# Adaptive area-preserving parameterization of open and closed anatomical surfaces

arXiv: https://arxiv.org/abs/2111.04265 · HTML: https://mlsanity.com/paper/2111.04265/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 77 references that are in the map.
- First posted on arXiv 8 Nov 2021.

## Builds on (most cited first, 16 of 16)

- [3DPeople: Modeling the Geometry of Dressed Humans](https://mlsanity.com/paper/1904.04571/) (2019)
- [Boundary First Flattening](https://mlsanity.com/paper/1704.06873/) (2017)
- [TEMPO: Feature-Endowed Teichm"uller Extremal Mappings of Point Clouds](https://mlsanity.com/paper/1511.06624/) (2015)
- [Fast Disk Conformal Parameterization of Simply-connected Open Surfaces](https://mlsanity.com/paper/1408.6974/) (2014)
- [Spherical Conformal Parameterization of Genus-0 Point Clouds for Meshing](https://mlsanity.com/paper/1508.07569/) (2015)
- [Shape analysis via inconsistent surface registration](https://mlsanity.com/paper/2003.01357/) (2020)
- [Parallelizable global conformal parameterization of simply-connected surfaces via partial welding](https://mlsanity.com/paper/1903.12359/) (2019)
- [Density-equalizing maps for simply-connected open surfaces](https://mlsanity.com/paper/1704.02525/) (2017)
- [Tooth morphometry using quasi-conformal theory](https://mlsanity.com/paper/1901.01651/) (2019)
- [A Linear Formulation for Disk Conformal Parameterization of Simply-Connected Open Surfaces](https://mlsanity.com/paper/1508.00396/) (2015)
- [Area-preserving mapping of 3D ultrasound carotid artery images using density-equalizing reference map](https://mlsanity.com/paper/1812.03434/) (2018)
- [Fast Spherical Quasiconformal Parameterization of Genus-0 Closed Surfaces with Application to Adaptive Remeshing](https://mlsanity.com/paper/1608.01140/) (2016)
- [A Novel Stretch Energy Minimization Algorithm for Equiareal Parameterizations](https://mlsanity.com/paper/1708.07391/) (2017)
- [Spherical Parameterization Balancing Angle and Area Distortions](https://mlsanity.com/paper/1810.09031/) (2018)
- [Conformal Mesh Parameterization Using Discrete Calabi Flow](https://mlsanity.com/paper/1807.08486/) (2018)
- [LMap: Shape-Preserving Local Mappings for Biomedical Visualization](https://mlsanity.com/paper/1809.06442/) (2018)

## Abstract (arXiv)

The parameterization of open and closed anatomical surfaces is of fundamental importance in many biomedical applications. Spherical harmonics, a set of basis functions defined on the unit sphere, are widely used for anatomical shape description. However, establishing a one-to-one correspondence between the object surface and the entire unit sphere may induce a large geometric distortion in case the shape of the surface is too different from a perfect sphere. In this work, we propose adaptive area-preserving parameterization methods for simply-connected open and closed surfaces with the target of the parameterization being a spherical cap. Our methods optimize the shape of the parameter domain along with the mapping from the object surface to the parameter domain. The object surface will be globally mapped to an optimal spherical cap region of the unit sphere in an area-preserving manner while also exhibiting low conformal distortion. We further develop a set of spherical harmonics-like basis functions defined over the adaptive spherical cap domain, which we call the adaptive harmonics. Experimental results show that the proposed parameterization methods outperform the existing methods for both open and closed anatomical surfaces in terms of area and angle distortion. Surface description of the object surfaces can be effectively achieved using a novel combination of the adaptive parameterization and the adaptive harmonics. Our work provides a novel way of mapping anatomical surfaces with improved accuracy and greater flexibility. More broadly, the idea of using an adaptive parameter domain allows easy handling of a wide range of biomedical shapes.