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arxiv: 2604.16512 · v1 · submitted 2026-04-15 · 💻 cs.CV · cs.CG· cs.GR· cs.LG· cs.NA· math.NA

Recognition: unknown

Medial Axis Aware Learning of Signed Distance Functions

Samuel Weidemaier , Christoph Norden-Smoch , Martin Rumpf
Authors on Pith no claims yet

Pith reviewed 2026-05-10 13:10 UTC · model grok-4.3

classification 💻 cs.CV cs.CGcs.GRcs.LGcs.NAmath.NA
keywords signed distance functionmedial axispoint cloudvariational methodphase fieldneural networkeikonal equation
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The pith

A variational method computes highly accurate global signed distance functions from point clouds by modeling their medial axis.

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

The authors develop a variational technique to find signed distance functions for surfaces represented by point clouds. They incorporate the medial axis directly into the formulation because it marks where the gradient of the distance function becomes discontinuous. A higher-order term in the variational energy enforces that the function increases linearly in the direction of the gradient away from this axis. The eikonal equation is imposed as a constraint along with the zero level set condition. To solve this, they use a phase field approximation that implicitly locates the medial axis and approximate both functions with neural networks. Experiments confirm improved accuracy near and far from the surface.

Core claim

By taking the jump set of the gradient of the signed distance function, which is the medial axis, explicitly into account in a higher-order variational formulation that enforces linear growth along the gradient direction, and using a phase field approximation of Ambrosio-Tortorelli type to make the problem tractable, the method produces a global SDF that satisfies the eikonal equation and accurately represents the distance to the point cloud surface.

What carries the argument

The higher-order variational formulation enforcing linear growth along the gradient away from the medial axis discontinuity, approximated by a phase field function that implicitly describes the medial axis.

If this is right

  • The SDF satisfies the eikonal equation as a hard constraint.
  • Linear growth is enforced away from the medial axis, improving global accuracy.
  • Neural networks can jointly approximate the SDF and the phase field for unoriented point clouds.
  • Quantitative comparisons demonstrate better performance than existing methods in both near-field and far-field regions.

Where Pith is reading between the lines

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

  • Such medial axis awareness could enhance other distance-based computations in computer vision and geometry processing.
  • Extending the method to time-dependent or deformable surfaces might yield more robust tracking.
  • Applying similar higher-order terms to other variational problems involving discontinuities could improve solutions in related fields.

Load-bearing premise

The phase field approximation of the medial axis is accurate enough that the higher-order term can enforce linear growth without creating artifacts or breaking the eikonal constraint.

What would settle it

Compute the medial axis from a known closed surface, train the method on a point cloud sampled from it, and check if the learned function's gradient discontinuity set matches the true medial axis and if the growth is exactly linear.

Figures

Figures reproduced from arXiv: 2604.16512 by Christoph Norden-Smoch, Martin Rumpf, Samuel Weidemaier.

