arxiv: 2604.28159 · v1 · submitted 2026-04-30 · 💻 cs.CV

Recognition: unknown

Continuous-tone Simple Points: An ell₀-Norm of Cyclic Gradient for Topology-Preserving Data-Driven Image Segmentation

Wenxiao Li , Faqiang Wang , Yuping Duan , Li Cui , Liqiang Zhang , Jun Liu

Authors on Pith no claims yet

Pith reviewed 2026-05-07 07:43 UTC · model grok-4.3

classification 💻 cs.CV

keywords topology preservationsimple pointscontinuous-valued imagesimage segmentationdifferentiable topologyvariational modeldeep neural networksskeleton extraction

0 comments

The pith

An ℓ₀-norm of the cyclic gradient detects simple points directly on continuous-valued images.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to compute simple points on continuous-valued images, which are points that can be removed without altering the topology. This addresses the limitation that traditional simple point detection works only on binary images and is non-differentiable, making it unusable in deep learning. The key is defining an ℓ₀-norm on the cyclic gradient that serves as a differentiable indicator of topological simplicity. Using this, a variational model is created that preserves non-simple points to enforce topology in segmentation networks with continuous outputs like softmax. Experiments show improved topological consistency in image segmentation tasks.

Core claim

We propose a novel method that directly computes simple points on continuous-valued images by means of the ℓ₀-norm of the cyclic gradient, enabling differentiable topological inference. This leads to an efficient skeleton extraction algorithm that preserves topological structures in both binary and continuous-valued images. Furthermore, we design a variational model that enforces topological constraints by preserving topologically non-removable points, which integrates seamlessly into deep neural network segmentation models with softmax or sigmoid outputs.

What carries the argument

The ℓ₀-norm of the cyclic gradient operator, which identifies topologically non-removable (non-simple) points in continuous images and allows gradient flow for topology preservation.

If this is right

Any deep neural network for segmentation with softmax or sigmoid outputs can incorporate topological constraints during training.
The approach yields segmentations with improved topological integrity and structural accuracy on benchmarks.
Skeleton extraction becomes possible for continuous-valued images while maintaining topology.
Topology preservation no longer requires separate binary post-processing steps.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar cyclic gradient norms might apply to other topological features beyond simple points in image processing.
This differentiable formulation could support end-to-end training for tasks like medical image analysis where topology is critical.
Verification on synthetic images with controlled topology changes would confirm equivalence to discrete simple point criteria.

Load-bearing premise

That the ℓ₀-norm of the cyclic gradient on continuous images correctly identifies non-simple points in a way equivalent to discrete binary simple point theorems.

What would settle it

An experiment on an image with a known hole or loop where the trained segmentation network using this model produces an output with altered topology, such as a missing connection.

Figures

Figures reproduced from arXiv: 2604.28159 by Faqiang Wang, Jun Liu, Li Cui, Liqiang Zhang, Wenxiao Li, Yuping Duan.

**Figure 1.** Figure 1: Comparison of segmentation results from SAM2 [1] without and with view at source ↗

**Figure 2.** Figure 2: Examples of non-simple and simple points. The solid grids means view at source ↗

**Figure 3.** Figure 3: Overview of proposed TCSP-SAM network Nθ defined in Eq. (6). TCSP denotes the topology-preserving variational model (5), and CSPS corresponds to the skeleton extraction algorithm 1. model to focus more on pixels that encode topological properties, thereby enhancing its ability to perceive and represent the global topological structure of the image. D. General Topology Preserving for Segmentation Network A… view at source ↗

**Figure 4.** Figure 4: Visualization of results with the regularization term view at source ↗

**Figure 5.** Figure 5: Visualization of skeleton extraction results from binary and continuous-valued images using three methods. SRSP* denotes the addition of logical view at source ↗

