# Extracting a functional representation from a dictionary for non-rigid shape matching

arXiv: https://arxiv.org/abs/2305.10332 · HTML: https://mlsanity.com/paper/2305.10332/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 49 references that are in the map.
- First posted on arXiv 17 May 2023.

## Builds on (most cited first, 13 of 13)

- [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)
- [Correspondence Learning via Linearly-invariant Embedding](https://mlsanity.com/paper/2010.13136/) (2020)
- [ZoomOut: Spectral Upsampling for Efficient Shape Correspondence](https://mlsanity.com/paper/1904.07865/) (2019)
- [Structured Regularization of Functional Map Computations](https://mlsanity.com/paper/2009.14624/) (2020)
- [Complex Functional Maps : a Conformal Link Between Tangent Bundles](https://mlsanity.com/paper/2112.09546/) (2021)
- [On the optimality of shape and data representation in the spectral domain](https://mlsanity.com/paper/1409.4349/) (2014)
- [Localized Manifold Harmonics for Spectral Shape Analysis](https://mlsanity.com/paper/1707.02596/) (2017)
- [MapTree: Recovering Multiple Solutions in the Space of Maps](https://mlsanity.com/paper/2006.02532/) (2020)
- [Point-wise Map Recovery and Refinement from Functional Correspondence](https://mlsanity.com/paper/1506.05603/) (2015)
- [Wavelet-based Heat Kernel Derivatives: Towards Informative Localized Shape Analysis](https://mlsanity.com/paper/2007.11632/) (2020)
- [Non-Isometric Shape Matching via Functional Maps on Landmark-Adapted Bases](https://mlsanity.com/paper/2205.04800/) (2022)

## Abstract (arXiv)

Shape matching is a fundamental problem in computer graphics with many applications. Functional maps translate the point-wise shape-matching problem into its functional counterpart and have inspired numerous solutions over the last decade. Nearly all the solutions based on functional maps rely on the eigenfunctions of the Laplace-Beltrami Operator (LB) to describe the functional spaces defined on the surfaces and then convert the functional correspondences into point-wise correspondences. However, this final step is often error-prone and inaccurate in tiny regions and protrusions, where the energy of LB does not uniformly cover the surface. We propose a new functional basis Principal Components of a Dictionary (PCD) to address such intrinsic limitation. PCD constructs an orthonormal basis from the Principal Component Analysis (PCA) of a dictionary of functions defined over the shape. These dictionaries can target specific properties of the final basis, such as achieving an even spreading of energy. Our experimental evaluation compares seven different dictionaries on established benchmarks, showing that PCD is suited to target different shape-matching scenarios, resulting in more accurate point-wise maps than the LB basis when used in the same pipeline. This evidence provides a promising alternative for improving correspondence estimation, confirming the power and flexibility of functional maps.