# A skeletonization algorithm for gradient-based optimization arXiv: https://arxiv.org/abs/2309.02527 · HTML: https://mlsanity.com/paper/2309.02527/ ## Key facts - Cited by 17 papers in the map (data through 14 Oct 2026). - Builds on 14 of its 48 references that are in the map. - First posted on arXiv 5 Sep 2023. ## Builds on (most cited first, 14 of 14) - [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014) - [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015) - [Domain-Adversarial Training of Neural Networks](https://mlsanity.com/paper/1505.07818/) (2015) - [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016) - [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013) - [The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables](https://mlsanity.com/paper/1611.00712/) (2016) - [clDice -- A Novel Topology-Preserving Loss Function for Tubular Structure Segmentation](https://mlsanity.com/paper/2003.07311/) (2020) - [A Review of the Gumbel-max Trick and its Extensions for Discrete Stochasticity in Machine Learning](https://mlsanity.com/paper/2110.01515/) (2021) - [Dynamic Feature Integration for Simultaneous Detection of Salient Object, Edge and Skeleton](https://mlsanity.com/paper/2004.08595/) (2020) - [DeepSkeleton: Learning Multi-task Scale-associated Deep Side Outputs for Object Skeleton Extraction in Natural Images](https://mlsanity.com/paper/1609.03659/) (2016) - [AirLab: Autograd Image Registration Laboratory](https://mlsanity.com/paper/1806.09907/) (2018) - [SkelNetOn 2019: Dataset and Challenge on Deep Learning for Geometric Shape Understanding](https://mlsanity.com/paper/1903.09233/) (2019) - [Object Skeleton Extraction in Natural Images by Fusing Scale-associated Deep Side Outputs](https://mlsanity.com/paper/1603.09446/) (2016) - [SkeletonNet: Shape Pixel to Skeleton Pixel](https://mlsanity.com/paper/1907.01683/) (2019) ## Built on by (most cited first, 17 of 17) - [The TopCoW Challenge -- Topology-Aware Circle of Willis Segmentation for CT and MR Angiography](https://mlsanity.com/paper/2312.17670/) (2023) - [Skeleton Recall Loss for Connectivity Conserving and Resource Efficient Segmentation of Thin Tubular Structures](https://mlsanity.com/paper/2404.03010/) (2024) - [Centerline Boundary Dice Loss for Vascular Segmentation](https://mlsanity.com/paper/2407.01517/) (2024) - [Topograph: An efficient Graph-Based Framework for Strictly Topology Preserving Image Segmentation](https://mlsanity.com/paper/2411.03228/) (2024) - [RoBus: A Multimodal Dataset for Controllable Road Networks and Building Layouts Generation](https://mlsanity.com/paper/2407.07835/) (2024) - [GraphMorph: Tubular Structure Extraction by Morphing Predicted Graphs](https://mlsanity.com/paper/2502.11731/) (2025) - [NeuroSeg Meets DINOv3: Transferring 2D Self-Supervised Visual Priors to 3D Neuron Segmentation via DINOv3 Initialization](https://mlsanity.com/paper/2603.23104/) (2026) - [Skelite: Compact Neural Networks for Efficient Iterative Skeletonization](https://mlsanity.com/paper/2503.07369/) (2025) - [Parametric shape models for vessels learned from segmentations via differentiable voxelization](https://mlsanity.com/paper/2507.02576/) (2025) - [Does the Skeleton-Recall Loss Really Work?](https://mlsanity.com/paper/2508.11374/) (2025) - [Evaluation of Blood Vessel Segmentation Methods on Hard-to-Detect Vascular Structures](https://mlsanity.com/paper/2406.13128/) (2024) - [Enhancing the automatic segmentation and analysis of 3D liver vasculature models](https://mlsanity.com/paper/2411.15778/) (2024) - [MorphoSkel3D: Morphological Skeletonization of 3D Point Clouds for Informed Sampling in Object Classification and Retrieval](https://mlsanity.com/paper/2501.12974/) (2025) - [ContextLoss: Context Information for Topology-Preserving Segmentation](https://mlsanity.com/paper/2506.11134/) (2025) - [Continuous-tone Simple Points: An ℓ0-Norm of Cyclic Gradient for Topology-Preserving Data-Driven Image Segmentation](https://mlsanity.com/paper/2604.28159/) (2026) - [RoadGIE: Towards A Global-Scale Aerial Benchmark for Generalizable Interactive Road Extraction](https://mlsanity.com/paper/2605.26862/) (2026) - [Sparse cubical complexes for efficient topology-preservation in image data](https://mlsanity.com/paper/2609.37177/) (2026) ## Abstract (arXiv) The skeleton of a digital image is a compact representation of its topology, geometry, and scale. It has utility in many computer vision applications, such as image description, segmentation, and registration. However, skeletonization has only seen limited use in contemporary deep learning solutions. Most existing skeletonization algorithms are not differentiable, making it impossible to integrate them with gradient-based optimization. Compatible algorithms based on morphological operations and neural networks have been proposed, but their results often deviate from the geometry and topology of the true medial axis. This work introduces the first three-dimensional skeletonization algorithm that is both compatible with gradient-based optimization and preserves an object's topology. Our method is exclusively based on matrix additions and multiplications, convolutional operations, basic non-linear functions, and sampling from a uniform probability distribution, allowing it to be easily implemented in any major deep learning library. In benchmarking experiments, we prove the advantages of our skeletonization algorithm compared to non-differentiable, morphological, and neural-network-based baselines. Finally, we demonstrate the utility of our algorithm by integrating it with two medical image processing applications that use gradient-based optimization: deep-learning-based blood vessel segmentation, and multimodal registration of the mandible in computed tomography and magnetic resonance images.