# Learning Spectral Unions of Partial Deformable 3D Shapes

arXiv: https://arxiv.org/abs/2104.00514 · HTML: https://mlsanity.com/paper/2104.00514/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 64 references that are in the map.
- First posted on arXiv 31 Mar 2021.

## Builds on (most cited first, 20 of 20)

- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [PartNet: A Large-scale Benchmark for Fine-grained and Hierarchical Part-level 3D Object Understanding](https://mlsanity.com/paper/1812.02713/) (2018)
- [Learning from Synthetic Humans](https://mlsanity.com/paper/1701.01370/) (2017)
- [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)
- [Unsupervised Deep Learning for Structured Shape Matching](https://mlsanity.com/paper/1812.03794/) (2018)
- [Partial Functional Correspondence](https://mlsanity.com/paper/1506.05274/) (2015)
- [Deformable Shape Completion with Graph Convolutional Autoencoders](https://mlsanity.com/paper/1712.00268/) (2017)
- [Correspondence Learning via Linearly-invariant Embedding](https://mlsanity.com/paper/2010.13136/) (2020)
- [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)
- [Isospectralization, or how to hear shape, style, and correspondence](https://mlsanity.com/paper/1811.11465/) (2018)
- [Non-Rigid Puzzles](https://mlsanity.com/paper/2011.13076/) (2020)
- [Correspondence-Free Region Localization for Partial Shape Similarity via Hamiltonian Spectrum Alignment](https://mlsanity.com/paper/1906.06226/) (2019)
- [Instant recovery of shape from spectrum via latent space connections](https://mlsanity.com/paper/2003.06523/) (2020)
- [The Whole Is Greater Than the Sum of Its Nonrigid Parts](https://mlsanity.com/paper/2001.09650/) (2020)
- [Unsupervised Dense Shape Correspondence using Heat Kernels](https://mlsanity.com/paper/2010.12682/) (2020)
- [High-Resolution Augmentation for Automatic Template-Based Matching of Human Models](https://mlsanity.com/paper/2009.09312/) (2020)
- [Learning Fuzzy Set Representations of Partial Shapes on Dual Embedding Spaces](https://mlsanity.com/paper/1807.01519/) (2018)

## Built on by (most cited first, 4 of 4)

- [Generating Graphs via Spectral Diffusion](https://mlsanity.com/paper/2402.18974/) (2024)
- [Latent Spectral Regularization for Continual Learning](https://mlsanity.com/paper/2301.03345/) (2023)
- [DifAttack++: Query-Efficient Black-Box Adversarial Attack via Hierarchical Disentangled Feature Space in Cross-Domain](https://mlsanity.com/paper/2406.03017/) (2024)
- [Partial Shape Similarity via Alignment of Multi-Metric Hamiltonian Spectra](https://mlsanity.com/paper/2207.03018/) (2022)

## Abstract (arXiv)

Spectral geometric methods have brought revolutionary changes to the field of geometry processing. Of particular interest is the study of the Laplacian spectrum as a compact, isometry and permutation-invariant representation of a shape. Some recent works show how the intrinsic geometry of a full shape can be recovered from its spectrum, but there are approaches that consider the more challenging problem of recovering the geometry from the spectral information of partial shapes. In this paper, we propose a possible way to fill this gap. We introduce a learning-based method to estimate the Laplacian spectrum of the union of partial non-rigid 3D shapes, without actually computing the 3D geometry of the union or any correspondence between those partial shapes. We do so by operating purely in the spectral domain and by defining the union operation between short sequences of eigenvalues. We show that the approximated union spectrum can be used as-is to reconstruct the complete geometry [MRC*19], perform region localization on a template [RTO*19] and retrieve shapes from a database, generalizing ShapeDNA [RWP06] to work with partialities. Working with eigenvalues allows us to deal with unknown correspondence, different sampling, and different discretizations (point clouds and meshes alike), making this operation especially robust and general. Our approach is data-driven and can generalize to isometric and non-isometric deformations of the surface, as long as these stay within the same semantic class (e.g., human bodies or horses), as well as to partiality artifacts not seen at training time.