Continuous-tone Simple Points: An \ell_0-Norm of Cyclic Gradient for Topology-Preserving Data-Driven Image Segmentation
Authors: Wenxiao Li, Faqiang Wang, Yuping Duan, Li Cui, Liqiang Zhang, Jun Liu
Abstract
Topological features play an essential role in ensuring geometric plausibility and structural consistency in image analysis tasks such as segmentation and skeletonization. However, integrating topology-preserving learning based on simple points into deep learning tasks remains challenging, as existing simple point detection methods are confined to binary images and are non-differentiable, rendering them incompatible with gradient-based optimization in modern deep learning. Moreover, morphological and purely data-driven approaches often fail to guaranty topological consistency. To address these limitations, we propose a novel method that directly computes simple points on continuous-valued images, enabling differentiable topological inference. Building on this theory, we develop an efficient skeleton extraction algorithm that preserves topological structures in binary and continuous-valued images. Furthermore, we design a variational model that enforces topological constraints by preserving topologically non-removable (i.e., non-simple) points, which can be seamlessly integrated into any deep neural network segmentation with softmax or sigmoid outputs. Experimental results demonstrate that the proposed approach effectively improves topological integrity and structural accuracy across multiple benchmarks. The codes are available in https://github.com/levnsio/CSP.
In natural images, object skeletons are used to represent geometric shapes. However, even slight variations in pose or movement can cause noticeable changes in skeleton structure, increasing the difficulty of detecting the skeleton and often resulting in discontinuous skeletons. Existing methods primarily focus on point-level skeleton point detection and overlook the importance of structural continuity in recovering complete skeletons. To address this issue, we propose Lighthouse-Skel, a topology-aware skeleton detection method via lighthouse-guided structured inference. Specifically, we introduce a dual-branch collaborative detection framework that jointly learns skeleton confidence field and structural anchors, including endpoints and junction points. The spatial distributions learned by the point branch guide the network to focus on topologically vulnerable regions, which improves the accuracy of skeleton detection. Based on the learned skeleton confidence field, we further propose a lighthouse-guided topology completion strategy, which uses detected junction points and breakpoints as lighthouses to reconnect discontinuous skeleton segments along low-cost paths, thereby improving skeleton continuity and structural integrity. Experimental results on four public datasets demonstrate that the proposed method achieves competitive detection accuracy while substantially improving skeleton connectivity and structural integrity.
Segmenting slender curvilinear structures such as retinal vessels, cracks, and roads demands topological correctness, as even a single-pixel discontinuity can fragment a continuous network and invalidate downstream analysis. Under standard binary-mask supervision, models optimized for pixel-level overlap frequently produce topologically broken predictions. We trace this to a fundamental mismatch: pixel-wise losses distribute gradients uniformly, yet connectivity hinges on a sparse set of bottleneck pixels. These pixels are vastly outnumbered by thick structures and background, rendering their aggregate gradient contribution negligible. We term this phenomenon topological gradient starvation (TGS). To address it, we propose Widest-Path Reachability Fields (WPRF), a differentiable Max-Min reachability objective that redirects gradient flow to connectivity bottlenecks. The module is plug-and-play, backbone-agnostic, and incurs no inference overhead. WPRF implements a differentiable Max-Min objective via dynamic programming on a domain-restricted graph, coupled with a bottleneck-aware observation term that balances gradient contributions across varying structures. Compared to prior topology-aware losses that rely on post-hoc skeletonization or homology computation, WPRF directly optimizes end-to-end reachability via differentiable Max-Min algebra, enabling gradient flow to concentrate on connectivity bottlenecks without auxiliary structures. We introduce OMVIS, a new oral microvessel segmentation dataset. Experiments across nine architectures and six datasets validate the bottleneck-focused gradient routing mechanism. WPRF improves 87% of experiments with fixed hyperparameters and achieves clDice gains of 7.2 percentage points on structurally fragile datasets.
A levelset free but a hybrid image segmentation approach based on a modified version of the piece wise constant shape gradient of an Mumford Shah shape functional and a repulsive function is considered. The segmentation is performed a non-local shape based through an evolution of discrete curves driven by a non local shape based energy to segment images containing disjoint regions and multiple boundaries. This formulation has a novel additional component as a multivariable function dependent on a few sampled points of the curves that handles the occurrence of self intersection during boundary curves evolution. The method is applied to a few gray scale and color images, including images with nested structures and astronomical objects. The results indicate effective segmentation in complex scenarios with absolute control on the topology of the segments and self-intersections of the boundaries