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基于严格的数学方法开发矢量和张量场的通用数据分析技术。该工作系统地探索了二维和三维矢量场和张量场的直接可视化和特征提取。最近，我们一直在开发可视化工作的下一个阶段：自动数据比较。矢量场比较的传统技术分为三大类：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也存在根本性的局限性。基于图像的比较在表示二维向量场以外的数据集方面存在困难，基于数据的比较存在网格对齐问题，而基于特征的比较虽然提供了特定特征的出色位置，但可能无法显示该领域的所有全局信息。我们解决这个问题的新方法本质上是一种基于特征的比较技术，有一个重要的规定，即我们的特征试图表示字段的拓扑结构。这确保我们不会忽视该领域的任何重要结构。我们的范式是分析地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将这些场可视化。我们已经成功地实现了二维矢量场和三维矢量场的想法。我们现在打算研究与白质脑功能相关的张量场。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个领域，从太阳的磁场到机翼上的气流。此外，还可以通过将固定时刻的场与稍后时刻的场进行比较来研究时变场。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，它是定量研究核磁共振张量脑数据的唯一方法。我们的张量和矢量数据集分析方法可以更好地设计飞行器，更好地理解电磁问题，并在处理大型实验数据集时节省大量时间。对脑白质功能的定量了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.