# Cyclic Functional Mapping: Self-supervised correspondence between non-isometric deformable shapes

arXiv: https://arxiv.org/abs/1912.01249 · HTML: https://mlsanity.com/paper/1912.01249/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Dec 2019.

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

- [Deep Orientation-Aware Functional Maps: Tackling Symmetry Issues in Shape Matching](https://mlsanity.com/paper/2204.13453/) (2022)
- [Learning Multi-resolution Functional Maps with Spectral Attention for Robust Shape Matching](https://mlsanity.com/paper/2210.06373/) (2022)
- [Spatially and Spectrally Consistent Deep Functional Maps](https://mlsanity.com/paper/2308.08871/) (2023)
- [Understanding and Improving Features Learned in Deep Functional Maps](https://mlsanity.com/paper/2303.16527/) (2023)
- [NCP: Neural Correspondence Prior for Effective Unsupervised Shape Matching](https://mlsanity.com/paper/2301.05839/) (2023)
- [Complex Functional Maps : a Conformal Link Between Tangent Bundles](https://mlsanity.com/paper/2112.09546/) (2021)
- [Point2SSM: Learning Morphological Variations of Anatomies from Point Cloud](https://mlsanity.com/paper/2305.14486/) (2023)
- [Integrating Efficient Optimal Transport and Functional Maps For Unsupervised Shape Correspondence Learning](https://mlsanity.com/paper/2403.01781/) (2024)
- [SRIF: Semantic Shape Registration Empowered by Diffusion-based Image Morphing and Flow Estimation](https://mlsanity.com/paper/2409.11682/) (2024)
- [On Unsupervised Partial Shape Correspondence](https://mlsanity.com/paper/2310.14692/) (2023)
- [MvDeCor: Multi-view Dense Correspondence Learning for Fine-grained 3D Segmentation](https://mlsanity.com/paper/2208.08580/) (2022)
- [Neural Intrinsic Embedding for Non-rigid Point Cloud Matching](https://mlsanity.com/paper/2303.01038/) (2023)
- [Self-supervised 3D Human Mesh Recovery from Noisy Point Clouds](https://mlsanity.com/paper/2107.07539/) (2021)
- [Deep Virtual Markers for Articulated 3D Shapes](https://mlsanity.com/paper/2108.09000/) (2021)
- [Shape Non-rigid Kinematics (SNK): A Zero-Shot Method for Non-Rigid Shape Matching via Unsupervised Functional Map Regularized Reconstruction](https://mlsanity.com/paper/2403.06804/) (2024)
- [Generalizable Local Feature Pre-training for Deformable Shape Analysis](https://mlsanity.com/paper/2303.15104/) (2023)
- [Surface-Aware Distilled 3D Semantic Features](https://mlsanity.com/paper/2503.18254/) (2025)
- [Learning Spectral Unions of Partial Deformable 3D Shapes](https://mlsanity.com/paper/2104.00514/) (2021)
- [Mesh Draping: Parametrization-Free Neural Mesh Transfer](https://mlsanity.com/paper/2110.05433/) (2021)
- [Unsupervised Dense Deformation Embedding Network for Template-Free Shape Correspondence](https://mlsanity.com/paper/2108.11609/) (2021)
- [Deformation-Guided Unsupervised Non-Rigid Shape Matching](https://mlsanity.com/paper/2311.15668/) (2023)
- [Deep Confidence Guided Distance for 3D Partial Shape Registration](https://mlsanity.com/paper/2201.11379/) (2022)
- [Learning Canonical Embeddings for Unsupervised Shape Correspondence with Locally Linear Transformations](https://mlsanity.com/paper/2209.02152/) (2022)
- [Self-supervised Shape Completion via Involution and Implicit Correspondences](https://mlsanity.com/paper/2409.15939/) (2024)
- [Learning Canonical Embedding for Non-rigid Shape Matching](https://mlsanity.com/paper/2110.02994/) (2021)
- [U-mesh: Human Correspondence Matching with Mesh Convolutional Networks](https://mlsanity.com/paper/2108.06695/) (2021)
- [DPC: Unsupervised Deep Point Correspondence via Cross and Self Construction](https://mlsanity.com/paper/2110.08636/) (2021)
- [Joint Symmetry Detection and Shape Matching for Non-Rigid Point Cloud](https://mlsanity.com/paper/2112.02713/) (2021)
- [Matching Shapes Under Different Topologies: A Topology-Adaptive Deformation Guided Approach](https://mlsanity.com/paper/2509.06862/) (2025)
- [NeuralSRNF: Neural Square Root Normal Fields for the Statistical Shape Analysis and Generation of Nonrigid 3D and 4D Objects](https://mlsanity.com/paper/2609.27728/) (2026)

## Abstract (arXiv)

We present the first utterly self-supervised network for dense correspondence mapping between non-isometric shapes. The task of alignment in non-Euclidean domains is one of the most fundamental and crucial problems in computer vision. As 3D scanners can generate highly complex and dense models, the mission of finding dense mappings between those models is vital. The novelty of our solution is based on a cyclic mapping between metric spaces, where the distance between a pair of points should remain invariant after the full cycle. As the same learnable rules that generate the point-wise descriptors apply in both directions, the network learns invariant structures without any labels while coping with non-isometric deformations. We show here state-of-the-art-results by a large margin for a variety of tasks compared to known self-supervised and supervised methods.