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

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## Abstract

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 naïve 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.

## Keywords

Bilateral filtering High-dimensional filtering Non-local means Lattices Gaussian filtering Permutohedral lattice## Notes

### Acknowledgements

We would like to thank Marc Levoy for his advice and support, as well as Nokia Research.

Jongmin Baek acknowledges support from Nokia Research, as well as Lucent Technologies Stanford Graduate Fellowship; Andrew Adams is supported by a Reed-Hodgson Stanford Graduate Fellowship; Jennifer Dolson acknowledges support from an NDSEG Graduate Fellowship from the United States Department of Defense.

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