# Representing three-dimensional cross fields using 4th order tensors

arXiv: https://arxiv.org/abs/1808.03999 · HTML: https://mlsanity.com/paper/1808.03999/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Aug 2018.

## Built on by (most cited first, 1 of 1)

- [Neural Octahedral Field: Octahedral prior for simultaneous smoothing and sharp edge regularization](https://mlsanity.com/paper/2408.00303/) (2024)

## Abstract (arXiv)

This paper presents a new way of describing cross fields based on fourth order tensors. We prove that the new formulation is forming a linear space in ℝ9. The algebraic structure of the tensors and their projections on mboxSO(3) are presented. The relationship of the new formulation with spherical harmonics is exposed. This paper is quite theoretical. Due to pages limitation, few practical aspects related to the computations of cross fields are exposed. Nevetheless, a global smoothing algorithm is briefly presented and computation of cross fields are finally depicted.