Some Useful Properties of the Permutohedral Lattice for Gaussian Filtering

The rest of the paper is organized as follows: in Section 1, we give preliminary definitions of relevant terms and notation; in Section 2, we formally define the permutohedral lattice in several ways and prove useful structural properties stemming from the definition; in Section 3, we suggest criteria for picking the ideal lattice with which to perform Gaussian filtering, and present an argument that the permutohedral lattice is the most appropriate choice within a large family of lattices; in Section 4, we discuss how to perform each step of the Gaussian filter using the permutohedral lattice. Finally, in Section 5, we analyze the overall algorithm and compare it to the implementation on Z in [PD09].