A Fuzzy Edge Detector Driven Telegraph Total Variation Model For Image Despeckling

Ananta K Majee; Rajendra K Ray; Subit K Jain; Sudeb Majee

REVIEW 4 major objections 4 minor 2 cited by

A Fuzzy Edge Detector Driven Telegraph Total Variation Model For Image Despeckling

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A fuzzy edge indicator drives a telegraph total variation despeckling model that beats five existing PDE filters on natural and SAR images, and whose regularized form has a unique weak solution.

desk verdict A useful engineering hybrid with a real theorem, but the theorem doesn't cover the implemented scheme and the variational derivation needs fixing. read the letter →

arxiv 1908.01134 v2 pith:DGKV2ZPN submitted 2019-08-03 eess.IV cs.NAmath.APmath.NA

Sudeb Majee , Subit K Jain , Rajendra K Ray , Ananta K Majee This is my paper · ORCID

classification eess.IVcs.NAmath.APmath.NA MSC 35K5565M0668U1094A08

keywords imagedespecklingspecklenoisetelegraphtotalvariationfuzzyedgedetectionintuitionisticdivergenceweaksolutionSchauderfixedpointtheoremSAR

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Speckle noise is multiplicative, so at high noise levels gradient-based edge detectors can mistake noise for edges. This paper proposes replacing the gradient-only smoothing coefficient in a telegraph total variation equation with a fuzzy edge indicator $\theta(I)=1-F(I)$, where $F$ is the intuitionistic fuzzy divergence between pixel neighborhoods and fixed edge templates. The paper claims this hybrid removes speckle while preserving edges better than five existing PDE-based despeckling models on natural and real SAR images, as judged by PSNR, MSSIM, speckle index, BRISQUE, ratio images, and line profiles. It further claims that the regularized version of the equation has one and only one weak solution, proved through Schauder fixed-point arguments. If right, this gives a principled route for folding fuzzy edge/noise classification into hyperbolic total variation despeckling.

What carries the argument

The load-bearing object is the fuzzy edge indicator $\theta(I)=1-F(I)$, where $F$ is the intuitionistic fuzzy divergence (IFD) obtained by matching each pixel neighborhood against a set of edge templates; $\theta$ is small near edges, close to 1 in homogeneous regions, and it replaces the gradient-only coefficient in the total variation term. In the well-posedness proof the operative object is the regularized coefficient $g_w = \theta(G_\xi*w)/(1+|\nabla G_\xi*w|)$. Its uniform lower bound $\delta/(1+C(G_\xi,\|I_0\|_{H^1}))$ and time-derivative bound make the linearized problem amenable to Galerkin energy estimates, and Schauder's fixed-point theorem yields a fixed point of $w\mapsto I_w$. Uniqueness is shown by testing the difference of two solutions with a time-integrated test function and applying Gronwall's lemma on short time intervals.

What would settle it

Compute $\theta(I)=1-F(I)$ on the actual test images at the noise levels used in the experiments and check whether $\delta\le\theta\le 1$ and $|\theta(x)-\theta(y)|\le C_\theta|x-y|$ hold for some positive $\delta$; alternatively, solve equation (4) and scheme (28) on the same image and compare. If $\theta$ touches zero or the discrete scheme diverges from the regularized PDE, Theorem 4.1 no longer applies to the implemented despeckling filter.

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Extended reading notes

Core claim

The paper's central claim is that the telegraph total variation equation $I_{tt}+\gamma I_t = \mathrm{div}(\theta(I)\nabla I/|\nabla I|) - \lambda(1-I_0/I)$, with $\theta(I)=1-F(I)$ computed from intuitionistic fuzzy divergence, suppresses multiplicative speckle noise while preserving edges better than the compared diffusion and variational models. The reported experiments show higher PSNR and MSSIM, lower speckle index and BRISQUE, and cleaner ratio images for noise looks $L=1,3,5,10,33$, with better edge contrast in line profiles, contours, and 3D surfaces on natural and real SAR images. The theoretical result is Theorem 4.1: the regularized equation (4), where $\theta(I)$ is replaced by $\theta(G_\xi*I)/(1+|\nabla G_\xi*I|)$, admits one and only one weak solution under assumptions A.1 and A.2. The authors present this as the first use of a fuzzy edge detector inside a telegraph total variation framework for multiplicative noise removal.