Figure 1
Figure 1. Figure 1: We compute a global neural network approximation of the signed distance function (SDF) by taking into account a simultaneously learned neural network phase field representing the medial axis, i.e. the SDF gradient discontinuities. Abstract We propose a novel variational method to compute a highly accurate global signed distance function (SDF) to a given point cloud. To this end, the jump set of the gradien… view at source ↗
Figure 2
Figure 2. Figure 2: Left: level sets of the neural SDF φ of a hexaeder. Mid￾dle: zero-level set of the SDF. Right: phase field approximation of the jump set of the SDF gradient, which coincides with the hex￾aeder’s medial axis. gether with an eikonal loss, to compute the SDF of a surface given by a point cloud. In [22], the PHASE method is introduced to com￾pute a neural implicit representation of a surface using an occu￾panc… view at source ↗
Figure 3
Figure 3. Figure 3: Left: two graphs of one-dimensional functions satisfy￾ing the eikonal equation a.e., with zero boundary conditions at the green dots. The measure of the jump set (red dots) of the gradient J∇φ is 3 at the top, and 1 for the viscosity solution displayed at the bottom. Right: SDF of a square (zero-level set in green) with the jump set J∇φ (in red), also denoted as the medial axis. Note that the equally space… view at source ↗
Figure 4
Figure 4. Figure 4: Isolines of different solutions to the eikonal equation vanishing on the surface contour (in green) together with the gra￾dient jump set (in red). Left: viscosity solution, Right: solution with shorter length of J∇φ (example from [4]). To increase the preference for the maximality property of the viscosity solution we add a further penalty Lexp[φ] := 3 ∑ p=1 ˆ Ω exp(−αp|φ| p )dx , favouring high values of … view at source ↗
Figure 5
Figure 5. Figure 5: On a slice in direction normal to J∇φ (red) the con￾structed φ ε and vε are depicted for a given solution φ of the eikonal equation with φ = 0 on S (in green). 4 Implementation Throughout all experiments, we consider surfaces S, represented by unoriented point clouds contained in the computational domain Ω := [−1.2,1.2] d . For the phase field parameter ε we choose ε = 10−3 for d = 2 and ε = 10−4 for d = 3… view at source ↗
Figure 6
Figure 6. Figure 6: Left: regular grid, where cells with |φθ| < τsdf or vη < τpf are marked. Right: Subdivision of marked cells and sampling of points on reference cell. Algorithm 1 Training based on adaptive sampling Require: Batch sizes N, M, s.t. N ≪ M, initial grid size h := |Ω| 1 d m −1 , thresholds τsdf, τpf, grid depth k. 1: Initialize uniform rectangular grid G0 on Ω with cell size h 2: Sample test set P ⊂ Ω with |P| … view at source ↗
Figure 7
Figure 7. Figure 7: From top to bottom: Ground truth SDF solution, Com￾puted SDF using the Hotspot method [32], Computed SDF us￾ing our method, Computed SDF using our method with the 0.25- sublevel set of the phase field (PF, in red). 5.2 Experiments in 3D 5.2.1 Evaluation metrics Both high-detail surface reconstruction and accurate distance es￾timation are crucial for a good SDF approximation. We employ different metrics to … view at source ↗
Figure 8
Figure 8. Figure 8: Quantitative evaluation of the Eikonal error (top) and the SDF error (bottom) averaged across the five shapes from the SRB dataset and evaluated on ground truth distance bands of width 0.05 to the surface (horizontal axes). Triangles denote the maxi￾mum error on the respective narrow band. (⋆) denotes a grid based method [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of our higher order loss with and without phase field on a 2D house geometry. Left: Ours without phase field regularization (i.e. constant vη ≡ 1). Right: Ours with phase field (red). While both methods capture the general shape, the level sets in the left figure are significantly less equally spaced and less sharp. Additionally, fine geometric details are lost in the modified version, which can… view at source ↗
Figure 10
Figure 10. Figure 10: Results for the SRB dataset for five different neural SDF reconstruction methods and one grid-based method (GSD). Across all five shapes, the zero-level sets of the Hessian method and Ours are closest to the groundtruth geometry (left), showing both high details and a good reconstruction of flat regions. The HotSpot method captures a similar level of details, while struggling with flat regions and sharp e… view at source ↗
Figure 13
Figure 13. Figure 13: Example of a shape from the Thingy10k dataset, where our method does not reconstruct the correct SDF. We believe that our second-order loss, in combination with the Ambrosio–Tortorelli functional, can be integrated into exist￾ing SDF methods to improve both surface reconstruction quality and the accuracy of the signed distance approximation. As a proof of concept, we introduce a variation of the two-step … view at source ↗
Figure 11
Figure 11. Figure 11: Sphere tracing rendering results (in black and white) on a torus with rectangular cross section and a hand shape, for our method (bottom row), HotSpot (middle row), and Hessian (top row). The color plots show the required number of iterations per pixel. These experiments where conducted using the implementa￾tion from [32]. First-order loss functionals, which involve the non-convex eikonal loss suffer from… view at source ↗
Figure 12
Figure 12. Figure 12: Left: level sets of the neural SDF φθ of a torus with rect￾angular cross section. Middle: zero-level set of φθ. Right: phase field approximation of J∇φ. In addition to the co-dimension 1 inte￾rior segments of the jump set, the phase field depicts a segment of the torus’s rotational axis, a co-dimension 2 component of J∇φ. fosters that minimizers φ are viscosity solutions, but there is so far no analytical… view at source ↗
Figure 15
Figure 15. Figure 15: First three rows: Computed solutions for the 2D￾dataset of [32]. Zero-level set of SDF in green, 0.25-sublevel set of phase field in red. Bottom row: Ground truth solution. Zero-level set of SDF in green, jump set J∇φ in red. C 3D Evaluation For the experiments in 3D, we start the training with the weights (γHO, γAT, γrecon, γeik, γexp)Phase 1 = (1,0.02,0.01,0.05,500). Following the schedule described in … view at source ↗
Figure 16
Figure 16. Figure 16: Marching Cubes results for the zero-level sets of the computed SDFs for a selection of shapes from the Thingy10k dataset. The first seven shapes are from those without tag specification. The last six shapes are with the tag ’sculpture’ or ’scan’. Runs that did not converge to an extractable geometry have been marked with an "X". 13 [PITH_FULL_IMAGE:figures/full_fig_p013_16.png] view at source ↗
read the original abstract

