New Mesh-Free Loss Function Improves Neural Field Topology

Researchers have developed a new, highly efficient auxiliary loss function for implicit neural representations, designed to enhance the topological accuracy of neural fields. This mesh-free method leverages Minkowski functionals—specifically area, boundary measure, and Euler characteristic—to describe the morphology of a field's level sets. Unlike traditional approaches that rely on grids or complex structures, this technique uses smooth Monte-Carlo estimators and autodifferentiation, allowing it to be evaluated at scattered points. The new estimator is significantly faster, achieving a 250x speedup compared to persistent homology losses on cubical grids, with accuracy within 1-3% in 2D and 3D. The paper outlines four critical design rules for successful implementation, including using a dense level ladder, a C^2 backbone, the full Minkowski vector, and proper sampling-scale coverage. While highly effective in 2D for repairing topology and preserving fidelity, a limitation emerged in 3D neural-SDF fitting: gradient descent can adversarially hide topological noise below the sampling density, making the estimator blind to it. This suggests that while promising, the method faces challenges in fully capturing fine-grained 3D topology without a cubic increase in sampling points, which would negate its cost advantage.