# π3: Permutation-Equivariant Visual Geometry Learning

arXiv: https://arxiv.org/abs/2507.13347 · HTML: https://mlsanity.com/paper/2507.13347/

## Key facts

- Cited by 319 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 51 references that are in the map.
- First posted on arXiv 17 Jul 2025.

## Builds on (most cited first, 30 of 38)

- [DINOv2: Learning Robust Visual Features without Supervision](https://mlsanity.com/paper/2304.07193/) (2023)
- [ScanNet: Richly-annotated 3D Reconstructions of Indoor Scenes](https://mlsanity.com/paper/1702.04405/) (2017)
- [DUSt3R: Geometric 3D Vision Made Easy](https://mlsanity.com/paper/2312.14132/) (2023)
- [VGGT: Visual Geometry Grounded Transformer](https://mlsanity.com/paper/2503.11651/) (2025)
- [Depth Anything V2](https://mlsanity.com/paper/2406.09414/) (2024)
- [Habitat: A Platform for Embodied AI Research](https://mlsanity.com/paper/1904.01201/) (2019)
- [ORB-SLAM: a Versatile and Accurate Monocular SLAM System](https://mlsanity.com/paper/1502.00956/) (2015)
- [Grounding Image Matching in 3D with MASt3R](https://mlsanity.com/paper/2406.09756/) (2024)
- [ScanNet++: A High-Fidelity Dataset of 3D Indoor Scenes](https://mlsanity.com/paper/2308.11417/) (2023)
- [MegaDepth: Learning Single-View Depth Prediction from Internet Photos](https://mlsanity.com/paper/1804.00607/) (2018)
- [Common Objects in 3D: Large-Scale Learning and Evaluation of Real-life 3D Category Reconstruction](https://mlsanity.com/paper/2109.00512/) (2021)
- [Taskonomy: Disentangling Task Transfer Learning](https://mlsanity.com/paper/1804.08328/) (2018)
- [Hypersim: A Photorealistic Synthetic Dataset for Holistic Indoor Scene Understanding](https://mlsanity.com/paper/2011.02523/) (2020)
- [Continuous 3D Perception Model with Persistent State](https://mlsanity.com/paper/2501.12387/) (2025)
- [TartanAir: A Dataset to Push the Limits of Visual SLAM](https://mlsanity.com/paper/2003.14338/) (2020)
- [BlendedMVS: A Large-scale Dataset for Generalized Multi-view Stereo Networks](https://mlsanity.com/paper/1911.10127/) (2019)
- [ARKitScenes: A Diverse Real-World Dataset For 3D Indoor Scene Understanding Using Mobile RGB-D Data](https://mlsanity.com/paper/2111.08897/) (2021)
- [MonST3R: A Simple Approach for Estimating Geometry in the Presence of Motion](https://mlsanity.com/paper/2410.03825/) (2024)
- [Fast3R: Towards 3D Reconstruction of 1000+ Images in One Forward Pass](https://mlsanity.com/paper/2501.13928/) (2025)
- [Neural RGB-D Surface Reconstruction](https://mlsanity.com/paper/2104.04532/) (2021)
- [MoGe: Unlocking Accurate Monocular Geometry Estimation for Open-Domain Images with Optimal Training Supervision](https://mlsanity.com/paper/2410.19115/) (2024)
- [3D Reconstruction with Spatial Memory](https://mlsanity.com/paper/2408.16061/) (2024)
- [MoGe-2: Accurate Monocular Geometry with Metric Scale and Sharp Details](https://mlsanity.com/paper/2507.02546/) (2025)
- [Global Structure-from-Motion Revisited](https://mlsanity.com/paper/2407.20219/) (2024)
- [Project Aria: A New Tool for Egocentric Multi-Modal AI Research](https://mlsanity.com/paper/2308.13561/) (2023)
- [MatrixCity: A Large-scale City Dataset for City-scale Neural Rendering and Beyond](https://mlsanity.com/paper/2309.16553/) (2023)
- [ReFusion: 3D Reconstruction in Dynamic Environments for RGB-D Cameras Exploiting Residuals](https://mlsanity.com/paper/1905.02082/) (2019)
- [FLARE: Feed-forward Geometry, Appearance and Camera Estimation from Uncalibrated Sparse Views](https://mlsanity.com/paper/2502.12138/) (2025)
- [PoseDiffusion: Solving Pose Estimation via Diffusion-aided Bundle Adjustment](https://mlsanity.com/paper/2306.15667/) (2023)
- [MV-DUSt3R+: Single-Stage Scene Reconstruction from Sparse Views In 2 Seconds](https://mlsanity.com/paper/2412.06974/) (2024)

