Discrete Total Variation of the Normal Vector Field as Shape Prior with Applications in Geometric Inverse Problems

Jos\'e Vidal N\'u\~nez; Marc Herrmann; Roland Herzog; Ronny Bergmann; Stephan Schmidt

REVIEW 3 major objections 4 minor 6 references

Discrete Total Variation of the Normal Vector Field as Shape Prior with Applications in Geometric Inverse Problems

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper introduces a discrete total variation of the normal-vector field on triangulated surfaces, proves it equals discrete total mean curvature, and shows it recovers polyhedral shapes in PDE-constrained inverse problems where…

desk verdict The paper's genuine novelty is the mollification limit and the first PDE-inverse-problem application of discrete total mean curvature; the numerical algorithm has a specification gap that needs addressing. read the letter →

arxiv 1908.07916 v1 pith:FXUZ25CI submitted 2019-08-20 math.NA cs.NA

Ronny Bergmann , Marc Herrmann , Roland Herzog , Stephan Schmidt , José Vidal Núñez This is my paper · ORCID

classification math.NAcs.NA MSC 49Q1065N2165K10

keywords totalvariationofthenormaldiscretemeancurvatureshapepriorsplitBregmaniterationRiemannianoptimizationgeometricinverseproblemelectricalimpedancetomographypiecewiseflatshapes

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

The paper proposes the edge-based sum $|n|_{\mathrm{DTV}}(\Gamma_h)=\sum_E d(n^+_E,n^-_E)\,|E|$, the geodesic distance between adjacent facet normals weighted by edge length, as the discrete substitute for the smooth total variation of the normal. It proves this functional coincides with the discrete total mean curvature and shows analytically that it becomes the smooth TV in the limit of mollified surfaces; numerically it stays within a factor of $\sqrt{2}$ of the smooth value on spheres, suggesting an anisotropic limit. Used as a regularizer in shape optimization of type (1.3), the functional penalizes curvature concentrated at edges rather than surface area, so its minimizers can be piecewise flat. A Riemannian split-Bregman algorithm is developed, and experiments on mesh denoising and on an electrical-impedance-tomography inclusion problem show that polyhedral objects—flat sides and sharp edges—are reconstructed accurately, in contrast to surface-area regularization which rounds and shrinks the result.

What carries the argument

The load-bearing object is the edge functional $|n|_{\mathrm{DTV}}(\Gamma_h)=\sum_E \arccos((n^+_E)^\top n^-_E)\,|E|$, an $\ell^1$-type sum of exterior dihedral angles weighted by edge length; it doubles as the discrete total mean curvature. Theorem 2.1 justifies it as the limit of smooth normal-TV under mollification, and numerical evidence on spheres indicates it tracks the anisotropic integral $\int_\Gamma(|k_1|+|k_2|)\,ds$. On the optimization side, the paper adapts the split Bregman/ADMM scheme to the sphere: the jump across each edge is represented by the logarithmic map $d_E=\log_{n^+_E} n^-_E$ in the tangent space $T_{n^+_E}S^2$, multipliers are carried between iterates by parallel transport, and the $d$-update reduces to an explicit vectorial shrinkage step (3.4). This machinery is what makes the nonconvex, nonsmooth shape problem (3.1) tractable numerically.

What would settle it

Construct the noise-free cube-reconstruction experiment from an initial sphere and track both the objective (3.1) and the Bregman residual $\|d-\log n-b\|$ across iterations; if the iteration stops at a rounded shape with positive dihedral angles, or if the residual fails to decrease, the paper's claim that the prior helps identify polyhedral shapes is not supported by its own algorithm.

