# Articulated Shape Matching Using Laplacian Eigenfunctions and Unsupervised Point Registration

arXiv: https://arxiv.org/abs/2012.07340 · HTML: https://mlsanity.com/paper/2012.07340/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Dec 2020.

## Built on by (most cited first, 6 of 6)

- [Leveraging Intrinsic Properties for Non-Rigid Garment Alignment](https://mlsanity.com/paper/2308.09519/) (2023)
- [Coupled Laplacian Eigenmaps for Locally-Aware 3D Rigid Point Cloud Matching](https://mlsanity.com/paper/2402.17372/) (2024)
- [Symmetric Volume Maps: Order-Invariant Volumetric Mesh Correspondence with Free Boundary](https://mlsanity.com/paper/2202.02568/) (2022)
- [3D Shape Registration Using Spectral Graph Embedding and Probabilistic Matching](https://mlsanity.com/paper/2106.11166/) (2021)
- [Casting graph isomorphism as a point set registration problem using a simplex embedding and sampling](https://mlsanity.com/paper/2111.09696/) (2021)
- [EA-RAS: Towards Efficient and Accurate End-to-End Reconstruction of Anatomical Skeleton](https://mlsanity.com/paper/2409.01555/) (2024)

## Abstract (arXiv)

Matching articulated shapes represented by voxel-sets reduces to maximal sub-graph isomorphism when each set is described by a weighted graph. Spectral graph theory can be used to map these graphs onto lower dimensional spaces and match shapes by aligning their embeddings in virtue of their invariance to change of pose. Classical graph isomorphism schemes relying on the ordering of the eigenvalues to align the eigenspaces fail when handling large data-sets or noisy data. We derive a new formulation that finds the best alignment between two congruent K-dimensional sets of points by selecting the best subset of eigenfunctions of the Laplacian matrix. The selection is done by matching eigenfunction signatures built with histograms, and the retained set provides a smart initialization for the alignment problem with a considerable impact on the overall performance. Dense shape matching casted into graph matching reduces then, to point registration of embeddings under orthogonal transformations; the registration is solved using the framework of unsupervised clustering and the EM algorithm. Maximal subset matching of non identical shapes is handled by defining an appropriate outlier class. Experimental results on challenging examples show how the algorithm naturally treats changes of topology, shape variations and different sampling densities.