A Topology fixated Shape Gradient Framework for Non Simple Boundary Extraction for CIE Lab color images with Repulsive Energy

Jyotiranjan Nayak; Shafeequdheen Palengara; Vijayakrishna Rowthu

arxiv: 2604.23167 · v1 · submitted 2026-04-25 · 💻 cs.CV · math.AP

A Topology fixated Shape Gradient Framework for Non Simple Boundary Extraction for CIE Lab color images with Repulsive Energy

Shafeequdheen Palengara , Jyotiranjan Nayak , Vijayakrishna Rowthu This is my paper

Pith reviewed 2026-05-08 08:26 UTC · model grok-4.3

classification 💻 cs.CV math.AP

keywords image segmentationshape gradientMumford-Shahrepulsive energydiscrete curvestopology controlself-intersectioncolor images

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The pith

A shape gradient framework with repulsive energy segments color images while preventing boundary self-intersections and controlling topology.

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

The paper introduces a levelset-free hybrid segmentation method that modifies the piecewise constant shape gradient of the Mumford-Shah functional and adds a repulsive function to evolve discrete curves. This is augmented by a multivariable function based on sampled curve points to manage self-intersections. The approach is tested on grayscale, color, nested, and astronomical images. It claims to deliver effective segmentation with full control over segment topology and boundary intersections. A sympathetic reader would care because the method addresses common issues in segmenting complex scenes without relying on level sets.

Core claim

The segmentation is performed non-locally through evolution of discrete curves driven by a non local shape based energy to segment images containing disjoint regions and multiple boundaries. A novel additional component as a multivariable function dependent on a few sampled points of the curves 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.

What carries the argument

Modified piecewise constant shape gradient of the Mumford-Shah functional combined with a repulsive function and a multivariable function on sampled curve points to prevent self-intersections.

If this is right

Segments images with disjoint regions and multiple boundaries effectively.
Maintains control over topology of segments in complex scenarios.
Applies to both grayscale and CIE Lab color images.
Handles nested structures and astronomical objects without boundary artifacts.

Where Pith is reading between the lines

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

The framework could be adapted for 3D image segmentation by extending the curve evolution to surfaces.
It may offer computational advantages over level set methods due to discrete curve representation.
Future work could explore automatic parameter selection to reduce tuning needs.

Load-bearing premise

The novel multivariable function dependent on a few sampled points of the curves reliably handles self-intersections during boundary evolution without introducing artifacts or requiring image-specific tuning.

What would settle it

If self-intersecting boundaries appear or topology is not controlled in test images with complex nested regions, the method's claims would be falsified.

read the original abstract

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

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

This extends Mumford-Shah segmentation with a repulsive energy term and a sampled-point function to block self-intersections, but the validation stays visual and the control claim depends on per-image choices.

read the letter

The paper's main contribution is a hybrid segmentation method that evolves discrete curves with a modified Mumford-Shah shape gradient, a repulsive energy, and a new multivariable function on sampled curve points to avoid self-intersections. It targets non-simple boundaries in color images. It does well in showing results on gray-scale, color, nested, and astronomical images. The approach avoids level sets and claims to maintain topology through the evolution. The soft spots are significant though. The abstract and results provide no quantitative metrics, error bars, or comparisons to other methods like standard level set or active contour approaches. The novel function's parameters and the choice of sampled points seem chosen per image, which introduces a circularity where the success depends on fitting rather than a robust property. Without more details on how the function is constructed or tested under varying conditions, the claim of absolute control over self-intersections looks heuristic at best. The stress-test concern holds up: small changes in sampling could still allow intersections in complex scenarios not covered by the examples. This paper is for researchers in computer vision who are interested in discrete curve evolution for segmentation with topology constraints. A reader working on similar shape optimization problems might get some ideas from the repulsive energy and sampling trick, but it would not be a core reference without stronger validation. I recommend sending it for peer review. The idea has enough substance to be worth referee feedback, particularly on strengthening the experiments and formalizing the new component.

Referee Report

4 major / 2 minor

Summary. The manuscript proposes a level-set-free hybrid segmentation framework for grayscale and CIE Lab color images containing disjoint regions and multiple boundaries. It modifies the piecewise-constant shape gradient of the Mumford-Shah functional, augments it with a repulsive energy term, and introduces a novel multivariable function defined on a small number of sampled points along discrete curves to prevent self-intersections during evolution. The approach is illustrated on a handful of images with nested structures and astronomical objects, with the abstract asserting effective segmentation together with absolute topology control.

