Elastic Shape Analysis of Surfaces with Second-Order Sobolev Metrics: A Comprehensive Numerical Framework

Area of Science:

• Computational geometry
• Differential geometry
• Computer vision

Background:

• Riemannian shape analysis is crucial for understanding 3D surface variations.
• Existing methods often struggle with reparametrization independence and varying mesh structures.
• Comparing and analyzing populations of 3D surfaces requires robust computational tools.

Purpose of the Study:

• To introduce numerical methods for Riemannian shape analysis of 3D surfaces using invariant second-order Sobolev metrics.
• To enable the computation of geodesics and geodesic distances for both parametrized and unparametrized surfaces.
• To develop tools for statistical analysis of surface populations, including Karcher means and tangent PCA.

Main Methods:

• Utilizing a relaxed variational formulation for geodesic matching with varifold fidelity terms.
• Developing algorithms for computing geodesics and distances on 3D meshes.
• Extending the framework to handle partially observed surface data.

Main Results:

• Achieved reparametrization independence in geodesic computations for unparametrized surfaces.
• Developed versatile algorithms capable of comparing surfaces with different sampling and mesh structures.
• Demonstrated the framework's applicability to partially observed data through synthetic and real examples.

Conclusions:

• The proposed numerical methods offer a robust and versatile approach to Riemannian shape analysis of 3D surfaces.
• The relaxed variational framework effectively addresses challenges in geodesic matching and statistical analysis.
• The methods are applicable to a wide range of surface comparison tasks, including those with incomplete data.