# Product Manifold Filter: Non-Rigid Shape Correspondence via Kernel Density Estimation in the Product Space

arXiv: https://arxiv.org/abs/1701.00669 · HTML: https://mlsanity.com/paper/1701.00669/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jan 2017.

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

- [NeuroMorph: Unsupervised Shape Interpolation and Correspondence in One Go](https://mlsanity.com/paper/2106.09431/) (2021)
- [Unsupervised Learning of Robust Spectral Shape Matching](https://mlsanity.com/paper/2304.14419/) (2023)
- [Unsupervised Deep Multi-Shape Matching](https://mlsanity.com/paper/2207.09610/) (2022)
- [Smooth Non-Rigid Shape Matching via Effective Dirichlet Energy Optimization](https://mlsanity.com/paper/2210.02870/) (2022)
- [Spectral Meets Spatial: Harmonising 3D Shape Matching and Interpolation](https://mlsanity.com/paper/2402.18920/) (2024)
- [Bending Graphs: Hierarchical Shape Matching using Gated Optimal Transport](https://mlsanity.com/paper/2202.01537/) (2022)
- [Self-Supervised Learning for Multimodal Non-Rigid 3D Shape Matching](https://mlsanity.com/paper/2303.10971/) (2023)
- [Partial-to-Partial Shape Matching with Geometric Consistency](https://mlsanity.com/paper/2404.12209/) (2024)
- [Memory-Scalable and Simplified Functional Map Learning](https://mlsanity.com/paper/2404.00330/) (2024)
- [SRFeat: Learning Locally Accurate and Globally Consistent Non-Rigid Shape Correspondence](https://mlsanity.com/paper/2209.07806/) (2022)
- [Geometrically Consistent Partial Shape Matching](https://mlsanity.com/paper/2309.05013/) (2023)
- [SRIF: Semantic Shape Registration Empowered by Diffusion-based Image Morphing and Flow Estimation](https://mlsanity.com/paper/2409.11682/) (2024)
- [S3M: Scalable Statistical Shape Modeling through Unsupervised Correspondences](https://mlsanity.com/paper/2304.07515/) (2023)
- [GenCorres: Consistent Shape Matching via Coupled Implicit-Explicit Shape Generative Models](https://mlsanity.com/paper/2304.10523/) (2023)
- [Self-supervised 3D Human Mesh Recovery from Noisy Point Clouds](https://mlsanity.com/paper/2107.07539/) (2021)
- [Revisiting Map Relations for Unsupervised Non-Rigid Shape Matching](https://mlsanity.com/paper/2310.11420/) (2023)
- [DiscoMatch: Fast Discrete Optimisation for Geometrically Consistent 3D Shape Matching](https://mlsanity.com/paper/2310.08230/) (2023)
- [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)
- [Synchronous Diffusion for Unsupervised Smooth Non-Rigid 3D Shape Matching](https://mlsanity.com/paper/2407.08244/) (2024)
- [Fast Globally Optimal and Geometrically Consistent 3D Shape Matching](https://mlsanity.com/paper/2504.06385/) (2025)
- [SeedGNN: Graph Neural Networks for Supervised Seeded Graph Matching](https://mlsanity.com/paper/2205.13679/) (2022)
- [Lifting the Convex Conjugate in Lagrangian Relaxations: A Tractable Approach for Continuous Markov Random Fields](https://mlsanity.com/paper/2107.06028/) (2021)
- [Searching Dense Point Correspondences via Permutation Matrix Learning](https://mlsanity.com/paper/2210.14897/) (2022)
- [ARC-Flow : Articulated, Resolution-Agnostic, Correspondence-Free Matching and Interpolation of 3D Shapes Under Flow Fields](https://mlsanity.com/paper/2503.02606/) (2025)
- [DcMatch: Unsupervised Multi-Shape Matching with Dual-Level Consistency](https://mlsanity.com/paper/2509.01204/) (2025)
- [Hyper-Network Neural Functional Maps for Unsupervised Robust 3D Shape Matching](https://mlsanity.com/paper/2606.30131/) (2026)
- [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)
- [Deep Frequency-Aware Functional Maps for Robust Shape Matching](https://mlsanity.com/paper/2402.03904/) (2024)
- [Matching Shapes Under Different Topologies: A Topology-Adaptive Deformation Guided Approach](https://mlsanity.com/paper/2509.06862/) (2025)

## Abstract (arXiv)

Many algorithms for the computation of correspondences between deformable shapes rely on some variant of nearest neighbor matching in a descriptor space. Such are, for example, various point-wise correspondence recovery algorithms used as a post-processing stage in the functional correspondence framework. Such frequently used techniques implicitly make restrictive assumptions (e.g., near-isometry) on the considered shapes and in practice suffer from lack of accuracy and result in poor surjectivity. We propose an alternative recovery technique capable of guaranteeing a bijective correspondence and producing significantly higher accuracy and smoothness. Unlike other methods our approach does not depend on the assumption that the analyzed shapes are isometric. We derive the proposed method from the statistical framework of kernel density estimation and demonstrate its performance on several challenging deformable 3D shape matching datasets.