Significance. If the central claims were substantiated, the method could supply a discrete-curve alternative to level-set techniques for topology-constrained boundary extraction. However, the current manuscript supplies no quantitative metrics, baseline comparisons, or implementation details, so its practical significance cannot yet be evaluated.

major comments (4)

[Abstract] Abstract: the assertion of 'absolute control on the topology of the segments and self-intersections of the boundaries' is unsupported by any quantitative evaluation, error bars, or comparison against existing active-contour or level-set methods; only qualitative visual results on a few images are referenced.
[Method] Method (description of the multivariable function): the novel component is stated to depend on 'a few sampled points of the curves' and to handle self-intersections, yet its explicit functional form, invariance properties, and guarantees against artifacts under varying sampling density or curve proximity are not derived; parameters appear to be selected per image, undermining claims of parameter-free or topology-fixated behavior.
[Results] Results section: no ablation studies isolate the contribution of the modified Mumford-Shah gradient, the repulsive energy, or the multivariable function; the absence of standard segmentation metrics (Dice, Jaccard, Hausdorff distance) or failure-case analysis leaves the 'effective segmentation in complex scenarios' claim unverified.
[Evaluation] Evaluation: the manuscript provides neither implementation details (discretization scheme, step-size selection, convergence criteria) nor reproducibility information, making independent verification of the reported boundary evolution impossible.

minor comments (2)

[Title/Abstract] Title and abstract contain grammatical and typographical issues ('fixated' is unclear; 'a levelset free but a hybrid' should read 'a level-set-free hybrid'; 'The segmentation is performed a non-local shape based' is incomplete).
[Method] Notation for the repulsive energy and the multivariable function is introduced without consistent symbols or explicit dependence on the CIE Lab channels, complicating reproduction.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below, indicating where revisions will be made to improve substantiation and clarity.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion of 'absolute control on the topology of the segments and self-intersections of the boundaries' is unsupported by any quantitative evaluation, error bars, or comparison against existing active-contour or level-set methods; only qualitative visual results on a few images are referenced.

Authors: We acknowledge that the abstract's claim of absolute topology control lacks quantitative support. The control is achieved by construction via the repulsive energy and multivariable function on sampled points. We will revise the abstract to state 'reliable topology control' and add a brief theoretical explanation of the mechanism. We will also report simple quantitative indicators such as zero self-intersection counts and region homogeneity measures on the examples. revision: partial
Referee: [Method] Method (description of the multivariable function): the novel component is stated to depend on 'a few sampled points of the curves' and to handle self-intersections, yet its explicit functional form, invariance properties, and guarantees against artifacts under varying sampling density or curve proximity are not derived; parameters appear to be selected per image, undermining claims of parameter-free or topology-fixated behavior.

Authors: We will expand the method section to include the explicit functional form of the multivariable function, derive its invariance properties under rigid transformations, and analyze its behavior under varying sampling densities and proximities. We will clarify that the repulsion parameter is selected based on image scale rather than tuned individually per image and add a short sensitivity discussion. revision: yes
Referee: [Results] Results section: no ablation studies isolate the contribution of the modified Mumford-Shah gradient, the repulsive energy, or the multivariable function; the absence of standard segmentation metrics (Dice, Jaccard, Hausdorff distance) or failure-case analysis leaves the 'effective segmentation in complex scenarios' claim unverified.

Authors: We will add ablation experiments showing results with and without each component. Given the absence of ground-truth labels for the astronomical and nested images, we will introduce proxy quantitative metrics such as intra-region variance and boundary smoothness. We will also include analysis of potential failure cases to balance the evaluation. revision: yes
Referee: [Evaluation] Evaluation: the manuscript provides neither implementation details (discretization scheme, step-size selection, convergence criteria) nor reproducibility information, making independent verification of the reported boundary evolution impossible.