Load-bearing premise

The load-bearing premise is that the fuzzy divergence edge detector satisfies Assumption A.2, namely $\delta\le\theta\le 1$ and Lipschitz continuity, and that the regularized coefficient in equation (4) faithfully represents the discrete $\theta$ computed by template matching in the numerical scheme (28); the paper demonstrates neither for the implemented filter.

Editorial extensions

If this is right

At the reported noise levels ($L=1$ to $33$), the model attains the highest PSNR and MSSIM and the lowest speckle index among the five compared PDE-based models on the tested natural images.
On the single-look real SAR image, the model attains the lowest speckle index and BRISQUE, indicating stronger speckle suppression without reference ground truth.
Because the energy functional remains convex and the regularized PDE is well-posed, the fuzzy edge indicator can be combined with the existing total-variation fidelity term without losing the guarantee of a unique weak solution.
The telegraph (hyperbolic) structure keeps edge profiles sharper than parabolic diffusion alone, which the paper states as the reason for adopting this framework.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The existence proof covers the regularized equation (4), not the explicit finite-difference scheme (28); showing that the discrete iterates converge to the weak solution would require an additional stability and consistency analysis, which the paper leaves implicit.
The IFD edge measure $F$ is defined from fixed templates, so the edge indicator is not rotation- or scale-invariant by construction; enriching the template set or learning templates is a natural testable extension.
A direct empirical check of Assumption A.2 on real noisy images would reveal whether the theoretical theorem governs the actual experiments, since the paper does not report such measurements.
The same construction could be applied to other multiplicative-noise fidelities or to texture-preserving regularization, a direction the conclusion identifies as future work.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The paper combines a telegraph total variation equation with an intuitionistic fuzzy divergence edge indicator for multiplicative speckle removal. That specific combination is new, and the numerical experiments are reasonably thorough: five PDE baselines, five natural images plus a real SAR image, and several quality metrics. The reported gains are modest but consistent, which is what you'd expect from an incremental but sensible filter design.

The well-posedness theorem for the regularized version—existence and uniqueness of a weak solution for θ evaluated on a Gaussian-smoothed image—is a genuine formal contribution, assuming the Lipschitz and positivity bounds on θ. The proof via Schauder fixed point is standard but not sloppy, and the energy estimates check out.

The soft spots are real though. First, the Euler-Lagrange equation (2) ignores that θ depends on I. The first variation of ∫θ(I)|∇I| includes a term θ'(I)|∇I|, so (2) is not the correct descending flow, and the claim that the energy is globally convex is unsupported. Second, and more load-bearing, the numerical scheme (28) is not a discretization of the regularized model (4). In (4) the diffusion coefficient is θ(Gξ*I)/(1+|∇Gξ*I|); in (28) it is the template-based IFD coefficient θ(I^n) with no Gaussian convolution and no normalization. Theorem 4.1 therefore applies to a different model. The authors would need a consistency argument linking the two, and they also never verify Assumption A.2 for the IFD-based θ. That assumption can actually fail: the IFD value F can exceed 1 near strong edges, making θ=1-F negative.

None of this kills the empirical contribution. The engineering heuristic is reasonable and the experiments support it. But as it stands, the theory and the numerics are disconnected, and the variational derivation needs correction.

Who should read it: anyone working on PDE-based despeckling or fuzzy edge indicators. It deserves peer review because the core idea is worth pursuing, but it needs substantial revision—fix the variational statement, align the numerical scheme with the analyzed equation, verify or replace A.2, and release parameters and code.

