Generalized Topological Simplification of Scalar Fields on Surfaces

Area of Science:

• Computational geometry
• Computer graphics
• Applied mathematics

Background:

• Scalar fields on surfaces often contain topological noise from numerical inaccuracies.
• Simplifying these fields is crucial for applications like geometry processing and data analysis.
• Existing methods may be restrictive in feature selection or removal strategies.

Purpose of the Study:

• To develop a general and robust algorithm for topological simplification of scalar fields on surfaces.
• To ensure simplified fields retain data fidelity and have controlled critical points.
• To provide a method that is efficient, easy to implement, and broadly applicable.

Main Methods:

• A combinatorial algorithm based on iterative reconstruction of sub- and sur-level sets.
• The algorithm allows arbitrary removal of critical points and is independent of feature selection strategies.
• Handles triangulated surfaces with or without boundaries, robust to multi-saddles.

Main Results:

• The algorithm provably simplifies scalar fields, guaranteeing a small distance to the original data.
• Achieves convergence in typically 2-5 iterations, with practical O(n log n) time performance.
• Demonstrates robustness and efficiency in extensive experiments, eliminating numerical noise.

Conclusions:

• The presented algorithm offers a general, efficient, and robust solution for topological simplification of scalar fields.
• Enables arbitrary topological pruning and noise removal without threshold parameters, useful for scalar field design and certified geometry processing.
• The method is simple to implement and validated for speed and accuracy, with a provided C++ implementation.