# Varifold Moment Invariants for Sustainable and Explainable Contour Feature Extraction

arXiv: https://arxiv.org/abs/2606.07333 · HTML: https://mlsanity.com/paper/2606.07333/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 29 of its 108 references that are in the map.
- First posted on arXiv 5 Jun 2026.

## Builds on (most cited first, 29 of 29)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Roto-Translation Covariant Convolutional Networks for Medical Image Analysis](https://mlsanity.com/paper/1804.03393/) (2018)
- [Rotation-invariant convolutional neural networks for galaxy morphology prediction](https://mlsanity.com/paper/1503.07077/) (2015)
- [geomstats: a Python Package for Riemannian Geometry in Machine Learning](https://mlsanity.com/paper/1805.08308/) (2018)
- [PDE-based Group Equivariant Convolutional Neural Networks](https://mlsanity.com/paper/2001.09046/) (2020)
- [The varifold representation of non-oriented shapes for diffeomorphic registration](https://mlsanity.com/paper/1304.6108/) (2013)
- [A Novel Geometric Framework on Gram Matrix Trajectories for Human Behavior Understanding](https://mlsanity.com/paper/1807.00676/) (2018)
- [A Survey of Orthogonal Moments for Image Representation: Theory, Implementation, and Evaluation](https://mlsanity.com/paper/2103.14799/) (2021)
- [3D Shape Sequence of Human Comparison and Classification using Current and Varifolds](https://mlsanity.com/paper/2207.12485/) (2022)
- [A Novel Space-Time Representation on the Positive Semidefinite Con for Facial Expression Recognition](https://mlsanity.com/paper/1707.06440/) (2017)
- [The Selective G-Bispectrum and its Inversion: Applications to G-Invariant Networks](https://mlsanity.com/paper/2407.07655/) (2024)
- [A General Framework for Robust G-Invariance in G-Equivariant Networks](https://mlsanity.com/paper/2310.18564/) (2023)
- [Basis restricted elastic shape analysis on the space of unregistered surfaces](https://mlsanity.com/paper/2311.04382/) (2023)
- [ShapeEmbed: a self-supervised learning framework for 2D contour quantification](https://mlsanity.com/paper/2507.01009/) (2025)
- [Partial Matching in the Space of Varifolds](https://mlsanity.com/paper/2103.12441/) (2021)
- [The fshape framework for the variability analysis of functional shapes](https://mlsanity.com/paper/1404.6039/) (2014)
- [On length measures of planar closed curves and the comparison of convex shapes](https://mlsanity.com/paper/2010.13964/) (2020)
- [Leaf Recognition Using Convolutional Neural Networks Based Features](https://mlsanity.com/paper/2108.01808/) (2021)
- [Projection-based Classification of Surfaces for 3D Human Mesh Sequence Retrieval](https://mlsanity.com/paper/2111.13985/) (2021)
- [Automated Classification of Cell Shapes: A Comparative Evaluation of Shape Descriptors](https://mlsanity.com/paper/2411.00561/) (2024)
- [Geometric Learning of Canonical Parameterizations of 2D-curves](https://mlsanity.com/paper/2509.26070/) (2025)
- [Training-free Spatially Grounded Geometric Shape Encoding (Technical Report)](https://mlsanity.com/paper/2604.07522/) (2026)

## Abstract (arXiv)

We introduce Varifold Moments Invariants (VMI) as a unifying framework for many previously introduced Moment Invariants. These invariants are deeply related to other contour features that are invariant under translations and rotations, like Extended Gaussian Image, Elliptic Fourier Descriptors or Shape Distributions. The advantage of the varifold approach to moments consists in being able to combine the geometry of the region, its boundary, and the family of lines tangent to it, in order to create a substantial number of invariant features with high discriminating power and clear geometric meaning. By coupling our VMI feature extraction with the light feature classifiers Random Forest or Multi-Layer-Perceptron, we outperform state-of-the-art approaches based on contours, while decreasing drastically the computational cost to the point of allowing our algorithm to run on light devices. We tested our approach on classification tasks on a large number of widely-used datasets of various types (leaves, objects, cells) and achieved high accuracy with a low number of geometrically interpretable features.