# Representing Topological Self-Similarity Using Fractal Feature Maps for Accurate Segmentation of Tubular Structures

arXiv: https://arxiv.org/abs/2407.14754 · HTML: https://mlsanity.com/paper/2407.14754/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 62 references that are in the map.
- First posted on arXiv 20 Jul 2024.

## Builds on (most cited first, 20 of 20)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fully Convolutional Networks for Semantic Segmentation](https://mlsanity.com/paper/1411.4038/) (2014)
- [TransUNet: Transformers Make Strong Encoders for Medical Image Segmentation](https://mlsanity.com/paper/2102.04306/) (2021)
- [UNet++: A Nested U-Net Architecture for Medical Image Segmentation](https://mlsanity.com/paper/1807.10165/) (2018)
- [Deformable Convolutional Networks](https://mlsanity.com/paper/1703.06211/) (2017)
- [Deep High-Resolution Representation Learning for Visual Recognition](https://mlsanity.com/paper/1908.07919/) (2019)
- [Dilated Residual Networks](https://mlsanity.com/paper/1705.09914/) (2017)
- [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)
- [DUNet: A deformable network for retinal vessel segmentation](https://mlsanity.com/paper/1811.01206/) (2018)
- [Dynamic Snake Convolution based on Topological Geometric Constraints for Tubular Structure Segmentation](https://mlsanity.com/paper/2307.08388/) (2023)
- [ROSE: A Retinal OCT-Angiography Vessel Segmentation Dataset and New Model](https://mlsanity.com/paper/2007.05201/) (2020)
- [Topology-Aware Segmentation Using Discrete Morse Theory](https://mlsanity.com/paper/2103.09992/) (2021)
- [Deep Distance Transform for Tubular Structure Segmentation in CT Scans](https://mlsanity.com/paper/1912.03383/) (2019)
- [Topology-Aware Uncertainty for Image Segmentation](https://mlsanity.com/paper/2306.05671/) (2023)
- [Directional Connectivity-based Segmentation of Medical Images](https://mlsanity.com/paper/2304.00145/) (2023)
- [Synthetic optical coherence tomography angiographs for detailed retinal vessel segmentation without human annotations](https://mlsanity.com/paper/2306.10941/) (2023)
- [Topological Similarity Index and Loss Function for Blood Vessel Segmentation](https://mlsanity.com/paper/2107.14531/) (2021)
- [Affinity Feature Strengthening for Accurate, Complete and Robust Vessel Segmentation](https://mlsanity.com/paper/2211.06578/) (2022)

## Built on by (most cited first, 13 of 13)

- [RUN: Reversible Unfolding Network for Concealed Object Segmentation](https://mlsanity.com/paper/2501.18783/) (2025)
- [Reversible Unfolding Network for Concealed Visual Perception with Generative Refinement](https://mlsanity.com/paper/2508.15027/) (2025)
- [GLCP: Global-to-Local Connectivity Preservation for Tubular Structure Segmentation](https://mlsanity.com/paper/2507.21328/) (2025)
- [HarmonySeg: Tubular Structure Segmentation with Deep-Shallow Feature Fusion and Growth-Suppression Balanced Loss](https://mlsanity.com/paper/2504.07827/) (2025)
- [Topology-Guaranteed Image Segmentation: Enforcing Connectivity, Genus, and Width Constraints](https://mlsanity.com/paper/2601.11409/) (2026)
- [COMMA: Coordinate-aware Modulated Mamba Network for 3D Dispersed Vessel Segmentation](https://mlsanity.com/paper/2503.02332/) (2025)
- [Opportunistic Osteoporosis Diagnosis via Texture-Preserving Self-Supervision, Mixture of Experts and Multi-Task Integration](https://mlsanity.com/paper/2506.20282/) (2025)
- [Leveraging Persistence Image to Enhance Robustness and Performance in Curvilinear Structure Segmentation](https://mlsanity.com/paper/2601.18045/) (2026)
- [Towards High-Quality Image Segmentation: Improving Topology Accuracy by Penalizing Neighbor Pixels](https://mlsanity.com/paper/2603.18671/) (2026)
- [LUMOS: Latent Universal Medical Priors for Segmentation](https://mlsanity.com/paper/2603.01115/) (2026)
- [Bridging the Geometry Mismatch: Frequency-Aware Anisotropic Serialization for Thin-Structure SSMs](https://mlsanity.com/paper/2603.28503/) (2026)
- [Continuous-tone Simple Points: An ℓ0-Norm of Cyclic Gradient for Topology-Preserving Data-Driven Image Segmentation](https://mlsanity.com/paper/2604.28159/) (2026)
- [Widest-Path Reachability Fields for Connectivity-Preserving Slender Structure Segmentation](https://mlsanity.com/paper/2607.07123/) (2026)

## Abstract (arXiv)

Accurate segmentation of long and thin tubular structures is required in a wide variety of areas such as biology, medicine, and remote sensing. The complex topology and geometry of such structures often pose significant technical challenges. A fundamental property of such structures is their topological self-similarity, which can be quantified by fractal features such as fractal dimension (FD). In this study, we incorporate fractal features into a deep learning model by extending FD to the pixel-level using a sliding window technique. The resulting fractal feature maps (FFMs) are then incorporated as additional input to the model and additional weight in the loss function to enhance segmentation performance by utilizing the topological self-similarity. Moreover, we extend the U-Net architecture by incorporating an edge decoder and a skeleton decoder to improve boundary accuracy and skeletal continuity of segmentation, respectively. Extensive experiments on five tubular structure datasets validate the effectiveness and robustness of our approach. Furthermore, the integration of FFMs with other popular segmentation models such as HR-Net also yields performance enhancement, suggesting FFM can be incorporated as a plug-in module with different model architectures. Code and data are openly accessible at https://github.com/cbmi-group/FFM-Multi-Decoder-Network.