Automatic Generation of Interpolants for Lattice Samplings: Part I -- Theory and Analysis

TL;DR

This paper introduces a general framework for efficiently generating interpolants for lattice-sampled data using algebraic analysis, enabling fast evaluation on CPU and GPU for various spline types and lattice structures.

Contribution

It presents a novel algebraic framework for automatic code generation of interpolants on different lattice structures, improving evaluation speed and broadening applicability.

Findings

Efficient implementations for box splines on BCC and FCC lattices.

Fast algorithms for Voronoi splines not previously documented.

Framework extends to non-Cartesian lattices in 4D.

Abstract

Interpolation is a fundamental technique in scientific computing and is at the heart of many scientific visualization techniques. There is usually a trade-off between the approximation capabilities of an interpolation scheme and its evaluation efficiency. For many applications, it is important for a user to be able to navigate their data in real time. In practice, the evaluation efficiency (or speed) outweighs any incremental improvements in reconstruction fidelity. In this two-part work, 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. In the sequel, we detail how we generate efficient implementations via automatic code generation on both CPU and GPU architectures. Specifically, in this paper, 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.