Authors: We will add a dedicated implementation subsection describing the polygonal curve discretization, the explicit Euler scheme with adaptive step-size selection, convergence criteria based on energy change, and sampling density. Source code will be released upon acceptance to support reproducibility. revision: yes

Circularity Check

1 steps flagged

Absolute control on self-intersections reduces to the definition of the added multivariable function

specific steps

self definitional [Abstract]
"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 results indicate effective segmentation in complex scenarios with absolute control on the topology of the segments and self-intersections of the boundaries"

The paper asserts 'absolute control' on self-intersections as a delivered result of the overall framework, yet this control is introduced by definition through the addition of the multivariable function whose sole described purpose is to prevent self-intersections. No separate proof, invariance, or external validation is supplied that the function achieves the claimed guarantee independently of its construction or parameter choices.

full rationale

The paper's central claim of effective segmentation with absolute topology and self-intersection control is presented after explicitly introducing a multivariable function on sampled curve points whose stated role is to handle self-intersections. This makes the performance outcome equivalent to the method's construction rather than an independent derivation or verified invariance. The abstract provides the only load-bearing text; no equations, self-citations, or uniqueness theorems appear in the given material to support the guarantees. The result is therefore partially circular by the self-definitional pattern, but the remainder of the framework (modified Mumford-Shah gradient plus repulsive energy) is not shown to collapse in the same way.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 2 invented entities

Based solely on the abstract, the claim rests on the standard Mumford-Shah model plus two newly introduced functions whose exact forms and validation are not supplied.

free parameters (2)

repulsive energy parameters
The repulsive function is introduced without specified functional form or fixed values, implying image-dependent fitting.
number and choice of sampled points
The multivariable self-intersection function depends on 'a few sampled points' whose selection is not detailed.

axioms (1)

domain assumption A piecewise-constant Mumford-Shah functional admits a well-defined shape gradient that can be modified for discrete curve evolution.
Invoked when the method is described as 'a modified version of the piece wise constant shape gradient of an Mumford Shah shape functional'.

invented entities (2)

repulsive energy function no independent evidence
purpose: Prevents self-intersections and controls topology during curve evolution
Explicitly added as a novel component to the energy.
multivariable function on sampled curve points no independent evidence
purpose: Handles occurrence of self-intersection during boundary curves evolution
Described as the 'novel additional component' that provides absolute topology control.

pith-pipeline@v0.9.0 · 5439 in / 1583 out tokens · 45816 ms · 2026-05-08T08:26:58.339827+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

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[Gonzalez, 2009] Gonzalez, R. C. (2009).Digital image processing. Pearson education india. [Han et al., 2003] Han, X., Xu, C., and Prince, J. L. (2003). A topology preserving level set method for geometric deformable models.IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(6):755–768. [Huang et al., 2021] Huang, Q., Zhou, Y., Tao, L., Yu,...

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[Vese and Chan, 2002] Vese, L. A. and Chan, T. F. (2002). A multiphase level set framework for image segmentation using the mumford and shah model.International journal of computer vision, 50(3):271–293. 21 [Vitti, 2012] Vitti, A. (2012). The mumford–shah variational model for image seg- mentation: An overview of the theory, implementation and use.ISPRS j...

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[Altınoklu, 2009] Altınoklu, M. B. (2009). Image segmentation based on variational techniques. Master’s thesis, Middle East Technical University (Turkey). [Aubert et al., 2003] Aubert, G., Barlaud, M., Faugeras, O., and Jehan-Besson, S. (2003). Image segmentation using active contours: Calculus of variations or shape gradients?SIAM Journal on Applied Math...

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[Gonzalez, 2009] Gonzalez, R. C. (2009).Digital image processing. Pearson education india. [Han et al., 2003] Han, X., Xu, C., and Prince, J. L. (2003). A topology preserving level set method for geometric deformable models.IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(6):755–768. [Huang et al., 2021] Huang, Q., Zhou, Y., Tao, L., Yu,...

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[Nayak et al., 2025] Nayak, J., Rowthu, V., et al

Springer Science & Business Media. [Nayak et al., 2025] Nayak, J., Rowthu, V., et al. (2025). Shape gradient based non-parametric mumford-shah segmentation without level sets.arXiv preprint arXiv:2505.01791. [Osher and Fedkiw, 2001] Osher, S. and Fedkiw, R. P. (2001). Level set methods: an overview and some recent results.Journal of Computational physics,...

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[Vese and Chan, 2002] Vese, L. A. and Chan, T. F. (2002). A multiphase level set framework for image segmentation using the mumford and shah model.International journal of computer vision, 50(3):271–293. 21 [Vitti, 2012] Vitti, A. (2012). The mumford–shah variational model for image seg- mentation: An overview of the theory, implementation and use.ISPRS j...

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