Lattice-Based High-Dimensional Gaussian Filtering and the Permutohedral Lattice

High-dimensional Gaussian filtering is a popular technique in image processing, geometry processing and computer graphics for smoothing data while preserving important features. For instance, the bilateral filter, cross bilateral filter and non-local means filter fall under the broad umbrella of high-dimensional Gaussian filters. Recent algorithmic advances therein have demonstrated that by relying on a sampled representation of the underlying space, one can obtain speed-ups of orders of magnitude over the naive approach. The simplest such sampled representation is a lattice, and it has been used successfully in the bilateral grid and the permutohedral lattice algorithms. In this paper, we analyze these lattice-based algorithms, developing a general theory of lattice-based high-dimensional Gaussian filtering. We consider the set of criteria for an optimal lattice for filtering, as it offers a good tradeoff of quality for computational efficiency, and evaluate the existing lattices under the criteria. In particular, we give a rigorous exposition of the properties of the permutohedral lattice and argue that it is the optimal lattice for Gaussian filtering. Lastly, we explore further uses of the permutohedral-lattice-based Gaussian filtering framework, showing that it can be easily adapted to perform mean shift filtering and yield improvement over the traditional approach based on a Cartesian grid.