Convex hull algorithms based on some variational models

Jiang Yang; Lingfeng Li; Shousheng Luo; Xue-Cheng Tai

REVIEW 4 major objections 6 minor 34 references

Convex hull algorithms based on some variational models

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that the exact convex hull of one or multiple binary objects is the zero sublevel set of a minimizer of an area-based energy, subject to a signed-distance constraint and a Laplacian convexity constraint, and that the…

desk verdict The outlier model is a genuinely useful contribution, but the exact model's multi-object convexity claim rests on an unproven converse that fails for disconnected sets. read the letter →

arxiv 1908.03323 v1 pith:2WPE6AR4 submitted 2019-08-09 cs.CV

Lingfeng Li , Shousheng Luo , Xue-Cheng Tai , Jiang Yang This is my paper · ORCID

classification cs.CV MSC 68U0568U1035A15

keywords convexhulllevelsetmethodsigneddistancefunctionvariationalmodelADMMoutlierrejectionbinaryimageconvexityconstraint

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

This paper proposes two variational models that find convex hulls of 2D binary images by searching for a level-set function rather than polygon vertices. The first model minimizes the area of the zero sublevel set while forcing the signed distance function to satisfy $|\nabla\varphi|=1$, to be nonpositive on the object, and to have nonnegative Laplacian inside its sublevel sets; the paper claims this returns the exact convex hull. The second model replaces the enclosure constraint with a penalty on the distance from each point to the zero sublevel set, so outliers and noise can be excluded. Numerical experiments report relative Hausdorff errors under 2% for the exact model and under about 10% for the outlier model, and the method also handles multiple objects and isolated point sets.

What carries the argument

The load-bearing object is the periodic signed distance function $\varphi$, whose zero sublevel set $\{x\mid \varphi(x)\le 0\}$ represents the hull. The identity doing the work is that, for a convex region, the curvature of level-set curves is nonnegative, and with $|\nabla\varphi|=1$ this curvature equals the Laplacian $\Delta\varphi=\mathrm{div}(\nabla\varphi/|\nabla\varphi|)$; hence the constraint $\Delta\varphi\ge 0$ inside the sublevel set is used to enforce convexity. The alternating direction method of multipliers splits the problem: $z_1$ is a normalized gradient projection, $z_2$ clips the Laplacian to be nonnegative inside the sublevel set, $z_3$ enforces nonpositivity on the object, and the $\varphi$ update solves a fourth-order linear PDE by two second-order FFT inversions under periodic boundary conditions.

What would settle it

Give Algorithm 4.1 a binary image whose exact convex hull is known, run it with the paper's parameters, and compare the returned zero sublevel set with quickhull; a relative Hausdorff error above 2% would contradict the reported bound. More directly, construct a periodic function with $|\nabla\varphi|=1$, $\varphi\le 0$ on a nonconvex set, and $\Delta\varphi\ge 0$ in $\mathrm{slev}_c^\varphi$; if such a function exists for any $c>0$, the convexity criterion (2.5) is insufficient and the exact-model claim collapses.

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Extended reading notes

Core claim

The central claim is that the zero sublevel set of the minimizer of the exact model (EC), with objective $\int_\Omega(-\omega\varphi+\mu|\nabla H(\varphi)|)\,dx$, constraints $|\nabla\varphi|=1$, $\Delta\varphi\ge 0$ in $\mathrm{slev}_c^\varphi$, and $\varphi\le 0$ on the input object, is exactly $\mathrm{Conv}(\Omega_0)$. Because the convex hull is the smallest convex set containing the object, minimizing its area under these constraints selects it. The relaxed problem (OC) adds $\lambda(m\varphi)_+$ to the objective and drops the enclosure constraint, so the hull can leave outliers outside when enclosing them is too costly. The paper reports that Algorithm 4.1 computes exact hulls with relative errors under 2% compared with quickhull, and Algorithm 4.2 filters most outliers with errors under roughly 10%.

Load-bearing premise

The whole construction rests on the imported claim that requiring the Laplacian of a periodic signed distance function to be nonnegative inside its sublevel sets is enough to guarantee the zero sublevel set is convex; if that equivalence fails for some $c$ or for periodic functions, the minimizer of (EC) need not be a convex hull.