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Extended reading notes

Core claim

The central claim is that the functional $|n|_{\mathrm{DTV}}(\Gamma_h) = \sum_E d(n^+_E, n^-_E)\,|E|$, formed by summing, over every edge, the geodesic distance between the two adjacent facet normals times the edge length, is the discrete counterpart of the smooth total variation of the normal. The paper proves that this sum coincides exactly with the discrete total mean curvature and that, for a family of mollified smooth surfaces converging to $\Gamma_h$, the smooth total variation of the normal converges to $|n|_{\mathrm{DTV}}(\Gamma_h)$ as the smoothing width goes to zero. As a shape prior it penalizes curvature concentrated at edges rather than surface area, so flat facets and sharp creases become admissible optimizers; numerical experiments on mesh denoising and on an electrical-impedance-tomography problem show that a cube-shaped inclusion is reconstructed with flat lateral faces and sharp edges, whereas surface-area regularization smooths and shrinks the result. For spheres the discrete value exceeds the smooth one by a factor close to $\sqrt{2}$, which the paper conjectures reflects convergence to $\int_\Gamma (|k_1|+|k_2|)\,ds$ rather than $\int_\Gamma\sqrt{k_1^2+k_2^2}\,ds$.

Load-bearing premise

The numerical demonstrations assume the proposed Riemannian split-Bregman iteration, for which the paper supplies no convergence analysis, actually reaches a useful minimizer of the nonconvex, nonsmooth problem (3.1).

Editorial extensions

If this is right

Shape optimization problems (1.3) gain a prior that explicitly permits flat facets and sharp edges, so polyhedral inclusions in PDE-constrained inverse problems can be recovered without the rounding produced by surface-area terms.
Because $|n|_{\mathrm{DTV}}$ equals discrete total mean curvature, results from discrete differential geometry—such as the stationarity of the icosahedron and crossed-diagonal cube under an area constraint—transfer directly to the prior.
The Riemannian split-Bregman algorithm extends ADMM to manifold-valued data and, as the paper notes, can be applied to other nonsmooth total-variation problems with values in $S^2$ or other manifolds.
Uniform scaling of $\Gamma_h$ scales $|n|_{\mathrm{DTV}}$ linearly, so any unconstrained use of the prior must be paired with an area constraint or a tracking term to prevent collapse to a point.
The observed $\sqrt{2}$ gap on spheres points to the conjecture that refined-mesh discrete values converge to the anisotropic measure $\int_\Gamma (|k_1|+|k_2|)\,ds$, implying the discrete prior is an $\ell^1$ analogue of the smooth $\ell^2$ functional.

Reading between the lines

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

A natural testable extension is an isotropic variant that measures the joint dihedral angles around each vertex; if it recovers the smooth value on spheres for all meshes, it would eliminate the $\sqrt{2}$ artifact while keeping piecewise-flat minimizers.
If Algorithm 3.1's convergence were established, the same prior could be applied to other severely ill-posed geometric inverse problems—crack detection, cavity imaging, interface reconstruction—where the ground truth is known to be piecewise flat.
The connectivity dependence of the minimizers (cube with crossed diagonals is stationary) suggests that using this prior with a fixed triangulation biases the reconstruction toward that triangulation's flat patterns; treating connectivity as a design variable could change the recovered shapes.
The prior's effectiveness likely depends on the ratio of the Bregman penalty to the regularization strength and on the number of inner gradient steps; a systematic parameter study on synthetic polyhedral targets would provide a practical calibration rule the paper does not give.

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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

Here's my take: the paper does not claim a brand new functional—it is openly identified as discrete total mean curvature (Sullivan 2005; Wu et al. 2015; Pellis et al. 2019). The actual new material is the mollification consistency theorem, the stationarity analysis, and the first application of this functional as a prior in a PDE-constrained inverse problem. That is a legitimate and useful contribution.

The paper does several things well. It is honest about prior work. Theorem 2.1, while not fully proved, gives a plausible argument that the edge-based sum is the limit of the smooth TV of the normal for mollified surfaces; the decomposition into edge strips and vertex caps is reasonable, though the monotone-geodesic and O(ε) vertex-cap estimates are asserted rather than derived. Theorem 2.2 is a solid, explicit stationarity check for the icosahedron and cube, including the non-smooth case via Clarke derivatives. The EIT experiment is a meaningful demonstration: the functional really does preserve flat faces and sharp edges, unlike surface area regularization.

