Anisotropic Distributions on Manifolds: Template Estimation and Most Probable Paths

Area of Science:

• Computational geometry
• Manifold learning
• Statistical modeling

Background:

• Generalizing normal distributions to manifolds is challenging.
• Existing methods often rely on linearization and tangent space approximations.
• Estimating template and covariance structure from manifold data requires robust techniques.

Purpose of the Study:

• To develop a framework for likelihood estimation of template and covariance structure from manifold-valued data using anisotropic diffusion.
• To avoid linearization issues inherent in traditional manifold data analysis.
• To propose a novel approach for modeling anisotropic distributions on manifolds.

Main Methods:

• Utilizing anisotropic diffusion processes to generalize normal distributions.
• Deriving flow equations for most probable paths to data points.
• Employing non-geodesic paths for likelihood estimation.
• Applying the framework to sphere and LDDMM manifolds.

Main Results:

• Anisotropic diffusion provides a robust method for manifold data analysis.
• The proposed framework avoids linearization, offering improved accuracy.
• The approach is demonstrated on sphere and LDDMM manifolds for landmark matching.
• Results show that accounting for anisotropy leads to algorithms not based on geodesic distances.

Conclusions:

• Anisotropic diffusion offers a powerful generalization of normal distributions to manifolds.
• The developed framework provides a novel and effective method for likelihood estimation of template and covariance structure from manifold data.
• This approach has potential applications in areas like landmark matching and other manifold-based statistical problems.