# Learning-based density-equalizing map

arXiv: https://arxiv.org/abs/2506.10027 · HTML: https://mlsanity.com/paper/2506.10027/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 33 references that are in the map.
- First posted on arXiv 10 Jun 2025.

## Builds on (most cited first, 10 of 10)

- [Spherical Density-Equalizing Map for Genus-0 Closed Surfaces](https://mlsanity.com/paper/2401.11795/) (2024)
- [Density-equalizing maps for simply-connected open surfaces](https://mlsanity.com/paper/1704.02525/) (2017)
- [Bijective Density-Equalizing Quasiconformal Map for Multiply-Connected Open Surfaces](https://mlsanity.com/paper/2308.05579/) (2023)
- [Area-preserving mapping of 3D ultrasound carotid artery images using density-equalizing reference map](https://mlsanity.com/paper/1812.03434/) (2018)
- [Volumetric density-equalizing reference map with applications](https://mlsanity.com/paper/2003.09725/) (2020)
- [The State of the Art in Cartograms](https://mlsanity.com/paper/1605.08485/) (2016)
- [Fast flow-based algorithm for creating density-equalizing map projections](https://mlsanity.com/paper/1802.07625/) (2018)
- [Ellipsoidal Density-Equalizing Map for Genus-0 Closed Surfaces](https://mlsanity.com/paper/2410.12331/) (2024)
- [Toroidal density-equalizing map for genus-one surfaces](https://mlsanity.com/paper/2410.16833/) (2024)
- [Hemispheroidal parameterization and harmonic decomposition of simply connected open surfaces](https://mlsanity.com/paper/2407.15417/) (2024)

## Built on by (most cited first, 2 of 2)

- [Resolution-free neural surrogates for geometric parameterization and mapping with spatially varying fields](https://mlsanity.com/paper/2605.28551/) (2026)
- [Learning Task-Aware Sampling with Shared Saliency through Density-Equalizing Mappings](https://mlsanity.com/paper/2606.12869/) (2026)

## Abstract (arXiv)

Density-equalizing map (DEM) serves as a powerful technique for creating shape deformations with the area changes reflecting an underlying density function. In recent decades, DEM has found widespread applications in fields such as data visualization, geometry processing, and medical imaging. Traditional approaches to DEM primarily rely on iterative numerical solvers for diffusion equations or optimization-based methods that minimize handcrafted energy functionals. However, these conventional techniques often face several challenges: they may suffer from limited accuracy, produce overlapping artifacts in extreme cases, and require substantial algorithmic redesign when extended from 2D to 3D, due to the derivative-dependent nature of their energy formulations. In this work, we propose a novel learning-based density-equalizing mapping framework (LDEM) using deep neural networks. Specifically, we introduce a loss function that enforces density uniformity and geometric regularity, and utilize a hierarchical approach to predict the transformations at both the coarse and dense levels. Our method demonstrates superior density-equalizing and bijectivity properties compared to prior methods for a wide range of simple and complex density distributions, and can be easily applied to surface remeshing with different effects. Also, it generalizes seamlessly from 2D to 3D domains without structural changes to the model architecture or loss formulation. Altogether, our work opens up new possibilities for scalable and robust computation of density-equalizing maps for practical applications.