Editorial extensions

If this is right

If the exact model is correct, the convex hull of any binary mask can be obtained as a rasterized region $\{x\mid\varphi(x)\le 0\}$ without extracting polygon vertices.
The same parameter $c$ in the constraint $\Delta\varphi\ge 0$ in $\mathrm{slev}_c^\varphi$ controls whether separated objects receive individual hulls or one merged hull, since larger $c$ merges nearby convex regions.
The outlier model gives a convex hull that can ignore noise; adjusting $\lambda$ trades off excluding outliers against including thin protruding parts of the object.
The method also applies to sets of isolated points, not only connected regions, as demonstrated in Section 5.3.

Reading between the lines

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

We infer that the level-set formulation could serve as a differentiable convex-hull layer in learning pipelines, because the output $\varphi$ is a smooth function of the input mask through the PDE solve, unlike vertex-based hull algorithms.
The parameter $c$ can be read as a convexity scale: small $c$ yields the minimal hull of each component, while larger $c$ merges components, producing a family of approximate hulls between component-wise and global hulls.
A natural testable extension is to run the same models on 3D binary volumes; the convexity criterion $\Delta\varphi\ge 0$ in sublevel sets and the FFT solver extend formally, but the paper only claims 2D results.

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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

The one thing to know: the outlier-robust model and its ADMM-FFT solver are the real contribution here; the 'exact' model's headline claim that the zero sublevel set is the convex hull for one or multiple objects is not actually proved, and the stress-test example (two disks) shows why. For c=0, the constraint Δφ≥0 in slev_0 is satisfied by the disconnected union, so the minimizer of (EC) can be non-convex. The paper's own Section 5.1 admits c must be chosen larger than half the inter-object distance to merge objects, which is data-dependent and not derived from the model. That doesn't sink the method for single objects, but it does sink the unqualified abstract claim.

What's genuinely new: the λ(mφ)_+ outlier penalty gives a clean variational way to ignore isolated outliers in hull extraction, and the FFT-based ADMM solver is a plausible extension of the authors' convexity-prior machinery. The experiments on the Weizmann silhouettes show errors under 2% for the exact model and under 10% for the outlier model, and the multi-object demonstrations (cars, Pokemon) are visually sensible. The paper is clearly written and the discretization is spelled out.

The soft spots are proportionate. First, the convexity characterization is imported from [25,32] and only the forward direction is stated; the converse 'Δφ≥0 in slev_c implies convex' is asserted, not proved, and is false for disconnected sets. Second, the switch from ∫(1−H(φ)) to ∫−ωφ+μ|∇H(φ)| is justified 'based on numerical test' rather than by a proof that the minimizer is still the convex hull. Third, the periodic boundary assumption is an extra modeling choice that changes the distance geometry and is not fully interrogated. Fourth, there are a lot of free parameters (ω, μ, ν, λ, c, ρ's, δ) and the choice of c does real work in the multi-object case. No code or runtime comparison against, say, a robust CGAL-style hull with preprocessing is provided, so the practical claim of advantage is not fully quantified.

Who this is for: people working on level-set shape priors and image-based hull extraction from noisy masks. It deserves a serious referee—the outlier model is worth engaging with—but the exactness claims need to be either proved or substantially qualified before publication.

Referee Report

4 major / 6 minor

Summary. The manuscript proposes two level-set variational models for computing the convex hull of binary objects in 2D. The exact model (EC) minimizes a weighted objective in the signed distance function φ subject to the Eikonal constraint |∇φ|=1, a sublevel-set Laplacian constraint Δφ≥0, and the enclosure constraint φ≤0 on the input set; the outlier-robust model (OC) replaces the hard enclosure by a hinge penalty λ(mφ)_+. Both models are solved by ADMM with FFT-based solvers under a periodic boundary assumption. Numerical tests on nine images report relative Hausdorff errors below 2% against quickhull for the exact model and below about 10% for the outlier model, with additional experiments on multiple objects, noisy images, and isolated point sets.

