Extremal Contours: Gradient-driven contours for compact visual attribution

Albert Alonso; Bulat Ibragimov; Frans Zdyb; Julius B. Kirkegaard; Reza Karimzadeh

arxiv: 2511.01411 · v1 · submitted 2025-11-03 · 💻 cs.CV · cs.LG· eess.IV

Extremal Contours: Gradient-driven contours for compact visual attribution

Reza Karimzadeh , Albert Alonso , Frans Zdyb , Julius B. Kirkegaard , Bulat Ibragimov This is my paper

Pith reviewed 2026-05-18 01:22 UTC · model grok-4.3

classification 💻 cs.CV cs.LGeess.IV

keywords visual attributionexplainable AIcontour optimizationFourier parameterizationstar-convex masksgradient-based explanationImageNet classifiers

0 comments

The pith

Smooth star-convex contours optimized by gradients deliver compact, faithful visual attributions that match dense masks in fidelity while using far fewer parameters.

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

The paper proposes replacing fragmented dense perturbation masks with smooth, tunable contours for explaining image classifier decisions. A star-convex region is defined by a truncated Fourier series and refined directly from classifier gradients under an extremal preserve-or-delete objective. This restriction to low-dimensional smooth shapes cuts the parameter count by orders of magnitude, guarantees a single connected region, and removes the need for post-processing cleanup. The resulting attributions maintain high relevance mass and run-to-run stability on ImageNet models, with larger gains reported on self-supervised DINO backbones. Explicit control over contour area further allows construction of importance profiles that trade fidelity against mask size.

Core claim

A training-free method parameterizes attribution regions as star-convex contours via truncated Fourier series and optimizes them under an extremal preserve/delete objective driven by classifier gradients, producing single simply-connected masks that achieve extremal fidelity comparable to dense masks yet with substantially lower complexity and improved consistency.

What carries the argument

Truncated Fourier series parameterization of star-convex contours optimized under an extremal preserve/delete objective using classifier gradients.

Load-bearing premise

Restricting the solution space to low-dimensional smooth star-convex contours will preserve faithfulness to the model's decision while preventing adversarial masking artifacts during gradient-based optimization.

What would settle it

A side-by-side evaluation on the same ImageNet images where the contour method yields lower preserve or delete scores than an unconstrained dense mask baseline optimized to the same objective.

Figures

Figures reproduced from arXiv: 2511.01411 by Albert Alonso, Bulat Ibragimov, Frans Zdyb, Julius B. Kirkegaard, Reza Karimzadeh.

**Figure 2.** Figure 2: Qualitative results on ImageNet images. Each column: input image, our [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Robustness of the method. (Top) Red circles denote different initial positions [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: Area-fidelity trade-off. (Left) Single closed contours at target areas [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: Examples of multiple contour optimization with [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

**Figure 6.** Figure 6: Convergence behavior of the Extremal Contour method across 10 images. [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗

read the original abstract

Faithful yet compact explanations for vision models remain a challenge, as commonly used dense perturbation masks are often fragmented and overfitted, needing careful post-processing. Here, we present a training-free explanation method that replaces dense masks with smooth tunable contours. A star-convex region is parameterized by a truncated Fourier series and optimized under an extremal preserve/delete objective using the classifier gradients. The approach guarantees a single, simply connected mask, cuts the number of free parameters by orders of magnitude, and yields stable boundary updates without cleanup. Restricting solutions to low-dimensional, smooth contours makes the method robust to adversarial masking artifacts. On ImageNet classifiers, it matches the extremal fidelity of dense masks while producing compact, interpretable regions with improved run-to-run consistency. Explicit area control also enables importance contour maps, yielding a transparent fidelity-area profiles. Finally, we extend the approach to multi-contour and show how it can localize multiple objects within the same framework. Across benchmarks, the method achieves higher relevance mass and lower complexity than gradient and perturbation based baselines, with especially strong gains on self-supervised DINO models where it improves relevance mass by over 15% and maintains positive faithfulness correlations.

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 uses a truncated Fourier series to parameterize star-convex contours for gradient-based extremal attribution, cutting parameters sharply while enforcing compact connected masks.

read the letter

The main point is that they replace dense perturbation masks with a low-dimensional star-convex contour whose radius is a truncated Fourier series, then optimize the coefficients directly with classifier gradients under a preserve/delete objective. This setup guarantees a single simply connected region, drops the free parameters by orders of magnitude, and removes the need for post-processing cleanup. They also add explicit area control so you can trace fidelity-area curves and extend the same framework to multiple contours for localizing several objects at once. On the positive side, the reported results show matching extremal fidelity to dense masks on ImageNet classifiers, higher relevance mass than gradient and perturbation baselines, and especially clear gains on DINO models with over 15% better relevance mass plus positive faithfulness correlations. The smoothness constraint is presented as a built-in guard against adversarial masking artifacts, and run-to-run consistency improves as a result. Those are concrete practical upsides for anyone who needs interpretable, controllable explanations without heavy tuning. The softer part is the fidelity claim. Restricting to smooth star-convex shapes means the method can at best find the best approximation within that class; if the truly optimal region has concavities or disconnected high-relevance pixels, the achieved score should be lower. The abstract asserts matching performance but does not show side-by-side final objective values or relevance-mass numbers between the contour method and an unconstrained dense mask optimized under identical loss and steps. Experimental details on baselines, error bars, and validation of the post-hoc claims are also light in the summary. This work is aimed at explainable-AI researchers who care about compact, geometrically constrained attributions for vision models, especially in applied settings where consistency and area control matter. A reader looking for efficient alternatives to noisy dense masks would get useful ideas from the parameterization. It is worth sending for peer review because the technical approach is distinct enough and the practical motivations are clear, even if some quantitative comparisons need tightening.

