Manifold curvature learning from hypersurface integral invariants
Manifold curvature learning from hypersurface integral invariants
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DOI：
10.1016/j.laa.2020.05.020
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发表时间：
2020-10
影响因子：
1.1
通讯作者：
Javier Álvarez-Vizoso;M. Kirby;C. Peterson
中科院分区：
文献类型：
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作者：
Javier Álvarez-Vizoso;M. Kirby;C. Peterson
从子流形的小核域上的主成分分析得到的积分不变量编码了重要的几何信息，这些信息经典地用微分几何项来定义。我们将空间中已知曲面的主要结果推广到任意维超曲面上，进而给出一种估计一般余维嵌入黎曼子流形的外曲率张量和内禀曲率张量的方法。具体地说，积分不变量由区域的体积、重心和协方差矩阵的EVD定义。对于由超曲面界定的球体分量和由球面相交产生的超曲面面片，我们得到了这些不变量的渐近展开式，表明这些特征值和特征向量可用作主曲率和主方向的多尺度估计器。这种方法可以解释为对子流形的底层点集进行统计分析，以获得尺度上的几何描述符，并可能应用于点云的流形学习和几何处理。
Integral invariants obtained from Principal Component Analysis on a small kernel domain of a submanifold encode important geometric information classically defined in differential-geometric terms. We generalize to hypersurfaces in any dimension major results known for surfaces in space, which in turn yield a method to estimate the extrinsic and intrinsic curvature tensor of an embedded Riemannian submanifold of general codimension. In particular, integral invariants are defined by the volume, barycenter, and the EVD of the covariance matrix of the domain. We obtain the asymptotic expansion of such invariants for a spherical volume component delimited by a hypersurface and for the hypersurface patch created by ball intersections, showing that the eigenvalues and eigenvectors can be used as multi-scale estimators of the principal curvatures and principal directions. This approach may be interpreted as performing statistical analysis on the underlying point-set of a submanifold in order to obtain geometric descriptors at scale with potential applications to Manifold Learning and Geometry Processing of point clouds.