**Figure 6.** Figure 6: Segmentation results of different loss functions. “Baseline” denotes view at source ↗

**Figure 7.** Figure 7: Visualization results of the CSP loss and the topology-preserving view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines continuous simple points via l0-norm of cyclic gradient to make topology preservation differentiable for segmentation networks, but whether this exactly reproduces the classical discrete simple-point test is the part that still needs verification.

read the letter

The main advance is a definition of simple points that works directly on continuous-valued images instead of requiring binarization first. They use the l0-norm of the cyclic gradient to identify these points and then build a variational loss that keeps non-simple points fixed during training. This loss plugs into any network with softmax or sigmoid outputs, which is the practical part for data-driven segmentation. They also give a skeleton extraction routine that runs on both binary and continuous data and release the code, so the method can be tested without reimplementation from scratch. That combination addresses a clear gap: prior simple-point detectors were binary-only and non-differentiable, so they could not enter gradient-based pipelines. The experiments on benchmarks are reported to show gains in topological integrity, which is the kind of evidence that matters for this subfield. The soft spot is the equivalence claim. The continuous operator is presented as identifying the same topologically removable points as the discrete neighborhood rules that preserve Euler characteristic. The abstract and stress-test note do not include a formal proof that the two coincide for every intensity pattern, nor exhaustive checks on small synthetic neighborhoods where a mismatch would show up as extra holes or breaks. If the continuous version only approximates, the variational model could either protect too many points or fail to protect critical ones. A reader would want to see those counter-example tests and an ablation that isolates the effect of the continuous operator versus a post-hoc binarization baseline. This work is aimed at groups already doing topology-aware medical image segmentation or skeletonization. Someone building differentiable topology constraints would get concrete value from the formulation and the released implementation. It deserves peer review because the targeted fix is clear and the code lets referees check the claims directly; the equivalence question is the main item that needs referee scrutiny rather than a reason to desk-reject.

Referee Report

2 major / 2 minor

Summary. The paper claims to introduce continuous-tone simple points defined as the ℓ₀-norm of the cyclic gradient on continuous-valued images. This allows for a differentiable way to detect simple points, leading to a skeleton extraction algorithm that preserves topology and a variational model that can be integrated into deep learning segmentation networks to enforce topological constraints by preserving non-simple points. Experiments on benchmarks show improved topological integrity and structural accuracy.

Significance. Should the equivalence to discrete simple points hold and the model enforce topology correctly, this would be significant for enabling topology-preserving deep learning in image segmentation, a common need in fields like medical imaging. The provision of code enhances the potential impact by allowing reproduction and extension. It bridges discrete topology with continuous differentiable models.

major comments (2)

[§3 (Method - Continuous-tone Simple Points)] The definition of continuous-tone simple points via the ℓ₀-norm of the cyclic gradient is presented without a formal proof that it coincides with the discrete binary simple point definition for arbitrary intensity values. This is a load-bearing issue for the central claim, as the variational model in the subsequent section relies on this to guarantee topology preservation equivalent to classical methods. A theorem or at least exhaustive verification on synthetic neighborhoods would be necessary.
[§4 (Variational Model)] The variational model enforces constraints by preserving non-simple points, but it is not clear from the formulation how this local constraint ensures global topological properties (such as correct Betti numbers) are maintained throughout the optimization. An analysis or additional experiments demonstrating no introduction of new topological artifacts would address this.

minor comments (2)

[Abstract] The abstract is clear but could mention the specific benchmarks used for experiments to give readers a better sense of the scope.
[Notation] The cyclic gradient is introduced but its exact mathematical definition (e.g., how the cycle is defined in 2D/3D) should be clarified earlier in the method section for better readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and insightful comments, which help strengthen the theoretical and empirical aspects of our work. We address each major comment below and will incorporate revisions to provide additional proofs, clarifications, and experiments in the revised manuscript.

read point-by-point responses

Referee: [§3 (Method - Continuous-tone Simple Points)] The definition of continuous-tone simple points via the ℓ₀-norm of the cyclic gradient is presented without a formal proof that it coincides with the discrete binary simple point definition for arbitrary intensity values. This is a load-bearing issue for the central claim, as the variational model in the subsequent section relies on this to guarantee topology preservation equivalent to classical methods. A theorem or at least exhaustive verification on synthetic neighborhoods would be necessary.