We propose a novel variational method to compute a highly accurate global signed distance function (SDF) to a given point cloud. To this end, the jump set of the gradient of the SDF, which coincides with the medial axis of the surface, is explicitly taken into account through a higher-order variational formulation that enforces linear growth along the gradient direction away from this discontinuity set. The eikonal equation and the zero-level set of the SDF are enforced as constraints. To make this variational problem computationally tractable, a phase field approximation of Ambrosio-Tortorelli type is employed. The associated phase field function implicitly describes the medial axis. The method is implemented for surfaces represented by unoriented point clouds using neural network approximations of both the SDF and the phase field. Experiments demonstrate the method's accuracy both in the near field and globally. Quantitative and qualitative comparisons with other approaches show the advantages of the proposed method.

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.

Referee Report

2 major / 2 minor

Summary. The paper proposes a novel variational method to compute a highly accurate global signed distance function (SDF) from an unoriented point cloud by explicitly incorporating the medial axis (the jump set of the SDF gradient) via a higher-order formulation that enforces linear growth along gradient directions away from this set. The eikonal equation and zero level-set are enforced as constraints, with an Ambrosio-Tortorelli phase-field approximation used to implicitly describe the medial axis; both the SDF and phase field are represented by neural networks, and experiments claim improved accuracy in near-field and global regimes relative to prior approaches.

Significance. If the central construction holds, the work would offer a meaningful advance in neural implicit geometry by making the medial axis an explicit part of the variational objective rather than an emergent property. This could yield more reliable global SDFs for downstream tasks such as collision detection, path planning, and surface reconstruction from sparse or noisy data. The combination of higher-order regularization with a phase-field relaxation and neural parameterization is technically interesting and, if supported by rigorous validation, would strengthen the literature on constrained implicit representations.