Significance. The outlier-robust formulation is a useful idea: classical convex hull algorithms do not tolerate outliers, and the proposed continuous level-set relaxation offers a differentiable alternative that can be extended to higher dimensions. The reported experiments are reasonably convincing at the level of the examples shown, and the parameter stability across images is a practical strength. However, the central theoretical claim of exactness is not established in the manuscript: the convexity characterization is imported from prior work without a self-contained proof, the objective functional is modified heuristically, and the multiple-object exactness claim is conditional on a data-dependent choice of the parameter c, as the paper's own Section 5.1 demonstrates.

major comments (4)

[Section 2, Theorem 2.1 and Eq. (2.5); Section 3.1, Eq. (3.7)] The constraint Δφ≥0 in slev_c^φ is used as a convexity-enforcing condition, but Theorem 2.1 proves only the forward direction: convexity of Ω1 implies Δφ≥0 in Ω. The converse is not proved in this manuscript and, as stated, is false for multi-component sets. For Ω0 consisting of two disjoint disks, the SDF of the union satisfies |∇φ|=1 and, inside each disk, Δφ=1/r>0, so Δφ≥0 holds in slev_0^φ even though the zero sublevel set is disconnected and nonconvex. Since the union has smaller area than the true convex hull, the minimizer of (EC) with c=0 is not the convex hull. Section 5.1 (Figure 7) implicitly acknowledges this by requiring c to exceed half the inter-object distance to merge objects, a data-dependent choice that is not derived from the model. The abstract's claim of an exact model for one or multiple objects is therefore only conditional; the authors should either prove the needed converse with precise conditions on c, Ω0, and the periodic setting, or substantially restate the claim.
[Section 3.1, Eq. (3.5)] The objective is switched from the area functional (or the equivalent ∫-φ functional) to the weighted combination ∫(-ωφ + μ|∇H(φ)|) with the justification "Based on our numerical test." No theorem or parameter analysis shows that the minimizer of this weighted functional under constraints (3.6)-(3.8) coincides with the SDF of the convex hull. For a pair of distant disks, the perimeter term can favor the disconnected union over the hull depending on the relative values of ω and μ, so the exactness claim depends on an unexamined parameter regime. The authors should either prove a range of (ω,μ) for which the minimizer is the convex hull, or explicitly present (EC) as a heuristic model whose empirical accuracy is measured in Section 5.
[Section 2, periodic extension paragraph] The periodic boundary assumption changes the notion of distance in the SDF (2.1): the distance is to the boundary of the periodic replication of the object, not to the original object's boundary. Thus for objects whose support is close to the image boundary, the model computes a hull of the periodically replicated union, which can differ from the convex hull of the input object. The suggestion to pad zeros is practical but is not analyzed; an object near the boundary of the padded domain still interacts with its periodic copies. This limitation should be stated explicitly and tested, or the model should be restricted to objects with sufficient margin.
[Section 4.1, Eqs. (4.21)-(4.22)] The φ update in Algorithm 4.1 is described as an approximation: Eq. (4.21) is obtained by moving nonlinear terms to the right-hand side and evaluating them at φ^t, and Eq. (4.22) further adds a proximity term. No proof is given that the fixed point of this approximate update solves the original optimality condition (4.20), nor that the ADMM scheme converges to a minimizer of (EC). Since the paper claims an exact model, the solver should be shown to converge to a solution of the stated constrained problem; otherwise the reported 2% errors are only evidence about the specific discretized algorithm, not about the variational model.

minor comments (6)

[Section 4.1, Eq. (4.19)] The terms ρ1/2||Δφ−z_1^{t+1}||² and ρ2/2||∇φ−z_2^{t+1}||² appear to be swapped; they should read ρ1/2||∇φ−z_1||² and ρ2/2||Δφ−z_2||² to match the definitions z1=∇φ and z2=Δφ.
[Section 5.1, Figure 7 caption] The phrase "we loot at the level-set curves" should be "we look at the level-set curves."
[Table 1] The table header "F rog" contains an unintended space; it should read "Frog."
[Section 1, introduction] The algorithm attributed to Gram in reference [12] is conventionally called the Graham scan; the spelling should be corrected.
[Section 4.1, paragraph after Eq. (4.21)] "Here we use the same technique with [25]" should be "the same technique as in [25]" or "the same technique used in [25]."
[Section 5.3, Figure 15 caption] "star fish" is more standardly written as "starfish."

