# Symmetric Volume Maps: Order-Invariant Volumetric Mesh Correspondence with Free Boundary

arXiv: https://arxiv.org/abs/2202.02568 · HTML: https://mlsanity.com/paper/2202.02568/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 115 references that are in the map.
- First posted on arXiv 5 Feb 2022.

## Builds on (most cited first, 11 of 11)

- [Thingi10K: A Dataset of 10,000 3D-Printing Models](https://mlsanity.com/paper/1605.04797/) (2016)
- [Fast Tetrahedral Meshing in the Wild](https://mlsanity.com/paper/1908.03581/) (2019)
- [Efficient Deformable Shape Correspondence via Kernel Matching](https://mlsanity.com/paper/1707.08991/) (2017)
- [Reversible Harmonic Maps between Discrete Surfaces](https://mlsanity.com/paper/1801.02453/) (2018)
- [Articulated Shape Matching Using Laplacian Eigenfunctions and Unsupervised Point Registration](https://mlsanity.com/paper/2012.07340/) (2020)
- [HexaLab.net: an online viewer for hexahedral meshes](https://mlsanity.com/paper/1806.06639/) (2018)
- [Volumetric Parameterization of the Placenta to a Flattened Template](https://mlsanity.com/paper/2111.07900/) (2021)
- [Foldover-free maps in 50 lines of code](https://mlsanity.com/paper/2102.03069/) (2021)
- [Interactive All-Hex Meshing via Cuboid Decomposition](https://mlsanity.com/paper/2109.06279/) (2021)
- [Algebraic Representations for Volumetric Frame Fields](https://mlsanity.com/paper/1908.05411/) (2019)
- [A Splitting Scheme for Flip-Free Distortion Energies](https://mlsanity.com/paper/2107.05200/) (2021)

## Built on by (most cited first, 2 of 2)

- [Variational Barycentric Coordinates](https://mlsanity.com/paper/2310.03861/) (2023)
- [Unified Brain Surface and Volume Registration](https://mlsanity.com/paper/2512.19928/) (2025)

## Abstract (arXiv)

Although shape correspondence is a central problem in geometry processing, most methods for this task apply only to two-dimensional surfaces. The neglected task of volumetric correspondence--a natural extension relevant to shapes extracted from simulation, medical imaging, and volume rendering--presents unique challenges that do not appear in the two-dimensional case. In this work, we propose a method for mapping between volumes represented as tetrahedral meshes. Our formulation minimizes a distortion energy designed to extract maps symmetrically, i.e., without dependence on the ordering of the source and target domains. We accompany our method with theoretical discussion describing the consequences of this symmetry assumption, leading us to select a symmetrized ARAP energy that favors isometric correspondences. Our final formulation optimizes for near-isometry while matching the boundary. We demonstrate our method on a diverse geometric dataset, producing low-distortion matchings that align closely to the boundary.