Computer Science Department and Computation Institute, University of Chicago, USA.
IEEE Trans Vis Comput Graph. 2010 Nov-Dec;16(6):1595-604. doi: 10.1109/TVCG.2010.199.
Symmetric second-order tensor fields play a central role in scientific and biomedical studies as well as in image analysis and feature-extraction methods. The utility of displaying tensor field samples has driven the development of visualization techniques that encode the tensor shape and orientation into the geometry of a tensor glyph. With some exceptions, these methods work only for positive-definite tensors (i.e. having positive eigenvalues, such as diffusion tensors). We expand the scope of tensor glyphs to all symmetric second-order tensors in two and three dimensions, gracefully and unambiguously depicting any combination of positive and negative eigenvalues. We generalize a previous method of superquadric glyphs for positive-definite tensors by drawing upon a larger portion of the superquadric shape space, supplemented with a coloring that indicates the quadratic form (including eigenvalue sign). We show that encoding arbitrary eigenvalue magnitudes requires design choices that differ fundamentally from those in previous work on traceless tensors that arise in the study of liquid crystals. Our method starts with a design of 2-D tensor glyphs guided by principles of scale-preservation and symmetry, and creates 3-D glyphs that include the 2-D glyphs in their axis-aligned cross-sections. A key ingredient of our method is a novel way of mapping from the shape space of three-dimensional symmetric second-order tensors to the unit square. We apply our new glyphs to stress tensors from mechanics, geometry tensors and Hessians from image analysis, and rate-of-deformation tensors in computational fluid dynamics.
对称二阶张量场在科学和生物医学研究以及图像分析和特征提取方法中起着核心作用。显示张量场样本的实用性推动了可视化技术的发展，这些技术将张量的形状和方向编码到张量符号的几何形状中。除了一些例外，这些方法仅适用于正定张量（即具有正特征值，如扩散张量）。我们将张量符号的范围扩展到二维和三维中的所有对称二阶张量，以优雅和明确的方式描绘任何正、负特征值的组合。我们通过利用超二次曲面形状空间的更大部分，以及表示二次形式（包括特征值符号）的颜色，对以前用于正定张量的超二次曲面符号的方法进行了扩展。我们表明，对任意特征值大小进行编码需要进行根本的设计选择，这些选择与在研究液晶时出现的无迹张量的先前工作中的选择不同。我们的方法从由比例保持和对称原理指导的二维张量符号设计开始，并创建包括其轴对齐截面中的二维符号的三维符号。我们的方法的一个关键组成部分是一种将三维对称二阶张量的形状空间映射到单位正方形的新方法。我们将我们的新符号应用于力学中的应力张量、图像分析中的几何张量和 Hessian 以及计算流体动力学中的变形率张量。
