Computing Smooth and Integrable Cross Fields via Iterative Singularity Adjustment.

Area of Science:

• Computational Geometry
• Computer Graphics
• Applied Mathematics

Background:

• Smooth and integrable cross fields are crucial for various applications in computer graphics and geometry processing.
• Existing methods for computing cross fields often rely on complex optimization or integer programming techniques.
• Challenges remain in ensuring integrability, especially in multiply connected domains.

Purpose of the Study:

• To develop a new, automatic method for computing smooth and integrable cross fields on 2D and 3D surfaces.
• To address limitations of existing approaches by providing a more robust and user-friendly solution.
• To enable applications such as seamless conformal parameterization and T-junction-free quadrangulation.

Main Methods:

• Minimize Dirichlet energy to compute initial smooth cross fields.
• Iteratively adjust singularities (position, merge, split) to determine their configuration.
• Construct a vector field to guide singularity optimization for guaranteed integrability, especially in multiply connected domains.

Main Results:

• A fully automatic method for computing smooth cross fields with controllable singularity configurations.
• Guaranteed integrability in simply connected domains and a method to achieve it in multiply connected domains.
• The method avoids specialized numerical solvers and is well-suited for smooth models requiring precise boundary alignment.

Conclusions:

• The proposed method offers a significant advancement in computing integrable cross fields.
• It provides a more direct and automatic alternative to existing optimization-based and integer programming approaches.
• The technique facilitates high-quality surface parameterization and meshing, with source code to be released.