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This research introduces a variational method that explicitly incorporates the medial axis to compute highly accurate global signed distance functions for point clouds using neural networks. By employing a higher-order formulation and phase field approximation, the approach effectively handles gradient discontinuities while maintaining computational tractability. The method demonstrates superior accuracy in both near-field and global contexts compared to existing techniques, offering a robust solution for geometric modeling tasks.
Medial Axis Aware Learning of Signed Distance Functions
arXiv:2604.16512v2 Announce Type: replace-cross 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.
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.
Computer Vision and Pattern Recognition (cs.CV); Computational Geometry (cs.CG); Graphics (cs.GR); Machine Learning (cs.LG); Numerical Analysis (math.NA)
G.1.6; I.3.5; I.3.7; I.4.5
arXiv:2604.16512 [cs.CV]
(or arXiv:2604.16512v2 [cs.CV] for this version)
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