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Description
We present a variational neural approach for computing global signed distance functions (SDFs) from unoriented point clouds, focusing on the medial axis as the unknown jump set of the SDF gradient. The method is based on the observation that the SDF gradient is smooth away from the medial axis, but jumps where the nearest-point projection onto the surface is not well-defined. We formulate SDF reconstruction as a higher-order variational problem that enforces linear growth in the gradient direction away from the jump set, together with classical eikonal and zero-level set constraints. To model this (free) discontinuity set, we use an Ambrosio-Tortorelli type phase-field approximation, represented by a second neural network, to implicitly describe the medial axis and locally deactivate the higher-order regularization. Simultaneous optimization of the SDF and phase field yields accurate surface reconstruction, reliable far-field distances, and an interpretable phase-field representation of the medial axis, with numerical experiments showing improved performance over existing neural SDF methods.