The soft spots are real. The proof of Theorem 2.1 rests on two unverified assumptions about the mollification geometry; a referee should ask for details. More serious is Algorithm 3.1. During the vertex update in Step 4, d^(k) and b^(k) are frozen vectors in the tangent spaces of the previous shape. As vertices move, those tangent spaces change, so the augmented Lagrangian (3.3) is not well-defined at the current iterate without a parallel transport that is only performed for b, after the update, and never for d. The algorithm as written is therefore not a fully specified Riemannian split Bregman method; the Step 4 subproblem is ambiguous. No convergence analysis is supplied, and the paper explicitly defers it. With no code and hand-picked regularization parameters, the reconstructions in Figure 5.2 are suggestive, not robust evidence.

None of this kills the paper. The central mathematical content—that this functional is a natural discrete analogue and that it promotes piecewise flat shapes—holds up. But the numerical utility claim needs a well-defined algorithm, and the experiments need more systematic parameter studies or code. This paper is for researchers in shape optimization, geometric inverse problems, and discrete differential geometry. It deserves a serious peer review, but with major revision expected.

My recommendation: send it to review. Ask the authors to fix the algorithm specification, add details to Theorem 2.1's proof, and provide code or more thorough numerical validation.

Referee Report

3 major / 4 minor

Summary. The paper introduces a discrete total variation functional for the unit normal vector field on piecewise flat surfaces, defined as the sum over edges of the geodesic distance between adjacent facet normals weighted by edge length. It claims this functional is the discrete analogue of the smooth total variation of the normal, agrees with discrete total mean curvature, admits piecewise flat minimizers, and can serve as a shape prior in geometric inverse problems. The paper analyzes simple stationary shapes, proposes a Riemannian split Bregman algorithm, and presents mesh denoising and EIT inclusion-detection experiments.

Significance. The functional (2.1) is natural and the connection to discrete total mean curvature via Sullivan (2005) is a real and useful observation. If the utility claim is properly supported, the paper offers an alternative to surface-area regularization that avoids rounding and shrinkage of polyhedral shapes. The paper is honest about open points, including the incomplete characterization of minimizers and the deferred convergence analysis of the ADMM scheme. However, the proof of the smooth-to-discrete limit is currently a sketch with unverified geometric assumptions, and Algorithm 3.1 has a tangent-space consistency gap that affects the interpretation of the numerical results. The central idea is promising, but the evidence presented for the main utility claim is not yet decisive.

major comments (3)

[Section 2, Theorem 2.1] The proof of the convergence (2.3) rests on two assumptions that are not established: that in each stripe of the transition region IE,ε the normal vector changes monotonically along the geodesic between n+E and n−E, and that the integrand on the vertex caps BV,ε has order ε−1 while the cap area has order ε2. The sentence "can be easily evaluated as an iterated integral" does not supply the required geometric estimates for the mollified surface. Since Theorem 2.1 underpins the claim that (2.1) is the discrete analogue of (1.1), this step needs either a rigorous proof or an explicit statement that it is to be read as a formal calculation.
[Section 3, Algorithm 3.1] Step 4 minimizes L(Ωh, d(k), b(k)) over vertex positions while d(k) and b(k) are held fixed, but these vectors are elements of T_{n+E^{(k)}} S2. As the vertices move during the gradient steps, n+E changes, so d(k) and b(k) are no longer tangent vectors at the current normal. Step 5 transports b(k) only after the shape update and d(k) is never transported during the subproblem. Consequently the object minimized in Step 4 is not the augmented Lagrangian (3.3) evaluated at the current iterate, and the algorithm is not fully specified as a Riemannian split Bregman method. This gap affects the validity of the numerical results in Section 5.
[Section 5, Figure 5.2] The reconstructions are presented visually with hand-selected regularization parameters, a single mesh size, and no quantitative error metric or parameter sensitivity study. In combination with the absent convergence analysis for Algorithm 3.1 (Section 6 explicitly defers it), the experiments do not yet robustly demonstrate that |n|DTV "can help to identify polyhedral shapes" as claimed in the Abstract.

minor comments (4)

[Section 2.2.1] In the proof of Theorem 2.2, "all facets are unilaterial triangles" should read "equilateral triangles".
[Section 2.2.2] The word "sucessively" should be "successively" in the discussion of refined meshes.
[Section 5] The sentence "For the surface area regularization, β|n|TV(Γ1) is replaced by γ∫Γ1 ds" is confusingly worded, since the left-hand side refers to the prior term in (4.1) rather than to the TV functional itself.
[Section 3, Algorithm 3.1] In Step 5, "Parallely transport" should be "transport in parallel" or "Parallel transport".

