# Pairwise-Constrained Implicit Functions for 3D Human Heart Modelling

arXiv: https://arxiv.org/abs/2307.08716 · HTML: https://mlsanity.com/paper/2307.08716/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 22 references that are in the map.
- First posted on arXiv 16 Jul 2023.

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

- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [TotalSegmentator: robust segmentation of 104 anatomical structures in CT images](https://mlsanity.com/paper/2208.05868/) (2022)
- [nnU-Net: Self-adapting Framework for U-Net-Based Medical Image Segmentation](https://mlsanity.com/paper/1809.10486/) (2018)
- [Neural Unsigned Distance Fields for Implicit Function Learning](https://mlsanity.com/paper/2010.13938/) (2020)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [Grasping Field: Learning Implicit Representations for Human Grasps](https://mlsanity.com/paper/2008.04451/) (2020)
- [Evaluation of Algorithms for Multi-Modality Whole Heart Segmentation: An Open-Access Grand Challenge](https://mlsanity.com/paper/1902.07880/) (2019)
- [Topology-Aware Segmentation Using Discrete Morse Theory](https://mlsanity.com/paper/2103.09992/) (2021)
- [Voxel2Mesh: 3D Mesh Model Generation from Volumetric Data](https://mlsanity.com/paper/1912.03681/) (2019)
- [A Deep-Learning Approach For Direct Whole-Heart Mesh Reconstruction](https://mlsanity.com/paper/2102.07899/) (2021)
- [Learning Topological Interactions for Multi-Class Medical Image Segmentation](https://mlsanity.com/paper/2207.09654/) (2022)
- [An Implicit Parametric Morphable Dental Model](https://mlsanity.com/paper/2211.11402/) (2022)
- [Neural Vector Fields for Implicit Surface Representation and Inference](https://mlsanity.com/paper/2204.06552/) (2022)

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

- [Neural Implicit Heart Coordinates: 3D cardiac shape reconstruction from sparse segmentations](https://mlsanity.com/paper/2512.19316/) (2025)
- [S2MDF: A Plug-And-Play Layer for Intersection-Free Multi-Object Signed Distance Fields](https://mlsanity.com/paper/2605.29761/) (2026)

## Abstract (arXiv)

Accurate 3D models of the human heart require not only correct outer surfaces but also realistic inner structures, such as the ventricles, atria, and myocardial layers. Approaches relying on implicit surfaces, such as signed distance functions (SDFs), are primarily designed for single watertight surfaces, making them ill-suited for multi-layered anatomical structures. They often produce gaps or overlaps in shared boundaries. Unsigned distance functions (UDFs) can model non-watertight geometries but are harder to optimize, while voxel-based methods are limited in resolution and struggle to produce smooth, anatomically realistic surfaces. We introduce a pairwise-constrained SDF approach that models the heart as a set of interdependent SDFs, each representing a distinct anatomical component. By enforcing proper contact between adjacent SDFs, we ensure that they form anatomically correct shared walls, preserving the internal structure of the heart and preventing overlaps, or unwanted gaps. Our method significantly improves inner structure accuracy over single-SDF, UDF-based, voxel-based, and segmentation-based reconstructions. We further demonstrate its generalizability by applying it to a vertebrae dataset, preventing unwanted contact between structures.