Transitions of the 3D Medial Axis under a One-Parameter Family of Deformations
Transitions of the 3D Medial Axis under a One-Parameter Family of Deformations
复制标题
一参数变形族下 3D 中轴的转变
DOI：
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发表时间：
2002
影响因子：
23.6
通讯作者：
B. Kimia
中科院分区：
文献类型：
--
作者：
P. Giblin;B. Kimia
变形下形状的中轴的不稳定性长期以来被认为是其在识别和其他应用中使用的主要障碍。当中轴图的结构在形状变形的情况下突然变化时，就会发生这些不稳定性或转变。最近对中轴和冲击图的这些转变的二维分类是对象识别系统开发的关键因素，其中分类的不稳定性被用来表示变形路径。 3D 中轴的通用过渡的分类同样可能导致 3D 中的类似表示。在本文中，通过检查球体与表面的接触顺序对这些转变进行分类，从而枚举可能的转变，然后根据具体情况进行检查。有些情况被排除为从未在任何变形族中发生，而其他情况则被证明在单参数变形族中是非通用的。最后，通过为每个案例开发一个具体的例子来证明其余案例的可行性。我们的工作受到 Bogaevsky 的启发，他在研究 Hamilton-Jacobi 方程的粘度解时获得了转变。我们的贡献是提供一种更实际的方法，使这项工作引起计算机视觉社区的注意，并使用简单的表面为各种转换提供明确的构造。我们相信，这些转变的分类对于中轴在实际应用中的成功规范化至关重要。
The instabilities of the medial axis of a shape under deformations have long been recognized as a major obstacle to its use in recognition and other applications. These instabilities, or transitions, occur when the structure of the medial axis graph changes abruptly under deformations of shape. The recent classification of these transitions in 2D for the medial axis and for the shock graph was a key factor in the development of an object recognition system where the classified instabilities were utilized to represent deformation paths. The classification of generic transitions of the 3D medial axis could likewise potentially lead to a similar representation in 3D. In this paper, these transitions are classified by examining the order of contact of spheres with the surface, leading to an enumeration of possible transitions which are then examined on a case-by-case basis. Some cases are ruled out as never occurring in any family of deformations, while others are shown to be nongeneric in a one-parameter family of deformations. Finally, the remaining cases are shown to be viable by developing a specific example for each. Our work is inspired by that of Bogaevsky, who obtained the transitions as part of an investigation of viscosity solutions of Hamilton-Jacobi equations. Our contribution is to give a more down-to-earth approach, bringing this work to the attention of the computer vision community, and to provide explicit constructions for the various transitions using simple surfaces. We believe that the classification of these transitions is vital to the successful regularization of the medial axis in its use in real applications.