A Riemannian framework for tensor computing
A Riemannian framework for tensor computing
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DOI：
10.1007/s11263-005-3222-z
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发表时间：
2006-01-01
影响因子：
19.5
通讯作者：
Ayache, N
中科院分区：
文献类型：
--
作者：
Pennec, X;Fillard, P;Ayache, N
张量现在是几何信息的常见来源。在本文中，我们建议赋予张量空间一个仿射不变的黎曼度量。我们证明，它导致强大的理论性质：锥正定对称矩阵被替换为一个经常和完整的流形没有边界（零特征值是在无穷远），两个张量之间的测地线和一组张量的平均值是唯一定义的，等等。我们以前已经表明，黎曼度量提供了一个强大的框架推广统计流形。在本文中，我们表明，它也可以推广到张量场许多重要的几何数据处理算法，如插值，滤波，扩散和恢复丢失的数据。例如，大多数插值和高斯滤波方案可以通过加权平均计算有效地处理。线性和各向异性扩散计划可以适应我们的黎曼框架，通过偏微分演化方程，只要考虑到张量空间的度量。为此，我们提供了内在的数值方案来计算梯度和Laplace-Beltrami算子。最后，为了加强对数据的保真度（稀疏分布的张量或完整的张量场），我们提出了基于不变黎曼距离的最小二乘准则，这些准则特别简单有效。
Tensors are nowadays a common source of geometric information. In this paper, we propose to endow the tensor space with an affine-invariant Riemannian metric. We demonstrate that it leads to strong theoretical properties: the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries (null eigenvalues are at the infinity), the geodesic between two tensors and the mean of a set of tensors are uniquely defined, etc.We have previously shown that the Riemannian metric provides a powerful framework for generalizing statistics to manifolds. In this paper, we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation, filtering, diffusion and restoration of missing data. For instance, most interpolation and Gaussian filtering schemes can be tackled efficiently through a weighted mean computation. Linear and anisotropic diffusion schemes can be adapted to our Riemannian framework, through partial differential evolution equations, provided that the metric of the tensor space is taken into account. For that purpose, we provide intrinsic numerical schemes to compute the gradient and Laplace-Beltrami operators. Finally, to enforce the fidelity to the data (either sparsely distributed tensors or complete tensors fields) we propose least-squares criteria based on our invariant Riemannian distance which are particularly simple and efficient to solve.