# The varifold representation of non-oriented shapes for diffeomorphic registration

arXiv: https://arxiv.org/abs/1304.6108 · HTML: https://mlsanity.com/paper/1304.6108/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Apr 2013.

## Built on by (most cited first, 24 of 24)

- [Accurate Point Cloud Registration with Robust Optimal Transport](https://mlsanity.com/paper/2111.00648/) (2021)
- [Elastic shape analysis of surfaces with second-order Sobolev metrics: a comprehensive numerical framework](https://mlsanity.com/paper/2204.04238/) (2022)
- [ResNet-LDDMM: Advancing the LDDMM Framework using Deep Residual Networks](https://mlsanity.com/paper/2102.07951/) (2021)
- [3D Shape Sequence of Human Comparison and Classification using Current and Varifolds](https://mlsanity.com/paper/2207.12485/) (2022)
- [BaRe-ESA: A Riemannian Framework for Unregistered Human Body Shapes](https://mlsanity.com/paper/2211.13185/) (2022)
- [RDA-INR: Riemannian Diffeomorphic Autoencoding via Implicit Neural Representations](https://mlsanity.com/paper/2305.12854/) (2023)
- [Diffeomorphic Mesh Deformation via Efficient Optimal Transport for Cortical Surface Reconstruction](https://mlsanity.com/paper/2305.17555/) (2023)
- [Toward Mesh-Invariant 3D Generative Deep Learning with Geometric Measures](https://mlsanity.com/paper/2306.15762/) (2023)
- [OT-Talk: Animating 3D Talking Head with Optimal Transportation](https://mlsanity.com/paper/2505.01932/) (2025)
- [Basis restricted elastic shape analysis on the space of unregistered surfaces](https://mlsanity.com/paper/2311.04382/) (2023)
- [Beyond Fixed Topologies: Unregistered Training and Comprehensive Evaluation Metrics for 3D Talking Heads](https://mlsanity.com/paper/2410.11041/) (2024)
- [VariGrad: A Novel Feature Vector Architecture for Geometric Deep Learning on Unregistered Data](https://mlsanity.com/paper/2307.03553/) (2023)
- [Partial Matching in the Space of Varifolds](https://mlsanity.com/paper/2103.12441/) (2021)
- [Self Supervised Networks for Learning Latent Space Representations of Human Body Scans and Motions](https://mlsanity.com/paper/2411.03475/) (2024)
- [A Bi-variant Variational Model for Diffeomorphic Image Registration with Relaxed Jacobian Determinant Constraints](https://mlsanity.com/paper/2308.02393/) (2023)
- [Neural varifolds: an aggregate representation for quantifying the geometry of point clouds](https://mlsanity.com/paper/2407.04844/) (2024)
- [Explainable Binary Classification of Separable Shape Ensembles](https://mlsanity.com/paper/2410.12994/) (2024)
- [ARC-Flow : Articulated, Resolution-Agnostic, Correspondence-Free Matching and Interpolation of 3D Shapes Under Flow Fields](https://mlsanity.com/paper/2503.02606/) (2025)
- [SVarM: Linear Support Varifold Machines for Classification and Regression on Geometric Data](https://mlsanity.com/paper/2506.01189/) (2025)
- [How to Register a Live onto a Liver ? Partial Matching in the Space of Varifolds](https://mlsanity.com/paper/2204.05665/) (2022)
- [AdamFlow: Adam-based Wasserstein Gradient Flows for Surface Registration in Medical Imaging](https://mlsanity.com/paper/2604.02290/) (2026)
- [Varifold Moment Invariants for Sustainable and Explainable Contour Feature Extraction](https://mlsanity.com/paper/2606.07333/) (2026)
- [Flow-based conditional cardiac anatomy generation for virtual cohorts](https://mlsanity.com/paper/2608.09460/) (2026)
- [A shape-similarity latent space for fluid interfaces: invertible reduced-order modelling of droplet morphology](https://mlsanity.com/paper/2609.37947/) (2026)

## Abstract (arXiv)

In this paper, we address the problem of orientation that naturally arises when representing shapes like curves or surfaces as currents. In the field of computational anatomy, the framework of currents has indeed proved very efficient to model a wide variety of shapes. However, in such approaches, orientation of shapes is a fundamental issue that can lead to several drawbacks in treating certain kind of datasets. More specifically, problems occur with structures like acute pikes because of canceling effects of currents or with data that consists in many disconnected pieces like fiber bundles for which currents require a consistent orientation of all pieces. As a promising alternative to currents, varifolds, introduced in the context of geometric measure theory by F. Almgren, allow the representation of any non-oriented manifold (more generally any non-oriented rectifiable set). In particular, we explain how varifolds can encode numerically non-oriented objects both from the discrete and continuous point of view. We show various ways to build a Hilbert space structure on the set of varifolds based on the theory of reproducing kernels. We show that, unlike the currents' setting, these metrics are consistent with shape volume (theorem 4.1) and we derive a formula for the variation of metric with respect to the shape (theorem 4.2). Finally, we propose a generalization to non-oriented shapes of registration algorithms in the context of Large Deformations Metric Mapping (LDDMM), which we detail with a few examples in the last part of the paper.