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the discrete normal-TV functional is explicitly identified with known discrete mean curvature, its smooth limit is an independent analytic claim (with a proof gap), and the numerical reconstructions are demonstrations, not fitted predictions.

full rationale

The paper's central claim is that the edge-based functional |n|_DTV(Γ_h) = Σ_E d(n^+_E, n^-_E)|E| is the discrete analogue of the smooth normal TV and is a useful shape prior. This functional is not derived from a fitted constant; it is proposed as a definition and then related to known quantities. The paper explicitly acknowledges prior use of the same expression, citing Sullivan (2005), Wu et al. (2015), and Pellis et al. (2019), so the identification with discrete total mean curvature is not presented as a new result and does not involve renaming a known result as an independent derivation. The self-citations to the companion paper (Bergmann, Herrmann, et al., 2019) supply the smooth TV definition and the continuous split-Bregman background, but the discrete analysis in Section 2, the stationary-point computation in Theorem 2.2, and the numerical implementation in Sections 3-5 stand on the present paper's own equations. Theorem 2.1's proof does contain an unproved assumption that the mollified normal changes monotonically along the geodesic in the edge transition bands, and it assumes the vertex-region contributions are O(ε); these are rigor gaps or correctness risks, not circular reductions, because the theorem's conclusion is not built into the definition of (2.1) by construction. The numerical section uses hand-selected regularization parameters, but this is standard parameter choice for demonstrating a regularizer, not a fitted input renamed as a prediction. The lack of convergence analysis for the nonconvex, nonsmooth ADMM scheme, explicitly admitted in Section 6, is a correctness concern about the numerical evidence, not a circularity of the derivation. Overall, no load-bearing step reduces to its own input, and the main analytical and numerical content is self-contained.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claims rest on the geometric definition, the mollification behavior in Theorem 2.1, the fixed-connectivity mesh assumption, and the unproven convergence of Algorithm 3.1. No invented physical entities are introduced; the auxiliary variable d_E is an optimization splitting variable, not a postulated entity.

free parameters (5)

Regularization weight β for discrete TV = 10^-6
Hand-selected for the EIT experiments (Table 5.1); no sensitivity study is reported; reconstructions depend on it.
Regularization weight γ for surface area = 5·10^-5 and 2·10^-5
Hand-selected comparison parameter for surface area regularization (Section 5, Table 5.1).
Split Bregman penalty λ = 10^-5
Hand-set in Table 5.1; controls the coupling between the auxiliary variable d and the log map in the augmented Lagrangian (3.3).
Robin coefficient α = 10^-5
Hand-set far-field and current injection parameter in the EIT model (4.1), Table 5.1.
Shape gradient step size = 10^2 initial with Armijo line search
Algorithmic step size in Section 5; afflicts the convergence trajectory and is not systematically tuned.

assumptions (6)

domain assumption Piecewise flat, compact, orientable surface without boundary, geometrically conforming mesh with no hanging nodes.
Assumed throughout Section 2 to define the edge-based functional; excludes open surfaces and nonconforming meshes.
domain assumption Fixed mesh connectivity; only triangular surface meshes considered.
Section 2.1; minimizers depend on connectivity (Pellis et al. 2019), so results are connectivity-dependent.
domain assumption Perfect conductor inclusion modeled by homogeneous Neumann condition on the unknown boundary Γ1.
Section 4.1; shields the interior, avoids meshing the inclusion, and is an approximation in the geological EIT model.
ad hoc to paper Mollified normal transitions along edges are monotone along geodesics and vertex cap contributions vanish at order ε.
Assumed in the proof of Theorem 2.1 without detailed estimates; load-bearing for the convergence to the discrete functional.
ad hoc to paper Riemannian split Bregman iteration converges to a useful minimizer.
Algorithm 3.1; Section 6 defers analysis; numerical results assume this convergence.
standard math Standard S^2 geometry formulas for logarithmic map and parallel transport (A.4) and (A.5).
Used in the algorithm; elementary Riemannian geometry on the sphere.

