# Complex Functional Maps : a Conformal Link Between Tangent Bundles

arXiv: https://arxiv.org/abs/2112.09546 · HTML: https://mlsanity.com/paper/2112.09546/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 64 references that are in the map.
- First posted on arXiv 17 Dec 2021.

## Builds on (most cited first, 16 of 16)

- [3D Menagerie: Modeling the 3D shape and pose of animals](https://mlsanity.com/paper/1611.07700/) (2016)
- [Deep Functional Maps: Structured Prediction for Dense Shape Correspondence](https://mlsanity.com/paper/1704.08686/) (2017)
- [Deep Geometric Functional Maps: Robust Feature Learning for Shape Correspondence](https://mlsanity.com/paper/2003.14286/) (2020)
- [Continuous and Orientation-preserving Correspondences via Functional Maps](https://mlsanity.com/paper/1806.04455/) (2018)
- [Smooth Shells: Multi-Scale Shape Registration with Functional Maps](https://mlsanity.com/paper/1905.12512/) (2019)
- [Partial Functional Correspondence](https://mlsanity.com/paper/1506.05274/) (2015)
- [Multi-directional Geodesic Neural Networks via Equivariant Convolution](https://mlsanity.com/paper/1810.02303/) (2018)
- [The Vector Heat Method](https://mlsanity.com/paper/1805.09170/) (2018)
- [ZoomOut: Spectral Upsampling for Efficient Shape Correspondence](https://mlsanity.com/paper/1904.07865/) (2019)
- [Cyclic Functional Mapping: Self-supervised correspondence between non-isometric deformable shapes](https://mlsanity.com/paper/1912.01249/) (2019)
- [FARM: Functional Automatic Registration Method for 3D Human Bodies](https://mlsanity.com/paper/1807.10517/) (2018)
- [Boundary First Flattening](https://mlsanity.com/paper/1704.06873/) (2017)
- [Spectral Generalized Multi-Dimensional Scaling](https://mlsanity.com/paper/1311.2187/) (2013)
- [MapTree: Recovering Multiple Solutions in the Space of Maps](https://mlsanity.com/paper/2006.02532/) (2020)
- [Robust Structure-based Shape Correspondence](https://mlsanity.com/paper/1710.05592/) (2017)
- [Functional Characterization of Deformation Fields](https://mlsanity.com/paper/1709.09701/) (2017)

## Built on by (most cited first, 17 of 17)

- [Unsupervised Learning of Robust Spectral Shape Matching](https://mlsanity.com/paper/2304.14419/) (2023)
- [Deep Orientation-Aware Functional Maps: Tackling Symmetry Issues in Shape Matching](https://mlsanity.com/paper/2204.13453/) (2022)
- [Spectral Meets Spatial: Harmonising 3D Shape Matching and Interpolation](https://mlsanity.com/paper/2402.18920/) (2024)
- [Hybrid Functional Maps for Crease-Aware Non-Isometric Shape Matching](https://mlsanity.com/paper/2312.03678/) (2023)
- [DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domains](https://mlsanity.com/paper/2402.07250/) (2024)
- [Memory-Scalable and Simplified Functional Map Learning](https://mlsanity.com/paper/2404.00330/) (2024)
- [Single Mesh Diffusion Models with Field Latents for Texture Generation](https://mlsanity.com/paper/2312.09250/) (2023)
- [DiffuMatch: Category-Agnostic Spectral Diffusion Priors for Robust Non-rigid Shape Matching](https://mlsanity.com/paper/2507.23715/) (2025)
- [G-MSM: Unsupervised Multi-Shape Matching with Graph-based Affinity Priors](https://mlsanity.com/paper/2212.02910/) (2022)
- [Revisiting Map Relations for Unsupervised Non-Rigid Shape Matching](https://mlsanity.com/paper/2310.11420/) (2023)
- [Non-Rigid 3D Shape Correspondences: From Foundations to Open Challenges and Opportunities](https://mlsanity.com/paper/2604.01274/) (2026)
- [RINO: Rotation-Invariant Non-Rigid Correspondences](https://mlsanity.com/paper/2603.27773/) (2026)
- [An Intrinsic Vector Heat Network](https://mlsanity.com/paper/2406.09648/) (2024)
- [SGSoft: Learning Fused Semantic-Geometric Features for 3D Shape Correspondence via Template-Guided Soft Signals](https://mlsanity.com/paper/2605.18039/) (2026)
- [Extracting a functional representation from a dictionary for non-rigid shape matching](https://mlsanity.com/paper/2305.10332/) (2023)
- [Deep Frequency-Aware Functional Maps for Robust Shape Matching](https://mlsanity.com/paper/2402.03904/) (2024)
- [MDND: Unsupervised Learning Guided by Non-Differentiable Refinement for Shape Correspondence](https://mlsanity.com/paper/2607.15887/) (2026)

## Abstract (arXiv)

In this paper, we introduce complex functional maps, which extend the functional map framework to conformal maps between tangent vector fields on surfaces. A key property of these maps is their orientation awareness. More specifically, we demonstrate that unlike regular functional maps that link functional spaces of two manifolds, our complex functional maps establish a link between oriented tangent bundles, thus permitting robust and efficient transfer of tangent vector fields. By first endowing and then exploiting the tangent bundle of each shape with a complex structure, the resulting operations become naturally orientationaware, thus favoring orientation and angle preserving correspondence across shapes, without relying on descriptors or extra regularization. Finally, and perhaps more importantly, we demonstrate how these objects enable several practical applications within the functional map framework. We show that functional maps and their complex counterparts can be estimated jointly to promote orientation preservation, regularizing pipelines that previously suffered from orientation-reversing symmetry errors.