## Built on by (most cited first, 30 of 319)

- [ViPE: Video Pose Engine for 3D Geometric Perception](https://mlsanity.com/paper/2508.10934/) (2025)
- [FastVGGT: Training-Free Acceleration of Visual Geometry Transformer](https://mlsanity.com/paper/2509.02560/) (2025)
- [SANA-WM: Efficient Minute-Scale World Modeling with Hybrid Linear Diffusion Transformer](https://mlsanity.com/paper/2605.15178/) (2026)
- [Efficiently Reconstructing Dynamic Scenes One D4RT at a Time](https://mlsanity.com/paper/2512.08924/) (2025)
- [G2VLM: Geometry Grounded Vision Language Model with Unified 3D Reconstruction and Spatial Reasoning](https://mlsanity.com/paper/2511.21688/) (2025)
- [LoGeR: Long-Context Geometric Reconstruction with Hybrid Memory](https://mlsanity.com/paper/2603.03269/) (2026)
- [LingBot-Map: Geometric Context Transformer for Streaming 3D Reconstruction](https://mlsanity.com/paper/2604.14141/) (2026)
- [WinT3R: Window-Based Streaming Reconstruction with Camera Token Pool](https://mlsanity.com/paper/2509.05296/) (2025)
- [V-DPM: 4D Video Reconstruction with Dynamic Point Maps](https://mlsanity.com/paper/2601.09499/) (2026)
- [SpaceMind: Camera-Guided Modality Fusion for Spatial Reasoning in Vision-Language Models](https://mlsanity.com/paper/2511.23075/) (2025)
- [WorldMark: A Unified Benchmark Suite for Interactive Video World Models](https://mlsanity.com/paper/2604.21686/) (2026)
- [VGGT-Omega](https://mlsanity.com/paper/2605.15195/) (2026)
- [Generative Video Motion Editing with 3D Point Tracks](https://mlsanity.com/paper/2512.02015/) (2025)
- [VideoGPA: Distilling Geometry Priors for 3D-Consistent Video Generation](https://mlsanity.com/paper/2601.23286/) (2026)
- [LongStream: Long-Sequence Streaming Autoregressive Visual Geometry](https://mlsanity.com/paper/2602.13172/) (2026)
- [Emergent Outlier View Rejection in Visual Geometry Grounded Transformers](https://mlsanity.com/paper/2512.04012/) (2025)
- [4RC: 4D Reconstruction via Conditional Querying Anytime and Anywhere](https://mlsanity.com/paper/2602.10094/) (2026)
- [DVGT: Driving Visual Geometry Transformer](https://mlsanity.com/paper/2512.16919/) (2025)
- [INSPATIO-WORLD: A Real-Time 4D World Simulator via Spatiotemporal Autoregressive Modeling](https://mlsanity.com/paper/2604.07209/) (2026)
- [RealX3D: A Physically-Degraded 3D Benchmark for Multi-view Visual Restoration and Reconstruction](https://mlsanity.com/paper/2512.23437/) (2025)
- [Vision-aligned Latent Reasoning for Multi-modal Large Language Model](https://mlsanity.com/paper/2602.04476/) (2026)
- [Thinking with Geometry: Active Geometry Integration for Spatial Reasoning](https://mlsanity.com/paper/2602.06037/) (2026)
- [VGG-T3: Offline Feed-Forward 3D Reconstruction at Scale](https://mlsanity.com/paper/2602.23361/) (2026)
- [SpatialClaw: Rethinking Action Interface for Agentic Spatial Reasoning](https://mlsanity.com/paper/2606.13673/) (2026)
- [Repurposing Geometric Foundation Models for Multi-view Diffusion](https://mlsanity.com/paper/2603.22275/) (2026)
- [WAM4D: Fast 4D World Action Model via Spatial Register Tokens](https://mlsanity.com/paper/2606.14048/) (2026)
- [Reconstructing 4D Spatial Intelligence: A Survey](https://mlsanity.com/paper/2507.21045/) (2025)
- [CoWTracker: Tracking by Warping instead of Correlation](https://mlsanity.com/paper/2602.04877/) (2026)
- [Track4World: Feedforward World-centric Dense 3D Tracking of All Pixels](https://mlsanity.com/paper/2603.02573/) (2026)
- [VGGT-World: Transforming VGGT into an Autoregressive Geometry World Model](https://mlsanity.com/paper/2603.12655/) (2026)

## Abstract (arXiv)

We introduce π3, a feed-forward neural network that offers a novel approach to visual geometry reconstruction, breaking the reliance on a conventional fixed reference view. Previous methods often anchor their reconstructions to a designated viewpoint, an inductive bias that can lead to instability and failures if the reference is suboptimal. In contrast, π3 employs a fully permutation-equivariant architecture to predict affine-invariant camera poses and scale-invariant local point maps without any reference frames. This design not only makes our model inherently robust to input ordering, but also leads to higher accuracy and performance. These advantages enable our simple and bias-free approach to achieve state-of-the-art performance on a wide range of tasks, including camera pose estimation, monocular/video depth estimation, and dense point map reconstruction. Code and models are available at https://github.com/yyfz/Pi3.