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Pith. "Pith review of Discrete Total Variation of the Normal Vector Field as Shape Prior with Applications in Geometric Inverse Problems." pith.science (2026). https://pith.science/paper/FXUZ25CI

@misc{pith2026190807916,
  author       = {Pith},
  title        = {Pith review of: Discrete Total Variation of the Normal Vector Field as Shape Prior with Applications in Geometric Inverse Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FXUZ25CI}},
  note         = {Machine review of arXiv:1908.07916}
}

read the original abstract

An analogue of the total variation prior for the normal vector field along the boundary of piecewise flat shapes in 3D is introduced. A major class of examples are triangulated surfaces as they occur for instance in finite element computations. The analysis of the functional is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. It is found to agree with the discrete total mean curvature known in discrete differential geometry. A split Bregman iteration is proposed for the solution of discretized shape optimization problems, in which the total variation of the normal appears as a regularizer. Unlike most other priors, such as surface area, the new functional allows for piecewise flat shapes. As two applications, a mesh denoising and a geometric inverse problem of inclusion detection type involving a partial differential equation are considered. Numerical experiments confirm that polyhedral shapes can be identified quite accurately.

Figures

Figures reproduced from arXiv: 1908.07916 by the authors.

**Figure 2.1.** Volume mesh of a cube domain Ωh consisting of tetrahedra (left) and corresponding triangular mesh of the boundary Γh (right). Since the surface Γh is non-smooth, the definition (1.1) of the total variation of the normal proposed in the companion paper Bergmann, Herrmann, et al., 2019 for smooth surfaces does not apply. Since the normal vector field n is piecewise constant here, its variation is concentrated in spon… view at source ↗

**Figure 4.** [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

Figures from the paper (13 more)

**Figure 2.2.** Figure 2.2: Illustration of the approximation of a portion of a triangulated surface Γh (left) by a family of smooth surfaces Γε (right). Two vertex caps BV,ε and one transition region along an edge IE,ε are highlighted, see the proof of Theorem 2.1. Theorem 2.1. Let {Γε} denote…

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png]

**Figure 2.3.** Figure 2.3: The icosahedron and the cube with crossed diagonals, two stationary surfaces for (2.5). The highlighted regions as well as the figure on the right illustrate the proof of Theorem 2.2. 2.2.2. Comparison of Discrete and Continuous Total Variation of the Normal. In this…

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png]

**Figure 2.4.** Figure 2.4: shows the numerical solutions of (2.8) and (2.9) for various choices of the regularization parameters β and γ, respectively. The initial guess for both problems is a sphere with the same connectivity as Γh. We can clearly see that our functional (2.1) achieves a very…

**Figure 4.1.** Figure 4.1: The left plot shows the domain Ω considered in the numerical example. Each color on the outer boundary represents the support of one out of r = 48 electric sources fi . The right figure shows a wireframe plot revealing the true inclusion Γ1, i.e., the boundary of the…

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 4.2.** Figure 4.2: The geodesic distance between normals n + E and n − E (shown in black) of two triangles T + E , T − E which share the edge E agrees with the geodesic distance between the co-normals µ + E and −µ − E (shown in orange). domain Ω unit sphere B1(0) \ [−0.4, 0.4]3 ⊂ R 3 m…

**Figure 5.** Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 5.** Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

**Figure 5.1.** Figure 5.1: Initial guess for the inclusion Γ1 on the left and the true inclusion on the right. As is expected and well known, the use of surface area regularization leads to results in which the identified inclusion Γ1 is smoothed out. This can be explained by the observation t…

**Figure 5.2.** Figure 5.2: Top row: setting without noise; left: total variation regularization, β = 10−6 and 90 iterations; middle: surface area regularization with γ = 5 · 10−5 and 1129 iterations; right: surface area regularization with γ = 2 · 10−5 and 978 iterations. Bottom row: setting w…

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Pith tools

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