The LIR Space Partitioning System Applied to Cartesian Grids

Abstract

We introduce a novel multi-dimensional space partitioning method. A new type of tree combines the advantages of the Octree and the KD-tree without having their disadvantages. We present in this paper a new data structure allowing local refinement, parallelization and proper restriction of transition ratios between cells. Our technique has no dimensional restrictions at all. The tree’s data structure is defined by a topological algebra based on the symbols A = { L, I, R } that encode the partitioning steps. The set of successors is restricted such that each cell has the partition of unity property to partition domains without overlap. With our method it is possible to construct a wide choice of spline spaces to compress or reconstruct scientific data such as pressure and velocity fields and multidimensional images. We present a generator function to build a tree that represents a voxel geometry. The space partitioning system is used as a framework to allow numerical computations. This work is triggered by the problem of representing, in a numerically appropriate way, huge three-dimensional voxel geometries that could have up to billions of voxels.