CIF: Small: Multidimensional Signal Processing With Box Splines

摘要提出了一种新的多元数据内插和逼近框架，介绍了一种为一般采样格量身定制的不可分多元样条线的构造方法，重点研究了球面填充和多维覆盖格点阵。这些多元样条族导致了多维信号处理工具的设计，这些工具证明了采样格的连续程度和逼近阶。所研究的样条族称为Voronoi样条线，它继承了任何采样格的Voronoi胞元的几何形状，并形成了重建的基础。此外，还研究了所提出的样条集与著名的多元盒样条之间的关系，并将其用于评价。与其他多面体样条线相比，Voronoi样条线形成了移位不变的逼近空间。这一特性使其特别适合于多维信号处理。

AbstractThis research develops a novel framework for interpolation and approximation of multivariate data.The research introduces a method for construction of non-separable multivariate splines that are geometricallytailored for general sampling lattices with emphasis on sphere packing and covering latticesin multidimensions. These multivariate splines lead to the design of multidimensional signal processingtools, that have proven degree of continuity and approximation order for sampling lattices.The splines under investigation, called Voronoi splines, are B-spline-like elements that inherit thegeometry of any sampling lattice from its Voronoi cell and form basis for reconstruction. Furthermore,the relationship of the proposed splines with the well-known multivariate box splines are investigatedand used for evaluation purposes. Compared to other polyhedral splines, Voronoi splines form shiftinvariantspaces for approximation. This property makes them uniquely suitable for multidimensionalsignal processing.