Gauge Invariant Framework for Shape Analysis of Surfaces.

Area of Science:

• Computational geometry
• Differential geometry
• Computer vision

Background:

• Calculating geodesic paths is essential for shape analysis.
• Existing methods often inherit metrics from pre-shape space, leading to complications.
• Elastic Riemannian metrics offer a powerful tool for measuring shape dissimilarities.

Purpose of the Study:

• To develop a novel framework for computing geodesic paths in shape spaces of spherical surfaces.
• To define an elastic Riemannian metric directly on the shape space.
• To achieve gauge invariance and provide a geometrical interpretation of elastic metrics.

Main Methods:

• Defining a Riemannian metric directly on the quotient (shape) space.
• Formulating a path energy based on normal velocity components.
• Solving for geodesics directly within the shape space.
• Demonstrating gauge invariance for arbitrary surface parameterizations.

Main Results:

• A comprehensive framework for computing geodesic paths on spherical surfaces under an elastic Riemannian metric.
• The framework is invariant to arbitrary parameterizations (gauge invariance).
• Established a link between computer science and mathematical literature on elastic metrics.
• Provided geometrical interpretations of involved terms.

Conclusions:

• The novel framework simplifies geodesic path computation in shape spaces.
• Directly defining the metric on shape space avoids quotient operation complexities.
• The approach offers a unified geometrical understanding of elastic metrics for 3D shape analysis.