Referee Report

1 major / 2 minor

Summary. The paper introduces Extremal Contours, a training-free visual attribution method that parameterizes a star-convex region via a truncated Fourier series on the radius function r(θ) and optimizes the coefficients under an extremal preserve/delete objective driven by classifier gradients. It claims this yields compact, simply-connected masks that match the fidelity of dense perturbation masks on ImageNet classifiers, improve relevance mass and run-to-run consistency, reduce complexity relative to gradient and perturbation baselines, and deliver especially large gains (>15% relevance mass) on DINO models while supporting explicit area control and multi-contour extensions.

Significance. If the central claims hold, the work provides a principled way to obtain faithful yet low-complexity attributions by restricting the feasible set to smooth, low-dimensional star-convex contours. The reduction from thousands of mask pixels to ~2N Fourier coefficients, the built-in guarantee of a single connected region, and the transparent fidelity-area profiles constitute concrete advances over fragmented dense-mask baselines. Strong reported gains on self-supervised DINO models and the absence of post-processing cleanup are additional strengths.

major comments (1)

[Abstract and experimental results] The central claim that the method 'matches the extremal fidelity of dense masks' (abstract) is load-bearing yet unsupported by a direct quantitative comparison. No table or section reports the final preserve/delete objective value (or relevance mass) achieved by the Fourier-contour optimizer versus an unconstrained dense mask optimized under identical loss, gradient steps, and initialization. Because the star-convex smooth parameterization is a strict subset of all possible masks, any reported fidelity match must be verified by showing that the restricted optimum reaches the same score; otherwise the restriction may silently incur a fidelity penalty when the true extremal region contains concavities or disconnected pixels.

minor comments (2)

[Abstract] The abstract states 'improved run-to-run consistency' and 'over 15% relevance mass' gains on DINO models but supplies no error bars, number of runs, or exact baseline implementations; these quantitative details should be added to the experimental section.
[Method] The truncation order N of the Fourier series and the area-control parameter are listed as free parameters; their chosen values and sensitivity analysis should be reported explicitly (e.g., in a table or appendix) to allow reproduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive review. The point raised about verifying the fidelity claim with a direct comparison is well-taken, and we address it below along with plans for revision.

read point-by-point responses

Referee: [Abstract and experimental results] The central claim that the method 'matches the extremal fidelity of dense masks' (abstract) is load-bearing yet unsupported by a direct quantitative comparison. No table or section reports the final preserve/delete objective value (or relevance mass) achieved by the Fourier-contour optimizer versus an unconstrained dense mask optimized under identical loss, gradient steps, and initialization. Because the star-convex smooth parameterization is a strict subset of all possible masks, any reported fidelity match must be verified by showing that the restricted optimum reaches the same score; otherwise the restriction may silently incur a fidelity penalty when the true extremal region contains concavities or disconnected pixels.

Authors: We agree that the manuscript would be strengthened by an explicit head-to-head comparison of the final preserve/delete objective (and relevance mass) between the Fourier-contour optimizer and an unconstrained dense mask under identical loss, gradient steps, and initialization. The current version reports that Extremal Contours match the fidelity of dense perturbation baselines on ImageNet classifiers and achieve higher relevance mass than gradient/perturbation methods, but does not include this specific controlled experiment against an optimized dense mask. In the revised manuscript we will add a new table (and corresponding text in the experimental section) that performs this direct comparison, reporting the achieved objective values for both approaches. This will allow readers to assess whether the star-convex restriction incurs any measurable fidelity penalty in practice. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper introduces a training-free method that parameterizes star-convex regions via truncated Fourier series and optimizes them directly under an extremal preserve/delete objective using standard classifier gradients. All load-bearing steps (parameterization, optimization, and reported fidelity gains) are defined from first principles and external benchmarks rather than reducing to self-fitted quantities, prior author results, or tautological redefinitions. No self-citation chains, ansatzes smuggled via citation, or uniqueness theorems imported from the same authors appear in the provided text; the central claims rest on empirical comparisons that remain independently falsifiable.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim depends on geometric constraints and gradient utility for a new parameterization; limited details available from abstract only.

free parameters (2)

Fourier series truncation order
Controls the number of coefficients and thus the smoothness versus expressiveness of the contour boundary; chosen to achieve orders-of-magnitude parameter reduction.
Area control parameter
Explicit scalar used to generate importance contour maps at varying region sizes.

axioms (2)

domain assumption Attribution regions can be adequately represented as star-convex sets
Invoked to guarantee a single simply connected mask without fragmentation.
domain assumption Classifier gradients supply reliable signals for boundary optimization under extremal preserve/delete objectives
Used to drive the training-free contour updates.

pith-pipeline@v0.9.0 · 5755 in / 1529 out tokens · 49276 ms · 2026-05-18T01:22:21.845171+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

A star-convex region is parameterized by a truncated Fourier series and optimized under an extremal preserve/delete objective using the classifier gradients.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The approach guarantees a single, simply connected mask, cuts the number of free parameters by orders of magnitude

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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