Computing Smooth and Integrable Cross Fields via Iterative Singularity Adjustment

We propose a new method for computing smooth and integrable cross fields on 2D and 3D surfaces. our approach first computes smooth cross fields by minimizing the Dirichlet energy. Unlike existing optimization-based methods, our technique determines the singularity configuration-i.e., the number, locations, and indices of singularities-by iteratively adjusting them. Singularities can move, merge and split, akin to the behavior of like charges repelling and unlike charges attracting. Once all singularities stop moving, we obtain a cross field with (locally) the lowest Dirichlet energy. In simply connected domains, this cross field is guaranteed to be integrable. However, this property does not hold in multiply connected domains. To make a smooth cross field integrable, we construct a vector field $\bf c$c that characterizes the deviation of the cross field from a curl-free field. We then optimize the locations of singularities by moving them along the field lines of $\bf c$c. Our method is fundamentally different from existing integer programming-based approaches, as it avoids combinatorial optimization. It is fully automatic and includes a parameter to control the number of singularities. Our method is well suited for smooth models where exact boundary alignment and sparse hard directional constraints are desired, and can guide seamless conformal parameterization and T-junction-free quadrangulation.