Reuter Martin, Wolter Franz-Erich, Shenton Martha, Niethammer Marc
Dept. of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge; A.A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, Boston.
Inst. für Mensch-Maschine-Kommunikation, Leibniz Universität Hannover, Germany.
Comput Aided Des. 2009 Oct 1;41(10):739-755. doi: 10.1016/j.cad.2009.02.007.
This paper proposes the use of the and the as shape descriptors for the comparison and analysis of shapes. These spectral measures are isometry invariant and therefore allow for shape comparisons with minimal shape pre-processing. In particular, no registration, mapping, or remeshing is necessary. The discriminatory power of the 2D surface and 3D solid methods is demonstrated on a population of female caudate nuclei (a subcortical gray matter structure of the brain, involved in memory function, emotion processing, and learning) of normal control subjects and of subjects with schizotypal personality disorder. The behavior and properties of the Laplace-Beltrami eigenvalues and -functions are discussed extensively for both the Dirichlet and Neumann boundary condition showing advantages of the Neumann vs. the Dirichlet spectra in 3D. Furthermore, topological analyses employing the Morse-Smale complex (on the surfaces) and the Reeb graph (in the solids) are performed on selected eigenfunctions, yielding shape descriptors, that are capable of localizing geometric properties and detecting shape differences by indirectly registering topological features such as critical points, level sets and integral lines of the gradient field across subjects. The use of these topological features of the Laplace-Beltrami eigenfunctions in 2D and 3D for statistical shape analysis is novel.
本文提出使用[具体内容1]和[具体内容2]作为形状描述符，用于形状的比较和分析。这些频谱度量是等距不变的，因此只需进行最少的形状预处理即可进行形状比较。特别是，无需配准、映射或重新网格化。在正常对照受试者和分裂型人格障碍受试者的一组女性尾状核（大脑的一种皮质下灰质结构，参与记忆功能、情绪处理和学习）上，展示了二维表面和三维实体方法的辨别能力。针对狄利克雷和诺伊曼边界条件，广泛讨论了拉普拉斯 - 贝尔特拉米特征值和特征函数的行为及性质，显示出在三维中诺伊曼频谱相对于狄利克雷频谱的优势。此外，对选定的特征函数进行了采用莫尔斯 - 斯梅尔复形（在表面上）和里布图（在实体中）的拓扑分析，得出能够定位几何特性并通过间接配准跨受试者的拓扑特征（如临界点、水平集和梯度场的积分线）来检测形状差异的形状描述符。在二维和三维中使用拉普拉斯 - 贝尔特拉米特征函数的这些拓扑特征进行统计形状分析是新颖的。