major comments (2)
  1. [variational formulation and phase-field approximation] The load-bearing assumption is that the jointly optimized Ambrosio-Tortorelli phase field φ recovers the (possibly non-unique) medial axis with sufficient fidelity that the higher-order linear-growth term can enforce u(x) ≈ u(y) + |x-y| along rays normal to the jump set while |∇u|=1 holds almost everywhere outside the diffuse interface. No derivation or consistency analysis of the coupled Euler-Lagrange equations is supplied to show that residual curvature or gradient jumps inside the ε-transition zone are prevented; this directly affects whether the eikonal constraint remains satisfied where the new penalty is active.
  2. [experiments and implementation] The experimental section reports qualitative and quantitative advantages but supplies neither the precise loss terms used to enforce the eikonal and zero-level-set constraints inside the neural optimization nor ablation studies that isolate the contribution of the higher-order medial-axis term. Without these, it is impossible to verify that observed improvements stem from the claimed mechanism rather than from network capacity or hyper-parameter tuning.
minor comments (2)
  1. [method] The notation distinguishing the phase-field variable φ from the SDF u should be introduced earlier and kept consistent when the Ambrosio-Tortorelli functional is first written.
  2. [figures] Figure captions would benefit from explicit statements of point-cloud density, sampling strategy, and the value of ε employed in each visualized result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. The comments highlight important aspects of the variational formulation and experimental validation that we will address in the revision. We provide detailed responses below.

read point-by-point responses

  1. Referee: [variational formulation and phase-field approximation] The load-bearing assumption is that the jointly optimized Ambrosio-Tortorelli phase field φ recovers the (possibly non-unique) medial axis with sufficient fidelity that the higher-order linear-growth term can enforce u(x) ≈ u(y) + |x-y| along rays normal to the jump set while |∇u|=1 holds almost everywhere outside the diffuse interface. No derivation or consistency analysis of the coupled Euler-Lagrange equations is supplied to show that residual curvature or gradient jumps inside the ε-transition zone are prevented; this directly affects whether the eikonal constraint remains satisfied where the new penalty is active.

    Authors: We agree that a more detailed analysis of the variational problem would strengthen the manuscript. The Ambrosio-Tortorelli approximation is chosen because it is known to Γ-converge to the Mumford-Shah functional, which in this context approximates the jump set of the gradient. The higher-order term is added to enforce the linear growth property characteristic of the signed distance function away from the medial axis. To address the concern regarding the Euler-Lagrange equations and consistency in the transition zone, we will include in the revised manuscript a derivation of the stationarity conditions for the coupled system and a discussion of how the eikonal constraint is preserved outside the ε-interface, supported by theoretical references and numerical verification. revision: yes

  2. Referee: [experiments and implementation] The experimental section reports qualitative and quantitative advantages but supplies neither the precise loss terms used to enforce the eikonal and zero-level-set constraints inside the neural optimization nor ablation studies that isolate the contribution of the higher-order medial-axis term. Without these, it is impossible to verify that observed improvements stem from the claimed mechanism rather than from network capacity or hyper-parameter tuning.

    Authors: We acknowledge that providing the exact loss formulations and ablation studies is essential for reproducibility and to validate the contribution of each component. To fully address this comment, we will add a dedicated subsection detailing all loss terms with their weights used in the neural optimization, and include ablation experiments that compare the full model against variants without the higher-order medial axis term. These additions will be incorporated in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity: formulation rests on standard eikonal and Ambrosio-Tortorelli phase-field concepts

full rationale

The paper proposes a variational method enforcing the eikonal equation and linear growth away from the medial axis (identified with the jump set of ∇u) via a higher-order term, approximated by an Ambrosio-Tortorelli phase field. These are established external techniques; the abstract and described construction do not reduce any prediction or central quantity to a fit on the same data, a self-citation chain, or a definitional renaming. Neural-network approximations of u and φ are standard function approximators, not a source of circularity. No load-bearing step equates an output to its input by construction. The method is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that the medial axis coincides with the jump set of the SDF gradient and that a phase-field relaxation can enforce the required linear growth; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (2)
  • domain assumption The jump set of the gradient of the SDF coincides with the medial axis of the surface
    Invoked in the abstract as the motivation for the higher-order variational term.
  • domain assumption The eikonal equation and zero-level set condition can be enforced as constraints in the variational problem
    Stated directly as part of the formulation.

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