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. We first compute smooth cross fields by minimizing the Dirichlet energy. Unlike the existing optimization based approaches, our method determines the singularity configuration, i.e., the number of singularities, their locations and indices, via iteratively adjusting singularities. The singularities can move, merge and split, as like charges repel and unlike charges attract. Once all singularities stop moving, we obtain a cross field with (locally) lowest Dirichlet energy. In simply connected domains, such a 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 c, which characterizes how far the cross field is away from a curl-free field. Then we optimize the locations of singularities by moving them along the field lines of c. Our method is fundamentally different from the existing integer programming-based approaches, since it does not require any special numerical solver. It is fully automatic and also has a parameter to control the number of singularities. Our method is well suited for smooth models in which exact boundary alignment and sparse hard directional constraints are desired, and can guide seamless conformal parameterization and T-junction-free quadrangulation. We will make the source code publicly available.