Interpolation is a foundational concept in scientific computing and is at the heart of many scientific visualization techniques. There is usually a tradeoff between the approximation capabilities of an interpolation scheme and its evaluation efficiency. For many applications, it is important for a user to navigate their data in real time. In practice, evaluation efficiency outweighs any incremental improvements in reconstruction fidelity. We first analyze, from a general standpoint, the use of compact piece-wise polynomial basis functions to efficiently interpolate data that is sampled on a lattice. We then detail our automatic code-generation framework on both CPU and GPU architectures. Specifically, we propose a general framework that can produce a fast evaluation scheme by analyzing the algebro-geometric structure of the convolution sum for a given lattice and basis function combination. We demonstrate the utility and generality of our framework by providing fast implementations of various box splines on the Body Centered and Face Centered Cubic lattices, as well as some non-separable box splines on the Cartesian lattice. We also provide fast implementations for certain Voronoi-splines that have not yet appeared in the literature. Finally, we demonstrate that this framework may also be used for non-Cartesian lattices in 4D.

插值是科学计算的基本概念，也是许多科学可视化技术的核心。插值方案的逼近能力与其评估效率之间通常存在权衡。对于许多应用程序来说，用户实时导航数据非常重要。在实践中，评估效率超过了重建保真度的任何增量改进。我们首先从一般的角度分析使用紧凑的分段多项式基函数来有效地插值在格子上采样的数据。然后，我们详细介绍了 CPU 和 GPU 架构上的自动代码生成框架。具体来说，我们提出了一个通用框架，可以通过分析给定格和基函数组合的卷积和的代数几何结构来生成快速评估方案。我们通过在体心和面心立方晶格上提供各种箱形样条的快速实现以及笛卡尔晶格上的一些不可分离的箱形样条来展示我们框架的实用性和通用性。我们还为某些尚未出现在文献中的 Voronoi 样条提供快速实现。最后，我们证明该框架也可用于 4D 中的非笛卡尔晶格。以及笛卡尔晶格上的一些不可分离的箱样条曲线。我们还为某些尚未出现在文献中的 Voronoi 样条提供快速实现。最后，我们证明该框架也可用于 4D 中的非笛卡尔晶格。以及笛卡尔晶格上的一些不可分离的箱样条曲线。我们还为某些尚未出现在文献中的 Voronoi 样条提供快速实现。最后，我们证明该框架也可用于 4D 中的非笛卡尔晶格。