Fast sparse kernel summation on cartesian grids: an on-chip algorithm for 3D implicit surface visualization

Abstract

This paper proposes a fast algorithm for evaluating summations of heterogenous sparse kernels of the form [EQUATION] points on an arbitrary fine Cartesian grid in Rd. The algorithm takes the advantage of sparsity and the structure of Cartesian grids. The sparsity admits operations only be done in some active subsets of the Cartesian grids; the structure of Cartesian grids reduce the storage for N points from O(dN) to O(1), a constant, and thus transforms costly memory intensive operations to cheap computationally intensive operations. This results in scalable algorithm with a complexity of O(N) and makes the postprocessing of large 3D implicit surface feasible on a PC or laptop. Numerical examples for 3D surface reconstruction are presented to illustrate the efficiency of the algorithm.