Medial Axis Aware Learning of Signed Distance Functions

Samuel Weidemaier; Christoph Norden-Smoch; Martin Rumpf

cs.CVApr 15, 2026

Medial Axis Aware Learning of Signed Distance Functions

Authors: Samuel Weidemaier, Christoph Norden-Smoch, Martin Rumpf

Organizations: Institute for Numerical Simulation, University of Bonn

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.

Figures & tables

Figure 2: Left: level sets of the neural SDF $\phi$ of a hexahedron. Middle: zero-level set of the SDF. Right: phase field approximation of the jump set of the SDF gradient, which coincides with the hexaeder’s medial axis.

Figure 3: Left: two graphs of one-dimensional functions satisfying the eikonal equation a.e., with zero boundary conditions at the green dots. The measure of the jump set (red dots) of the gradient $J_{\nabla\phi}$ 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 $J_{\nabla\phi}$ (in red), also denoted as the medial axis. Note that the equally spaced levelsets indicate the constant slope of the SDF in normal direction.

Figure 4: Isolines of different solutions to the eikonal equation vanishing on the surface contour (in green) together with the gradient jump set (in red). Left: viscosity solution, Right: solution with shorter length of $J_{\nabla\phi}$ (example from [ 4 ] ).

Figure 5: Left: regular grid, where cells with $|\phi_{\theta}|<\tau_{\mathrm{sdf}}$ or $v_{\eta}<\tau_{\mathrm{pf}}$ are marked. Right: Subdivision of marked cells and sampling of quadrature points on reference cell.

Panel of Figure 6 — Figure 6: From top to bottom: Ground truth SDF solution, Computed SDF using the Hotspot method [ 38 ] , Computed SDF using our method, Computed SDF using our method with the $0.25$ -sublevel set of the phase field (PF, in red).

	$\mathbf{d_{C}}$		$\mathbf{d_{H}}$		$\mathbf{E_{n}}$
Method	mean	std.	mean	std.	mean	std.
GSD $(\star)$	0.0302	0.0074	0.1171	0.0450	0.1249	0.1736
1-Lip	0.0573	0.0170	0.2468	0.0965	0.1188	0.0332
HeatSDF	0.0147	0.0030	0.1521	0.0609	0.0378	0.0255
Hessian	0.0102	0.0016	0.0521	0.0327	0.0116	0.0126
HotSpot	0.0105	0.0017	0.0704	0.0427	0.0114	0.0079

Table 1: Averaged surface reconstruction metrics for the SRB dataset. Mean and standard deviation are reported across all shapes. The best results are highlighted in bold and the second-best results are underlined.

(\star)

denotes a grid based method.

Figure 7: 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 maximum error on the respective narrow band. $(\star)$ denotes a grid based method.

	$\mathbf{E}_{\mathbf{eik}}^{\Omega}$		$\mathbf{E}_{\mathbf{eik}}^{\mathcal{N}}$		$\mathbf{E}_{\mathbf{SDF}}^{\Omega}$		$\mathbf{E}_{\mathbf{SDF}}^{\mathcal{N}}$		Timings
Method	mean	std.	mean	std.	mean	std.	mean	std.	avg. / shape
GSD $(\star)$	0.0375	0.0048	0.0673	0.0292	0.0321	0.0060	0.0772	0.0083	11 mins.
1-Lip	0.0429	0.0106	0.0553	0.0276	0.4806	0.1063	0.0917	0.0051	74 mins.
HeatSDF	0.2228	0.0654	0.14313	0.0366	0.1431	0.0358	0.0134	0.0038	32 mins.
Hessian	0.7693	0.0628	0.6389	0.0639	0.4945	0.0842	0.0283	0.0054	19 mins.
HotSpot	0.0905	0.0637	0.1415	0.0523	0.4261	0.0412	0.0712	0.0082	6 mins.

Table 2: Left: SDF evaluation metrics for the SRB dataset. Mean and standard deviation are reported across all shapes. The best results are highlighted in bold and the second-best results are underlined.

(\star)

denotes a grid based method. Right: Average timings per shape for the different methods.

Figure 8: 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 edges (see, e.g. top row example). The remaining methods (HeatSDF, 1-Lip, GSD), show a significant smoothing of the geometry and miss details.

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 $v_{\eta}\equiv 1$ ). 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 be seen, for example, at the door handle.

Figure 10: 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 implementation from [ 38 ] .

