# Conformable Convolution for Topologically Aware Learning of Complex Anatomical Structures

arXiv: https://arxiv.org/abs/2412.20608 · HTML: https://mlsanity.com/paper/2412.20608/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 33 of its 58 references that are in the map.
- First posted on arXiv 29 Dec 2024.

## Builds on (most cited first, 30 of 33)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation](https://mlsanity.com/paper/1606.04797/) (2016)
- [Deformable Convolutional Networks](https://mlsanity.com/paper/1703.06211/) (2017)
- [Swin UNETR: Swin Transformers for Semantic Segmentation of Brain Tumors in MRI Images](https://mlsanity.com/paper/2201.01266/) (2022)
- [Deformable ConvNets v2: More Deformable, Better Results](https://mlsanity.com/paper/1811.11168/) (2018)
- [VoxelMorph: A Learning Framework for Deformable Medical Image Registration](https://mlsanity.com/paper/1809.05231/) (2018)
- [InternImage: Exploring Large-Scale Vision Foundation Models with Deformable Convolutions](https://mlsanity.com/paper/2211.05778/) (2022)
- [clDice -- A Novel Topology-Preserving Loss Function for Tubular Structure Segmentation](https://mlsanity.com/paper/2003.07311/) (2020)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology](https://mlsanity.com/paper/1910.01877/) (2019)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [DUNet: A deformable network for retinal vessel segmentation](https://mlsanity.com/paper/1811.01206/) (2018)
- [Topological Graph Neural Networks](https://mlsanity.com/paper/2102.07835/) (2021)
- [Dynamic Snake Convolution based on Topological Geometric Constraints for Tubular Structure Segmentation](https://mlsanity.com/paper/2307.08388/) (2023)
- [Topology-Aware Segmentation Using Discrete Morse Theory](https://mlsanity.com/paper/2103.09992/) (2021)
- [Efficient Deformable ConvNets: Rethinking Dynamic and Sparse Operator for Vision Applications](https://mlsanity.com/paper/2401.06197/) (2024)
- [Topologically faithful image segmentation via induced matching of persistence barcodes](https://mlsanity.com/paper/2211.15272/) (2022)
- [Inverse Distance Aggregation for Federated Learning with Non-IID Data](https://mlsanity.com/paper/2008.07665/) (2020)
- [Deep Vessel Segmentation By Learning Graphical Connectivity](https://mlsanity.com/paper/1806.02279/) (2018)
- [Topology-Aware Uncertainty for Image Segmentation](https://mlsanity.com/paper/2306.05671/) (2023)
- [Learning Topological Interactions for Multi-Class Medical Image Segmentation](https://mlsanity.com/paper/2207.09654/) (2022)
- [Centerline Boundary Dice Loss for Vascular Segmentation](https://mlsanity.com/paper/2407.01517/) (2024)
- [Uncovering the Topology of Time-Varying fMRI Data using Cubical Persistence](https://mlsanity.com/paper/2006.07882/) (2020)
- [Structure-Aware Image Segmentation with Homotopy Warping](https://mlsanity.com/paper/2112.07812/) (2021)
- [Study Group Learning: Improving Retinal Vessel Segmentation Trained with Noisy Labels](https://mlsanity.com/paper/2103.03451/) (2021)
- [Y-Net: A Spatiospectral Dual-Encoder Networkfor Medical Image Segmentation](https://mlsanity.com/paper/2204.07613/) (2022)
- [TA-Net: Topology-Aware Network for Gland Segmentation](https://mlsanity.com/paper/2110.14593/) (2021)

## Abstract (arXiv)

While conventional computer vision emphasizes pixel-level and feature-based objectives, medical image analysis of intricate biological structures necessitates explicit representation of their complex topological properties. Despite their successes, deep learning models often struggle to accurately capture the connectivity and continuity of fine, sometimes pixel-thin, yet critical structures due to their reliance on implicit learning from data. Such shortcomings can significantly impact the reliability of analysis results and hinder clinical decision-making. To address this challenge, we introduce Conformable Convolution, a novel convolutional layer designed to explicitly enforce topological consistency. Conformable Convolution learns adaptive kernel offsets that preferentially focus on regions of high topological significance within an image. This prioritization is guided by our proposed Topological Posterior Generator (TPG) module, which leverages persistent homology. The TPG module identifies key topological features and guides the convolutional layers by applying persistent homology to feature maps transformed into cubical complexes. Our proposed modules are architecture-agnostic, enabling them to be integrated seamlessly into various architectures. We showcase the effectiveness of our framework in the segmentation task, where preserving the interconnectedness of structures is critical. Experimental results on three diverse datasets demonstrate that our framework effectively preserves the topology in the segmentation downstream task, both quantitatively and qualitatively.