# Isospectralization, or how to hear shape, style, and correspondence

arXiv: https://arxiv.org/abs/1811.11465 · HTML: https://mlsanity.com/paper/1811.11465/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Nov 2018.

## Built on by (most cited first, 18 of 18)

- [Rethinking Graph Transformers with Spectral Attention](https://mlsanity.com/paper/2106.03893/) (2021)
- [A Survey of Non-Rigid 3D Registration](https://mlsanity.com/paper/2203.07858/) (2022)
- [Universal Spectral Adversarial Attacks for Deformable Shapes](https://mlsanity.com/paper/2104.03356/) (2021)
- [SRIF: Semantic Shape Registration Empowered by Diffusion-based Image Morphing and Flow Estimation](https://mlsanity.com/paper/2409.11682/) (2024)
- [Generating Graphs via Spectral Diffusion](https://mlsanity.com/paper/2402.18974/) (2024)
- [Neural Human Deformation Transfer](https://mlsanity.com/paper/2109.01588/) (2021)
- [Latent Spectral Regularization for Continual Learning](https://mlsanity.com/paper/2301.03345/) (2023)
- [Log-Euclidean Signatures for Intrinsic Distances Between Unaligned Datasets](https://mlsanity.com/paper/2202.01671/) (2022)
- [Laplace-Beltrami Operator for Gaussian Splatting](https://mlsanity.com/paper/2502.17531/) (2025)
- [Learning Spectral Unions of Partial Deformable 3D Shapes](https://mlsanity.com/paper/2104.00514/) (2021)
- [Disentangling Geometric Deformation Spaces in Generative Latent Shape Models](https://mlsanity.com/paper/2103.00142/) (2021)
- [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)
- [LBONet: Supervised Spectral Descriptors for Shape Analysis](https://mlsanity.com/paper/2411.08272/) (2024)
- [Localized Shape Modelling with Global Coherence: An Inverse Spectral Approach](https://mlsanity.com/paper/2108.02161/) (2021)
- [Temporally-Consistent Surface Reconstruction using Metrically-Consistent Atlases](https://mlsanity.com/paper/2111.06838/) (2021)
- [Can A Neural Network Hear the Shape of A Drum?](https://mlsanity.com/paper/2203.08073/) (2022)
- [Iskra: A System for Inverse Geometry Processing](https://mlsanity.com/paper/2602.12105/) (2026)

## Abstract (arXiv)

The question whether one can recover the shape of a geometric object from its Laplacian spectrum ('hear the shape of the drum') is a classical problem in spectral geometry with a broad range of implications and applications. While theoretically the answer to this question is negative (there exist examples of iso-spectral but non-isometric manifolds), little is known about the practical possibility of using the spectrum for shape reconstruction and optimization. In this paper, we introduce a numerical procedure called isospectralization, consisting of deforming one shape to make its Laplacian spectrum match that of another. We implement the isospectralization procedure using modern differentiable programming techniques and exemplify its applications in some of the classical and notoriously hard problems in geometry processing, computer vision, and graphics such as shape reconstruction, pose and style transfer, and dense deformable correspondence.