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点数据的切割轨迹和拓扑
Cut Locus and Topology from Point Data |
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| 课程网址: |
http://videolectures.net/mlss09us_dey_cltpd/
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| 主讲教师: |
Tamal Dey
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| 开课单位: |
俄亥俄州立大学
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| 开课时间: |
2009-07-30
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| 课程语种: |
英语
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| 中文简介: |
紧致黎曼流形M中点P的切割轨迹定义为点集，其中从点P发出的测地线最小化是最小化的。已知切割轨迹包含M的大部分拓扑信息，我们的目标是利用切割轨迹的这一性质从点样本中解出M的拓扑。近年来研究表明，可以从m的点样本p系统地建立Rips配合物来计算Betti数，可以很容易地计算出m的同系物群的秩，因此，Rips配合物比其他配合物（如受限Delaunay、Alpha、Cech和见证配合物）更受欢迎。然而，RIPS复合物的尺寸往往较大。由于切割轨迹的维数低于流形M的维数，接近切割轨迹的P子样本的大小通常要小得多，因此可以容纳一个相对较小的RIPS复合体。在本文中，我们探讨了上述方法，从嵌入任何高维欧几里得空间的曲面上采样的点数据。我们提出了一种计算2流形样本p的子样本p'的算法，其中p'近似于一个切割轨迹。实验结果表明，M的第一个betti数可以从这些子样本上构建的rips复合物中计算出来。这些岩石复合物的大小比在原始M样品上建造的要小得多。
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| 课程简介: |
A cut locus of a point p in a compact Riemannian manifold M is defined as the set of points where minimizing geodesics issued from p stop being minimizing. It is known that a cut locus contains most of the topological information of M. Our goal is to utilize this property of cut loci to decipher the topology of M from a point sample. Recently it has been shown that Rips complexes can be built from a point sample P of M systematically to compute the Betti numbers, the rank of the homology groups of M. Rips complexes can be computed easily and therefore are favored over others such as restricted Delaunay, alpha, Cech, and witness complex. However, the sizes of the Rips complexes tend to be large. Since the dimension of a cut locus is lower than that of the manifold M, a subsample of P approximating the cut locus is usually much smaller in size and hence admits a relatively small Rips complex. In this talk we explore the above approach for point data sampled from surfaces embedded in any high dimensional Euclidean space. We present an algorithm that computes a subsample P' of a sample P of a 2-manifold where P' approximates a cut locus. Empirical results show that the first Betti number of M can be computed from the Rips complexes built on these subsamples. The sizes of these Rips complexes are much smaller than the one built on the original sample of M.
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| 关 键 词: |
拓扑信息; 点数据采样; 高维欧氏空间; 二维流形
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| 课程来源: |
视频讲座网
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| 最后编审: |
2020-05-29：吴雨秋(课程编辑志愿者)
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| 阅读次数: |
86
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