Referee Report

4 major / 4 minor

Summary. The manuscript proposes a despeckling model that combines a telegraph total variation diffusion term with a fuzzy edge indicator θ(I)=1−F(I), where F is an intuitionistic fuzzy divergence measure defined in Section 2.3. Starting from the energy functional (1), the authors derive the Euler-Lagrange equation (2) and the evolutionary telegraph total variation model (3), prove existence and uniqueness of a weak solution for the regularized version (4)–(6) under Assumptions A.1–A.2 (Theorem 4.1), and give an explicit finite-difference scheme (28) with a relative-error stopping criterion. Experiments compare the proposed method against five existing PDE-based despeckling models on five natural/synthetic images at looks 1, 3, 5, 10, and 33 and on a real single-look SAR image, using PSNR, MSSIM, SI, and BRISQUE metrics.

Significance. The idea of coupling a fuzzy edge detector with a hyperbolic telegraph total variation equation for multiplicative speckle removal is reasonable and relatively unexplored, and the experimental section is extensive and consistently favorable to the proposed method across images and metrics. The paper also makes a serious attempt at a well-posedness analysis with explicit assumptions and a Schauder fixed-point argument. However, the theoretical and numerical parts are not connected: Theorem 4.1 covers a regularized coefficient that is not what is implemented in (28), and the key regularity and positivity assumption A.2 is never verified for the proposed IFD-based edge indicator. These gaps directly affect the central claim that the proposed model is mathematically justified, although they do not by themselves disprove the empirical results. With substantial revision, the contribution could be publishable.

major comments (4)

[§3.1, Eqs. (1)–(2)] The Euler-Lagrange equation (2) is not the first variation of the energy (1): since the edge indicator θ depends explicitly on I, the variation of ∫_Ω θ(I)|∇I| dx contains the additional term θ'(I)|∇I|, which is absent from (2). Consequently the gradient flow (3) is not the descent equation associated with the stated energy, and the claim in Section 3.2 that the associated variational problem has a unique minimizer is not supported by the derivation. The authors should either correct the Euler-Lagrange equation and re-derive the evolution model, or reformulate θ as a coefficient that is fixed during the minimization, for example by computing θ from a pre-smoothed version of the observed image.
[§3.2, energy (1)] The assertion in Section 3.2 that “the energy functional (1) is globally convex” is unsupported and in general false. With a nonconstant θ(I), the term ∫_Ω θ(I)|∇I| dx is not necessarily convex in I, and the fidelity term ∫_Ω (I + I0 log(1/I)) dx is strictly convex only under a restricted range condition, as the discussion of the AA model in Section 2.2 indicates. The cited convexity result in [16] applies to a coefficient α(x) that is independent of the unknown I. This bullet point should be removed or replaced by a precise convexity statement with proof.
[§4, Assumption A.2 vs. §2.3] Assumption A.2 is never verified for the proposed fuzzy indicator θ(I)=1−F(I). In the IFD construction of Section 2.3, F can exceed 1: for a template with μ_P=1 and μ_Q=0, the divergence expression gives F=2−2e^{−1}>1, so θ=1−F can be negative. Thus the lower bound δ≤θ≤1 in A.2, which is used in the proof of Theorem 4.1 to control the coefficient g_w from below (see Eq. (10) and estimate (12)), is not established. The authors need either to prove A.2 for their θ, or to modify the definition of θ so that it provably lies in [δ,1] and is Lipschitz.
[§5, Eq. (28) vs. Eq. (4)] The numerical scheme (28) does not discretize the regularized equation (4) for which Theorem 4.1 is proved. In (4) the diffusion coefficient is θ(G_ξ*I)/(1+|∇G_ξ*I|), whereas in (28) the coefficient is θ(I^n_{i,j})=1−F(I^n_{i,j}), using the nonlocal max/min template divergence F of Section 2.3; no Gaussian convolution G_ξ*I or denominator 1+|∇G_ξ*I| appears, and no value of ξ is reported. Unless a consistency argument is supplied showing that the discrete scheme is a convergent approximation of (4), the existence-uniqueness theorem does not cover the implemented filter, and the experiments in Section 6 cannot be presented as validating the model analyzed in Section 4.