Circularity Check

1 steps flagged · score 4.0 of 10

Exactness claim rests on an unproved converse of Theorem 2.1 imported from the authors' own prior work [25,32].

self citation load bearing [Section 1 (Introduction); Section 2 after Theorem 2.1 and Eq. (2.5); load-bearing in Section 3.1 model (EC), constraints (3.6)-(3.8)]
"To impose the convexity constraint, we require the Laplacian of the SDF is non-negative at the given set. The equivalence of these two conditions was proved in [32] and [25]. ... It is enough to choose c = 0 to guarantee the represented region is convex."

The exact model (EC) uses Δφ ≥ 0 in slev_c^φ (constraint (3.7)) as the convexity-enforcing mechanism. Theorem 2.1 in this paper proves only the forward implication (convex Ω1 ⇒ Δφ ≥ 0). The converse — that Δφ ≥ 0 in the sublevel set is sufficient for the zero sublevel set to be convex — is not proved in this text and is attributed to [25] and [32], both prior works with overlapping authorship (Tai is a coauthor of both, Luo of [25]). This converse is load-bearing: without it, the minimizer of (EC) need not be convex (for example, the SDF of two disjoint disks satisfies |∇φ|=1 and Δφ≥0 in slev_0^φ, but the sublevel set is disconnected and not convex). Thus the central 'exact hull' claim reduces to an unverified self-citation rather than to an argument in this paper.

full rationale

The paper is not constructionally circular: it reformulates the convex hull as the minimizer of an area-type functional over signed distance functions and validates Algorithm 4.1 against the external quickhull benchmark (Table 1, errors under 2%). The area minimization and the φ ≤ 0 constraint on Ω0 are straightforward reformulations of the definition of convex hull, not circular steps. The single load-bearing circular element is the convexity constraint: model (EC) enforces convexity through Δφ ≥ 0 in the sublevel set, and the paper asserts that the equivalence between convexity and this Laplacian condition was proved in the authors' own prior papers [25] and [32]. This paper's Theorem 2.1 states only the forward direction; the converse needed for the model is neither proved nor stated as a theorem here. Because the converse is false for disconnected feasible sets, the exactness claim for multiple objects is conditional at best, and the theoretical derivation depends on an unverified self-citation. The multi-object discussion in Section 5.1 confirms this: the user must choose c based on inter-object distance and on whether separate or merged hulls are desired, so the model does not itself determine the claimed 'exact' hull without this externally supplied choice. This is a limitation rather than a constructional circularity, but it reinforces that the convexity criterion is an imported premise, not an established result. Overall, the numerical work is independently grounded, but the central theoretical guarantee is not, giving a circularity score of 4.

Assumptions & free parameters 7 free parameters · 4 assumptions · 0 invented entities

The model relies on the authors' prior convexity-prior theorems, a periodic-domain assumption that makes FFT applicable, a heuristic modification of the area objective, and the ADMM solver's unproved convergence. The free parameters omega, mu, nu, lambda, c, and rho's are all user-specified and several are tuned per experiment.

free parameters (7)

omega (weight of -phi term) = 0.01 (exact); 0.005 (outliers)
Weight of the -phi area surrogate in Eq. (3.5). Not derived; chosen 'based on numerical test'.
mu (weight of TV perimeter term) = 5 (exact); 3 (outliers)
Weight of TV perimeter term |grad H(phi)| in Eq. (3.5); affects whether the minimizer is truly the convex hull or a rounder set.
nu (weight of landmark penalty) = 10 or 20
Weight for landmark penalty sum phi(hat x_i)^2; only active if landmarks are provided.
lambda (outlier penalty weight) = 3 for most images; 4 for helicopter/boat
Outlier penalty weight in Eq. (3.9); directly controls how many outliers are excluded; tuned per image class.
c (convexity sublevel threshold) = 20 (single objects); 30 (merged hulls)
Requires Delta phi >= 0 only where phi <= c. Must be less than half the distance between objects for separate hulls; data-dependent.
rho0, rho2, rho3 (ADMM penalties) = rho0=1, rho2=15, rho3=1 (exact); rho2=20 sometimes
ADMM penalty parameters; affect convergence speed, not the optimum in exact ADMM, but the linearized solver's fixed point.
delta (Heaviside smoothing width)
Smoothing width in Heaviside approximation H_delta; value not specified.

