata would beapplicable to almost every field of science and engineering, ranging from the magnetic field ofthe sun to airflow over a wing. In addition, time varying fields can be studied by comparinga field at a fixed time with its state at later times. Our approach is unique in the field ofscientific visualization because it is based on rigorous mathematical analysis, and it is theonly approach available to quantitatively study NMR tensor brain data. Our methodologyin the analysis of tensor and vector datasets allows better design of air vehicles, betterunderstanding of electromagnetic problems, and provides substantial time savings whendealing with large experimental datasets. Quantitative understanding of white matter brainfunctions has the potential to open up a whole new area of medical research.