minor comments (4)

[§6, Figs. 2–4 and Tables 1–3] The baseline method is called “TPM” in the caption of Fig. 2 but “TDM” in Tables 1–3; please unify the notation.
[§6.1] The statement that “different parameters of considered models are optimized manually” means that the comparison partly reflects tuning effort; a sensitivity analysis or an automatic parameter-selection rule would make the empirical comparison more robust.
[§6.2, Tables 1–2] The quantitative results appear to be single realizations: no standard deviations or number of noise realizations are reported, so the statistical significance of the observed improvements is unclear.
[§4.3 and §2.3] There are minor typographical issues, e.g., “in the sence of distribution” in Section 4.3 and the inconsistent spelling “Attanassov” versus “Atanassov” in Section 2.3.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; theorem-experiment mismatch and parameter tuning are rigor/fairness issues, not circular derivation.

full rationale

The paper's central claims are (i) that the proposed telegraph total variation model with fuzzy edge indicator outperforms five baseline methods in despeckling benchmarks and (ii) that the regularized version (4)-(6) admits a unique weak solution. Neither claim reduces to its own inputs. The numerical comparisons in Section 6.3 are against independently corrupted test images using PSNR, MSSIM, SI, and BRISQUE; these metrics are not fitted into the model to produce the reported output. The manually optimized parameters in Section 6.1 affect the experimental comparison but are not fitted inputs renamed as predictions, so they are a fairness concern rather than circularity. The existence and uniqueness proof in Section 4 is an in-paper Schauder fixed-point argument that assumes A.1-A.2 and does not assume the conclusion; the cited Evans textbook is external, and the authors' self-citations [21]-[24] appear only in related work and numerical background, not as the load-bearing justification for well-posedness or superiority. The significant gaps are correctness/rigor defects, not circular ones: Assumption A.2 is never verified for the actual fuzzy indicator theta(I)=1-F(I), the Euler-Lagrange equation (2) omits the theta'(I) contribution, and the discretized filter (28) uses theta(I^n_{i,j}) without the Gaussian-regularized denominator of the analyzed model (4). These defects disconnect the theorem from the implementation, but they do not make the derivation equivalent to its inputs. No prediction in the paper is forced by construction, and no self-citation chain supplies the central result. Therefore no significant circularity is found.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central results depend on the stated regularity of I0 and θ, on the validity of the regularized model as a proxy for the discrete algorithm, and on the unproven convexity of the energy. No new physical entities are introduced; the free parameters are the manually tuned weights and the unspecified Gaussian scale.

free parameters (4)

damping coefficient γ = not reported
Section 6.1 says parameters of all models are manually optimized for best performance; γ is one of the three parameters on which the solution depends in Section 5(d), but its values are never listed.
fidelity weight λ = not reported
λ weights the fidelity term in equation (28) and is manually tuned per experiment without reporting values.
stopping threshold ε = 10^-4 or smaller
The stopping criterion in equation (29) uses ε ≤ 10^-4, which affects the number of iterations and the final output, although it is a standard convergence tolerance.
Gaussian scale ξ in Gξ = not reported
The regularized model and the theoretical section use the convolution scale ξ in Gξ*I; no value is given, and the numerical behavior may depend on it.

