A Riemannian framework for matching point clouds represented by the Schrödinger distance transform

Area of Science:

• Computational geometry
• Shape analysis
• Geometric deep learning

Background:

• Point cloud matching is crucial for 3D data analysis.
• Existing methods often struggle with rigid and non-rigid transformations.
• A novel shape representation is needed for robust matching.

Purpose of the Study:

• To develop a new point cloud matching algorithm using the Schrödinger distance transform (SDT).
• To apply a Riemannian framework to point cloud matching for the first time.
• To evaluate the algorithm's performance against state-of-the-art techniques.

Main Methods:

• Transforming point clouds into Schrödinger distance transform (SDT) representations by solving a static Schrödinger equation.
• Normalizing SDT representations to unit L2 norm, creating square-root densities on a unit Hilbert sphere.
• Utilizing the Fisher-Rao metric and Riemannian geometry for geodesic distance calculations.
• Applying nonlinear optimization techniques to solve for rigid and non-rigid transformations.

Main Results:

• The proposed algorithm, SDTM, effectively matches point clouds under various transformations.
• SDTM demonstrates superior performance on quantitative metrics compared to state-of-the-art registration algorithms.
• The Riemannian framework provides a robust approach for point cloud matching.

Conclusions:

• The Schrödinger distance transform offers a powerful shape representation for point cloud matching.
• The Riemannian framework combined with SDT is a promising direction for advanced point cloud registration.
• SDTM represents a significant advancement in point cloud matching technology.