# Rigidity Preserving Image Transformations and Equivariance in Perspective

arXiv: https://arxiv.org/abs/2201.13065 · HTML: https://mlsanity.com/paper/2201.13065/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 62 references that are in the map.
- First posted on arXiv 31 Jan 2022.

## Builds on (most cited first, 30 of 36)

- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [RandAugment: Practical automated data augmentation with a reduced search space](https://mlsanity.com/paper/1909.13719/) (2019)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [EfficientDet: Scalable and Efficient Object Detection](https://mlsanity.com/paper/1911.09070/) (2019)
- [AutoAugment: Learning Augmentation Policies from Data](https://mlsanity.com/paper/1805.09501/) (2018)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Making Convolutional Networks Shift-Invariant Again](https://mlsanity.com/paper/1904.11486/) (2019)
- [PVNet: Pixel-wise Voting Network for 6DoF Pose Estimation](https://mlsanity.com/paper/1812.11788/) (2018)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [DSAC - Differentiable RANSAC for Camera Localization](https://mlsanity.com/paper/1611.05705/) (2016)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [Back to the Feature: Learning Robust Camera Localization from Pixels to Pose](https://mlsanity.com/paper/2103.09213/) (2021)
- [Why do deep convolutional networks generalize so poorly to small image transformations?](https://mlsanity.com/paper/1805.12177/) (2018)
- [DPOD: 6D Pose Object Detector and Refiner](https://mlsanity.com/paper/1902.11020/) (2019)
- [Visual Camera Re-Localization from RGB and RGB-D Images Using DSAC](https://mlsanity.com/paper/2002.12324/) (2020)
- [How Much Position Information Do Convolutional Neural Networks Encode?](https://mlsanity.com/paper/2001.08248/) (2020)
- [On Translation Invariance in CNNs: Convolutional Layers can Exploit Absolute Spatial Location](https://mlsanity.com/paper/2003.07064/) (2020)
- [Universal approximations of invariant maps by neural networks](https://mlsanity.com/paper/1804.10306/) (2018)
- [Adversarial AutoAugment](https://mlsanity.com/paper/1912.11188/) (2019)
- [HybridPose: 6D Object Pose Estimation under Hybrid Representations](https://mlsanity.com/paper/2001.01869/) (2020)
- [Hierarchical Scene Coordinate Classification and Regression for Visual Localization](https://mlsanity.com/paper/1909.06216/) (2019)
- [Single-Stage 6D Object Pose Estimation](https://mlsanity.com/paper/1911.08324/) (2019)
- [Spherical CNNs on Unstructured Grids](https://mlsanity.com/paper/1901.02039/) (2019)
- [Tangent Images for Mitigating Spherical Distortion](https://mlsanity.com/paper/1912.09390/) (2019)
- [DeepSphere: a graph-based spherical CNN](https://mlsanity.com/paper/2012.15000/) (2020)
- [End-to-End Learnable Geometric Vision by Backpropagating PnP Optimization](https://mlsanity.com/paper/1909.06043/) (2019)
- [Ground-aware Monocular 3D Object Detection for Autonomous Driving](https://mlsanity.com/paper/2102.00690/) (2021)
- [Kernel Transformer Networks for Compact Spherical Convolution](https://mlsanity.com/paper/1812.03115/) (2018)

## Built on by (most cited first, 3 of 3)

- [Investigating how ReLU-networks encode symmetries](https://mlsanity.com/paper/2305.17017/) (2023)
- [Loss it right: Euclidean and Riemannian Metrics in Learning-based Visual Odometry](https://mlsanity.com/paper/2401.05396/) (2023)
- [Monocular visual simultaneous localization and mapping: (r)evolution from geometry to deep learning-based pipelines](https://mlsanity.com/paper/2503.02955/) (2025)

## Abstract (arXiv)

We characterize the class of image plane transformations which realize rigid camera motions and call these transformations `rigidity preserving'. In particular, 2D translations of pinhole images are not rigidity preserving. Hence, when using CNNs for 3D inference tasks, it can be beneficial to modify the inductive bias from equivariance towards translations to equivariance towards rigidity preserving transformations. We investigate how equivariance with respect to rigidity preserving transformations can be approximated in CNNs, and test our ideas on both 6D object pose estimation and visual localization. Experimentally, we improve on several competitive baselines.