A Riemannian Framework for Intrinsic Comparison of Closed Genus-Zero Shapes

Area of Science:

• Differential Geometry
• Computer Vision
• Medical Imaging Analysis

Background:

• Comparing surface shapes often relies on parameterization, which can introduce distortions.
• Existing methods for shape comparison may not be invariant to changes in surface parameterization.
• Intrinsic properties like metric structures and curvatures are crucial for accurate shape analysis.

Purpose of the Study:

• To develop a framework for intrinsic comparison of surface metric structures and curvatures.
• To achieve parameterization-invariant comparison of shapes, particularly genus zero surfaces.
• To establish a Riemannian framework for shape comparison and reconstruction.

Main Methods:

• Focusing on the first fundamental form and metric tensor fields.
• Utilizing a conjugation-invariant metric based on the L2 norm of symmetric positive definite matrices.
• Deriving a complete Riemannian framework by augmenting the metric with curvature information.
• Employing the fast spherical fluid algorithm for optimizing surface mapping.

Main Results:

• An intrinsic comparison framework for shape metric structure, independent of spherical mapping specifics.
• A geodesically complete metric manifold for metric tensor fields under fixed volume form.
• A by-product of near-isometric and curvature-preserving surface mapping.
• Validation using subcortical boundary surface models from the ADNI dataset.

Conclusions:

• The proposed Riemannian framework enables robust and intrinsic shape comparison and reconstruction.
• The method overcomes limitations of parameterization-dependent shape analysis.
• This approach has potential applications in medical image analysis and computer vision.