assumptions (4)

domain assumption A region is convex iff its SDF satisfies Delta phi >= 0 (weak sense) a.e. in the region
Stated as Theorem 2.1 and cited to [25] and [32], both overlapping with the present authors. The paper extends it to periodic domains and to slev_c^phi without proof.
ad hoc to paper Periodic extension of the image and restriction of Delta phi >= 0 to phi <= c preserves convexity of the represented zero sublevel set
Section 2, Eq. (2.5): to use FFT the authors assume periodic data and only enforce Delta phi >= 0 in the sublevel set. No proof is given that this yields a convex set in the periodic/discrete setting.
ad hoc to paper The minimizer of integral(-omega phi) + mu|grad H(phi)| under constraints equals the SDF of the convex hull
Section 3.1: the equivalence is sketched only for integral(-phi); the modification to the weighted combination is justified by numerical testing, not derivation.
ad hoc to paper The discrete ADMM scheme with proximity term and FFT solves the continuous constrained models; the proximity term does not change the optimal solution
Section 4, Eqs. (4.21)-(4.23): standard but unproved in this paper; no convergence analysis is given.

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Pith. "Pith review of Convex hull algorithms based on some variational models." pith.science (2026). https://pith.science/paper/2WPE6AR4

@misc{pith2026190803323,
  author       = {Pith},
  title        = {Pith review of: Convex hull algorithms based on some variational models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2WPE6AR4}},
  note         = {Machine review of arXiv:1908.03323}
}

read the original abstract

Seeking the convex hull of an object is a very fundamental problem arising from various tasks. In this work, we propose two variational convex hull models using level set representation for 2-dimensional data. The first one is an exact model, which can get the convex hull of one or multiple objects. In this model, the convex hull is characterized by the zero sublevel-set of a convex level set function, which is non-positive at every given point. By minimizing the area of the zero sublevel-set, we can find the desired convex hull. The second one is intended to get convex hull of objects with outliers. Instead of requiring all the given points are included, this model penalizes the distance from each given point to the zero sublevel-set. Literature methods are not able to handle outliers. For the solution of these models, we develop efficient numerical schemes using alternating direction method of multipliers. Numerical examples are given to demonstrate the advantages of the proposed methods.

Figures

Figures reproduced from arXiv: 1908.03323 by the authors.

**Figure 1.** An example of convex hull. Convex hull problems arise from different areas, such as data clustering [23], robot motion planning [14], collision detection [31], image segmentation [10], disease diagnosis [33] and robust estimation [24]. When the input data is in a 2-dimensional space, which is the most common case, the convex hull of a finite set is a polygon. When the input data is in an n-dimensional (n > 2) space,… view at source ↗

**Figure 2.** Convex hulls of objects polluted by noise. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** (a) shows periodic replications of a leaf image and their convex hulls, (b) shows [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: Original images explain this, we use an image of two cars as an example. In [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: Convex hulls found by Algorithm 4.1 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: (a), (b) and (c) plot the values of function [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

**Figure 7.** Figure 7: (a) shows the convex hulls result when c = 10. (b), (c) and (d) show the corresponding function values of φ, level-set curves of φ and value of ∆φ. (e) shows the convex hulls result when c = 30. (f), (g) and (h) show the corresponding function values of φ, level-set cu…

**Figure 8.** Figure 8: Convex hulls of multi-objects by chosing proper values of [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

**Figure 9.** Figure 9: Convex hulls in object detecting tasks. (a) Eggs (b) Frog (c) Helicopter (d) Moth (e) Tendrils (f) Owl (g) Boat (h) Castle (i) Cart [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

**Figure 10.** Figure 10: Images with outliers and their convex hulls. [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]

**Figure 11.** Figure 11: (a), (b), (c) and (d) plot the function value of [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]

**Figure 12.** Figure 12: Convex hulls of occluded objects with outliers. [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]

**Figure 13.** Figure 13: Convex hulls of camera with different λ. (a) λ = 3 (b) λ = 2 (c) λ = 1 (d) λ = 1 with landmarks [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]

**Figure 14.** Figure 14: Convex hulls of helicopter with different [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]

**Figure 15.** Figure 15: Convex hulls of a set of isolated points. [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]

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1990

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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