CIF: Small: Multidimensional Signal Processing With Box Splines

摘要本文提出了一种新的多元数据插值和逼近框架。该研究介绍了一种构造不可分离多元样条的方法，该方法在几何上适合于一般采样格，重点是球体填充和多维覆盖格。这些多元样条导致了多维信号处理工具的设计，这些工具已经证明了采样格的连续性程度和近似顺序。被研究的样条被称为Voronoi样条，是类似b样条的元素，它从Voronoi细胞继承任何采样晶格的几何形状，并形成重建的基础。此外，所提出的样条与众所周知的多元箱样条的关系进行了研究，并用于评估目的。与其他多面体样条相比，Voronoi样条形成平移不变空间进行逼近。这种特性使它们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.