Authors: We agree that a formal equivalence result for the binary case would strengthen the central claim. The continuous-tone definition is constructed so that the ℓ₀-norm of the cyclic gradient vanishes precisely when a point satisfies the classical discrete simple-point conditions (no change in local topology). In the current manuscript we demonstrate this via the motivating discrete case and selected examples, but we did not include an exhaustive verification or theorem statement. In the revision we will add (i) a theorem proving that, when the input is strictly binary, the proposed ℓ₀-norm is zero if and only if the point is simple in the sense of Bertrand (or equivalent discrete definitions), and (ii) an exhaustive table/figure checking all 256 possible 3×3 binary configurations (and the analogous 3-D case if space permits). For non-binary intensities the definition is intentionally a continuous relaxation; we will explicitly state that equivalence is claimed only in the binary limit and provide a short continuity argument showing that small perturbations around 0/1 values preserve the simple/non-simple classification. revision: yes
Referee: [§4 (Variational Model)] The variational model enforces constraints by preserving non-simple points, but it is not clear from the formulation how this local constraint ensures global topological properties (such as correct Betti numbers) are maintained throughout the optimization. An analysis or additional experiments demonstrating no introduction of new topological artifacts would address this.

Authors: The local constraint is grounded in the classical digital-topology result that a point is simple if and only if its removal (or addition) does not alter the local homotopy type; when only simple points are modified, the global Betti numbers are therefore preserved. Our variational term penalizes the removal of non-simple points, thereby restricting the optimizer to topologically safe updates at every iteration. To make this connection explicit we will insert a short paragraph in §4 that recalls the relevant theorem from digital topology and explains why the soft constraint inherits the global invariance. In addition, we will add a new experiment on synthetic images whose ground-truth Betti numbers are known a priori; we will report the evolution of Betti numbers (β₀, β₁, β₂) during optimization and show that no spurious components, holes, or tunnels are created. These results will be summarized in a new table and a supplementary figure. revision: yes

Circularity Check

0 steps flagged

No circularity: novel continuous operator proposed as independent construction

full rationale

The derivation introduces an ℓ₀-norm of cyclic gradient as a new continuous-valued detector for simple points, then builds a variational model that preserves non-simple points inside differentiable losses for segmentation networks. This operator is presented as a direct proposal rather than derived from or fitted to the classical discrete binary simple-point test; the paper does not reduce the continuous definition to the discrete one by construction, nor rename a known result, nor invoke self-citations for a uniqueness theorem that would force the choice. The skeleton extraction and topology-enforcing loss follow directly from the new operator without statistical forcing or self-referential closure. The chain is therefore self-contained as an original ansatz with external experimental checks, yielding no load-bearing circular step.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review based on abstract only. The central claim rests on a new definition of simple points for continuous images and the claim that the resulting variational model enforces topology. No explicit free parameters, axioms, or invented entities are stated in the abstract.

axioms (1)

domain assumption The ℓ₀-norm of the cyclic gradient on continuous images identifies topologically simple points equivalently to the discrete binary definition.
This equivalence is required for the method to preserve topology while remaining differentiable.

invented entities (1)

Continuous-tone simple points no independent evidence
purpose: Enable differentiable detection of topologically removable points in non-binary images
New concept introduced to bridge discrete topology with continuous deep-learning outputs.

pith-pipeline@v0.9.0 · 5509 in / 1382 out tokens · 80055 ms · 2026-05-07T07:43:59.509138+00:00 · methodology

discussion (0)

Reference graph

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