Ennis Daniel B, Kindlmann Gordon
Department of Radiology, Stanford University, Palo Alto, California 94305-5488, USA.
Magn Reson Med. 2006 Jan;55(1):136-46. doi: 10.1002/mrm.20741.
This paper outlines the mathematical development and application of two analytically orthogonal tensor invariants sets. Diffusion tensors can be mathematically decomposed into shape and orientation information, determined by the eigenvalues and eigenvectors, respectively. The developments herein orthogonally decompose the tensor shape using a set of three orthogonal invariants that characterize the magnitude of isotropy, the magnitude of anisotropy, and the mode of anisotropy. The mode of anisotropy is useful for resolving whether a region of anisotropy is linear anisotropic, orthotropic, or planar anisotropic. Both tensor trace and fractional anisotropy are members of an orthogonal invariant set, but they do not belong to the same set. It is proven that tensor trace and fractional anisotropy are not mutually orthogonal measures of the diffusive process. The results are applied to the analysis and visualization of diffusion tensor magnetic resonance images of the brain in a healthy volunteer. The theoretical developments provide a method for generating scalar maps of the diffusion tensor data, including novel fractional anisotropy maps that are color encoded for the mode of anisotropy and directionally encoded colormaps of only linearly anisotropic structures, rather than of high fractional anisotropy structures.
本文概述了两组解析正交张量不变量的数学推导及应用。扩散张量在数学上可分解为形状和方向信息，分别由特征值和特征向量确定。本文的推导利用一组三个正交不变量对张量形状进行正交分解，这三个不变量分别表征各向同性的大小、各向异性的大小以及各向异性的模式。各向异性的模式有助于分辨一个各向异性区域是线性各向异性、正交各向异性还是平面各向异性。张量迹和分数各向异性都是一个正交不变量集的成员，但它们不属于同一集合。已证明张量迹和分数各向异性并非扩散过程的相互正交度量。研究结果应用于一名健康志愿者大脑的扩散张量磁共振图像的分析和可视化。理论推导提供了一种生成扩散张量数据标量图的方法，包括针对各向异性模式进行颜色编码的新型分数各向异性图，以及仅针对线性各向异性结构而非高分数各向异性结构的方向编码彩色图。