Figure 11: Left: level sets of the neural SDF $\phi_{\theta}$ of a torus with rectangular cross section. Middle: zero-level set of $\phi_{\theta}$ . Right: phase field approximation of $J_{\nabla\phi}$ . In addition to the co-dimension 1 interior segments of the jump set, the phase field depicts a segment of the torus’s rotational axis, a co-dimension 2 component of $J_{\nabla\phi}$ .

Figure 12: Example of a shape from the Thingy10k dataset, where our method does not reconstruct the correct SDF.

Panel of Figure 13 — Figure 13: HeatSDF method (left) and modified HeatSDF method with higher-order loss functional and phase field regularization (right). Top row shows improved level set quality near the surface (right) compared to the original approach (left) for a slice ( $y=0$ ) of the 3D anchor shape from the SRB data set [ 6 ] . The bottom row displays renderings of both results, where the right rendering (modified HeatSDF) exhibits reduced surface noise compared to the left (original HeatSDF).

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 14: On a slice in direction normal to

J_{\nabla\phi}

(red) the constructed

\phi^{\varepsilon}

and

v^{\varepsilon}

are sketched for a given solution

\phi

of the eikonal equation with

\phi=0

\mathcal{S}

(in green).

Panel of Figure 15 — Figure 15: First three rows: Computed solutions for the 2D-dataset of [ 38 ] . 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 $J_{\nabla\phi}$ in red.

	$\mathbf{d_{C}}$		$\mathbf{d_{H}}$		$\mathbf{E_{n}}$		$\mathbf{E}_{\mathbf{eik}}^{\Omega}$		$\mathbf{E}_{\mathbf{eik}}^{\mathcal{N}}$		$\mathbf{E}_{\mathbf{SDF}}^{\Omega}$		$\mathbf{E}_{\mathbf{SDF}}^{\mathcal{N}}$
Method	mean	std.	mean	std.	mean	std.	mean	std.	mean	std.	mean	std.	mean	std.
1-Lip	0.0536	0.0146	0.1576	0.0938	0.1413	0.1115	0.0728	0.0326	0.1109	0.0602	0.8607	0.1761	0.1109	0.0602
HeatSDF	0.0448	0.0208	0.1689	0.1206	0.0893	0.0947	0.2136	0.0209	0.1588	0.0724	0.3215	0.0942	0.0172	0.0103
Hessian	0.0110	0.0043	0.0472	0.0744	0.0753	0.1106	0.7615	0.0598	0.8185	0.2042	0.4841	0.1089	0.0318	0.0090
HotSpot	0.0130	0.0089	0.0566	0.0665	0.0818	0.1049	0.1069	0.0742	0.2143	0.0774	0.4266	0.0613	0.0671	0.0010
Ours	0.0124	0.0073	0.0847	0.1125	0.0741	0.1024	0.0769	0.0829	0.0189	0.0164	0.0253	0.0079	0.0871	0.0174

Appendix

Table 3: Error analysis for the Thingy10k dataset. Mean and standard deviation are reported across all shapes. The best results are highlighted in bold and the second-best results are underlined.

Panel of 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".

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Signed distance functions encode surfaces as signed distance values, enabling 3D reconstruction, geometric queries, and collision-aware planning.

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Signed Distance Functions 3D Reconstruction Point Cloud Surface Reconstruction

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Neural Signed Distance Functions (SDFs) excel at reconstructing watertight manifolds but fail on thin structures and open boundaries due to strict inside--outside constraints. Conversely, Unsigned Distance Fields (UDFs) accommodate general geometries but suffer from gradient singularities at the zero-level set, hindering optimization and extraction. We introduce Metric--Phase Fields (MPFs), a decoupled implicit representation that separates metric proximity from topological phase. Given an unoriented point cloud, MPFs learn (i) an unsigned metric field

r

and (ii) a smooth phase field

θ

, for which we derive a bounded phase indicator

P=\tanh(βθ)

that provides soft inside--outside cues where they are meaningful. We couple the two fields via a gated-metric formulation with a residual phase injection to obtain a signed implicit function with stable near-surface gradients. The phase coefficient

β

is learnable, allowing MPFs to adaptively control the sharpness of the phase transition and the degree of saturation of the soft sign indicator. Experiments on both synthetic and scanned thin-shell and thin-plate shapes demonstrate that MPFs preserve thin and layered structures more faithfully than recent SDF-based methods, while also enabling more robust training and more reliable surface extraction than UDF-based approaches. Check out MPFs-GitHub for source code and test models.

Abstract

Signed Distance Functions

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