  Quadrilateral mesh generation plays a crucial role in numerical simulations
within Computer-Aided Design and Engineering (CAD/E). The quality of the cross
field is essential for generating a quadrilateral mesh. In this paper, we
propose a self-supervised neural representation of the cross field, named
NeurCross, comprising two modules: one to fit the signed distance function
(SDF) and another to predict the cross field. Unlike most existing approaches
that operate directly on the given polygonal surface, NeurCross takes the SDF
as a bridge to allow for SDF overfitting and the prediction of the cross field
to proceed simultaneously. By utilizing a neural SDF, we achieve a smooth
representation of the base surface, minimizing the impact of piecewise planar
discretization and minor surface variations. Moreover, the principal curvatures
and directions are fully encoded by the Hessian of the SDF, enabling the
regularization of the overall cross field through minor adjustments to the SDF.
Compared to state-of-the-art methods, NeurCross significantly improves the
placement of singular points and the approximation accuracy between the input
triangular surface and the output quad mesh, as demonstrated in the teaser
figure.
