Vectorial Slepian Functions on the Ball
Vectorial Slepian Functions on the Ball
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DOI：
10.1080/01630563.2018.1465953
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发表时间：
2017-07
影响因子：
1.2
通讯作者：
S. Leweke;V. Michel;N. Schneider
中科院分区：
文献类型：
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作者：
S. Leweke;V. Michel;N. Schneider
由于不确定性原理，一个函数不能同时在空间和频率上受到限制。一般来说，Slepian函数的思想是找到至少在空间光谱上是最优定域的函数。在这里，我们正在寻找适合于表示三维球上的实值向量场的Slepian函数。我们在球上使用各种向量基，它们都由雅可比多项式和矢量球谐波组成。这种基函数出现在地球物理和医学成像中一些层析反演问题的奇异值分解中。我们的目标是找到在顶点位于原点的锥的一部分中定位良好的带限制向量场。按照Slepian函数的原始方法，优化问题可以转化为有限维的代数特征值问题。对相应矩阵的元素尽可能进行解析处理。对于其余的积分，必须应用数值积分公式。特征值问题解耦成一个法向和一个切向问题。良好定域向量场的数量可用香农数估计，香农数主要取决于基函数的最大径向度和角度以及定域区域的大小。我们给出了球上的向量Slepian函数的数值例子，证明了这些函数的良好局部化和香农数的准确估计。
Abstract Due to the uncertainty principle, a function cannot be simultaneously limited in space as well as in frequency. The idea of Slepian functions, in general, is to find functions that are at least optimally spatio-spectrally localized. Here, we are looking for Slepian functions which are suitable for the representation of real-valued vector fields on a three-dimensional ball. We work with diverse vectorial bases on the ball which all consist of Jacobi polynomials and vector spherical harmonics. Such basis functions occur in the singular value decomposition of some tomographic inverse problems in geophysics and medical imaging. Our aim is to find band-limited vector fields that are well-localized in a part of a cone whose apex is situated in the origin. Following the original approach towards Slepian functions, the optimization problem can be transformed into a finite-dimensional algebraic eigenvalue problem. The entries of the corresponding matrix are treated analytically as far as possible. For the remaining integrals, numerical quadrature formulae have to be applied. The eigenvalue problem decouples into a normal and a tangential problem. The number of well-localized vector fields can be estimated by a Shannon number which mainly depends on the maximal radial and angular degree of the basis functions as well as the size of the localization region. We show numerical examples of vectorial Slepian functions on the ball, which demonstrate the good localization of these functions and the accurate estimate of the Shannon number.