assumptions (5)

domain assumption Initial image I0 lies in H2(Ω) and satisfies 0<α≤I0≤β with finite α, β (Assumption A.1).
Used to define the solution space W and control the fidelity term 1-I0/I in Sections 4.1 and 4.2.
ad hoc to paper The fuzzy edge indicator θ satisfies δ≤θ≤1 and is Lipschitz with constant Cθ (Assumption A.2).
The theorem requires this regularity, but the actual θ(I)=1-F(I) built from the IFD max/min template divergence is never checked against it.
ad hoc to paper The regularized equation (4), with θ(Gξ*I)/(1+|∇Gξ*I|), is a valid continuous surrogate for the discrete fuzzy edge model implemented in (28).
The numerical θ uses a nonlocal sliding-window Div_measure, while the analysis uses a local regularized coefficient; no equivalence or convergence result is supplied.
ad hoc to paper The energy functional (1) is globally convex in I.
Section 3.2 asserts convexity by citing [16], but [16] treats a fixed gray-level indicator α(x); here θ(I) depends on I, so the functional is not shown to be convex and the Euler-Lagrange derivation omits θ'(I) terms.
standard math Compact Sobolev embedding and the Schauder fixed point theorem are applicable.
Used to pass to the limit and obtain a fixed point in Section 4.2.2.

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Cite this review

Pith. "Pith review of A Fuzzy Edge Detector Driven Telegraph Total Variation Model For Image Despeckling." pith.science (2026). https://pith.science/paper/DGKV2ZPN

@misc{pith2026190801134,
  author       = {Pith},
  title        = {Pith review of: A Fuzzy Edge Detector Driven Telegraph Total Variation Model For Image Despeckling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DGKV2ZPN}},
  note         = {Machine review of arXiv:1908.01134}
}

read the original abstract

Speckle noise suppression is a challenging and crucial pre-processing stage for higher-level image analysis. In this work, a new attempt has been made using telegraph total variation equation and fuzzy set theory for speckle noise suppression. The intuitionistic fuzzy divergence (IFD) function has been used to distinguish between edges and noise. To the best of the author's knowledge, most of the studies on multiplicative speckle noise removal process focus on only diffusion-based filters, and little attention has been paid to the study of fuzzy set theory. The proposed approach enjoy the benefits of both telegraph total variation equation and fuzzy edge detector, which is not only robust to noise but also preserves image structural details. Moreover, we establish the existence and uniqueness of a weak solution of the regularized version of the proposed model using Schauder fixed point theorem. With the proposed model, despeckling is carried out on natural and Synthetic Aperture Radar (SAR) images. The experimental results of the proposed model are reported, which found better in terms of noise suppression and detail/edge preservation, with respect to the existing approaches.

Figures

Figures reproduced from arXiv: 1908.01134 by the authors.

**Figure 1.** Test Images. in terms of PSNR[18], MSSIM[40], speckle index (SI)[15] and blind/referenceless image spatial quality evaluator (BRISQUE)[30] are shown with existing models. A higher value of MSSIM and PSNR confirm that the recovered output is closer to the ground truth information. Whereas, for the optimal filtering, computed values of SI and BRISQUE should be minimum. Another typical qualitative measures is also com… view at source ↗

**Figure 2.** (a) Original (b) Noisy: L = 10 (c) Line profile (d-f) TPM (g-i) Dong (j-l) DDDM (m-o) Proposed. First column: Images. Middle column: Ratio images. Last column: Line profile showing 1D details. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** Comparison of contours and 3D Surface Plots. (a-b) [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: (a) Image1: One look radar image [2], (b) Restored i [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Well-posedness study of a non-linear hyperbolic-parabolic coupled system applied to image speckle reduction
math.AP 2019-08 conditional novelty 5.0 of 10

A telegraph-diffusion model coupled to an edge-variable reaction-diffusion equation is proven to have a unique weak solution, and it modestly improves despeckling metrics over two comparison models on three test images.
A Gray Level Indicator-Based Regularized Telegraph Diffusion Equation Applied to Image Despeckling
math.NA 2019-08 conditional novelty 4.0 of 10

A gray level indicator based telegraph diffusion model for speckle noise removal is proposed, with a well-posedness proof and modest numerical gains over one baseline on three test images.

Reference graph

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