The Varifold Representation of Nonoriented Shapes for Diffeomorphic Registration
The Varifold Representation of Nonoriented Shapes for Diffeomorphic Registration
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DOI：
10.1137/130918885
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发表时间：
2013-01-01
影响因子：
2.1
通讯作者：
Trouve, Alain
中科院分区：
文献类型：
--
作者：
Charon, Nicolas;Trouve, Alain
在本文中，我们解决的问题，自然出现的方向时，表示的形状，如曲线或曲面电流。在计算解剖学领域，电流的框架确实被证明是非常有效的建模各种各样的形状。然而，在这种方法中，形状的方向是一个基本问题，这可能导致在处理某些类型的数据集的几个缺点。更具体地说，由于电流的抵消效应，或者由于存在于许多断开的片段（如纤维束）中的数据（电流需要所有片段的一致取向），所以像尖锐的尖峰（acute pike）这样的结构会出现问题。作为一个有前途的替代电流，varifolds，介绍了在上下文中的几何测量理论的Almgren，允许表示任何非定向流形（更一般地说，任何非定向可求长集）。特别是，我们解释了varifolds可以编码数字无方向的对象从离散和连续的观点。我们展示了基于再生核理论在变分集合上构建Hilbert空间结构的各种方法。我们表明，与电流的设置不同，这些度量与形状体积一致（定理4.1），并且我们推导出度量相对于形状的变化公式（定理4.2）。最后，我们提出了一个推广的无方向形状的注册算法的背景下，大变形几何度量映射（LDDMM），我们详细的几个例子在最后一部分的文件。
In this paper, we address the problem of orientation that naturally arises when representing shapes such as curves or surfaces as currents. In the field of computational anatomy, the framework of currents has indeed proved very efficient in modeling a wide variety of shapes. However, in such approaches, orientation of shapes is a fundamental issue that can lead to several drawbacks in treating certain kinds of datasets. More specifically, problems occur with structures like acute pikes because of canceling effects of currents or with data that consists in many disconnected pieces like fiber bundles for which currents require a consistent orientation of all pieces. As a promising alternative to currents, varifolds, introduced in the context of geometric measure theory by Almgren, allow the representation of any nonoriented manifold (more generally any nonoriented rectifiable set). In particular, we explain how varifolds can encode numerically nonoriented objects both from the discrete and the continuous points of view. We show various ways to build a Hilbert space structure on the set of varifolds based on the theory of reproducing kernels. We show that, unlike the currents' setting, these metrics are consistent with shape volume (Theorem 4.1), and we derive a formula for the variation of metrics with respect to the shape (Theorem 4.2). Finally, we propose a generalization to nonoriented shapes of registration algorithms in the context of large deformation diffeomorphic metric mapping (LDDMM), which we detail with a few examples in the last part of the paper.